On Identification Issues in Business Cycle Accounting Models

Pedro Brinca (Nova School of Business and Economics, Universidade Nova de Lisboa, Lisboa, Portugal)

Nikolay Iskrev (Banco de Portugal, Lisboa, Portugal)

Francesca Loria (Board of Governors of the Federal Reserve System, Washington, DC, USA)

Essays in Honour of Fabio Canova

ISBN: 978-1-80382-636-3, eISBN: 978-1-80382-635-6

ISSN: 0731-9053

Publication date: 16 September 2022

pdf (4.5 MB) ePub (10.9 MB)

Abstract

Since its introduction by Chari, Kehoe, and McGrattan (2007), Business Cycle Accounting (BCA) exercises have become widespread. Much attention has been devoted to the results of such exercises and to methodological departures from the baseline methodology. Little attention has been paid to identification issues within these classes of models. In this chapter, the authors investigate whether such issues are of concern in the original methodology and in an extension proposed by Šustek (2011) called Monetary Business Cycle Accounting. The authors resort to two types of identification tests in population. One concerns strict identification as theorized by Komunjer and Ng (2011) while the other deals both with strict and weak identification as in Iskrev (2010). Most importantly, the authors explore the extent to which these weak identification problems affect the main economic takeaways and find that the identification deficiencies are not relevant for the standard BCA model. Finally, the authors compute some statistics of interest to practitioners of the BCA methodology.

Keywords

Citation

Brinca, P., Iskrev, N. and Loria, F. (2022), "On Identification Issues in Business Cycle Accounting Models", Dolado, J.J., Gambetti, L. and Matthes, C. (Ed.) Essays in Honour of Fabio Canova (Advances in Econometrics, Vol. 44A), Emerald Publishing Limited, Leeds, pp. 55-138. https://doi.org/10.1108/S0731-90532022000044A004

Publisher

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Emerald Publishing Limited

1 Introduction

The business cycle accounting (BCA) procedure developed by Chari et al. (2007) and recently revived by Brinca, Chari, Kehoe, and McGrattan (2016) has sparked great interest among quantitative and theoretical economists. This tool allows to look the data through the lens of a standard business cycle model and, most importantly, to detect and quantitatively assess to which extent and in which equilibrium conditions the model performs better or worse. The so-called ‘wedges’ – which in practical terms are whatever makes the equilibrium conditions not hold – can be mapped into frictions of richer models and vice versa. In this context, the BCA exercise can thus be thought of as some ex ante diagnosis tool for researchers to know in which broad classes of models it is worth or not worth investing time if one wants to explain fluctuations in macroeconomic aggregates such as GDP, investment and working hours during a particular economic episode.

BCA has become a standard tool of business cycle analysis and, since its inception, literally hundreds of applications of the original methodology, have been performed.

Examples can be found in Kobayashi and Inaba (2006) for Japan, Simonovska and Söderling (2008) for Chile and Lama (2009) for Argentina, Mexico and Brazil. The results seem to conclude, much in line with Chari et al. (2007), that total factor productivity and distortions to the labour choice are relevant, whereas distortions to the savings decision are considerably less important. Some authors focus their analysis to one type of distortions as in Restrepo-Echavarria and Cheremukhin (2010) or Cociuba and Ueberfedt (2010), where the focus is on the labour-leisure margin, or other numerous studies which deal with total factor productivity, such as Islam, Dai, and Sakamoto (2006). Another line of work looks into a selected sample of countries and a specific periods of fluctuations such as output drops (see, e.g., Dooyeon & Doblas-Madrid, 2012). Brinca (2014) instead provides a comprehensive exercise for 22 OECD countries covering the period from 1970 to 2012. It looks at the quantitative relevance of each distortion over the whole business cycle and not just booms or busts. The results confirm the findings of Chari et al. (2007) regarding the Great Depression and the 1981 recession in the United States, while stressing the relevance of the international channels of transmission of these distortions.

In terms of methodological departures, Otsu (2009) extends the methodology to a two-country setting. Šustek (2011) includes a Taylor-type nominal interest rate setting rule and an extra asset, government bonds, to study the behaviour of nominal variables such as the nominal interest rate on bonds and the inflation rate, and gives it the name of monetary business cycle accounting (MBCA). These departures have also been explored. For instance, Brinca (2013) applies Šustek (2011) model to perform a MBCA exercise for Sweden, comparing the 1990s crisis with the period of the Great Recession.

While much attention has been devoted to the BCA methodology, no efforts have been made in investigating whether the parameter estimation procedure associated with this tool is affected by identification deficiencies. The question of identifiability in dynamic stochastic general equilibrium (DSGE) is an important one as it might jeopardize the consistency and adversely affect the precision of parameter estimates. This issue becomes even more important in light of the fact that, in the past years, DSGE models have become a standard and important asset within the toolkit of economic policy-makers to make quantitative statements about real and nominal variables. When these models are brought to the data researchers should be cautious in taking for granted the empirical credibility of their estimated parameters and, thus, of the economic implications that the latter entail. Indeed, due to identifiability issues inherent to DSGE models, it is far from obvious that parameters can be inferred successfully even when one has an infinite sample of observed data and when full-information methods such as maximum likelihood (ML) are employed in estimation. Most importantly, as pointed out by Canova and Sala (2009), in some cases these identification deficiencies can result in significantly different economic inference from the theoretical models of interest. To the extent that the BCA exercises by Chari et al. (2007) and Šustek (2011) draw quantitative conclusions from their respective DSGE models of reference it is of outmost importance that their identification potential is carefully analyzed.

This chapter builds up on the literature which studies local sample and population identification issues specific to linearized DSGE models. Our methodological approach to the analysis of such issues is closely related to the work by Canova and Sala (2009) whose contribution was to provide: (i) a working language which allows researchers to classify identification problems; (ii) formal graphical inspection tools to detect those problems; and (iii) possible ways to obviate them. Due to the high number of parameters to be estimated in both models graphical inspection quickly becomes unmanageable and dispersive. This leads us to bring into our toolkit the formal identification tests developed by Komunjer and Ng (2011) and Iskrev (2010).

The results presented in this chapter suggest that the standard and MBCA model do not suffer from strict identification failures when estimation is restricted to the parameters governing the law of motion of the latent variables and that this is not true anymore once one extends the estimation to the deep parameters of the model. We show that restricting estimation of some deep parameters can obviate these strict identification failures. We also find that both models are affected by weak identification deficiencies and that these are induced by several parameters of the model not exerting a distinct effect on our objective function of interest, namely the likelihood function of the model. Finally, we explore to which extent this type of identification failure affects the main conclusions to be drawn from (M)BCA exercises. We find that the main takeaways from a standard BCA exercise are not overturned when one explicitly takes into account the weak identifiability of the model’s parameters. We are still investigating whether this result holds through in the MBCA framework.

Finally, we analyze how overall identification strength of the estimated parameter vector varies across sample sizes. To do so, we compute empirical distance measures as in Qu and Tkachenko (2017). We find that in both models the parameter set is overall well identified even if a practitioner is faced with a data set of only 20 data points per observable.

This chapter is organized as follows. Section 2 presents the prototype MBCA economy. The methodology to run the identification tests is illustrated in Section 3. Their results are reported in Section 4 while Section 5 discusses their economic relevance. In Section 6, we present statistics of interest to practitioners and then conclude in Section 7.

2 The Prototype (M)BCA Economy

We start by introducing the MBCA prototype economy as in Šustek (2011), which is an extension of the prototype economy in Chari et al. (2007), a neoclassical growth model with labour-leisure choice. The extension consists in allowing for an extra asset (government bonds) and introducing a nominal interest rate setting rule.

2.1 Description of the Economy

There is an infinitely lived representative agent that maximizes expected discounted utility and a representative firm, both price-takers in all markets. The economy experiences one of finitely many events s_t, where st=(s0,…,st) is the history of events up to period t which occur with probability πt(st). There are six exogenous stochastic variables which are all function of the random variable s_t. Four of them are the same as in Chari et al. (2007). These are the efficiency wedge Zt(st), the labour wedge 1−τl,t(st), the investment wedge 1/[1+τx,t(st)] and the government wedge gt(st). With the Šustek (2011) extension, two more stochastic variables are added: an asset market wedge 1/[1+τb,t(st)] and a monetary policy wedge R˜t(st).

The representative household chooses how much to consume ct(st) and how much labour to supply lt(st). Given the discount factor β it solves the following maximization problem:

(2.1)max{ct(st),lt(st),kt+1(st),bt(st)}E0Σt=0∞βtUct(st),1−l(st)(1+gn)t,

subject to the budget constraint:

(2.2)ct(st)+[1+τx(st)]xt(st)+[1+τb(st)](1+gn)bt(st)[1+Rt(st)]pt(st)−bt−1(st−1)pt(st)=[1−τl(st)]wt(st)lt(st)+rt(st)kt(st−1)+Tt(st)

where xt(st) is investment, g_n the population growth rate, bt(st) are bond holdings paying a net nominal rate of return Rt(st) in all states of the world, pt(st) is the nominal price of goods in terms of a numeraire, wt(st) the wage rate, rt(st) the real rental rate of return on capital kt(st) held at the beginning of period and Tt(st) lump-sum transfers from the government. Capital accumulation follows:

(2.3)(1+gn)kt+1(st)=(1−δ)kt(st)+xt(st)

where δ is capital’s depreciation rate. The production function for the representative firm is given by

(2.4)yt(st)=Fkt(st−1),Zt(st)lt(st),

which is assumed to exhibit constant returns to scale (CRS). The aggregate resource constraint is then given by

(2.5)yt(st)=ct(st)+gt(st)+xt(st).

There is a monetary authority who reacts to deviations from steady-state output y and inflation π by setting the nominal interest rate Rt(st) according to:

(2.6)Rt(st)=(1−ρR)R+ωy(lnyt(st)−lny)+ωπ(πt(st)−π)+ρRRt−1(st−1)+R˜t(st)

where ρR∈[0,1) and πt(st)≡lnpt(st)−lnpt−1(st−1) is the inflation rate. In addition, it is assumed that ωπ>1, thus eliminating explosive paths for inflation.

Just like in Chari et al. (2007) it is assumed that the state s_t follows a Makov process of the type

(2.7)st+1=P0+Pst+Qεs,t+1,

where εs,t+1∼N(0,I). Moreover, the mapping between this process of the underlying event st=(sZt,slt,sxt,sgt,sbt,sR˜t) and the wedges Zt,1−τl,t,11+τx,t,gt,11+τb,t,Rt˜ is one-to-one and onto. This setup is equivalent to assuming that agents use only past realizations of wedges to forecast future ones. Note that in the case where the matrix P is diagonal then, irrespectively of whether the covariance matrix QQ′ QQ′is diagonal or not (i.e. whether the shocks are allowed to be correlated or not), the MBCA model is block-recursive in the sense that shocks to the wedges of the standard BCA setup only affect real variables while leaving the model’s nominal variables – interest rate R_t and inflation rate π_t– unaffected. In this sense, we can think of the MBCA theoretical framework as nesting the plain-vanilla BCA one. This is why we avoid a detailed exposition of the latter.

2.2 Equilibrium Conditions

Equilibrium allocations are pinned down by the production function in (2.4), the aggregate resource constraint in (2.5) and the first-order conditions with respect to labour, capital and bond holdings below:

(2.8)−Ul,t(st)Uc,t(st)=[1−τl,t(st)]wt(st),

(2.9)1=βEt{Uc,t+1(st+1)Uc,t(st)[[1+τx,t+1(st+1)](1−δ)+rt+1(st+1)1+τx,t(st)]},

(2.10)1=βEt{Uc,t+1(st+1)Uc,t(st)1+τb,t+1(st+1)1+τb,t(st)pt(st)pt+1(st+1)[1+Rt(st)]}.

In Appendix B, we derive these conditions and present the full-fledged model.

The notation v^≡vt(1+gz)t≡VtNt(1+gz)t refers to model variables V_t which are not only expressed in per-capita terms v_t but also detrended v^t.

2.3 Operational Model

In the operational version of the model which we bring to the data we consider quantities which are not only expressed in per-capita terms but also detrended (see Appendix B.5 for derivations). To highlight the differences between this model’s and the previous model’s variables we introduce the notation v^≡vt(1+gz)t≡VtNt(1+gz)t.

The model is given by the CRS Production Function:

(2.11)y^t(st)=k^t(st−1)αztlt(st)1−α,

the aggregate resource constraint:

(2.12)y^t(st)=c^t(st)+g^t+x^t(st),

the capital accumulation law:

(2.13)(1+gn)(1+gz)k^t+1(zt)=(1−δ)k^t(zt−1)+x^t(zt),

the Taylor rule:

(2.14)Rt(st)=(1−ρR)R+ωy(lny^t(st)−lny^)+ωπ(πt(st)−π)+ρRRt−1(st−1)+R˜t,

the F.O.C. for labour:

(2.15)ψc^t(st)1−lt(st)=(1−τl,t)(1−α)k^t(st−1)αzt1−αlt(st)−α,

the F.O.C. for capital:

(2.16)1=β˜Etc^t+1c^t−σ1−lt+11−ltψ(1−σ)(1+τx,t+1)(1−δ)+αk^t+1(st)α−1zt+1lt+1(st+1)1−α1+τx,t

and the F.O.C. Bonds:

(2.17)1=β˜Etc^t(st)c^t+1(st+1)1+τb,t+11+τb,tpt(st)pt+1(st+1)[1+Rt(st)],

where β˜=β/(1+gz).

Notice that the operational model the efficiency wedge is thus given by zt=Zt(1+gn)t and the government wedge by g^t=gt(1+gn)t.

The structural parameters of the BCA model are: (i) g_n, population growth rate of labour-augmenting technological process; (ii) g_z, growth rate of labour-augmenting technological process; (iii) α, parameter which determines the share of (and weight on) net capital stock in the Cobb-Douglas CRS production function; (iv) β, subjective discount factor, reflecting the time preference of the household; (v) δ, depreciation rate of net capital stock; and (vi) ψ, Frisch elasticity of labour supply. In the MBCA model the deep parameters also include (vii) ρ_R, weight on lagged nominal interest rate in the Taylor rule (extent of ‘interest rate smoothing’); (viii) ωπ, coefficient on deviations of inflation from its steady-state value in Taylor rule; (ix) ω_y, coefficient on deviations of output from its steady-state value in Taylor rule; and (x) π_ss, steady-state inflation.

3 Methodology

The problem of identification emerges when a researcher seeks to infer the parameters of his theoretical model from a sample of observed data. In general and loosely speaking, the requirements for ‘successful estimation’ are that (i) the objective function (e.g. the log-likelihood function) has a unique maximum; (ii) the Hessian at the mode is negative definite and has full rank; and that (iii) the objective function exhibits ‘sufficient’ curvature. Local identifiability is crucial as it guarantees consistent parameter estimation and that the estimator has the usual asymptotic properties, where by ‘local’ we mean that the maximum in condition (i) above is at least locally unique. It is thus of outmost importance to investigate thoroughly whether the conditions for local identifiability are satisfied within the standard and MBCA framework.

In this analysis, we employ strict and weak identification tests. Both are performed in population, that is, given a theoretical sample of infinite length. The former provides a yes or no answer to the question whether a parameter is identifiable or not and, thus, whether condition (ii) above is satisfied or not. The latter investigates condition (iii) above and is thus informative about whether the likelihood exhibits small curvature, where small is in relation to an economically relevant parameter range. The weak identification test also instructs about whether the small curvature is due to the fact that the parameters have no effect on the objective function or to the fact that the variation which they induce on the objective function is not distinct enough from other parameters which are being estimated.

As to the strict identification tests in population, consider the case where the researcher has a sample of length T generated by (3.4) with θ=θ0. In this context, one can ask the following question: If the sample was infinitely large, that is, T→∞, under which conditions would it be possible to uncover the value θ₀ and the model that generated the data? Problems specific to the dynamic nature of DSGE models make it difficult to test whether the conditions for local identifiability mentioned above hold given a sample of data. Indeed, as pointed out by Canova and Sala (2009), the mapping from the structural parameters to the solution coefficients is typically unknown and the latter, in turn, usually appear in a non-linear way in the objective function. These are some of the reasons why the rank and order conditions of Rothenberg (1971) derived for simultaneous system of equations cannot be applied. As pointed out by Komunjer and Ng (2011) other reasons are that classical conditions: (i) are derived for static models whose reduced-form errors are orthogonal to the regressors, an assumption which is implausible for DSGE models given that they are dynamic; (ii) do not recognize the fact that reduced-form parameters themselves can be not identifiable unless all state variables are observed: and (iii) are for simultaneous system of equations which is not the form of DSGE model solutions.

Alternative rank and order conditions for strict identification are derived by Komunjer and Ng (2011) using the spectral density matrix and by Iskrev (2010) using the likelihood function of the DSGE model. Both tests treat parameter identification as a property of the underlying structural model. This is motivated by the fact that DSGE models completely characterize the data generating process. This is in contrast with other types of models where the mapping from the model to the data is only partially known. Therefore, the economic model is the origin of identification problems which appear in a particular data set. It is then straightforward to see that identification problems may occur as an intrinsic property of the model when, for instance, the restrictions that the model imposes on the joint distribution of the observed variables do not contain sufficient information about some parameters of interest. It is important to recognize the fact that, in general, these restrictions are a function of the parameters. Hence, also the data crucially contributes to identify those parameter values for which the model can account well for the movements in the data.

An advantage of the Iskrev (2010) approach over Komunjer and Ng (2011) is that key objects of interest are obtained analytically rather than numerically. Furthermore, it can detect not only strict but also weak identification problems. A central tool in his analysis is the expected Fisher information matrix, as first suggested by Rothenberg (1971). It is intuitive to understand why this matrix comes handy to study identification problems. The information matrix measures the curvature of the expected log-likelihood surface and, as pointed out by Rothenberg (1971), it is informative about the (degree of) informational content available in the sample about the unknown parameters. For instance, one should expect identification deficiencies to arise when the log-likelihood surface is flat or nearly flat with respect to the parameters to be estimated. The degree of ‘flatness’ can be detected and quantified via the information matrix.

There are two main reasons why parameters might be unidentifiable or just weakly identifiable; they can be broadly classified as a ‘sensitivity’ and a ‘collinearity’ factor. They originate from the fact that the economic features which operate via the problematic parameters may be nearly or completely irrelevant with respect to the variables of the model used in estimation. The ‘sensitivity’ factor signals that the identification problem occurs because the features are inherently unimportant while the ‘collinearity’ factor attributes it to the nearly redundant nature of these features given others present in the model. The information matrix can be used to assess the importance of the two factors via a simple decomposition. This analysis thus allows not only to flag problematic parameters but also to quantify the strength and discern the nature of their identification deficiencies.

3.1 State Space Form

Following Chari et al. (2007) the state space form of the model is given by:

(3.1)Xt+1=A(θ)Xt+B(θ)εt+1,Yt=C^(θ)Xt+ωt,ωt=D^(θ)ωt−1+ηt,

where, in the standard BCA model, the vectors of state variables and the vector of observables are, respectively, given by Xt=[log(k^t)−log(k),log(zt)−log(z),τlt−τl,τxt−τx,log(g^t)−log(g^),1]′ Yt=[logy^t−log(y^),logx^t−log(x^),loglt−log(l),logg^t−log(g^)]′, whereas in the MBCA model Xt=[log(k^t)−log(k),log(zt)−log(z),τlt−τl,τxt−τx, log(g^t)−log(g^),τbt−τb,R˜t−R˜,1]′,Yt=[logy^t−log(y^),logx^t−log(x^),loglt−log(l),logg^t−log(y^),Rt−R,πt−π]′ .

The parameters of the model are collected in a vector θ and belong to a set Θ⊆ℝnθ. The matrices A,B,C^ and D^ are, respectively, the ones describing: (i) the transition of the states; (ii) the variance covariance matrix of the shocks to the wedges ut≡B(θ)εt given by Ω≡BB′ since it is assumed that E[εtεt′]=I; (iii) the mapping from the states to the observables; and (iv) the serial correlation of the measurement errors (set to 0 here).

For both models, the state space matrices A(θ) and B(θ) are obtained by solving the rational expectations systems using the Gensys algorithm developed by Sims (2002) (see Appendix C for more details).

Let us assume that Eηtη′t=R and Eεtη′s=0 for all periods t and s. Next, define Y¯t≡Yt+1−D^Yt. Then we can rewrite (3.1) as

(3.2)Xt+1=A(θ)Xt+B(θ)εt+1,Y¯t=C¯(θ)Xt+C^(θ)B(θ)εt+1+ηt+1,

where C¯(θ)=C^(θ)A(θ)−D^(θ)C^(θ).

Stacking the vector of innovations and measurement errors into a nϵ×1 vector ϵt=ε′t,η′t yields the following representation:

(3.3)Xt+1=A(θ)Xt+B(θ)εt+1,Y¯t=C¯(θ)Xt+D¯(θ)ϵt+1.

In all periods and all identification tests we set the measurement errors equal to 0 so that D^=R=04×4 and

(3.4)Xt+1︸nX×1=A(θ)︸nX×nXXt︸nX×1+B(θ)︸nX×nεεt+1︸nε×1,Yt+1︸nY×1=C(θ)︸nY×nXXt︸nX×1+D(θ)︸nY×nεεt+1︸nε×1,

where C(θ)=C^(θ)A(θ) and D(θ)=C^(θ)B(θ).

3.2 Estimated Parameters

The estimated parameters are those governing the stochastic process:

(3.5)st+1=P0+Pst+Qεs,t+1

underlying the wedges and are thus the ones appearing in the matrices P₀, P and Q. More specifically, for the standard BCA model the stochastic process of the wedge shocks takes the form:

(3.6)logzt+1τlt+1τxt+1logg^t+1︸st+1=z¯τ¯lτ¯xg¯︸P0+ρzρz,τlρz,τxρz,gρτl,zρτlρτl,τxρτl,gρτx,zρτx,τlρτxρτx,gρg,zρg,τlρg,τxρg︸Plogztτltτxtlogg^t︸st+q11000q21q2200q31q32q330q41q42q43q44︸Qεz,t+1ετl,t+1ετx,t+1εg,t+1︸εs,t+1,

The estimated steady-state vector of wedge shocks [logzt+1,τlt+1,τxt+1,logg^t+1]′ are given by (I4×4−P)−1P0=[log(z)τlτxlog(g^)]′ which we define as [zssτlssτxssgss]′ for convenience.

In the MBCA model the stochastic process of the wedge shocks takes the form:

(3.7)logzt+1τlt+1τxt+1logg^t+1τbt+1R˜t+1︸st+1=z¯τ¯lτ¯xg¯τ¯bR¯︸P0+ρzρz,τlρz,τxρz,gρz,τbρz,R˜ρτl,zρτlρτl,τxρτl,gρτl,τbρτl,R˜ρτx,zρτx,τlρτxρτx,gρτx,τbρτx,R˜ρg,zρg,τlρg,τxρgρg,τbρg,R˜ρτb,zρτb,τlρτb,τxρτb,gρτbρτb,R˜ρR˜,zρR˜,τlρR˜,τxρR˜,gρR˜,τbρR˜︸Plogztτltτxtlogg^tτbtR˜t︸st+q1100000q21q220000q31q32q33000q41q42q43q4400q51q52q53q54q550q61q62q63q64q65q66︸Qεz,t+1ετl,t+1ετx,t+1εg,t+1ετb,t+1εR˜,t+1︸εs,t+1.

The estimated steady-state vector of wedge shocks [logzt+1,τlt+1,τxt+1,logg^t+1,τbt+1,R˜t+1] is given by (I6×6−P)−1P0=[log(z)τlτxlog(g^)τbR˜] which, again for convenience, we define as [zssτlssτxssgssτbssR˜ss]′. Šustek (2011) imposes τbss=0 and R˜ss=0 in estimation. In the identification tests we explore, inter alia, the case where these two parameters are estimated as well.

In Section 4, we also present results for the case where the structural parameters of the respective models are included in the identification analysis.

3.3 Komunjer and Ng (2011) Test for Strict Identification

In a DSGE model two parameter vectors θ₀ and θ₁ are observationally equivalent if the spectral density matrix evaluated at θ₀ is equal to the spectral density matrix evaluated at θ₁. A parameter set θ0∈Θ is defined to be locally identifiable from the autocovariances of Y_t if there exists an open neighboorhood of θ₀ such that θ₀ and θ₁ being observationally equivalent necessarily implies θ1=θ0.

Formally, Komunjer and Ng (2011) state that two triples (θ0,InX,Inε) and (θ1,T,U) are observationally equivalent if

(3.8)A(θ1)=TA(θ0)T−1

(3.9)B(θ1)=TB(θ0)U

(3.10)C(θ1)=C(θ0)T−1

(3.11)D(θ1)=D(θ0)U

(3.12)Σ(θ1)=UΣ(θ0)U−1

where A(·),B(·),C(·),D(·) and Σ(·) are the matrices of the state space model described in (3.4) with T and U being full rank matrices.

A necessary sufficient condition for identification is thus checking that the mapping

(3.13)δS(θ,T,U)=vec(TA(θ)T−1)vec(TB(θ)U)vec(C(θ)T−1)vec(D(θ)U),vec(UΣ(θ)U−1)

has full rank. The rank condition for local identification at θ₀ when the state space system is square (i.e. nε=nY) is thus given by

(3.14)rankΔS(θ0)=nθ+nX2+nε2,

where

(3.15)ΔS(θ0)≡ΔΛS(θ0),ΔTS(θ0),ΔUS(θ0)≡∂δ(θ0,InX,Inε)∂θ,∂δ(θ0,InX,Inε)∂vecT,∂δ(θ0,InX,Inε)∂vecU.

They also establish the following necessary order condition for identification:

(3.16)nθ+nX2+nε2≤nΛS=(nX+nY)(nX+nε)+nε(nε+1)/2.

It requires the number of equations defined by δ^S to be at least as large as the number of unknowns in those equations and can be rewritten as:

(3.17)nθ≤nYnX+nε(nX+nY−nε)+nε(nε+1)2≡nδ.

Minimality and left-invertibility of the state space system are maintained assumptions of these conditions and we thus verify that they hold in our analysis. The first condition gets fulfilled by rewriting (3.4) in the particular form:

(3.18)X˜t+1=X1,t+1X2,t+1=A˜1(θ)0A˜2(θ)0X1,tX2,t+B˜1(θ)B˜2(θ)εt+1,

(3.19)Yt+1=C˜1(θ)C˜2(θ)X1,t+1X2,t+1,

so that

(3.20)X1,t+1=A˜1(θ)︸A(θ)X1,t+B˜1(θ)︸B(θ)εt+1,

(3.21)Yt+1=C˜1(θ)A˜1(θ)+C˜2(θ)C˜2(θ)︸C(θ)X1,t+C˜1(θ)B˜1(θ)+C˜2(θ)B˜2(θ)︸D(θ)εt+1.

Left-invertibility is ensured by the following assumption: For every θ∈Θ, rank P(z;θ)=nX+nε in |z|>1, where P(z;θ)≡zInX−A(θ)B(θ)−C(θ)D(θ),z∈ℂ.

3.4 Iskrev (2010) Test for Strict and Weak Identification

In the following exposition, we closely follow Iskrev (2010).

3.4.1 Preliminaries

The log-likelihood function of a sample YT of data can be obtained using the sequence of one-step ahead prediction errors et|t−1=Yt−C^(θ)X^t|t−1. The latter can be easily constructed using the one-step ahead forecasts of the state vector X^t|t−1 returned by the Kalman filter. Assuming that the structural shocks are Gaussian implies that the conditional distribution of et|t−1 is Gaussian as well, with mean 0 and covariance matrix St|t−1=C^(θ)Pt|t−1C^(θ)′, where Pt|t−1=EXt−X^t|t−1Xt−X^t|t−1′ can also be obtained from the Kalman filter recursions and is the covariance matrix of the one-step ahead forecasts conditional on information up to time t − 1. The log-likelihood of the sample is then given by

(3.22)lT(θ)=const.−12Σt=1Tlog|St|t−1|−12Σt=1Tet|t−1′St|t−1−1et|t−1,

Under some regularity conditions the ML estimator θ˜T is consistent, asymptotically efficient and asymptotically normally distributed with

(3.23)T(θ˜T−θ0)→dℕ(0,I0−1),

where I0 is the asymptotic Fisher information matrix evaluated at the true value of θ. More formally,

(3.24)I0≡limT→∞1TIT,

where IT is the finite sample Fisher information matrix given by

(3.25)IT≡E∂lT(θ)∂θ′′∂lT(θ)∂θ′.

3.4.2 General Principles of Identification Analysis

Suppose that inference about the parameters of the model collected in the vector θ is made using a sample with T observations of a random vector Y with a known probability density function p(YT;θ), where YT=[Y1′,…,YT′]′. The latter, when considered as a function of θ, contains all available sample information about θ associated with the observed data. It is then straightforward to see that a prerequisite for successful inference about θ is that its values imply distinct values of the density function p(YT;θ). More formally, a point θ0∈Θ is said to be identified if:

(3.26)Prp(YT;θ)=p(YT;θ0)=1⇒θ=θ0.

That is, if the density function yields the same value when evaluated at θ and at θ₀ this implies that θ is equal to θ₀. It is possible to rewrite this condition in terms of the log-likelihood function lT(θ)≡logP(YT;θ):

(3.27)E0lT(θ0)≥E0lT(θ),∀θ.

This follows from Jensen’s inequality, see Rao (1971), and the logarithmic function being concave. It implies that the function H(θ0,θ)≡E0lT(θ)−lT(θ0) achieves a maximum at θ=θ0, and that θ₀ is identified if and only if the maximum is unique. The conditions for local uniqueness of a maximum at θ₀ are that (i) ∂H(θ0,θ)∂θ|θ=θ0=0 and (ii) ∂2H(θ0,θ)∂θ∂θ′|θ=θ0 is negative definite. If the maximum at θ₀ is locally unique then θ₀ is locally identified, that is, there exists an open neighborhood of θ₀ where (3.26) holds ∀θ.¹ One can show that, see Bowden (1973), the condition in (i) is always fulfilled and that the Hessian matrix in (ii) is equal to the negative of the Fisher information matrix. This leads to the following result by Rothenberg (1971): Let θ₀ be a regular point² of the information matrix IT(θ). Then θ₀ is locally identifiable if and only if IT(θ0) is non-singular.

In general, non-singularity of the Fisher information matrix is both necessary and sufficient for local identification.³ The information matrix is singular whenever the expected log-likelihood function is flat at θ₀. In this case, due to the lack of the variability induced by the parameters on the log-likelihood function, it is impossible to make inference about the parameters even with an infinite sample of data. There are two reasons why this might occur, following Iskrev (2010) we call them the ‘sensitivity’ and ‘collinearity’ factors. Either the parameters have no effect on the expected log-likelihood (‘lack of sensitivity’), or different parameter values induce the same changes in the expected log-likelihood (‘perfect collinearity’). It is thus useful to formalize ideas in order to investigate to which extent the two channels are at work. This can be done by using the fact that the information matrix is equal to the covariance matrix of the scores and can thus be expressed as:

(3.28)IT(θ0)=Δ1/2RT(θ0)Δ1/2,

where Δ=diagIT(θ0) is a diagonal matrix containing the variances of the elements of the score vector, and RT(θ0) is the correlation matrix of the score vector. Thus a parameter θi∈θ is locally unidentifiable if:

(I)
Lack of sensitivity: The expected log-likelihood is not affected by small changes in θ_i, that is,
(3.29)Δi≡E∂lT(θ0)∂θi2=−E∂2lT(θ0)∂θi2=0
(II)
Perfect collinearity: The effect of small changes in θ_i on the expected log-likelihood can be offset by varying other parameters, that is,
(3.30)ϱi≡1−1/RTii=1,

where RTii is the ith diagonal element of the inverse of RT. As Iskrev (2010) puts its

[t]he intution about the meaning of ϱi comes from a well-known property of the correlation matrix Tucker, Cooper, and Meredith (1972), which implies that ϱi is the coefficient of multiple correlation between the partial derivative of the log-likelihood with respect to θ_i and the partial derivatives of the log-likelihood with respect to the other elements of θ.

Conditions (I) and (II) characterize the case in which the expected log-likelihood is completely flat and the parameters are thus not identifiable in a strict sense. The case of weak identification, on the other hand, arises when the expected log-likelihood features little curvature with respect to some parameters. We delve further into this issue next.

3.4.3 Identification Strength

Local identifiability guarantees, in general, consistent estimation of θ. The precision with which θ is estimated is governed by the curvature of the expected log-likelihood function in the neighborhood of θ₀, of which the rank conditions above are not informative. Identification is weak whenever small changes in θ do not induce sufficiently large changes in lT(θ) or, equivalently, when small changes in lT(θ) are associated with large changes in θ. By weak we mean that the estimates are prone to be imprecisely estimated even in the presence of an infinitely large sample of data. The degree of ‘weakness’ is thus related to the degree of precision. The latter is not an absolute but rather a relative concept which varies according to the application at hand.

We already saw that a central tool when investigating whether a parameter is locally identifiable or not is the Fisher information matrix. This is because, as shown in Rothenberg (1971), the latter is indicative of the degree of curvature of the expected log-likelihood function. To understand the next logical step in the analysis, namely the relationship between the curvature and the precision of the ML estimator θ^T it is useful to recall its asymptotic distribution described in (3.23). It is then straightforward to see that IT−1(θ0)/T is the sample counterpart of the covariance matrix of θ^T and, analogously, ITii−1(θ0)/T is the sample counterpart of the variance of θ^i.

Asymptotic efficiency of ML estimation implies that θ^T has the smallest asymptotic covariance matrix within the class of consistent estimators. This follows directly from the fact that, according to the Cramér-Rao theorem, the lower bound of the asymptotic covariance of any consistent estimator θ is given by the inverse of its asymptotic information matrix I0. Concurrently, the covariance matrix of any unbiased estimator is bounded below by the inverse of the sample information matrix IT so that bi≡ITii−1 represents the lower bound on the variance of any unbiased estimator θ_i. To measure identification strength we can thus construct bounds on 1 standard deviation intervals for the individual parameters.

As shown in Iskrev (2010), it is possible to relate the size of the bounds to the potential roots of identification deficiencies. Indeed, using the decomposition of IT(θ) in (3.25) and the properties of the correlation matrix one obtains:

(3.31)bi=1Δi(1−ρi)2.

When can one ascribe the identification problem to the ‘sensitivity’ or ‘correlation’ factor? In the first case, we know that the parameter does exert an irrelevant or weak effect on the likelihood, in which case Δi≈0. In the second case, the effects induced on the likelihood by one parameter are compensated, and thus made redundant, by changes in other parameters, in which case ρi≈1. In both cases, the sources of weak identification lead to a large b_i and make inference about a parameter value challenging at best.

Notice that the sensitivity factor alone cannot guarantee successful parameter identification. Indeed, even if Δ_i is large, nearly perfect collinear effects of a parameter θ_i with respect to the other parameters θ−i lead to values of ρ_i close to 1, in which case identification remains weak. This example illustrates well the difference between the information about θ_i contained in the likelihood when the other parameters are known, see Δ_i, and when they are not known, see b_i. This second source of information is smaller and the difference is increasing in the ‘correlation’ factor, see ϱi.

4 Results

In this section, we report results for the tests by Komunjer and Ng (2011) and Iskrev (2010). Results for the case where investment adjustment costs are introduced can be found in Appendix A. In the BCA and MBCA models, strict and local identifications are checked at the parameter set estimated (or fixed, as it is the case for the deep parameters) in Chari et al. (2007) and Šustek (2011), respectively.

4.1 Komunjer and Ng (2011)

We explore whether a parameter is identifiable in a strict sense. To answer this question in population we show results of the test by Komunjer and Ng (2011) first. Since this test requires computing the rank of the matrix ΔS(θ0) in (3.15) to check whether the rank condition for strict identification holds we report results for different tolerance levels. Indeed, the rank of a matrix is equal to the number of its non-zero eigenvalues which are found by numerical routines using a cut-off to establish whether they are sufficiently small. Matlab, for instance, uses the tolerance Tol = max(size(M))EPS ||M||, where EPS is the float point precision of M. As pointed out by Komunjer and Ng (2011) this default tolerance does not take into account the fact that the matrix ΔS(θ0) is often sparse and can thus be misleading. Results are thus reported for 11 tolerance values ranging from a maximum tolerance of 1e−2 to a minimum of 1e−11, along with the Matlab default one.

To isolate the parameters which are not strictly identifiable even with an infinite sample of data combinations of parameters which cause full rank failures in the ΔS(θ0) are searched by inspecting its change in rank and its null space. This is first done at a high tolerance level of 1e−3 to flag the most difficult parameters to identify and then at the lowest tolerance level for which identification fails in order to find additional problematic parameters. Komunjer and Ng (2011) choose the highest tolerance level of 1e−3 ‘on the grounds that the numerical derivatives are computed using a step size of 1e−3’. In Appendix D, we report results for the case where a Matlab selected measure of the step size is being used when computing numerical derivatives. When this other measure is used the results suggest more identification power in both models.

In general, the lower the tolerance level the higher the rank of the matrix since more of the smallest eigenvalues are considered to be numerically different from 0. Then, according to the rank-nullity theorem – which states that the sum of the rank and the nullity of a matrix is equal to its number of columns – the null space will be smaller as well. This would lead to think that the set of parameters which are flagged as troublesome should be larger the lower the tolerance level. However, the way the set of problematic parameters is found by identifying the number of columns of the orthonormal basis for the null space of the matrix ΔTS – obtained via singular value decomposition – whose (absolute) sum of elements is greater than the tolerance value (i.e. numerically larger than 0). This is because the vectors contained in a basis must be linearly independent and a null vector would always be linearly dependent with the other vectors. Thus, the lower the tolerance level, the more columns (and associated parameters) will fall within this set, the lower is the null space of the matrix ΔTS and, hence, the smaller is the set of parameters which are not strictly identifiable.

Finally, we perform conditional identification tests. More specifically, when the rank condition for strict identification fails, we check which parameters, when restricted, can enable identification of the remaining parameters.

4.1.1 Chari et al. (2007) BCA Model

Table 1 reveals that the model’s parameters are strictly identifiable at Tol = 1e−11 and at the Matlab default tolerance level. When inspecting the null space of ΔS(θ0) no problematic parameters are found. This result might be explained by the fact that the lower is the tolerance level, the smaller is the set of problematic parameters, as explained above.

Table 1.

Komunjer and Ng Test Results BCA Model.

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	29	25	15	51	40	62	0
e−03	29	25	16	54	45	69	0
e−04	29	25	16	a	45	69	0
e−05	29	25	16	54	45	69	0
e−06	29	25	16	54	45	69	0
e−07	29	25	16	54	45	69	0
e−08	30	25	16	54	46	70	0
e−09	30	25	16	54	46	70	0
e−10	30	25	16	55	46	70	0
e−11	30	25	16	55	46	71	1
Default = 2.756906e−12	30	25	16	55	46	71	1
Required	30	25	16	55	46	71	1

Summary: nθ=30,nX=5,nε=4.

Order condition: nθ=30,nδ=50.

Moving to the case where also the deep parameters of the model are included in the identification test reveals a different picture, as reported in Table 2. Indeed, the extended parameter set does not pass the test at any tolerance value and several problematic parameters are found. At Tol = 1e−3 the latter mainly concern some steady-state values of the wedge shocks, off-diagonal elements of the matrix P and all deep parameters. Once the tolerance is further lowered to Tol = Default also several off-diagonal elements of the Q matrix are flagged as troublesome. At the default tolerance level, the model’s parameters might be strictly identifiable if at least two restrictions are imposed. Indeed, all matrices are full rank with the exception of ΛS which is short rank by 2. We thus check whether fixing some parameters alleviates the strict identification deficiencies. As emerges from Table 3 there are several sets of restricted parameter combinations which allow to do so.

Table 2.

Komunjer and Ng Test Results BCA Model (Deep Parameters Estimated).

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	33	25	15	52	44	64	0
e−03	35	25	16	57	51	71	0
e−04	35	25	16	57	51	71	0
e−05	35	25	16	58	51	71	0
e−06	35	25	16	58	51	71	0
e−07	36	25	16	58	52	72	0
e−08	36	25	16	59	52	74	0
e−09	37	25	16	60	53	75	0
e−10	37	25	16	60	53	76	0
e−11	37	25	16	60	53	76	0
Default = 5.513812e−12	37	25	16	61	53	76	0
Required	37	25	16	62	53	78	1

Summary: nθ=37,nX=5,nε=4.

Order condition: nθ=37,nδ=50.

Problematic parameters at Tol = 1e−3: z_ss, τlss, g_ss, ρτl,z, ρz,τl, ρτx,τl, ρg,τl, ρτl,τx, ρτl,g, q₂₂, g_n, g_z, β, ψ, σ.

Problematic parameters at Tol = 5.513812e−12: z_ss, τlss, τxss, g_ss, ρ_z, ρτl,z, ρτx,z, ρg,z, ρz,τl, ρτl, ρτx,τl, ρg,τl, ρz,τx, ρτl,τx, ρτx, ρg,τx, ρz,g, ρτl,g, ρτx,g, ρ_g, q₂₁, q₃₁, q₂₂, q₃₂, q₄₂, q₃₃, q₄₃, q₄₄, g_n, g_z, β, δ, ψ, σ, α.

Table 3.

Komunjer and Ng Conditional Test Results BCA Model, Tol = 5.513812e−12.

Fixed	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
τlss z_ss	37	25	16	62	53	78	1
ρτl,z z_ss	37	25	16	62	53	78	1
ρz,τl z_ss	37	25	16	62	53	78	1
ρτx,τl z_ss	37	25	16	62	53	78	1
ρg,τl z_ss	37	25	16	62	53	78	1
ρτl,τx z_ss	37	25	16	62	53	78	1
ρτl,g z_ss	37	25	16	62	53	78	1
q₂₁ z_ss	37	25	16	62	53	78	1
q₂₂ z_ss	37	25	16	62	53	78	1
ψz_ss	37	25	16	62	53	78	1
g_ss τlss	37	25	16	62	53	78	1
g_n τlss	37	25	16	62	53	78	1
g_z τlss	37	25	16	62	53	78	1
β τlss	37	25	16	62	53	78	1
ρτl,z g_ss	37	25	16	62	53	78	1
ρz,τl g_ss	37	25	16	62	53	78	1
ρτx,τl g_ss	37	25	16	62	53	78	1
ρg,τl g_ss	37	25	16	62	53	78	1
ρτl,τx g_ss	37	25	16	62	53	78	1
ρτl,g g_ss	37	25	16	62	53	78	1
q₂₁ g_ss	37	25	16	62	53	78	1
q₂₂ g_ss	37	25	16	62	53	78	1
ψg_ss	37	25	16	62	53	78	1
g_n ρτl,z	37	25	16	62	53	78	1
g_z ρτl,z	37	25	16	62	53	78	1
β ρτl,z	37	25	16	62	53	78	1
g_n ρz,τl	37	25	16	62	53	78	1
g_z ρz,τl	37	25	16	62	53	78	1
β ρz,τl	37	25	16	62	53	78	1
g_n ρτx,τl	37	25	16	62	53	78	1
g_z ρτx,τl	37	25	16	62	53	78	1
β ρτx,τl	37	25	16	62	53	78	1
g_n ρg,τl	37	25	16	62	53	78	1
g_z ρg,τl	37	25	16	62	53	78	1
β ρg,τl	37	25	16	62	53	78	1
ρτl,g ρτl,τx	37	25	16	62	53	78	1
g_n ρτl,τx	37	25	16	62	53	78	1
g_z ρτl,τx	37	25	16	62	53	78	1
β ρτl,τx	37	25	16	62	53	78	1
g_n ρτl,g	37	25	16	62	53	78	1
g_z ρτl,g	37	25	16	62	53	78	1
β ρτl,g	37	25	16	62	53	78	1
g_n q₂₁	37	25	16	62	53	78	1
g_z q₂₁	37	25	16	62	53	78	1
β q₂₁	37	25	16	62	53	78	1
g_n q₂₂	37	25	16	62	53	78	1
g_z q₂₂	37	25	16	62	53	78	1
β q₂₂	37	25	16	62	53	78	1
ψ g_n	37	25	16	62	53	78	1
ψ g_z	37	25	16	62	53	78	1
ψ β	37	25	16	62	53	78	1
Required	37	25	16	62	53	78	1

Summary: nθ=37,nX=5,nε=4.

Order condition: nθ=37,nδ=50.

4.1.2 Šustek (2011) MBCA Model

The results for the MBCA model by Šustek (2011) resemble the ones just reported for the standard BCA model. Indeed the MBCA model fulfils the rank condition for strict identification established by Komunjer and Ng (2011) at Tol = 1e−11 and at the Matlab default tolerance when the baseline set of estimated parameters is considered (Table 4) or when the latter contains also [τbss,R˜ss] (Table 5). This is no longer true once the deep parameters are included in estimation. Indeed, also in this case, several steady-state wedge shocks and off-diagonal elements of the P and Q matrix are not strictly identifiable, though this is only true when both [τbss,R˜ss] and the deep parameters of the model are included in estimation (Table 6). Indeed, when only the deep parameters of the model are included in estimation on top of the baseline set of parameters considered in Šustek (2011) then this is only true at a low Tol = Default since at Tol = 1e−3 only a few steady-state wedge shocks are found to not meet the requirements for strict identifiability (Table 7). A similar pattern emerges once investment adjustment costs are introduced in the model (see Appendix A).

Table 4.

Komunjer and Ng Test Results MBCA Model.

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	60	49	33	101	89	130	0
e−03	60	49	36	108	96	143	0
e−04	60	49	36	109	96	144	0
e−05	60	49	36	109	96	144	0
e−06	60	49	36	109	96	144	0
e−07	60	49	36	109	96	145	0
e−08	60	49	36	109	96	145	0
e−09	61	49	36	109	97	145	0
e−10	61	49	36	109	97	145	0
Default = 1.165290e−11	61	49	36	110	97	146	1
e−11	61	49	36	110	97	146	1
Required	61	49	36	110	97	146	1

Summary: nθ=61,nX=7,nε=6.

Order condition: nθ=61,nδ=105.

Table 5.

Komunjer and Ng Test Results MBCA Model (τbss and R˜ss Estimated).

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	62	49	33	103	91	131	0
e−03	62	49	36	110	98	144	0
e−04	62	49	36	111	98	146	0
e−05	62	49	36	111	98	146	0
e−06	62	49	36	111	98	146	0
e−07	62	49	36	111	98	147	0
Default = 1.193257e−08	62	49	36	111	98	147	0
e−08	62	49	36	111	98	147	0
e−09	63	49	36	111	99	147	0
e−10	63	49	36	112	99	147	0
e−11	63	49	36	112	99	148	1
Required	63	49	36	112	99	148	1

Summary: nθ=63,nX=7,nε=6.

Order condition: nθ=63,nδ=105.

Problematic parameters at Tol = 1e−3: z_ss, g_ss.

Problematic parameters at Tol = 1.000000e−10: z_ss, τlss, τxss, g_ss, ρ_z, ρτl,z, ρτx,z, ρg,z, ρτb,z, ρR˜,z, ρz,τl, ρτl, ρτx,τl, ρg,τl, ρτb,τl, ρR˜,τl, ρz,τx, ρτl,τx, ρτx, ρg,τx, ρτb,τx, ρR˜,τx, ρz,g, ρτl,g, ρτx,g, ρ_g, ρτb,g, ρR˜,g, ρz,τb, ρτl,τb, ρτx,τb, ρg,τb, ρτb, ρR˜,τb, ρz,R˜, ρτl,R˜, ρτx,R˜, ρg,R˜, ρτb,R˜, ρR˜, q₂₁, q₃₁, q₅₁, q₂₂, q₃₂, q₄₂, q₅₂, q₃₃, q₅₃, q₆₃, q₄₄.

Table 6.

Komunjer and Ng Test Results MBCA Model (τbss,R˜ss and Deep Parameters Estimated).

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	70	49	35	106	100	134	0
e−03	71	49	36	114	107	147	0
e−04	71	49	36	118	107	149	0
e−05	71	49	36	118	107	150	0
e−06	71	49	36	118	107	150	0
e−07	72	49	36	119	108	151	0
Default = 2.386514e−08	72	49	36	120	108	153	0
e−08	72	49	36	120	108	154	0
e−09	73	49	36	120	109	154	0
e−10	73	49	36	121	109	156	0
e−11	74	49	36	122	110	157	0
Required	74	49	36	123	110	159	1

Summary: nθ=74,nX=7,nε=6.

Order condition: nθ=74,nδ=105.

Problematic parameters at Tol = 1e−3: z_ss, τlss, τxss, g_ss, ρ_z, ρτl,z, ρτx,z, ρg,z, ρτb,z, ρR˜,z, ρz,τl, ρτl, ρτx,τl, ρg,τl, ρτb,τl, ρR˜,τl, ρz,τx, ρτl,τx, ρτx, ρg,τx, ρτb,τx, ρR˜,τx, ρz,g, ρτl,g, ρτx,g, ρ_g, ρτb,g, ρR˜,g, ρz,τb, ρτl,τb, ρτx,τb, ρg,τb, ρτb, ρR˜,τb, ρz,R˜, ρτl,R˜, ρτx,R˜, ρg,R˜, ρτb,R˜, ρR˜, q₂₁, q₃₁, q₅₁, q₂₂, q₃₂, q₄₂, q₅₂, q₃₃, q₄₃, q₅₃, q₆₃, q₄₄, q₅₄, q₆₄, q₅₅, g_n, g_z, β, δ, ψ, σ, ρ_R, ωπ, ω_y, π_ss.

Problematic parameters at Tol = 1.000000e−11: z_ss, τlss, τxss, g_ss, τbss, R˜ss, ρ_z, ρτl,z, ρτx,z, ρg,z, ρτb,z, ρR˜,z, ρz,τl, ρτl, ρτx,τl, ρg,τl, ρτb,τl, ρR˜,τl, ρz,τx, ρτl,τx, ρτx, ρg,τx, ρτb,τx, ρR˜,τx, ρz,g, ρτl,g, ρτx,g, ρ_g, ρτb,g, ρR˜,g, ρz,τb, ρτl,τb, ρτx,τb, ρg,τb, ρτb, ρR˜,τb, ρz,R˜, ρτl,R˜, ρτx,R˜, ρg,R˜, ρτb,R˜, ρR˜, q₁₁, q₂₁, q₃₁, q₄₁, q₅₁, q₆₁, q₂₂, q₃₂, q₄₂, q₅₂, q₆₂, q₃₃, q₄₃, q₅₃, q₆₃, q₄₄, q₅₄, q₆₄, q₅₅, q₆₅, q₆₆, g_n, g_z, β, δ, ψ, σ, α, ρ_R, ωπ, ω_y, π_ss.

Table 7.

Komunjer and Ng Test Results MBCA Model (Deep Parameters Estimated).

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS
e−02	68	49	35	104	98	133
e−03	69	49	36	113	105	146
e−04	69	49	36	116	105	148
e−05	69	49	36	116	105	148
e−06	69	49	36	116	105	148
e−07	70	49	36	117	106	149
e−08	70	49	36	118	106	152
e−09	71	49	36	118	107	153
e−10	71	49	36	119	107	154
Default = 2.330580e−11	72	49	36	120	108	155
e−11	72	49	36	120	108	155	Required	72	49	36	121	108	157	1

Summary: nθ=72,nX=7,nε=6.

Order condition: nθ=72,nδ=105.

Problematic parameters at Tol = 1e−3: z_ss, τlss, τxss, g_ss, ρ_z, ρτl,z, ρτx,z, ρg,z, ρτb,z, ρR˜,z, ρz,τl, ρτx,τl, ρg,τl, ρτb,τl, ρR˜,τl, ρz,τx, ρτl,τx, ρg,τx, ρτb,τx, ρR˜,τx, ρz,g, ρτl,g, ρτx,g, ρτb,g, ρR˜,g, ρz,τb, ρτl,τb, ρτx,τb, ρg,τb, ρR˜,τb, ρz,R˜, ρτl,R˜, ρτx,R˜, ρg,R˜, ρτb,R˜, ρR˜, q₂₂, q₅₃, g_n, g_z, β, ψ, σ, ρ_R, ωπ, ω_y, π_ss.

Problematic parameters at Tol = 1.000000e−11: z_ss, τlss, τxss, g_ss, ρ_z, ρτl,z, ρτx,z, ρg,z, ρτb,z, ρR˜,z, ρz,τl, ρτl, ρτx,τl, ρg,τl, ρτb,τl, ρR˜,τl, ρz,τx, ρτl,τx, ρτx, ρg,τx, ρτb,τx, ρR˜,τx, ρz,g, ρτl,g, ρτx,g, ρ_g, ρτb,g, ρR˜,g, ρz,τb, ρτl,τb, ρτx,τb, ρg,τb, ρτb, ρR˜,τb, ρz,R˜, ρτl,R˜, ρτx,R˜, ρg,R˜, ρτb,R˜, ρR˜, q₁₁, q₂₁, q₃₁, q₅₁, q₆₁, q₂₂, q₃₂, q₄₂, q₅₂, q₆₂, q₃₃, q₄₃, q₅₃, q₆₃, q₄₄, q₅₄, q₆₄, q₅₅, q₆₅, q₆₆, g_n, g_z, β, δ, ψ, σ, α, ρ_R, ωπ, ω_y, π_ss.

As to the conditional identification tests, we find that when the parameter set also includes the deep parameters of the model it is possible to fix some model parameters so as to make the rest identifiable (see Table 8) only when τbss and R˜SS are not included in estimation. The sets which restrict the least number of parameters are two and are found at Tol = 1e−11, namely (i) {πSS,zSS} and (ii) {πSS,gSS}

Table 8.

Komunjer and Ng Conditional Test Results MBCA Model, Tol = 2.330580e−11.

Fixed	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
π_ss z_ss	72	49	36	121	108	157	1
π_ss g_ss	72	49	36	121	108	157	1
Required	72	49	36	121	108	157	1

Summary: nθ=72,nX=7,nε=6.

Order condition: nθ=72,nδ=105.

4.2 Iskrev (2010)

In this subsection, we show the results of the parameter identification analysis of the baseline BCA and MBCA models. We only summarize the results on the strict identification tests and then focus on the weak identification analysis.

We start with the standard BCA model in the case where the deep parameters of the model are estimated on top of the ones governing the VAR(1) process of the innovations to the wedges, for a total of 37 parameters. The rank of the information matrix is 33. Using population data it is possible to obtain g_n and g_z can assumed to be known since in our detrending it set to a value such that average detrended output is equal to 0.⁴ With 35 parameters, the rank of the information matrix is still 33. By brute force, we check the rank of the information matrix for all possible combinations of 33 parameters out of the 35 (595 such combinations) being restricted and find that: (i) 21 combinations deliver a rank of 31; (ii) 355 combinations give rank of 32; and (iii) 219 combinations result in a rank of 33. Thus, we can fix 219 possible pairs of parameters and identify the ones left unrestricted in the respective cases.

As to the MBCA model, we start with 74 parameters which corresponds to the case where the deep parameter of the model as well as the steady-state innovations to the asset market and monetary policy wedge (τbssandR˜ss) are estimated on top of the ones governing the VAR(1) process. Fixing g_n and g_z as discussed above leaves 72 parameters and the rank of the information matrix is 68. There are 1,028,790 possible combinations of 68 out of 72 parameters which can be fixed, for 16,652 of them the resulting rank of the information matrix is 68. When all deep parameters are excluded from the analysis, and thus only the wedge parameters are considered 63 parameters are left. In this case, the rank of the information matrix is 62 and fixing any one of the following set of parameters gives a full rank of 62: {τbss,ρτb,z,ρτb,τl,ρτb,τx,ρτb,g,ρz,τb,ρτl,τb,ρτx,τb,ρg,τb,ρR˜,τb,ρτb,R˜,q51,q52,q53,q54,q55}. For instance, fixing τbss as done in Šustek (2011) gives a full rank of 62. If the non-wedge parameters are added 71 parameters are estimated and the rank of the information matrix is 68. There are 57,155 combinations of 68 out of 71 parameters and for 59 of them fixing the relevant parameters gives a rank of 68.

Next, we study whether the parameters are affected by weak identification problems. We report Cramér-Rao lower bounds (CRLBs) as a measure of parameter estimation uncertainty in absolute terms, and relative CRLBs, rCRLB(θi)=CRLB(θi)/abs(θi), as a measure of relative uncertainty. The CRLBs reflect the uncertainty which arises from both the ‘sensitivity’ and the ‘collinearity’ factors. This is because, as discussed in Section 3.4.2, the CRLB is the product of these two components. The first component concerns the sensitivity of the log-likelihood function with respect to θ_i, whereas the second relates to the degree of collinearity between the derivative of the likelihood with respect to θ_i and the derivatives of the likelihood with respect to all other free parameters θ−i.

In order to interpret them it is useful to recall the following facts. The sensitivity factor for a parameter reports the value of the conditional CRLB, that is, the lower bound of uncertainty given that all other parameters are known and no collinearity is present. As to the collinearity factor, it is indicative of the increase in the CRLB when the other parameters are unknown as well. The magnitude of the sensitivity component, unlike the collinearity one, depends on the scale of the parameters and is thus divided by the respective parameter values.

4.2.1 Chari et al. (2007) BCA Model

Table 9 deals with the standard BCA model and the case where the structural parameters are assumed to be known. The first column shows the values of the parameters governing the stochastic process of the wedges, the second one the CRLBs and the third one the relative CRLBs. The latter measure can be used to compare the relative identification strength across parameters which take different values. For instance, z_ss is much worse identified than g_ss. This is because the CRLB of z_ss relative to its value is 6.3701 while only 0.3617 for g_ss. Further examination of Table 9 suggests that the worst identified parameters, using (arbitrarily) a cut-off value of 1, are z_ss, ρτl,z,ρτx,z,ρg,z,ρz,τl,ρτx,τl,ρg,τl,ρz,τx,ρτl,τx,ρg,τx,ρz,g,ρτl,g, q₂₁, q₃₁, q₄₁, q₃₂, q₄₂ and q₄₄.

Table 9.

BCA, Parameter Identification (All Deep Parameters Fixed).

	Value	CRLB	rCRLB
z_ss	−0.0239	0.1524	6.3701
τlss	0.3279	0.2513	0.7662
τxss	0.4834	0.3763	0.7783
g_ss	−1.5344	0.5550	0.3617
ρ_z	0.9800	0.0484	0.0494
ρτl,z	−0.0330	0.0631	1.9123
ρτx,z	−0.0702	0.1135	1.6161
ρg,z	0.0048	0.0627	13.0322
ρz,τl	−0.0138	0.0353	2.5588
ρτl	0.9564	0.0675	0.0706
ρτx,τl	−0.0460	0.1346	2.9252
ρg,τl	−0.0081	0.0578	7.1346
ρz,τx	−0.0117	0.0825	7.0314
ρτl,τx	−0.0451	0.0768	1.7024
ρτx	0.8962	0.0994	0.1109
ρg,τx	0.0488	0.0964	1.9744
ρz,g	0.0192	0.0791	4.1117
ρτl,g	0.0569	0.0760	1.3357
ρτx,g	0.1041	0.0866	0.8321
ρ_g	0.9711	0.0907	0.0934
q₁₁	0.0116	0.0007	0.0578
q₂₁	0.0014	0.0028	2.0187
q₃₁	−0.0105	0.0111	1.0555
q₄₁	−0.0006	0.0013	2.2905
q₂₂	0.0064	0.0004	0.0633
q₃₂	0.0010	0.0099	9.6266
q₄₂	0.0061	0.0085	1.3946
q₃₃	0.0158	0.0110	0.6912
q₄₃	0.0142	0.0034	0.2412
q₄₄	0.0046	0.0047	1.0269

Table 10 considers the case where all parameters in the BCA model are estimated with the exception of four out of the seven structural parameters in the BCA model being held fixed so as to guarantee strict identification of the remaining parameters. As described above, there are many such combinations which would have guaranteed a full rank information matrix. On top of leaving out g_n and g_z which we can calculate using additional data we leave out β and ψ from the analysis. Not surprisingly, the relative uncertainty measures increase. Moreover, the set of worst identified parameters includes also τ_{l_ss}, τ_{x_ss}, ρ_{τ_x,g} and q₃₃.

Table 10.

BCA, Parameter Identification (Some Deep Parameters Free).

	Value	CRLB	rCRLB
z_ss	−0.0239	1.7607	73.6039
τlss	0.3279	0.4714	1.4376
τxss	0.4834	1.5796	3.2674
g_ss	−1.5344	0.7628	0.4972
ρ_z	0.9800	0.0609	0.0622
ρτl,z	−0.0330	0.0683	2.0712
ρτx,z	−0.0702	0.1159	1.6503
ρg,z	0.0048	0.0695	14.4480
ρz,τl	−0.0138	0.0571	4.1391
ρτl	0.9564	0.0750	0.0784
ρτx,τl	−0.0460	0.1441	3.1325
ρg,τl	−0.0081	0.0830	10.2360
ρz,τx	−0.0117	0.1214	10.3566
ρτl,τx	−0.0451	0.0798	1.7692
ρτx	0.8962	0.1288	0.1437
ρg,τx	0.0488	0.0970	1.9863
ρz,g	0.0192	0.1046	5.4386
ρτl,g	0.0569	0.1020	1.7917
ρτx,g	0.1041	0.1243	1.1947
ρ_g	0.9711	0.1141	0.1175
q₁₁	0.0116	0.0057	0.4902
q₂₁	0.0014	0.0040	2.8614
q₃₁	−0.0105	0.0149	1.4217
q₄₁	−0.0006	0.0013	2.3349
q₂₂	0.0064	0.0035	0.5488
q₃₂	0.0010	0.0123	11.8844
q₄₂	0.0061	0.0117	1.9119
q₃₃	0.0158	0.0215	1.3562
q₄₃	0.0142	0.0046	0.3233
q₄₄	0.0046	0.0068	1.4834
δ	0.0118	0.0014	0.1215
σ	1.0000	0.5176	0.5176
α	0.3500	0.3171	0.9059

The ‘sensitivity’ and ‘collinearity’ components of the parameters’ CRLBs for the case where the deep parameters are fixed are reported in Table 11 and labelled as ‘sens.’ and ‘coll.’. It is immediate to see that the channel which drives weak identification is the collinearity one. Indeed, while all parameters exert a strong effect on the likelihood, some of them induce a variation in the likelihood which is very similar to other parameters. For some parameters, the sensitivity factor is so strong that it outweighs the negative effect of the collinearity factor on their overall uncertainty. For others, however, the collinearity factor predominates and leads their relative uncertainty to be high. It is also interesting to take a look at the largest multiple correlation coefficients between ∂lT(θ)/∂θi and ∂lT(θ)/∂θ−i, namely ϱi(n). We report the single and pairwise correlation coefficients in the columns labelled by ϱi(1) and ϱi(2), respectively. The problematic parameters exert an effect on the likelihood which is mostly collinear with the elements of the matrices they belong to in the VAR(1) process of the innovations to the wedges (i.e. P₀, P and Q). The fact that the steady-state innovations to the wedges are strongly correlated with elements of both the P₀ and P matrix is due to the fact that they are obtained as (I−P)−1P0.

Table 11.

BCA, Information Matrix Decomposition (Observing 4 Standard Variables).

	CRLB/para.	sens/para.	coll.	ϱi	ϱi(1)	ϱi(2)
z_ss	6.370	0.575	11.082	0.995920	0.886 (g_ss)	0.931 (g_ss, ρg,z)
τlss	0.766	0.018	41.833	0.999714	0.739 (τxss)	0.905 (τxss,ρz,τx)
τxss	0.778	0.009	82.492	0.999927	0.886 (g_ss)	0.934 (τlss, g_ss)
g_ss	0.362	0.006	59.573	0.999859	0.886 (τxss)	0.971 (z_ss, τxss)
ρ_z	0.049	0.000	153.529	0.999979	0.988 (ρg,z)	0.991 (ρτl,z,ρg,z)
ρτl,z	1.912	0.030	64.792	0.999881	0.907 (ρ_z)	0.982 (ρτl,τx,ρτl,g)
ρτx,z	1.616	0.026	62.859	0.999873	0.908 (ρg,z)	0.974 (ρτx,ρτx,g)
ρg,z	13.032	0.097	134.773	0.999972	0.988 (ρ_z)	0.990 (ρ_z, ρτx,z)
ρz,τl	2.559	0.014	176.574	0.999984	0.988 (ρg,τl)	0.993 (ρg,τl,ρz,g)
ρτl	0.071	0.001	108.028	0.999957	0.978 (ρτl,g)	0.990 (ρτl,τx,ρτl,g)
ρτx,τl	2.925	0.026	112.667	0.999961	0.969 (ρτx,g)	0.986 (ρτx,ρτx,g)
ρg,τl	7.135	0.036	196.423	0.999987	0.988 (ρz,τl)	0.993 (ρg,τx, ρ_g)
ρz,τx	7.031	0.012	600.752	0.999999	0.988 (ρg,τx)	0.999 (ρ_z, ρz,g)
ρτl,τx	1.702	0.009	181.791	0.999985	0.983 (ρτl,g)	0.998 (ρτl,z,ρτl,g)
ρτx	0.111	0.001	126.383	0.999969	0.978 (ρτx,g)	0.997 (ρτx,z,ρτx,g)
ρg,τx	1.974	0.004	474.827	0.999998	0.988 (ρz,τx)	0.999 (ρg,z, ρ_g)
ρz,g	4.112	0.005	881.973	0.999999	0.988 (ρ_g)	0.999 (ρ_z, ρz,τx)
ρτl,g	1.336	0.005	272.329	0.999993	0.983 (ρτl,τx)	0.999 (ρτl,z,ρτl,τx)
ρτx,g	0.832	0.005	163.595	0.999981	0.978 (ρτx)	0.998 (ρτx,z,ρτx)
ρ_g	0.093	0.000	684.987	0.999999	0.988 (ρz,g)	0.999 (ρg,z,ρg,τx)
q₁₁	0.058	0.036	1.628	0.788970	0.780 (q₃₁)	0.788 (q₂₁, q₃₁)
q₂₁	2.019	0.247	8.160	0.992462	0.747 (q₄₁)	0.756 (q₁₁, q₄₁)
q₃₁	1.056	0.038	27.708	0.999349	0.951 (q₄₁)	0.960 (q₁₁, q₄₁)
q₄₁	2.291	0.654	3.504	0.958421	0.951 (q₃₁)	0.958 (q₂₁, q₃₁)
q₂₂	0.063	0.045	1.399	0.699484	0.622 (q₄₂)	0.632 (τlss, q₄₂)
q₃₂	9.627	0.388	24.829	0.999189	0.951 (q₄₂)	0.951 (τlss, q₄₂)
q₄₂	1.395	0.062	22.658	0.999026	0.951 (q₃₂)	0.955 (q₂₂, q₃₂)
q₃₃	0.691	0.024	28.643	0.999390	0.909 (q₄₃)	0.911 (ρτx, q₄₃)
q₄₃	0.241	0.027	9.087	0.993927	0.909 (q₃₃)	0.909 (z_ss, q₃₃)
q₄₄	1.027	0.058	17.694	0.998402	0.088 (τlss)	0.173 (τlss,τxss)

Table 12 presents the decomposition of parameter uncertainty into a sensitivity and collinearity component for the case where also some deep parameters are estimated. The conclusions drawn about identification patterns drawn above are largely unaffected. The only key difference is that steady-state parameters are mainly collinear with the deep parameters of the model and vice versa. This is due to the fact that the latter are directly informative about each other via the steady-state equations of the model.

Table 12.

BCA (Some Deep Parameters Estimated), Information Matrix Decomposition.

	CRLB/para.	sens/para.	coll.	ϱi	ϱi(1)	ϱi(2)
z_ss	73.604	0.575	128.045	0.999970	0.886 (g_ss)	0.955 (g_ss, α)
τlss	1.438	0.018	78.490	0.999919	0.887 (α)	0.909 (z_ss, α)
τxss	3.267	0.009	346.308	0.999996	0.900 (δ)	0.970 (ρ_g, α)
g_ss	0.497	0.006	81.884	0.999925	0.886 (τxss)	0.971 (z_ss, τxss)
ρ_z	0.062	0.000	193.365	0.999987	0.988 (ρg,z)	0.991 (ρτl,z,ρg,z)
ρτl,z	2.071	0.030	70.178	0.999898	0.907 (ρ_z)	0.982 (ρτl,τx,ρτl,g)
ρτx,z	1.650	0.026	64.190	0.999879	0.908 (ρg,z)	0.974 (ρτx,ρτx,g)
ρg,z	14.448	0.097	149.415	0.999978	0.988 (ρ_z)	0.990 (ρ_z, ρτx,z)
ρz,τl	4.139	0.014	285.622	0.999994	0.988 (ρg,τl)	0.993 (ρg,τl,ρz,g)
ρτl	0.078	0.001	120.051	0.999965	0.978 (ρτl,g)	0.990 (ρτl,τx,ρτl,g)
ρτx,τl	3.132	0.026	120.650	0.999966	0.969 (ρτx,g)	0.986 (ρτx,ρτx,g)
ρg,τl	10.236	0.036	281.808	0.999994	0.988 (ρz,τl)	0.993 (ρg,τx, ρ_g)
ρz,τx	10.357	0.012	884.848	0.999999	0.988 (ρg,τx)	0.999 (ρ_z, ρz,g)
ρτl,τx	1.769	0.009	188.922	0.999986	0.983 (ρτl,g)	0.998 (ρτl,z,ρτl,g)
ρτx	0.144	0.001	163.762	0.999981	0.978 (ρτx,g)	0.997 (ρτx,z,ρτx,g)
ρg,τx	1.986	0.004	477.688	0.999998	0.988 (ρz,τx)	0.999 (ρg,z, ρ_g)
ρz,g	5.439	0.005	1166.579	1.000000	0.988 (ρ_g)	0.999 (ρ_z, ρz,τx)
ρτl,g	1.792	0.005	365.308	0.999996	0.983 (ρτl,τx)	0.999 (ρτl,z,ρτl,τx)
ρτx,g	1.195	0.005	234.875	0.999991	0.978 (ρτx)	0.998 (ρτx,z,ρτx)
ρ_g	0.118	0.000	862.241	0.999999	0.988 (ρz,g)	0.999 (ρg,z,ρg,τx)
q₁₁	0.490	0.036	13.799	0.997371	0.780 (q₃₁)	0.788 (q₂₁, q₃₁)
q₂₁	2.861	0.247	11.566	0.996255	0.747 (q₄₁)	0.756 (q₁₁, q₄₁)
q₃₁	1.422	0.038	37.321	0.999641	0.951 (q₄₁)	0.960 (q₁₁, q₄₁)
q₄₁	2.335	0.654	3.572	0.960019	0.951 (q₃₁)	0.958 (q₂₁, q₃₁)
q₂₂	0.549	0.045	12.138	0.996600	0.622 (q₄₂)	0.632 (τlss, q₄₂)
q₃₂	11.884	0.388	30.653	0.999468	0.951 (q₄₂)	0.951 (τlss, q₄₂)
q₄₂	1.912	0.062	31.063	0.999482	0.951 (q₃₂)	0.955 (q₂₂, q₃₂)
q₃₃	1.356	0.024	56.198	0.999842	0.909 (q₄₃)	0.911 (ρτx, q₄₃)
q₄₃	0.323	0.027	12.183	0.996626	0.909 (q₃₃)	0.909 (z_ss, q₃₃)
q₄₄	1.483	0.058	25.559	0.999234	0.408 (σ)	0.660 (τxss, σ)
δ	0.122	0.022	5.444	0.982983	0.900 (τxss)	0.949 (z_ss, α)
σ	0.518	0.015	35.581	0.999605	0.752 (τxss)	0.868 (τxss, q₄₄)
α	0.906	0.003	292.762	0.999994	0.887 (τlss)	0.973 (τxss, ρ_g)

We thus conclude that weak identification can be attributed to the fact that parameters in the steady-state vector, in the autoregressive matrix and in standard deviations of the fundamental innovations have a similar effect on the likelihood as other parameters appearing within the matrices they belong to. In order to understand how this pattern comes about we perform the following experiment. First, we assume that the observed variables are not output, investment, hours and government consumption (i.e. logy^t,logx^t,loglt,logg^t) but rather the four innovations to the wedges (i.e.log(zt),τlt,τxt,log(g^t)). Second, we assume that the parameters governing the VAR(1) process of the innovations to the wedges are estimated in isolation of the BCA model. In other words, we perform identification analysis on the parameters assuming that a VAR is run directly on the wedges, which are treated as observable in this experiment. This allows us to shed light on the question whether the weak identification patterns described above are an intrinsic property of the model or of the VAR(1) process itself. Table 13 presents the information matrix decomposition for the case just described.

Table 13.

BCA Model, Information Matrix Decomposition (Observing 4 Wedges).

	CRLB/para.	sens/para.	coll.	ϱi	ϱi(1)	ϱi(2)
z_ss	8.528	0.137	62.163	0.999871	0.997 (g_ss)	0.998 (τlss,g_ss)
τlss	1.062	0.013	79.708	0.999921	0.985 (g_ss)	0.998 (τxss,g_ss)
τxss	1.007	0.004	251.762	0.999992	0.999 (g_ss)	1.000 (τlss,g_ss)
g_ss	0.501	0.001	390.297	0.999997	0.999 (τxss)	1.000 (τlss,τxss)
ρ_z	0.053	0.002	30.412	0.999459	0.937 (ρz,τl)	0.994 (ρz,τx,ρz,g)
ρτl,z	0.893	0.042	21.480	0.998916	0.938 (ρτl)	0.993 (ρτl,τx,ρτl,g)
ρτx,z	1.206	0.023	51.881	0.999814	0.935 (ρτx,τl)	0.993 (ρτx,ρτx,g)
ρg,z	15.015	0.323	46.481	0.999769	0.934 (ρg,τl)	0.993 (ρg,τx, ρ_g)
ρz,τl	2.922	0.074	39.262	0.999676	0.991 (ρz,g)	0.996 (ρz,τx,ρz,g)
ρτl	0.024	0.001	27.282	0.999328	0.990 (ρτl,g)	0.996 (ρτl,τx,ρτl,g)
ρτx,τl	1.433	0.022	65.457	0.999883	0.990 (ρτx,g)	0.996 (ρτx,ρτx,g)
ρg,τl	6.950	0.118	58.772	0.999855	0.989 (ρ_g)	0.995 (ρg,τx, ρ_g)
ρz,τx	7.522	0.061	123.966	0.999967	0.992 (ρz,g)	0.999 (ρ_z, ρz,g)
ρτl,τx	1.110	0.013	84.669	0.999930	0.992 (ρτl,g)	0.999 (ρτl,z,ρτl,g)
ρτx	0.161	0.001	204.424	0.999988	0.991 (ρτx,g)	0.999 (ρτx,z,ρτx,g)
ρg,τx	2.512	0.014	182.688	0.999985	0.991 (ρ_g)	0.999 (ρg,z, ρ_g)
ρz,g	4.432	0.024	185.122	0.999985	0.992 (ρz,τx)	1.000 (ρ_z, ρz,τx)
ρτl,g	0.851	0.007	127.227	0.999969	0.992 (ρτl,τx)	1.000 (ρτl,z,ρτl,τx)
ρτx,g	1.341	0.004	306.941	0.999995	0.991 (ρτx)	1.000 (ρτx,z,ρτx)
ρ_g	0.122	0.000	274.011	0.999993	0.991 (ρg,τx)	1.000 (ρg,z,ρg,τx)
q₁₁	0.058	0.035	1.630	0.789720	0.781 (q₃₁)	0.789 (q₂₁, q₃₁)
q₂₁	0.378	0.247	1.530	0.756825	0.748 (q₄₁)	0.757 (q₁₁, q₄₁)
q₃₁	0.137	0.038	3.593	0.960488	0.951 (q₄₁)	0.960 (q₁₁, q₄₁)
q₄₁	2.292	0.652	3.515	0.958676	0.951 (q₃₁)	0.959 (q₂₁, q₃₁)
q₂₂	0.058	0.045	1.282	0.625594	0.624 (q₄₂)	0.625 (q₃₂, q₄₂)
q₃₂	1.258	0.387	3.253	0.951570	0.951 (q₄₂)	0.952 (q₂₂, q₄₂)
q₄₂	0.208	0.061	3.384	0.955345	0.951 (q₃₂)	0.955 (q₂₂, q₃₂)
q₃₃	0.058	0.024	2.404	0.909374	0.909 (q₄₃)	0.909 (ρτl,τx, q₄₃)
q₄₃	0.064	0.026	2.404	0.909367	0.909 (q₃₃)	0.909 (ρτl,τx, q₃₃)
q₄₄	0.058	0.058	1.000	0.008064	0.002 (q₃₁)	0.003 (ρg,τl, ρ_g)

The main takeaways are that (i) the collinearity patterns are preserved and (ii) for almost all steady-state innovations to the wedges, collinearity is higher. The intuition for result (ii) is that by fixing deep parameters the model imposes additional restrictions on the steady states of the innovations to the wedges via the steady-state equations. This results in additional information and, therefore, improved identification strength.

To further develop intuition and visualize our findings, we plot pairwise correlations coefficients between ∂lT(θ)/∂θi and ∂lT(θ)/∂θ−i across the entire parameter space in Figs. 1 and 2. Clearly, correlation coefficients between the derivatives with respect to the same parameter are equal to 1, marked as dark red squares on the main diagonal. The following results emerge. First, the figures confirm the main finding that the steady-state innovations to the wedges (I−P)−1P0, the elements of the autoregressive matrix P and the Choleski decomposition of the variance covariance matrix Q induce variation in the likelihood which is mainly similar to that generated by other parameters in the matrix they belong to. Indeed, in Fig. 2 the strongest collinearity is among elements of the same matrix and only some elements of the P and Q have a weakly collinear effect. Second, collinearity is less pervasive when innovations to the wedges are assumed to be observed and the parameters are estimated using the VAR in isolation from the model. The additional links between parameters and their likelihood introduced by the model equations are thus not sufficiently rich to let each parameter in the VAR(1) matrices P0,P and Q raise its own, distinct voice. In other words, estimating the wedge parameters within a model does not obviate the collinearity patterns which emerge in isolation from it.

Fig. 1.

BCA, Pairwise Correlations (Observing 4 Standard Variables).

Fig. 2.

BCA, Pairwise Correlations (Observing 4 Wedges).

4.2.2 Šustek (2011) MBCA Model

Tables 14 and 15 contain information about the relative CRLBs as well as about the sensitivity and collinearity components for the MBCA model wedge parameters. Using again a cut-off value of 1 the set of worst identified parameters is given by z_ss, ρτl,z,ρτx,z,ρg,z,ρτb,z,ρR˜,z,ρz,τl,ρτx,τl,ρg,τl,ρτb,τl,ρR˜,τl,ρz,τx,ρτl,τx,ρg,τx,ρτb,τx,ρR˜,τx,ρz,g,ρτl,g,ρτx,g,ρτb,g,ρR˜,g,ρz,τb,ρτl,τb,ρτx,τb,ρg,τb,ρR˜,τb,ρτx,R˜,ρg,R˜,ρτb,R˜, q₄₁, q₅₁, q₆₁, q₃₂, q₅₂, q₅₃, q₆₃, q₅₄, q₆₄ and q₆₆. This amount to 39 out of 61 parameters, that is, roughly two thirds of the estimated parameters are weakly identified. As in the standard BCA model these parameters concern the steady-state innovation to the efficiency wedge z_ss as well as the off-diagonal elements of the P and Q matrices. Identification deficiencies can again be attributed to the strong collinearity between the effects that parameters within the same matrix exert on the likelihood. We focus on the case where some deep parameters are included in the identification analysis in Tables 16 and 17. Also in this case, we keep fixed the other deep parameters so as to guarantee a full rank information matrix. Among the many available choices we choose to leave out β, ψ as well as steady-state inflation π_ss and the weight on inflation in the Taylor rule ωπ. Again, relative CRLBs increase as expected. Additional parameters which are weakly identified are τlss,τxss,ρz,R˜, q₃₁, q₃₃, q₄₃, δ, σ, α, ωπ. The collinearity patterns are similar to the ones in the BCA model.

Table 14.

MBCA, Parameter Identification (All Deep Parameters, τbss and R˜ss Fixed).

	Value	CRLB	rCRLB
z_ss	−0.0246	0.1333	5.4109
τlss	0.1267	0.0345	0.2720
τxss	0.4649	0.1310	0.2818
g_ss	−1.5389	0.0592	0.0385
R˜ss	0.0000	0.0023	0.0023
ρ_z	0.8541	0.2833	0.3317
ρτl,z	−0.0437	0.1504	3.4424
ρτx,z	−0.0557	0.1393	2.5014
ρg,z	0.0533	0.1774	3.3285
ρτb,z	0.0633	0.4210	6.6544
ρR˜,z	−0.0141	0.0229	1.6232
ρz,τl	−0.1482	0.4083	2.7553
ρτl	1.0580	0.2108	0.1993
ρτx,τl	−0.0335	0.1629	4.8637
ρg,τl	0.0587	0.2139	3.6444
ρτb,τl	−0.2979	0.6935	2.3279
ρR˜,τl	0.0146	0.0295	2.0178
ρz,τx	0.2654	0.3643	1.3724
ρτl,τx	−0.0014	0.2048	146.3072
ρτx	1.0877	0.1840	0.1692
ρg,τx	−0.0974	0.2229	2.2881
ρτb,τx	0.0850	0.6202	7.2964
ρR˜,τx	0.0005	0.0358	71.6947
ρz,g	−0.0094	0.1084	11.4917
ρτl,g	0.0097	0.0619	6.3770
ρτx,g	0.0026	0.0326	12.5353
ρ_g	1.0053	0.0479	0.0477
ρτb,g	−0.0076	0.2253	29.6442
ρR˜,g	0.0004	0.0088	21.8878
ρz,τb	−0.0654	0.1679	2.5674
ρτl,τb	0.0465	0.0865	1.8607
ρτx,τb	−0.0116	0.0763	6.5772
ρg,τb	0.0241	0.0940	3.9020
ρτb	0.8263	0.2712	0.3282
ρR˜,τb	0.0063	0.0137	2.1719
ρz,R˜	0.7994	0.7968	0.9967
ρτl,R˜	−0.7219	0.4557	0.6313
ρτx,R˜	0.4016	0.5299	1.3194
ρg,R˜	0.3411	0.5499	1.6123
ρτb,R˜	0.1200	1.2617	10.5144
ρR˜	0.4412	0.1101	0.2495
q₁₁	0.0110	0.0006	0.0579
q₂₁	0.0037	0.0009	0.2430
q₃₁	0.0058	0.0024	0.4065
q₄₁	0.0009	0.0013	1.4043
q₅₁	0.0005	0.0100	19.9724
q₆₁	0.0003	0.0003	1.1280
q₂₂	0.0092	0.0007	0.0717
q₃₂	−0.0008	0.0028	3.5125
q₄₂	0.0050	0.0014	0.2832
q₅₂	−0.0175	0.0189	1.0819
q₆₂	0.0000	0.0003	0.0003
q₃₃	0.0029	0.0022	0.7422
q₄₃	0.0117	0.0050	0.4255
q₅₃	−0.0013	0.0237	18.2111
q₆₃	0.0001	0.0033	32.6605
q₄₄	0.0087	0.0067	0.7715
q₅₄	0.0014	0.0292	20.8730
q₆₄	−0.0002	0.0045	22.4300
q₅₅	0.0219	0.0116	0.5293
q₆₅	0.0040	0.0005	0.1246
q₆₆	0.0010	0.0012	1.1923

Table 15.

MBCA, Information Matrix Decomposition (Observing 6 Standard Variables).

	CRLB/para.	sens/para.	coll.	ϱi	ϱi(1)	ϱi(2)
z_ss	5.411	0.437	12.388	0.996736	0.923 (τxss)	0.945 (τxss,R˜ss)
τlss	0.272	0.079	3.436	0.956717	0.451 (g_ss)	0.630 (z_ss, g_ss)
τxss	0.282	0.017	16.867	0.998241	0.923 (z_ss)	0.955 (z_ss, R˜ss)
g_ss	0.038	0.009	4.104	0.969858	0.736 (τxss)	0.850 (z_ss, τlss)
R˜ss	0.002	0.000	5.086	0.980482	0.496 (τxss)	0.676 (z_ss, τxss)
ρ_z	0.332	0.001	420.889	0.999997	0.993 (ρz,τx)	0.999 (ρz,τx,ρz,g)
ρτl,z	3.442	0.018	187.343	0.999986	0.989 (ρτl,τx)	0.997 (ρτl,τx,ρτl,g)
ρτx,z	2.501	0.005	456.118	0.999998	0.992 (ρτx)	0.998 (ρτx,ρτx,g)
ρg,z	3.329	0.021	159.247	0.999980	0.990 (ρg,τx)	0.997 (ρg,τx, ρ_g)
ρτb,z	6.654	0.035	191.861	0.999986	0.979 (ρτb,τx)	0.989 (ρτb,τx,ρτb,g)
ρR˜,z	1.623	0.017	94.732	0.999944	0.990 (ρR˜,τx)	0.996 (ρR˜,τx,ρR˜,g)
ρz,τl	2.755	0.014	194.002	0.999987	0.917 (ρτx,τl)	0.982 (ρτx,τl,ρτb,τl)
ρτl	0.199	0.002	118.928	0.999965	0.790 (ρτb,τl)	0.965 (ρτl,τx,ρτl,τb)
ρτx,τl	4.864	0.024	201.703	0.999988	0.949 (ρg,τl)	0.986 (ρz,τl,ρτb,τl)
ρg,τl	3.644	0.045	80.899	0.999924	0.949 (ρτx,τl)	0.975 (ρg,τx,ρg,τb)
ρτb,τl	2.328	0.013	185.382	0.999985	0.790 (ρτl)	0.939 (ρτb,τx,ρτb)
ρR˜,τl	2.018	0.045	44.361	0.999746	0.675 (ρR˜,g)	0.931 (ρR˜,τx,ρR˜,τb)
ρz,τx	1.372	0.003	435.327	0.999997	0.993 (ρ_z)	0.999 (ρ_z, ρz,g)
ρτl,τx	146.307	0.696	210.110	0.999989	0.989 (ρτl,z)	0.997 (ρτl,z,ρτl,g)
ρτx	0.169	0.000	488.457	0.999998	0.992 (ρτx,z)	0.998 (ρτx,z,ρτx,g)
ρg,τx	2.288	0.014	163.487	0.999981	0.990 (ρg,z)	0.998 (ρg,z, ρ_g)
ρτb,τx	7.296	0.032	231.052	0.999991	0.979 (ρτb,z)	0.990 (ρτb,z,ρτb,g)
ρR˜,τx	71.695	0.605	118.481	0.999964	0.990 (ρR˜,z)	0.996 (ρR˜,z,ρR˜,g)
ρz,g	11.492	0.081	141.658	0.999975	0.929 (ρτx,g)	0.988 (ρτx,g,ρτb,g)
ρτl,g	6.377	0.072	88.102	0.999936	0.824 (ρτb,g)	0.921 (ρτb,g,ρR˜,g)
ρτx,g	12.535	0.120	104.264	0.999954	0.937 (ρ_g)	0.989 (ρz,g,ρτb,g)
ρ_g	0.048	0.001	47.038	0.999774	0.937 (ρτx,g)	0.952 (ρz,g,ρτb,g)
ρτb,g	29.644	0.222	133.747	0.999972	0.824 (ρτl,g)	0.920 (ρz,g,ρτx,g)
ρR˜,g	21.888	0.659	33.218	0.999547	0.675 (ρR˜,τl)	0.820 (ρR˜,z,ρR˜,τx)
ρz,τb	2.567	0.014	180.589	0.999985	0.918 (ρτx,τb)	0.982 (ρτx,τb,ρτb)
ρτl,τb	1.861	0.017	107.845	0.999957	0.767 (ρτb)	0.967 (ρτl,ρτl,τx)
ρτx,τb	6.577	0.031	212.675	0.999989	0.966 (ρg,τb)	0.986 (ρz,τb,ρg,τb)
ρg,τb	3.902	0.049	78.892	0.999920	0.966 (ρτx,τb)	0.977 (ρg,τl,ρg,τx)
ρτb	0.328	0.002	165.624	0.999982	0.767 (ρτl,τb)	0.947 (ρτb,τl,ρτb,τx)
ρR˜,τb	2.172	0.046	47.425	0.999778	0.528 (ρR˜,z)	0.932 (ρR˜,τl,ρR˜,τx)
ρz,R˜	0.997	0.038	25.915	0.999255	0.889 (ρτx,R˜)	0.953 (ρτl,R˜,ρτx,R˜)
ρτl,R˜	0.631	0.039	16.073	0.998063	0.500 (ρR˜)	0.871 (ρτb,R˜,ρR˜)
ρτx,R˜	1.319	0.038	34.906	0.999590	0.899 (ρg,R˜)	0.969 (ρz,R˜,ρg,R˜)
ρg,R˜	1.612	0.164	9.860	0.994844	0.899 (ρτx,R˜)	0.924 (ρz,R˜,ρτx,R˜)
ρτb,R˜	10.514	0.367	28.685	0.999392	0.481 (ρR˜)	0.868 (ρτl,R˜,ρR˜)
ρR˜	0.249	0.016	15.685	0.997965	0.846 (q₅₅)	0.941 (ρR˜,z, q₅₅)
q₁₁	0.058	0.016	3.584	0.960297	0.937 (q₃₁)	0.960 (q₂₁, q₃₁)
q₂₁	0.243	0.050	4.858	0.978583	0.939 (q₅₁)	0.968 (q₁₁, q₅₁)
q₃₁	0.407	0.021	19.169	0.998638	0.937 (q₁₁)	0.950 (q₁₁, q₄₁)
q₄₁	1.404	0.720	1.951	0.858718	0.840 (q₃₁)	0.842 (q₂₁, q₃₁)
q₅₁	19.972	0.870	22.955	0.999051	0.970 (q₆₁)	0.979 (q₂₁, q₆₁)
q₆₁	1.128	0.273	4.131	0.970260	0.970 (q₅₁)	0.970 (q₃₁, q₅₁)
q₂₂	0.072	0.020	3.675	0.962262	0.912 (q₅₂)	0.939 (q₄₂, q₅₂)
q₃₂	3.512	0.154	22.840	0.999041	0.840 (q₄₂)	0.884 (ρτb,τl, q₄₂)
q₄₂	0.283	0.129	2.187	0.889370	0.840 (q₃₂)	0.842 (q₂₂, q₃₂)
q₅₂	1.082	0.025	43.524	0.999736	0.970 (q₆₂)	0.976 (q₂₂, q₆₂)
q₆₂	0.000	0.000	4.237	0.971743	0.970 (q₅₂)	0.970 (q₃₂, q₅₂)
q₃₃	0.742	0.038	19.698	0.998711	0.745 (q₄₃)	0.786 (ρτl,g, q₄₃)
q₄₃	0.426	0.055	7.692	0.991514	0.745 (q₃₃)	0.756 (ρτl,g, q₃₃)
q₅₃	18.211	0.335	54.425	0.999831	0.970 (q₆₃)	0.973 (ρg,R˜, q₆₃)
q₆₃	32.661	0.819	39.874	0.999685	0.970 (q₅₃)	0.973 (ρg,R˜, q₅₃)
q₄₄	0.772	0.055	13.997	0.997445	0.515 (ρτl,g)	0.580 (ρz,g,ρτx,g)
q₅₄	20.873	0.311	67.113	0.999889	0.970 (q₆₄)	0.976 (ρg,R˜, q₆₄)
q₆₄	22.430	0.410	54.717	0.999833	0.970 (q₅₄)	0.976 (ρg,R˜, q₅₄)
q₅₅	0.529	0.019	27.447	0.999336	0.943 (q₆₅)	0.956 (ρR˜, q₆₅)
q₆₅	0.125	0.020	6.091	0.986429	0.943 (q₅₅)	0.943 (ρτl,R˜, q₅₅)
q₆₆	1.192	0.058	20.582	0.998819	0.242 (ρR˜)	0.454 (ρR˜, q₅₅)

Table 16.

MBCA, Parameter Identification (Some Deep Parameters Free).

	Value	CRLB	rCRLB
z_ss	−0.0246	2.1912	88.9226
τlss	0.1267	0.4744	3.7431
τxss	0.4649	6.2711	13.4892
g_ss	−1.5389	0.0694	0.0451
R˜ss	0.0000	0.0023	0.0023
ρ_z	0.8541	0.5182	0.6068
ρτl,z	−0.0437	0.3059	6.9991
ρτx,z	−0.0557	0.1738	3.1195
ρg,z	0.0533	0.3181	5.9690
ρτb,z	0.0633	0.7185	11.3585
ρR˜,z	−0.0141	0.0880	6.2419
ρz,τl	−0.1482	0.4659	3.1440
ρτl	1.0580	0.3470	0.3280
ρτx,τl	−0.0335	0.9381	28.0039
ρg,τl	0.0587	0.2177	3.7081
ρτb,τl	−0.2979	1.2796	4.2953
ρR˜,τl	0.0146	0.1146	7.8510
ρz,τx	0.2654	0.4307	1.6228
ρτl,τx	−0.0014	0.4689	334.9210
ρτx	1.0877	0.5609	0.5157
ρg,τx	−0.0974	0.2500	2.5668
ρτb,τx	0.0850	1.1771	13.8482
ρR˜,τx	0.0005	0.0920	184.0737
ρz,g	−0.0094	0.5517	58.5039
ρτl,g	0.0097	0.1280	13.1959
ρτx,g	0.0026	0.2396	92.1551
ρ_g	1.0053	0.2162	0.2151
ρτb,g	−0.0076	0.2403	31.6135
ρR˜,g	0.0004	0.0235	58.7368
ρz,τb	−0.0654	0.5120	7.8303
ρτl,τb	0.0465	0.0947	2.0375
ρτx,τb	−0.0116	0.6453	55.6283
ρg,τb	0.0241	0.1851	7.6786
ρτb	0.8263	0.5197	0.6289
ρR˜,τb	0.0063	0.0362	5.7450
ρz,R˜	0.7994	2.0086	2.5127
ρτl,R˜	−0.7219	0.5524	0.7652
ρτx,R˜	0.4016	7.3077	18.1964
ρg,R˜	0.3411	0.7627	2.2359
ρτb,R˜	0.1200	3.9863	33.2192
ρR˜	0.4412	0.1855	0.4204
q₁₁	0.0110	0.0046	0.4202
q₂₁	0.0037	0.0012	0.3225
q₃₁	0.0058	0.0188	3.2388
q₄₁	0.0009	0.0014	1.5813
q₅₁	0.0005	0.0187	37.4840
q₆₁	0.0003	0.0005	1.6802
q₂₂	0.0092	0.0061	0.6675
q₃₂	−0.0008	0.0189	23.6018
q₄₂	0.0050	0.0015	0.2959
q₅₂	−0.0175	0.0225	1.2844
q₆₂	0.0000	0.0004	0.0004
q₃₃	0.0029	0.0202	6.9657
q₄₃	0.0117	0.0815	6.9617
q₅₃	−0.0013	0.1007	77.4485
q₆₃	0.0001	0.0112	111.8444
q₄₄	0.0087	0.1096	12.5977
q₅₄	0.0014	0.0854	61.0234
q₆₄	−0.0002	0.0184	92.0608
q₅₅	0.0219	0.0165	0.7539
q₆₅	0.0040	0.0029	0.7225
q₆₆	0.0010	0.0022	2.2075
g_n	0.0037	0.0923	24.9454
g_z	0.0040	0.0112	2.8109
δ	0.0118	0.1014	8.5917
σ	1.0000	2.7391	2.7391
α	0.3500	0.3514	1.0039
ρ_R	0.7500	0.2378	0.3170
ωπ	1.5000	1.8906	1.2604

Table 17.

MBCA (Some Deep Parameters Estimated), Information Matrix Decomposition.

	CRLB/para.	sens/para.	coll.	ϱi	ϱi(1)	ϱi(2)
z_ss	88.923	0.437	203.582	0.999988	0.967 (α)	0.985 (δ, α)
τlss	3.743	0.079	47.283	0.999776	0.451 (g_ss)	0.675 (g_ss, α)
τxss	13.489	0.017	807.496	0.999999	0.981 (g_n)	0.990 (g_n, δ)
g_ss	0.045	0.009	4.807	0.978125	0.736 (τxss)	0.853 (τlss, α)
R˜ss	0.002	0.000	5.282	0.981918	0.566 (g_n)	0.835 (g_n, α)
ρ_z	0.607	0.001	769.983	0.999999	0.993 (ρz,τx)	0.999 (ρz,τx,ρz,g)
ρτl,z	6.999	0.018	380.905	0.999997	0.989 (ρτl,τx)	0.997 (ρτl,τx,ρτl,g)
ρτx,z	3.120	0.005	568.821	0.999998	0.992 (ρτx)	0.998 (ρτx,ρτx,g)
ρg,z	5.969	0.021	285.575	0.999994	0.990 (ρg,τx)	0.997 (ρg,τx, ρ_g)
ρτb,z	11.358	0.035	327.492	0.999995	0.979 (ρτb,τx)	0.989 (ρτb,τx,ρτb,g)
ρR˜,z	6.242	0.017	364.272	0.999996	0.990 (ρR˜,τx)	0.996 (ρR˜,τx,ρR˜,g)
ρz,τl	3.144	0.014	221.371	0.999990	0.917 (ρτx,τl)	0.982 (ρτx,τl,ρτb,τl)
ρτl	0.328	0.002	195.729	0.999987	0.790 (ρτb,τl)	0.965 (ρτl,τx,ρτl,τb)
ρτx,τl	28.004	0.024	1,161.364	1.000000	0.949 (ρg,τl)	0.986 (ρz,τl,ρτb,τl)
ρg,τl	3.708	0.045	82.311	0.999926	0.949 (ρτx,τl)	0.975 (ρg,τx,ρg,τb)
ρτb,τl	4.295	0.013	342.062	0.999996	0.790 (ρτl)	0.939 (ρτb,τx,ρτb)
ρR˜,τl	7.851	0.045	172.608	0.999983	0.675 (ρR˜,g)	0.931 (ρR˜,τx,ρR˜,τb)
ρz,τx	1.623	0.003	514.749	0.999998	0.993 (ρ_z)	0.999 (ρ_z, ρz,g)
ρτl,τx	334.921	0.696	480.977	0.999998	0.989 (ρτl,z)	0.997 (ρτl,z,ρτl,g)
ρτx	0.516	0.000	1,489.071	1.000000	0.992 (ρτx,z)	0.998 (ρτx,z,ρτx,g)
ρg,τx	2.567	0.014	183.404	0.999985	0.990 (ρg,z)	0.998 (ρg,z, ρ_g)
ρτb,τx	13.848	0.032	438.524	0.999997	0.979 (ρτb,z)	0.990 (ρτb,z,ρτb,g)
ρR˜,τx	184.074	0.605	304.196	0.999995	0.990 (ρR˜,z)	0.996 (ρR˜,z,ρR˜,g)
ρz,g	58.504	0.081	721.174	0.999999	0.929 (ρτx,g)	0.988 (ρτx,g,ρτb,g)
ρτl,g	13.196	0.072	182.309	0.999985	0.824 (ρτb,g)	0.921 (ρτb,g,ρR˜,g)
ρτx,g	92.155	0.120	766.509	0.999999	0.937 (ρ_g)	0.989 (ρz,g,ρτb,g)
ρ_g	0.215	0.001	212.213	0.999989	0.937 (ρτx,g)	0.952 (ρz,g,ρτb,g)
ρτb,g	31.613	0.222	142.631	0.999975	0.824 (ρτl,g)	0.920 (ρz,g,ρτx,g)
ρR˜,g	58.737	0.659	89.143	0.999937	0.675 (ρR˜,τl)	0.820 (ρR˜,z,ρR˜,τx)
ρz,τb	7.830	0.014	550.781	0.999998	0.918 (ρτx,τb)	0.982 (ρτx,τb,ρτb)
ρτl,τb	2.038	0.017	118.095	0.999964	0.767 (ρτb)	0.967 (ρτl,ρτl,τx)
ρτx,τb	55.628	0.031	1,798.754	1.000000	0.966 (ρg,τb)	0.986 (ρz,τb,ρg,τb)
ρg,τb	7.679	0.049	155.247	0.999979	0.966 (ρτx,τb)	0.977 (ρg,τl,ρg,τx)
ρτb	0.629	0.002	317.345	0.999995	0.767 (ρτl,τb)	0.947 (ρτb,τl,ρτb,τx)
ρR˜,τb	5.745	0.046	125.447	0.999968	0.528 (ρR˜,z)	0.932 (ρR˜,τl,ρR˜,τx)
ρz,R˜	2.513	0.038	65.329	0.999883	0.889 (ρτx,R˜)	0.953 (ρτl,R˜,ρτx,R˜)
ρτl,R˜	0.765	0.039	19.483	0.998682	0.500 (ρR˜)	0.871 (ρτb,R˜,ρR˜)
ρτx,R˜	18.196	0.038	481.414	0.999998	0.899 (ρg,R˜)	0.969 (ρz,R˜,ρg,R˜)
ρg,R˜	2.236	0.164	13.674	0.997322	0.899 (ρτx,R˜)	0.924 (ρz,R˜,ρτx,R˜)
ρτb,R˜	33.219	0.367	90.628	0.999939	0.593 (ρ_R)	0.868 (ρτl,R˜,ρR˜)
ρR˜	0.420	0.016	26.429	0.999284	0.850 (ρ_R)	0.950 (q₅₅, ωπ)
q₁₁	0.420	0.016	26.007	0.999260	0.937 (q₃₁)	0.960 (q₂₁, q₃₁)
q₂₁	0.323	0.050	6.449	0.987903	0.939 (q₅₁)	0.968 (q₁₁, q₅₁)
q₃₁	3.239	0.021	152.717	0.999979	0.937 (q₁₁)	0.950 (q₁₁, q₄₁)
q₄₁	1.581	0.720	2.197	0.890440	0.840 (q₃₁)	0.842 (q₂₁, q₃₁)
q₅₁	37.484	0.870	43.082	0.999731	0.970 (q₆₁)	0.979 (q₂₁, q₆₁)
q₆₁	1.680	0.273	6.153	0.986706	0.970 (q₅₁)	0.970 (q₅₁, ωπ)
q₂₂	0.668	0.020	34.190	0.999572	0.912 (q₅₂)	0.939 (q₄₂, q₅₂)
q₃₂	23.602	0.154	153.473	0.999979	0.840 (q₄₂)	0.884 (ρτb,τl, q₄₂)
q₄₂	0.296	0.129	2.285	0.899189	0.840 (q₃₂)	0.842 (q₂₂, q₃₂)
q₅₂	1.284	0.025	51.670	0.999813	0.970 (q₆₂)	0.976 (q₂₂, q₆₂)
q₆₂	0.000	0.000	4.512	0.975134	0.970 (q₅₂)	0.970 (q₃₂, q₅₂)
q₃₃	6.966	0.038	184.862	0.999985	0.745 (q₄₃)	0.786 (ρτl,g, q₄₃)
q₄₃	6.962	0.055	125.846	0.999968	0.745 (q₃₃)	0.756 (ρτl,g, q₃₃)
q₅₃	77.448	0.335	231.461	0.999991	0.970 (q₆₃)	0.973 (ρg,R˜, q₆₃)
q₆₃	111.844	0.819	136.547	0.999973	0.970 (q₅₃)	0.973 (ρg,R˜, q₅₃)
q₄₄	12.598	0.055	228.544	0.999990	0.515 (ρτl,g)	0.627 (ρτl,g, σ)
q₅₄	61.023	0.311	196.210	0.999987	0.970 (q₆₄)	0.976 (ρg,R˜, q₆₄)
q₆₄	92.061	0.410	224.580	0.999990	0.970 (q₅₄)	0.976 (ρg,R˜, q₅₄)
q₅₅	0.754	0.019	39.094	0.999673	0.943 (q₆₅)	0.956 (ρR˜, q₆₅)
q₆₅	0.722	0.020	35.319	0.999599	0.943 (q₅₅)	0.952 (q₅₅, ρ_R)
q₆₆	2.208	0.058	38.106	0.999656	0.535 (ωπ)	0.729 (ρ_R, ωπ)
g_n	24.945	0.037	666.842	0.999999	0.981 (τxss)	0.988 (τxss,ρg,z)
g_z	2.811	0.120	23.506	0.999095	0.961 (δ)	0.967 (g_n, δ)
δ	8.592	0.039	222.782	0.999990	0.961 (g_z)	0.979 (z_ss, g_z)
σ	2.739	0.016	174.691	0.999984	0.557 (z_ss)	0.723 (z_ss, q₄₄)
α	1.004	0.003	370.406	0.999996	0.974 (τxss)	0.990 (z_ss, τxss)
ρ_R	0.317	0.006	52.055	0.999815	0.850 (ρR˜)	0.902 (ρτl,R˜,ρR˜)
ωπ	1.260	0.020	63.537	0.999876	0.738 (ρ_R)	0.886 (q₆₆, ρ_R)

Finally, we carry out the identification analysis on the wedge parameters with the twist of treating the innovations to the wedges as observables for the MBCA model as well. We again find that (i) parameters of the same matrices having a collinear effect on the likelihood is a property of the VAR process itself (Table 18) and (ii) collinearity is less pervasive when the parameters are estimated in isolation to the model (Figs. 3 and 4).

Fig. 3.

MBCA, Pairwise Correlations (Observing 6 Standard Variables).

Fig. 4.

MBCA, Pairwise Correlations (Observing 6 Wedges).

Table 18.

MBCA Model, Information Matrix Decomposition (Observing 6 Wedges).

	CRLB/para.	sens/para.	coll.	ϱi	ϱi(1)	ϱi(2)
z_ss	5.486	0.099	55.563	0.999838	0.861 (τxss)	0.998 (τxss,τbss)
τlss	0.308	0.016	18.980	0.998611	0.977 (τbss)	0.995 (g_ss, τbss)
τxss	0.233	0.004	64.877	0.999881	0.904 (g_ss)	0.998 (z_ss, τbss)
g_ss	0.045	0.007	6.672	0.988704	0.904 (τxss)	0.914 (z_ss, τbss)
τbss	0.080	0.004	20.423	0.998801	0.977 (τlss)	0.996 (τlss, g_ss)
R˜ss	0.002	0.000	6.817	0.989181	0.838 (τlss)	0.908 (z_ss, τxss)
ρ_z	0.095	0.001	113.826	0.999961	0.994 (ρz,τx)	0.999 (ρz,τx,ρz,g)
ρτl,z	1.691	0.017	102.092	0.999952	0.994 (ρτl,τx)	0.999 (ρτl,τx,ρτl,g)
ρτx,z	0.875	0.009	102.811	0.999953	0.994 (ρτx)	0.999 (ρτx,ρτx,g)
ρg,z	2.179	0.048	45.056	0.999754	0.991 (ρg,τx)	0.999 (ρg,τx, ρ_g)
ρτb,z	3.307	0.027	122.830	0.999967	0.994 (ρτb,τx)	0.999 (ρτb,τx,ρτb,g)
ρR˜,z	2.189	0.023	95.026	0.999945	0.994 (ρR˜,τx)	0.999 (ρR˜,τx,ρR˜,g)
ρz,τl	0.575	0.016	35.237	0.999597	0.952 (ρτx,τl)	0.977 (ρτl,ρτx,τl)
ρτl	0.072	0.002	31.555	0.999498	0.937 (ρτb,τl)	0.977 (ρτl,τx,ρτl,τb)
ρτx,τl	1.511	0.049	30.748	0.999471	0.952 (ρz,τl)	0.976 (ρτx,ρτx,τb)
ρg,τl	2.055	0.127	16.157	0.998083	0.858 (ρ_g)	0.982 (ρg,τx,ρg,τb)
ρτb,τl	0.726	0.019	37.927	0.999652	0.963 (ρR˜,τl)	0.977 (ρτb,τx,ρτb)
ρR˜,τl	2.191	0.074	29.561	0.999428	0.963 (ρτb,τl)	0.976 (ρR˜,τx,ρR˜,τb)
ρz,τx	0.399	0.003	118.786	0.999965	0.994 (ρ_z)	0.999 (ρ_z, ρz,g)
ρτl,τx	68.331	0.642	106.404	0.999956	0.994 (ρτl,z)	0.999 (ρτl,z, ρτl,g)
ρτx	0.058	0.001	106.998	0.999956	0.994 (ρτx,z)	0.999 (ρτx,z, ρτx,g)
ρg,τx	1.542	0.033	46.985	0.999773	0.991 (ρg,z)	0.999 (ρg,z, ρ_g)
ρτb,τx	3.188	0.025	128.102	0.999970	0.994 (ρτb,z)	0.999 (ρτb,z, ρτb,g)
ρR˜,τx	79.866	0.807	98.976	0.999949	0.994 (ρR˜,z)	0.999 (ρR˜,z,ρR˜,g)
ρz,g	1.514	0.089	16.932	0.998255	0.953 (ρτx,z)	0.977 (ρτl,g, ρτx,g)
ρτl,g	1.310	0.087	15.082	0.997800	0.937 (ρτb,g)	0.964 (ρz,g, ρτb,g)
ρτx,g	3.269	0.221	14.792	0.997712	0.953 (ρz,g)	0.967 (ρz,g, ρτb,g)
ρ_g	0.020	0.002	8.356	0.992814	0.858 (ρg,τl)	0.947 (ρg,z, ρg,τb)
ρτb,g	4.759	0.262	18.154	0.998482	0.963 (ρR˜,g)	0.974 (ρτl,g, ρR˜,g)
ρR˜,g	13.415	0.945	14.195	0.997515	0.963 (ρτb,g)	0.967 (ρR˜,τl, ρτb,g)
ρz,τb	0.628	0.016	38.536	0.999663	0.952 (ρτx,τb)	0.979 (ρτl,τb,ρτx,τb)
ρτl,τb	0.792	0.023	33.771	0.999561	0.935 (ρτb)	0.977 (ρτl,z,ρτl)
ρτx,τb	2.103	0.061	34.475	0.999579	0.952 (ρz,τb)	0.979 (ρτx,z,ρτx,τl)
ρg,τb	2.409	0.144	16.731	0.998212	0.845 (ρτx,τb)	0.981 (ρg,z,ρg,τl)
ρτb	0.126	0.003	40.518	0.999695	0.968 (ρR˜,τb)	0.977 (ρτb,z,ρτb,τl)
ρR˜,τb	2.442	0.077	31.533	0.999497	0.968 (ρτb)	0.977 (ρR˜,z,ρR˜,τl)
ρz,R˜	0.380	0.037	10.192	0.995175	0.953 (ρτx,R˜)	0.979 (ρτl,R˜,ρτx,R˜)
ρτl,R˜	0.379	0.042	9.006	0.993817	0.937 (ρτb,R˜)	0.966 (ρz,R˜,ρτb,R˜)
ρτx,R˜	0.450	0.050	9.012	0.993824	0.953 (ρz,R˜)	0.968 (ρz,R˜,ρτl,R˜)
ρg,R˜	1.255	0.312	4.017	0.968518	0.836 (ρτx,R˜)	0.865 (ρg,z,ρτx,R˜)
ρτb,R˜	6.461	0.595	10.849	0.995743	0.968 (ρR˜)	0.976 (ρτl,R˜,ρR˜)
ρR˜	0.259	0.031	8.453	0.992978	0.968 (ρτb,R˜)	0.969 (ρR˜,z,ρτb,R˜)
q₁₁	0.058	0.016	3.585	0.960317	0.937 (q₃₁)	0.960 (q₂₁, q₃₁)
q₂₁	0.212	0.050	4.231	0.971666	0.939 (q₅₁)	0.968 (q₁₁, q₅₁)
q₃₁	0.072	0.021	3.389	0.955473	0.937 (q₁₁)	0.950 (q₁₁, q₄₁)
q₄₁	1.404	0.719	1.952	0.858840	0.840 (q₃₁)	0.842 (q₂₁, q₃₁)
q₅₁	4.598	0.870	5.286	0.981945	0.970 (q₆₁)	0.979 (q₂₁, q₆₁)
q₆₁	1.128	0.273	4.131	0.970265	0.970 (q₅₁)	0.970 (q₃₁, q₅₁)
q₂₂	0.058	0.020	2.964	0.941350	0.912 (q₅₂)	0.939 (q₄₂, q₅₂)
q₃₂	0.302	0.154	1.967	0.861055	0.840 (q₄₂)	0.861 (q₂₂, q₄₂)
q₄₂	0.246	0.129	1.899	0.850068	0.840 (q₃₂)	0.842 (q₂₂, q₃₂)
q₅₂	0.118	0.025	4.746	0.977547	0.970 (q₆₂)	0.976 (q₂₂, q₆₂)
q₆₂	0.000	0.000	4.127	0.970205	0.970 (q₅₂)	0.970 (q₃₂, q₅₂)
q₃₃	0.058	0.038	1.538	0.759661	0.746 (q₄₃)	0.759 (q₄₃, q₅₃)
q₄₃	0.084	0.055	1.520	0.752922	0.746 (q₃₃)	0.752 (q₃₃, q₆₃)
q₅₃	1.383	0.334	4.133	0.970290	0.970 (q₆₃)	0.970 (q₃₃, q₆₃)
q₆₃	3.379	0.819	4.127	0.970198	0.970 (q₅₃)	0.970 (q₄₃, q₅₃)
q₄₄	0.058	0.055	1.052	0.310266	0.307 (q₅₄)	0.309 (q₅₄, q₆₄)
q₅₄	1.282	0.311	4.126	0.970180	0.970 (q₆₄)	0.970 (q₄₄, q₆₄)
q₆₄	1.689	0.410	4.124	0.970155	0.970 (q₅₄)	0.970 (q₄₄, q₅₄)
q₅₅	0.058	0.019	2.993	0.942520	0.943 (q₆₅)	0.943 (ρ_g, q₆₅)
q₆₅	0.061	0.020	2.993	0.942515	0.943 (q₅₅)	0.943 (ρ_g, q₅₅)
q₆₆	0.058	0.058	1.000	0.007684	0.001 (ρτl)	0.002 (ρτl,ρτl,g)

5 Economic Relevance

To assess which wedge is most important in accounting for cyclical fluctuations in the data Brinca et al. (2016) use the statistic fiY given by:

fiY=1/Σt(Yt−Yit)2Σt,j(1/(Yt−Yjt)2),

where j={z,τl,τx,g} in the BCA and j={z,τl,τx,g,τb,R˜} in the MBCA model. Y_t is actual data of a given observable Y={y,l,x} (output, labour or investment)⁵ and Yi_t is the component of observable Y due to wedge i and with fiY∈[0,1], Σ fiY=1. For instance, fix roughly measures the fraction of movement and level in actual investment explained by wedge i. The statistic fiY ranges from 0 to 1 and is increasing in the explanatory power of a wedge. Indeed, when Y_t = Y_it, then fiY=1 since fjY=0 ∀j≠i, whereas, in the limit, fiY=0 when the deviation of actual data from its component due to a given wedge goes to infinity. Notice that since the MSE statistic is used a model can be penalized along both the variance and the bias dimension. In other words both missing the data by a constant and missing the variation in the data is penalized.

When simulating the observables, we follow Chari et al. (2007) and consider two classes of counterfactual economies, namely the ‘one-wedge-on’ and ‘one-wedge-off’ economies. These economies are constructed by feeding the estimated wedges back into the model either one at a time (‘one-wedge-on’) or all but one (‘one-wedge-off’). It is important to point out the fact that the wedges have both a direct and a forecasting effect on the model. In the experiments, we seek to retain the forecasting effect only. This is done by setting the inactive wedges equal to some constant value (typically their intercept) at time t while preserving the estimated stochastic process and the realization of the wedges at time t − 1 to forecast their future realizations. Notice that this procedure would not be necessary if the matrices P and Q in the VAR(1) law of motion of the wedge shocks were diagonal.

The intuition behind the ‘one-wedge-on’ and ‘one-wedge-off’ counterfactual economies is the following. On one hand, the first seeks to understand how far a single wedge channel can bring the model to replicate the movements in the data while turning off the other margins. On the other hand, the second asks how badly the model performs when freezing the same channel and while keeping the others active.

5.1 Chari et al. (2007) BCA Model

Table 19 shows the fiY statistic for different observables and counterfactual economies as well as its one standard error bands given by f_i + /− Sd(f_i), where the standard deviations are computed using the delta method and the parameter covariance matrix (inverse Fisher information matrix) for the case when only the wedges parameters are assumed unknown (and thus the deep parameters of the model are fixed). To calculate the statistics we focus on the 1982 recession episode as in Chari et al. (2007) and use their original data.

Table 19.

BCA Model, Uncertainty Around f_i Statistics.

Observable	Counterfactual Economy	f_i-Sd(f_i)	f_i	f_i-Sd(f_i)
Output	Efficiency wedge on	0.63	0.78	0.93
	Labour wedge on	0.05	0.13	0.20
	Investment wedge on	0.00	0.04	0.09
	Government wedge on	0.01	0.05	0.10
Hours	Efficiency wedge on	−0.07	0.06	0.20
	Labour wedge on	0.69	0.89	1.09
	Investment wedge on	−0.02	0.02	0.06
	Government wedge on	−0.01	0.02	0.06
Investment	Efficiency wedge on	0.57	0.68	0.79
	Labour wedge on	0.07	0.19	0.31
	Investment wedge on	−0.02	0.06	0.13
	Government wedge on	0.04	0.07	0.10
Output	Efficiency wedge off	−0.01	0.04	0.09
	Labour wedge off	−0.07	0.20	0.47
	Investment wedge off	0.11	0.46	0.80
	Government wedge off	0.00	0.31	0.62
Hours	Efficiency wedge off	0.43	0.90	1.36
	Labour wedge off	−0.06	0.02	0.10
	Investment wedge off	−0.18	0.05	0.28
	Government wedge off	−0.13	0.03	0.20
Investment	Efficiency wedge off	−0.02	0.08	0.19
	Labour wedge off	−0.06	0.28	0.62
	Investment wedge off	0.05	0.13	0.21
	Government wedge off	0.07	0.50	0.93

The one standard deviation bands around f_i statistics for the one-wedge-on counterfactual economies cover an economically non-negligible range. However, these bands never overlap and are always quite far from each other. This means that if one used this statistic to measure and rank the relative importance of the wedges in replicating movements in the data, the main conclusions would not be overturned.⁶ Thus, the main result of Chari et al. (2007) still holds trough: the labour and the efficiency wedge play primary roles in explaining business cycle fluctuations during the period covering 1982 recession, whereas the investment and government wedge are negligible.

This is no longer true if one considers the one-wedge-off economies since the statistics exhibit such a higher level of uncertainty that the bands around them overlap. Our analysis thus discourages from using this statistic only to evaluate the relative importance of the wedges in a BCA model.

The intuition behind this result is that while the one-wedge-on f_i statistics are a function of only one object subject to uncertainty, the one-wedge-off f_i statistics are a function of three such objects. Indeed, in the one-wedge-on counterfactual economies only one wedge is active whereas in the one-wedge-off experiments this is the case for three wedges.

5.2 Brinca et al. (2016) Multi-country BCA Analysis

We now explore whether the results above for the 1982 recession in the United States, focus of the Chari et al. (2007), change if one considers multiple countries and focuses on the 2008 recession, as in Brinca et al. (2016). Tables 20 and 21, respectively, show the f_i statistics for the one-wedge-on and one-wegde-off economies. Specifically, each table reports the f_i statistics for each country, observable and wedge. Numbers are marked in bold only if, for a given observable and country, the most important wedge’s f_i statistic is significantly different from the others (i.e. the bands do not overlap).⁷

Table 20.

ϕ Statistics for Observable‘s Components – Great Recession – One-Wedge-On.

Countries	Output				Labour				Investment
Australia	0.69	0.27	0.02	0.02	0.64	0.12	0.12	0.10	0.53	0.25	0.03	0.17
Austria	0.71	0.07	0.11	0.12	0.28	0.07	0.19	0.43	0.62	0.05	0.20	0.10
Belgium	0.87	0.05	0.05	0.03	0.19	0.56	0.15	0.06	0.57	0.18	0.15	0.08
Canada	0.49	0.13	0.16	0.20	0.17	0.16	0.27	0.35	0.40	0.09	0.39	0.05
Denmark	0.58	0.06	0.29	0.06	0.26	0.13	0.49	0.11	0.19	0.05	0.69	0.07
Finland	0.93	0.01	0.03	0.02	0.38	0.02	0.09	0.49	0.60	0.03	0.29	0.06
France	0.93	0.01	0.04	0.02	0.71	0.02	0.19	0.08	0.69	0.03	0.21	0.05
Germany	0.79	0.03	0.12	0.07	0.26	0.16	0.33	0.23	0.43	0.04	0.47	0.06
Iceland	0.25	0.14	0.51	0.09	0.36	0.25	0.26	0.12	0.02	0.02	0.94	0.03
Ireland	0.21	0.24	0.46	0.08	0.06	0.30	0.58	0.04	0.07	0.07	0.80	0.06
Israel	0.77	0.03	0.16	0.04	0.38	0.26	0.08	0.27	0.21	0.08	0.60	0.11
Italy	0.68	0.08	0.17	0.07	0.18	0.13	0.59	0.09	0.33	0.07	0.56	0.04
Japan	0.62	0.10	0.15	0.12	0.18	0.14	0.44	0.22	0.36	0.15	0.32	0.16
Korea	0.51	0.18	0.18	0.14	0.37	0.24	0.16	0.23	0.44	0.09	0.33	0.14
Luxembourg	0.96	0.01	0.01	0.02	0.61	0.16	0.14	0.07	0.38	0.12	0.07	0.42
Mexico	0.55	0.11	0.26	0.07	0.22	0.23	0.43	0.10	0.24	0.14	0.48	0.12
Netherlands	0.89	0.02	0.05	0.03	0.38	0.10	0.24	0.26	0.70	0.03	0.22	0.04
New Zealand	0.44	0.09	0.40	0.08	0.22	0.15	0.53	0.10	0.08	0.03	0.83	0.05
Norway	0.76	0.04	0.05	0.15	0.27	0.08	0.22	0.37	0.77	0.03	0.06	0.12
Spain	0.11	0.05	0.81	0.03	0.15	0.14	0.63	0.07	0.02	0.01	0.96	0.01
Sweden	0.98	0.00	0.01	0.01	0.63	0.02	0.19	0.15	0.74	0.02	0.22	0.02
Switzerland	0.89	0.02	0.06	0.02	0.86	0.03	0.03	0.07	0.05	0.02	0.91	0.02
Great Britain	0.65	0.12	0.14	0.09	0.15	0.26	0.39	0.10	0.33	0.14	0.38	0.12
USA	0.15	0.50	0.26	0.06	0.03	0.80	0.15	0.01	0.08	0.08	0.78	0.04

Table 21.

ϕ Statistics for Observable‘s Components – Great Recession – One-Wedge-Off.

Countries	Output				Labour				Investment
Australia	0.06	0.10	0.44	0.39	0.77	0.02	0.11	0.09	0.11	0.21	0.15	0.50
Austria	0.03	0.35	0.17	0.44	0.52	0.16	0.08	0.20	0.02	0.14	0.02	0.82
Belgium	0.01	0.16	0.36	0.40	0.64	0.05	0.12	0.14	0.07	0.16	0.25	0.43
Canada	0.04	0.58	0.21	0.13	0.59	0.22	0.08	0.06	0.03	0.71	0.03	0.18
Denmark	0.02	0.25	0.03	0.69	0.69	0.06	0.01	0.20	0.02	0.17	0.01	0.80
Finland	0.02	0.15	0.16	0.64	0.79	0.02	0.02	0.12	0.01	0.09	0.02	0.86
France	0.01	0.07	0.04	0.87	0.16	0.06	0.04	0.73	0.00	0.04	0.01	0.95
Germany	0.01	0.04	0.10	0.84	0.26	0.03	0.07	0.60	0.03	0.15	0.07	0.74
Iceland	0.54	0.31	0.02	0.14	0.83	0.10	0.01	0.05	0.49	0.24	0.00	0.23
Ireland	0.19	0.12	0.06	0.61	0.95	0.01	0.00	0.04	0.19	0.09	0.00	0.68
Israel	0.05	0.15	0.09	0.71	0.94	0.01	0.01	0.05	0.06	0.16	0.01	0.75
Italy	0.01	0.20	0.03	0.76	0.41	0.09	0.01	0.43	0.04	0.59	0.02	0.33
Japan	0.01	0.47	0.07	0.40	0.49	0.21	0.03	0.18	0.02	0.35	0.02	0.53
Korea	0.01	0.01	0.02	0.96	0.16	0.01	0.02	0.81	0.04	0.03	0.01	0.92
Luxembourg	0.02	0.35	0.23	0.38	0.90	0.04	0.02	0.04	0.02	0.29	0.03	0.65
Mexico	0.03	0.12	0.10	0.72	0.57	0.05	0.04	0.31	0.04	0.10	0.01	0.83
Netherlands	0.00	0.06	0.02	0.91	0.25	0.04	0.01	0.63	0.02	0.27	0.04	0.65
New Zealand	0.04	0.37	0.04	0.56	0.65	0.13	0.01	0.20	0.04	0.25	0.01	0.70
Norway	0.03	0.39	0.42	0.11	0.78	0.08	0.09	0.02	0.05	0.41	0.16	0.32
Spain	0.08	0.80	0.01	0.11	0.62	0.32	0.00	0.05	0.15	0.29	0.00	0.52
Sweden	0.01	0.12	0.21	0.65	0.48	0.06	0.10	0.33	0.02	0.19	0.09	0.69
Switzerland	0.07	0.15	0.06	0.73	0.71	0.05	0.02	0.22	0.19	0.49	0.03	0.28
Great Britain	0.00	0.04	0.03	0.90	0.66	0.01	0.01	0.29	0.02	0.14	0.01	0.77
United States	0.43	0.07	0.15	0.27	0.89	0.01	0.03	0.05	0.22	0.18	0.01	0.35

Two results stand out from Table 20. First, across the vast majority of countries, the f_i statistics for output point to the efficiency wedge being the most relevant wedge. This result can be established with precision, as the uncertainty bands around the f_i statistic for φAY do not overlap with those of other wedges. In other countries, the investment wedge takes a predominant role in explaining fluctuations in output (see Iceland, Ireland and Spain) and can also be distinguished from others. Second, the f_i statistics for the United States are no longer statistically different from each other, as it was the case for the 1982 recession. These two results are largely confirmed by the one-wedge-off results in Table 21, for which we again find less discriminatory power due to lower precision of the f_i statistics in the case where multiple wedges are active.

Notice that Chari et al. (2007) use a longer sample than Brinca et al. (2016) (specifically, 1959–2004 vs. 1980–2014) as the latter uses a balanced panel for 24 OECD countries for which the data availability is more limited. The larger uncertainty in the f_i statistics thus also points to the crucial role of sample size in BCA applications, which we discuss in the next section.

6 Statistics for Practitioners

In this section, we present some statistics that might be of special interest to (monetary) BCA practitioners. We show how the size of the sample available affects the overall strength of parameter identification.

6.1 Empirical Distance Measures

While the analysis in the previous sections has been carried out in the time domain using the state space representation of the DSGE model solution the methodology proposed by Qu and Tkachenko (2012) and Qu and Tkachenko (2017) takes a frequency domain perspective to study identification issues. In particular, rank conditions for local identification of dynamic parameters are established by studying the spectral density matrix which maps the structural parameters to functions in the Banach space. Local identifiability occurs if and only if a local change in the parameter values leads to a different image. This analysis thus requires to study the Jacobian of the spectral density matrix w.r.t. the deep parameters of the model. If steady-state-related parameters are estimated as well, like in our case, it is possible to incorporate in the analysis also the first-order properties of the model by considering an extra term involving the steady-state parameters. In both cases, under some regularity conditions, the rank conditions are necessary and sufficient for local identification also in the time domain. More specifically, it can be shown that when the model is non-singular, like in our case, analyzing the rank conditions in the frequency domain is equivalent to inspecting the rank of the information matrix in the time domain. This analysis is thus expected to deliver analogous results to Iskrev (2010) and Komunjer and Ng (2011). For a more detailed comparison of these tests we refer the reader to Qu and Tkachenko (2012).

In general, the solution of a DSGE model log-linearized around its steady state can be expressed in state space form. Assuming no measurement errors, as it is the case in the BCA and MBCA model, it is given by:

(6.1)Xt+1=A(θ)Xt+B(θ)εt+1,Yt=C^(θ)Xt

and can be rewritten as a vector moving average (VMA) representation using:

(6.2)Yt(θ)=C^(θ)A(θ)Xt−1+B(θ)εt=C^(θ)A(θ)A(θ)Xt−2+B(θ)εt−1+B(θ)εt=C^(θ)A(θ)2Xt−2+C^(θ)A(θ)B(θ)εt−1+B(θ)εt=…

where, following the exposition in Qu and Tkachenko (2012) we made explicit the dependence of Y_t on θ. More generally the process can be expressed as:

(6.3)Yt(θ)=C^(θ)A(θ)kXt−k+C^(θ)A(θ)k−1B(θ)εt−k+1+…+C^(θ)A(θ)B(θ)εt−1+B(θ)εt.

If all eigenvalues of A(θ) then limk→∞A(θ)kXt−k=0. It thus follows that

(6.4)Yt(θ)=Σj=0∞C^(θ)A(θ)jB(θ)εt−j

(6.5)=Σj=0∞hj(θ)εt−j

(6.6)=H(L;θ)εt,

where hj(θ)(j=0,…,∞) are nY×nε matrices and H(L;θ)=Σj=0∞hj(θ)Lj the matrix of lagged polynomials.

Equation 6.5 along with some standard assumption about the process of the error term ϵ implies that Yt(θ) is covariance stationary and has a spectral density matrix fθ(ω) that can be represented as:

(6.7)fθ(ω)=12πH(exp(−iω);θ)Σ(θ)H(exp(−iω);θ)∗,

where X∗ denotes the conjugate transpose of a generic complex matrix X.

In some cases, like in the BCA and MBCA methodological framework, not only dynamic but also steady-state parameters are estimated. In this case, it is useful to define the augmented parameter vector as θ¯=(θ′,α′)′ and make explicit the relationship between data observables Yto, model observables Yt(θ) and steady states μ(θ¯)

(6.8)Yto=μ(θ¯)+Yt(θ),

where it is important to recall the fact that the model observables are expressed as log-deviations to form their respective steady-state values. This representation is useful in the case where steady-state parameters are estimated since considering only the second-order properties would omit the full identification potential of these parameters. This is why, in this case, the identification of θ¯ will be examined on the properties of μ(θ¯) and fθ¯(ω) jointly.

Definition 6.1. The parameter vector θ¯ is said to be locally identifiable from the first and second order properties of {Yt} at θ¯=θ¯0 if there exists an open neighborhood of θ¯0 in which μ(θ¯1)=μ(θ¯0) and fθ¯1(ω)=fθ¯0(ω) for all ω∈[−π,π] necessarily implies θ¯1=θ¯0.

Now consider the following object:

(6.9)G¯(θ¯)=∫−ππ∂vec(fθ(ω))θ¯′′∂vec(fθ(ω))θ¯′dω+∂μ(θ¯)∂θ¯′′∂μ(θ¯)∂θ¯′.

Under some mild assumptions Qu and Tkachenko (2012) show that:

Theorem 6.1. θ¯ is locally identifiable from the first and second order properties of Yto at a point θ¯0 if and only if G¯(θ¯0) is nonsingular. Establishing whether the parameter set θ is identifiable thus reduces to checking that the matrix G¯(θ¯0) is full rank.

Analogously to Komunjer and Ng (2011) the eigenvalues of G(θ¯0) are obtained numerically, with the sole exception of the default Matlab tolerance level being used to determine its rank. This is due to the fact that the matrix is not as sparse as the one considered by Komunjer and Ng (2011). Also, the derivatives ∂fθ(ωj)θk¯ are computed numerically as [fθ0+ekhk(ωj)−fθ0(ωj)]/ where ek is a unit vector whose kth element is equal to 1 and h_k is the step size. Unlike Komunjer and Ng (2011), however, the step size is not set to 1e−3 but to 1e−7.

To estimate ∂fθ(ωj)θk¯ and to compute the integral in 6.9 we make use of the symmetric difference quotient and Gaussian quadrature, respectively. We find that both models are locally identifiable only when the first- and second-order properties of the data are used since we find the matrix G¯(θ¯0) in (6.9) to be full rank but the matrix G¯(θ¯)=∫−ππ∂vec(fθ(ω))θ¯′′∂vec(fθ(ω))θ¯′dω) to be rank deficient.

If θ¯ is shown to be locally unidentifiable, as it is the case here for both the BCA and MBCA models, it is possible to trace out a curve of points χ which are observationally equivalent to θ¯0 in a local neighborhood of the latter. Indeed Qu and Tkachenko (2012), following Rothenberg (1971), show that this curve can be defined using the function θ(v) such that:

(6.10)∂θ¯(v)∂v=c(θ¯),θ¯(0)=θ¯0,

where c(θ¯0) is the eigenvector which corresponds to the smallest eigenvalue of G¯(θ¯), v is a scalar which varies in a neighborhood of 0 such that θ¯(v)∈δ(θ¯0).⁸

As shown by Qu and Tkachenko (2012), along the non-identification curve χ the parameter set θ¯ is not identified at θ¯0 since ∂θ¯(v)∂v=0 and thus c(θ¯)=0, which implies:

(6.11)∂vec(fθ¯(v)(ω))∂v=∂vec(fθ¯(v)(ω))∂θ¯(v)′c(θ¯)=0

for all ω∈[−π,π].

The curve can be traced out recursively using the Euler method such that θ¯(vj+1)=θ¯(vj)+c(θ¯(vj))h, where h is the step size (fixed at 1e−04).

We apply two methods (the first due to Qu & Tkachenko, 2012 and the latter due to Qu & Tkachenko, 2017) can be used to robustify conclusions on both local and global identifications success or failure.

The first method recognizes the fact that it is important to make sure that the points on the curve result in identical spectral densities. It thus computes the maximum absolute and relative deviations of fθ¯1(ω) from fθ¯0(ω) across the frequencies: maxω∈[0,π]||fθ¯(ω)−fθ¯0(ω)||∞⁹ and maxω∈[0,π]||fθ¯(ω)−fθ¯0(ω)||∞\fθ¯0,hl(ω), where fθ¯0,hl(ω) denotes the (h, l) element of the spectral density matrix given the parameter set θ and is evaluated at the same frequency and element of fθ¯0(ω) that maximizes the numerator. In the BCA model we find that at norm (θ¯0−θ¯) = 0.1276 the maximum absolute deviation on the curve is 0.0015 whereas the maximum relative deviation in relative form is 1.3811e−04. When norm (θ¯0−θ¯) = is 1.30 the two measures are 0.0083 and 7.5130e−04, respectively. As to the MBCA model, when norm (θ¯0−θ¯) = 0.0251 the maximum absolute deviation on the curve is 1.0226e−04 and the maximum relative deviation in relative form is 3.2755e−05. When norm (θ¯0−θ¯) = 2.1728 the two measures are 2.3858e−04 and 7.4645e−05, respectively. In both models, the points on the curve θ¯0 for the smallest norm are chosen such that at least the absolute measure is bigger than the numerical errors associated with the Euler method, which Qu and Tkachenko (2017) say should remain near or below 1.0E−04.

The second method computes a so-called ‘empirical distance between DSGE models’ using a range of sample sizes. It is equal to the testing power of the likelihood ratio test of the null hypothesis that fθ¯(ω) is the true spectral density against the alternative hypothesis that hφ0(ω) is the true spectral density. In the context of our analysis, hφ0(ω) is taken to be fθ¯0(ω) but the test is general enough to accommodate different parameter vectors as well as different model structures. The empirical distance measure has the following properties. First, its value is between 0 and 1 for any sample size T and significance level α. A higher value thus means that it is easier to distinguish between the spectral densities implied by the two parameter vectors θ¯ and θ¯0 and thus argues in favour of stronger identification. Second, consider what Qu and Tkachenko (2017) refer to as the ‘Kullback–Leibler (KL) distance between two DSGE models (and, more generally, between two vector linear processes) with spectral densities fθ¯0(ω) and fθ¯0(ω) ’. It is given by

(6.12)KL(θ¯0,θ¯1)=KL(θ0,θ1)+14πμ(θ¯0)−μ(θ¯1)′fθ¯1−1(0)μ(θ¯0)−μ(θ¯1)′

Where

(6.13)KL(θ0,θ1)=14π∫−ππtr(fθ¯1−1(ω)fθ¯0(ω))−logdet(fθ¯1−1(ω)fθ¯0(ω))−nYdω

To provide intuition, let l(θ¯) denote the frequency domain approximate likelihood (or the time domain Gaussian likelihood) based on Yt(θ¯). Then, T−1EYt(θ¯),θ¯=θ¯0L(θ¯0)−L(θ¯1)→KL(θ¯0,θ¯1). If the Kullback–Leibler distance KL(θ¯,θ¯0) is non-zero its value increase consistently with T and approaches 1 as T→∞. As becomes evident from Tables 22 and 23 all empirical distance measures (p-values of the likelihood ratio tests) are well above the 5% significance level for all sample sizes considered, even for a small sample size of just T = 20, that is, five years of observations, and for the smallest normed differences between the points on the identification curve and the points at which local identification is checked in the respective models. This is reassuring evidence for practitioners since even with a relatively small sample size the overall identification strength is sufficiently high.

Table 22.

BCA Model, Wedges Parameters, Empirical Distance Measures.

Empirical Distance Measures on the Curve When Norm (θ¯0−θ¯) = 0.13
Sample	T = 20	T = 40	T = 80	T = 120	T = 160	T = 200	T = 1,000
Size	0.12	0.17	0.24	0.32	0.38	0.44	0.96
Empirical Distance Measures on the Curve When Norm (θ¯0−θ¯) = 1.30
Sample	T = 20	T = 40	T = 80	T = 120	T = 160	T = 200	T = 1,000
Size	1.00	1.00	1.00	1.00	1.00	1.00	1.00

Table 23.

MBCA Model, Wedges Parameters (no τbss,R˜ss), Empirical Distance Measures.

Empirical Distance Measures on the Curve When Norm (θ¯0−θ¯) = 0.03
Sample	T = 20	T = 40	T = 80	T = 120	T = 160	T = 200	T = 1,000
Size	0.25	0.40	0.6204	0.77	0.87	0.93	1.00
Empirical Distance Measures on the Curve When Norm (θ¯0−θ¯) = 2.17
Sample	T = 20	T = 40	T = 80	T = 120	T = 160	T = 200	T = 1,000
Size	1.00	1.00	1.00	1.00	1.00	1.00	1.00

7 Conclusion

In the past years, BCA exercises have sparked great interest among theoretical and applied macroeconomists insofar as they can shed light on which classes of models are able to explain fluctuations in macroeconomic aggregates during a particular economic episode. These exercises involve ML estimation of the stochastic process governing the latent variables. Even though they have been extensively performed, the methodology has yet to be properly scrutinized in terms of identification deficiencies. Given that quantitative recommendations are made the statistical and economic relevance of potential identification issues of the methodology is of the utmost importance.

In this chapter, we take seriously the notes of caution raised by the literature which investigates identification issues in DSGE models. Indeed, we first perform strict and weak local identification tests as theorized by Komunjer and Ng (2011) and Iskrev (2010) on the parameters vectors estimated in Chari et al. (2007) and Šustek (2011). We find that in both the standard and MBCA frameworks the model parameters are strictly identifiable. This is no longer true once one extends the estimation to the deep parameters of the model. In these cases, we show how to obviate such failures by imposing restrictions on the space of estimated parameters. Our analysis also points out that both models are affected by weak identification problems and thus suffer from a low degree of estimation precision. In particular, we find that the elements which suffer from weak identification are the off-diagonal elements of the VAR(1) law of motion of the latent variables. This is due to the fact that while these parameters do affect the likelihood, the effect which they exert on the latter is strongly collinear. We find that this is an inherent property of the VAR(1) process evaluated at the estimated parameter vectors. Indeed, we show that when innovations to the wedges are assumed to be observed and the parameters are estimated using the VAR in isolation from the model it is the off-diagonal elements which have a strongly collinear effect on the likelihood. Second, we investigate the economic severity of these identification problems by computing the uncertainty around a statistic which is used to rank which classes of models best explain business cycle fluctuations. We find that the main conclusion is not overturned for the standard BCA model. We are currently investigating whether this result also holds for the MBCA framework. Finally, we investigate a question of interest to practitioners, namely how identification strength varies across sample sizes. We show that even in small samples the parameter sets in both frameworks are overall well identified by computing empirical distance measures as in Qu and Tkachenko (2017).

A. Appendix – BCA and MBCA Model with Investment Adjustment Costs

A.1. Komunjer and Ng (2011)

A.1.1 Chari et al. (2007) BCA Model

We introduce the version of the BCA model by Chari et al. (2007) which allows for investment adjustment costs calibrated at the ‘normal’ intensity used by Bernanke et al. (1999), see Appendix C.2. The results are similar to the baseline case with the default parameters being strictly identifiable and several steady-state wedge shocks, off-diagonal elements of the P and Q matrix as well as deep parameters not being identifiable once the set of estimated parameters is extended to the latter (Tables A-1 and A-2).

Table A-1.

Komunjer and Ng Test Results BCA Model With Normal Adjustment Costs.

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	29	25	15	51	40	62	0
e−03	29	25	16	53	45	68	0
e−04	29	25	16	54	45	69	0
e−05	29	25	16	54	45	69	0
e−06	29	25	16	54	45	69	0
e−07	29	25	16	54	45	69	0
e−08	30	25	16	54	46	70	0
e−09	30	25	16	54	46	70	0
e−10	30	25	16	55	46	70	0
e−11	30	25	16	55	46	71	1
Default = 1.378453e−12	30	25	16	55	46	71	1
Required	30	25	16	55	46	71	1

Summary: nθ=30,nX=5,nε=4.

Order condition: nθ=30,nδ=50.

Table A-2.

Komunjer and Ng Test Results BCA Model With Normal Adjustment Costs (Deep Parameters Estimated).

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	34	25	15	53	45	64	0
e−03	36	25	16	57	52	71	0
e−04	36	25	16	57	52	71	0
e−05	36	25	16	58	52	71	0
e−06	36	25	16	58	52	71	0
e−07	36	25	16	58	52	71	0
e−08	37	25	16	60	53	74	0
e−09	37	25	16	61	53	75	0
e−10	38	25	16	61	54	76	0
e−11	38	25	16	63	54	77	0
Default = 1.378453e−12	38	25	16	63	54	78	0
Required	39	25	16	64	55	80	1

Summary: nθ=39,nX=5,nε=4.

Order condition: nθ=39,nδ=50.

Problematic parameters at Tol = 1e−3: z_ss, τlss, τxss, g_ss, ρτl,z, ρτx,z, ρg,z, ρz,τl, ρτx,τl, ρg,τl, ρz,τx, ρτl,τx, ρg,τx, ρτl,g, ρτx,g, q₂₁, q₂₂, q₃₂, q₄₂, q₃₃, q₄₃, β, ψ, σ, a, b.

Problematic parameters at Tol = 1.378453e−12: z_ss, τlss, τxss, g_ss, ρ_z, ρτl,z, ρτx,z, ρg,z, ρz,τl, ρτl, ρτx,τl, ρg,τl, ρz,τx, ρτl,τx, ρτx, ρg,τx, ρz,g, ρτl,g, ρτx,g, ρ_g, q₂₁, q₃₁, q₂₂, q₃₂, q₄₂, q₃₃, q₄₃, q₄₄, g_n, g_z, β, δ, ψ, σ, α, a, b.

In line with the analysis carried out for the baseline model, we check which restricted parameter combinations would allow the other parameters to be strictly identifiable. As reported in Table A-3 we find three such sets at the lowest tolerance level for which identification fails in A-2, mostly consisting of three parameters each. These sets are (i) {a,ρg}, (ii) {b,ρg} and (iii) {a,ψ} and thus involve the parameters governing the degree of investment adjustment costs (a and b), the autocorrelation of the government wedge and the inverse of the constant Frisch elasticity of labour supply ψ.

Table A-3.

Komunjer and Ng Conditional Test Results BCA Model With Normal Adjustment Costs, Tol = 1.378453e−12.

Fixed	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
a ρ_g	39	25	16	64	55	80	1
b ρ_g	39	25	16	64	55	80	1
a ψ	39	25	16	64	55	80	1
Required	39	25	16	64	55	80	1

Summary: nθ=39,nX=5,nε=4.

Order condition: nθ=39,nδ=50.

A.1.2 Šustek (2011) MBCA Model

An analogous pattern to the one found in the baseline MBCA model and reported in Section 4 emerges once investment adjustment costs are introduced in the model (see Tables A-4 and A-7).

Table A-4.

Komunjer and Ng Test Results MBCA Model With Normal Adjustment Costs.

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	60	49	33	101	89	130	0
e−03	60	49	36	107	96	142	0
e−04	60	49	36	109	96	144	0
e−05	60	49	36	109	96	144	0
e−06	60	49	36	109	96	144	0
e−07	60	49	36	109	96	145	0
e−08	60	49	36	109	96	145	0
e−09	61	49	36	109	97	145	0
e−10	61	49	36	110	97	145	0
e−11	61	49	36	110	97	146	1
Default = 5.826450e−12	61	49	36	110	97	146	1
Required	61	49	36	110	97	146	1

Summary: nθ=61,nX=7,nε=6.

Order condition: nθ=61,nδ=105.

Table A-5.

Komunjer and Ng Test Results MBCA Model with Normal Adjustment Costs (τbss and R˜ss Estimated).

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	62	49	33	103	91	132	0
e−03	62	49	36	109	98	143	0
e−04	62	49	36	111	98	146	0
e−05	62	49	36	111	98	146	0
e−06	62	49	36	111	98	146	0
e−07	62	49	36	111	98	147	0
e−08	62	49	36	111	98	147	0
Default = 2.983143e−09	62	49	36	111	98	147	0
e−09	63	49	36	111	99	147	0
e−10	63	49	36	111	99	147	0
e−11	63	49	36	112	99	148	1
Required	63	49	36	112	99	148	1

Summary: nθ=63,nX=7,nε=6.

Order condition: nθ=63,nδ=105.

Problematic parameters at Tol = 1e−3: z_ss, g_ss.

Problematic parameters at Tol = 1.000000e−10: z_ss, τlss, τxss, g_ss, ρ_z, ρτl,z, ρτx,z, ρg,z, ρτb,z, ρR˜,z, ρz,τl,ρτl, ρτx,τl, ρg,τl, ρτb,τl, ρR˜,τl, ρz,τx, ρτl,τx, ρτx, ρg,τx, ρτb,τx, ρR˜,τx, ρz,g, ρτl,g, ρτx,g, ρ_g, ρτb,g, ρR˜,g, ρz,τb, ρτl,τb, ρτx,τb, ρg,τb, ρτb, ρz,R˜, ρτl,R˜, ρτx,R˜, ρg,R˜, ρτb,R˜, ρR˜, q₂₁, q₃₁, q₅₁, q₂₂, q₃₂, q₄₂, q₃₃, q₄₃, q₅₃, q₄₄, q₅₄, q₆₄, q₅₅.

Table A-6.

Komunjer and Ng Test Results MBCA Model With Normal Adjustment Costs (Deep Parameters Estimated).

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	69	49	35	104	99	133	0
e−03	70	49	36	113	106	146	0
e−04	70	49	36	116	106	148	0
e−05	70	49	36	117	106	148	0
e−06	70	49	36	117	106	148	0
e−07	71	49	36	118	107	149	0
e−08	71	49	36	119	107	152	0
e−09	72	49	36	119	108	153	0
e−10	73	49	36	120	109	154	0
Default = 2.330580e−11	73	49	36	121	109	155	0
e−11	73	49	36	122	109	157	0
Required	74	49	36	123	110	159	1

Summary: nθ=74,nX=7,nε=6.

Order condition: nθ=74,nδ=105.

Problematic parameters at Tol = 1e−3: z_ss, τlss, τxss, g_ss, ρ_z, ρτl,z, ρτx,z, ρg,z, ρτb,z, ρR˜,z, ρz,τl, ρτx,τl, ρg,τl, ρτb,τl, ρR˜,τl, ρz,τx, ρτl,τx, ρg,τx, ρτb,τx, ρR˜,τx, ρz,g, ρτl,g, ρτx,g, ρτb,g, ρR˜,g, ρz,τb, ρτl,τb, ρτx,τb, ρg,τb, ρτb, ρR˜,τb, ρz,R˜, ρτl,R˜, ρτx,R˜, ρg,R˜, ρτb,R˜, ρR˜, q₂₂, q₄₂, q₄₃, q₅₃, g_n, g_z, β, ψ, σ, ρ_R, ωπ, ω_y, π_ss, a, b.

Problematic parameters at Tol = 1.000000e−11: z_ss, τlss, τxss, g_ss, ρ_z, ρτl,z, ρτx,z, ρg,z, ρτb,z, ρR˜,z, ρz,τl, ρτl, ρτx,τl, ρg,τl, ρτb,τl, ρR˜,τl, ρz,τx, ρτl,τx, ρτx, ρg,τx, ρτb,τx, ρR˜,τx, ρz,g, ρτl,g, ρτx,g, ρ_g, ρτb,g, ρR˜,g, ρz,τb, ρτl,τb, ρτx,τb, ρg,τb, ρτb, ρR˜,τb, ρz,R˜, ρτl,R˜, ρτx,R˜, ρg,R˜, ρτb,R˜, ρR˜, q₁₁, q₂₁, q₃₁, q₅₁, q₆₁, q₂₂, q₃₂, q₄₂, q₅₂, q₆₂, q₃₃, q₄₃, q₅₃, q₆₃, q₄₄, q₅₄, q₆₄, q₅₅, q₆₅, q₆₆, g_n, g_z, β, δ, ψ, σ, α, ρ_R, ωπ, ω_y, π_ss, a, b.

Table A-7.

Komunjer and Ng Test Results MBCA Model with Normal Adjustment Costs (τbss,R˜ss and Deep Parameters Estimated).

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	71	49	35	105	101	134	0
e−03	72	49	36	114	108	146	0
e−04	72	49	36	118	108	149	0
e−05	72	49	36	119	108	150	0
e−06	72	49	36	119	108	150	0
e−07	73	49	36	120	109	151	0
Default = 9.546056e−08	73	49	36	120	109	151	0
e−08	73	49	36	121	109	154	0
e−09	74	49	36	121	110	154	0
e−10	74	49	36	122	110	156	0
e−11	75	49	36	123	111	158	0
Required	76	49	36	125	112	161	1

Summary: nθ=76,nX=7,nε=6.

Order condition: nθ=76,nδ=105.

Problematic parameters at Tol = 1e−3: z_ss, τlss, τxss, g_ss, ρ_z, ρτl,z, ρτx,z, ρg,z, ρτb,z, ρR˜,z, ρz,τl, ρτl, ρτx,τl, ρg,τl, ρτb,τl, ρR˜,τl, ρz,τx, ρτl,τx, ρτx, ρg,τx, ρτb,τx, ρR˜,τx, ρz,g, ρτl,g, ρτx,g, ρ_g, ρτb,g, ρR˜,g, ρz,τb, ρτl,τb, ρτx,τb, ρg,τb, ρτb, ρR˜,τb, ρz,R˜, ρτl,R˜, ρτx,R˜, ρg,R˜, ρτb,R˜, ρR˜, q₂₁, q₃₁, q₅₁, q₂₂, q₃₂, q₄₂, q₅₂, q₃₃, q₄₃, q₅₃, q₆₃, q₄₄, q₅₄, q₆₄, q₅₅, g_n, g_z, β, δ, ψ, σ, ρ_R, ωπ, ω_y, π_ss, a, b.

Problematic parameters at Tol = 1.000000e−11: z_ss, τlss, τxss, g_ss, τbss, R˜ss, ρ_z, ρτl,z, ρτx,z, ρg,z, ρτb,z, ρR˜,z, ρz,τl, ρτl, ρτx,τl, ρg,τl, ρτb,τl, ρR˜,τl, ρz,τx, ρτl,τx, ρτx, ρg,τx, ρτb,τx, ρR˜,τx, ρz,g, ρτl,g, ρτx,g, ρ_g, ρτb,g, ρR˜,g, ρz,τb, ρτl,τb, ρτx,τb, ρg,τb, ρτb, ρR˜,τb, ρz,R˜, ρτl,R˜, ρτx,R˜, ρg,R˜, ρτb,R˜, ρR˜, q₁₁, q₂₁, q₃₁, q₄₁, q₅₁, q₆₁, q₂₂, q₃₂, q₄₂, q₅₂, q₆₂, q₃₃, q₄₃, q₅₃, q₆₃, q₄₄, q₅₄, q₆₄, q₅₅, q₆₅, q₆₆, g_n, g_z, β, δ, ψ, σ, α, ρ_R, ωπ, ω_y, π_ss, a, b.

As to the conditional identification tests, we find that only for the MBCA model with normal adjustment costs where τbss and R˜SS not included in estimation it is possible to reparameterize the model in a way which makes the other parameters identifiable. We find 27 parameter combinations which restrict only four parameters at the lowest tolerance level for which identification fails in A-6 (see Table A-8). These sets mainly feature the steady-state innovations to the efficiency and government wedge (z_ss and g_ss) as well as steady-state inflation π_ss and some deep parameters. The parameter b which governs, with a, the degree of investment adjustment costs also shows up prominently in these sets. This is because this parameter is a function of other deep parameters and fixing it thus provides an additional restriction which can help to identify the latter.

Table A-8.

Komunjer and Ng Conditional Test Results MBCA Model with Normal Adjustment Costs, Tol = 2.330580e−11.

Fixed	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
z_ss τlss π_ss b	74	49	36	123	110	159	1
z_ss ρ_z π_ss b	74	49	36	123	110	159	1
z_ss ρτl,τx π_ss b	74	49	36	123	110	159	1
z_ss ρτb π_ss b	74	49	36	123	110	159	1
z_ss ρτb,R˜ π_ss b	74	49	36	123	110	159	1
z_ss q₃₁ π_ss b	74	49	36	123	110	159	1
z_ss q₅₁ π_ss b	74	49	36	123	110	159	1
z_ss q₅₄ π_ss b	74	49	36	123	110	159	1
z_ss ψ π_ss b	74	49	36	123	110	159	1
z_ss σ π_ss b	74	49	36	123	110	159	1
z_ss π_ss a b	74	49	36	123	110	159	1
τlss g_ss π_ss b	74	49	36	123	110	159	1
g_ss ρ_z π_ss b	74	49	36	123	110	159	1
g_ss ρτl,τx π_ss b	74	49	36	123	110	159	1
g_ss ρτb π_ss b	74	49	36	123	110	159	1
g_ss ρτb,R˜ π_ss b	74	49	36	123	110	159	1
g_ss q₃₁ π_ss b	74	49	36	123	110	159	1
g_ss q₅₁ π_ss b	74	49	36	123	110	159	1
g_ss q₅₄ π_ss b	74	49	36	123	110	159	1
g_ss ψ π_ss b	74	49	36	123	110	159	1
g_ss σ π_ss b	74	49	36	123	110	159	1
g_ss π_ss a b	74	49	36	123	110	159	1
ρ_z α π_ss b	74	49	36	123	110	158	0
ρτb,R˜ α π_ss b	74	49	36	123	110	158	0
q₅₄ α π_ss b	74	49	36	123	110	158	0
ψ α π_ss b	74	49	36	123	110	158	0
σ α π_ss b	74	49	36	123	110	158	0
Required	74	49	36	123	110	159	1

Summary: nθ=74,nX=7,nε=6.

Order condition: nθ=74,nδ=105.

A.2 Iskrev (2010)

A.2.1 Chari et al. (2007) BCA Model

We first study strict identification in the standard BCA model featuring investment adjustment costs of normal intensity. We start with the 38 parameters case (only a is considered in estimation since b is de facto a function of other structural parameters). Fixing g_n and g_z in the standard BCA case discussed above leaves 36 parameters and the rank of the information matrix is found to be 33. There are 7,140 combinations of 33 parameters out of 36 and as many as 2,406 combinations of parameters which, when fixed, deliver a rank of 33 and thus enable identification of the ones left unrestricted.

Table A-9 is instructive about weak identification problems. Interestingly, relative uncertainty decreases such that q31 and q44 no longer feature among the set of worst identified parameters. This is no longer true for q31 once also the deep parameters of the model are considered in the identification analysis (see Tables A-10 and A-11). The same results of the no adjustment costs counterpart hold through, with the exception of α also being one of the most badly identified parameters.

Table A-9.

BCA Model with Normal Adjustment Costs, Parameter Identification (All Deep Parameters Fixed).

	Value	CRLB	rCRLB
z_ss	−0.0239	0.0955	3.9926
τlss	0.3279	0.1450	0.4423
τxss	0.4834	0.2142	0.4431
g_ss	−1.5344	0.3212	0.2093
ρ_z	0.9800	0.0488	0.0498
ρτl,z	−0.0330	0.0514	1.5588
ρτx,z	−0.0702	0.1022	1.4548
ρz,g	0.0048	0.0596	12.3854
ρz,τl	−0.0138	0.0351	2.5492
ρτl	0.9564	0.0528	0.0552
ρτx,τl	−0.0460	0.1147	2.4941
ρg,τl	−0.0081	0.0470	5.7929
ρz,τx	−0.0117	0.0836	7.1323
ρτl,τx	−0.0451	0.0699	1.5497
ρτx	0.8962	0.0941	0.1050
ρg,τx	0.0488	0.0995	2.0369
ρz,g	0.0192	0.0802	4.1698
ρτl,g	0.0569	0.0663	1.1650
ρτx,g	0.1041	0.0949	0.9114
ρ_g	0.9711	0.0926	0.0953
q₁₁	0.0116	0.0007	0.0578
q₂₁	0.0014	0.0015	1.0792
q₃₁	−0.0105	0.0078	0.7389
q₄₁	−0.0006	0.0013	2.2903
q₂₂	0.0064	0.0004	0.0636
q₃₂	0.0010	0.0067	6.5323
q₄₂	0.0061	0.0050	0.8153
q₃₃	0.0158	0.0075	0.4712
q₄₃	0.0142	0.0023	0.1619
q₄₄	0.0046	0.0036	0.7903

Table A-10.

BCA Model with Normal Adjustment Costs, Wedges Parameter Identification.

	Value	CRLB	rCRLB
z_ss	−0.0239	2.3406	97.8465
τlss	0.3279	0.5711	1.7414
τxss	0.4834	1.9206	3.9729
g_ss	−1.5344	0.7576	0.4937
ρ_z	0.9800	0.0554	0.0565
ρτl,z	−0.0330	0.0545	1.6526
ρτx,z	−0.0702	0.1190	1.6937
ρz,g	0.0048	0.0654	13.5906
ρz,τl	−0.0138	0.0548	3.9733
ρτl	0.9564	0.0586	0.0612
ρτx,τl	−0.0460	0.1305	2.8372
ρg,τl	−0.0081	0.0660	8.1379
ρz,τx	−0.0117	0.1096	9.3467
ρτl,τx	−0.0451	0.0876	1.9431
ρτx	0.8962	0.1101	0.1229
ρg,τx	0.0488	0.1026	2.1003
ρz,g	0.0192	0.0934	4.8530
ρτl,g	0.0569	0.0795	1.3964
ρτx,g	0.1041	0.1715	1.6475
ρ_g	0.9711	0.1042	0.1073
q₁₁	0.0116	0.0064	0.5536
q₂₁	0.0014	0.0035	2.4901
q₃₁	−0.0105	0.0163	1.5549
q₄₁	−0.0006	0.0013	2.2904
q₂₂	0.0064	0.0046	0.7156
q₃₂	0.0010	0.0113	10.9783
q₄₂	0.0061	0.0116	1.8940
q₃₃	0.0158	0.0244	1.5433
q₄₃	0.0142	0.0045	0.3185
q₄₄	0.0046	0.0045	0.9858
δ	0.0118	0.0015	0.1285
σ	1.0000	0.4395	0.4395
α	0.3500	0.4199	1.1996

Table A-11.

BCA Model with Normal Adjustment Costs (Deep Parameters Estimated), Information Matrix Decomposition.

	CRLB/para.	sens/para.	coll.	ϱi	ϱi(1)	ϱi(2)
z_ss	97.846	0.410	238.923	0.999991	0.926 (g_ss)	0.980 (g_ss, α)
τlss	1.741	0.019	90.135	0.999938	0.801 (α)	0.861 (τxss,ρτx)
τxss	3.973	0.006	681.977	0.999999	0.946 (g_ss)	0.983 (ρg,τl, α)
g_ss	0.494	0.003	151.290	0.999978	0.946 (τxss)	0.989 (z_ss, α)
ρ_z	0.057	0.000	211.820	0.999989	0.992 (ρz,g)	0.994 (ρτl,z,ρz,g)
ρτl,z	1.653	0.038	43.825	0.999740	0.900 (ρτl)	0.987 (ρτl,τx,ρτl,g)
ρτx,z	1.694	0.019	89.376	0.999937	0.948 (ρz,g)	0.985 (ρτx,ρτx,g)
ρz,g	13.591	0.074	184.275	0.999985	0.992 (ρ_z)	0.993 (ρ_z, ρτx,z)
ρz,τl	3.973	0.012	336.677	0.999996	0.991 (ρg,τl)	0.995 (ρg,τl,ρz,g)
ρτl	0.061	0.001	75.356	0.999912	0.983 (ρτl,g)	0.992 (ρτl,τx,ρτl,g)
ρτx,τl	2.837	0.018	155.079	0.999979	0.980 (ρτx,g)	0.991 (ρτx,ρτx,g)
ρg,τl	8.138	0.027	298.583	0.999994	0.991 (ρz,τl)	0.994 (ρz,τl, ρ_g)
ρz,τx	9.347	0.010	979.814	0.999999	0.992 (ρg,τx)	0.999 (ρ_z, ρz,g)
ρτl,τx	1.943	0.012	166.317	0.999982	0.987 (ρτl,g)	0.999 (ρτl,z,ρτl,g)
ρτx	0.123	0.001	195.368	0.999987	0.986 (ρτx,g)	0.998 (ρτx,z,ρτx,g)
ρg,τx	2.100	0.003	673.543	0.999999	0.992 (ρz,τx)	0.999 (ρz,g, ρ_g)
ρz,g	4.853	0.004	1,280.821	1.000000	0.992 (ρ_g)	0.999 (ρ_z, ρz,τx)
ρτl,g	1.396	0.006	229.345	0.999990	0.987 (ρτl,τx)	0.999 (ρτl,z,ρτl,τx)
ρτx,g	1.648	0.004	459.104	0.999998	0.986 (ρτx)	0.999 (ρτx,z,ρτx)
ρ_g	0.107	0.000	1,051.673	1.000000	0.992 (ρz,g)	0.999 (ρz,g,ρg,τx)
q₁₁	0.554	0.036	15.585	0.997939	0.780 (q₃₁)	0.788 (q₂₁, q₃₁)
q₂₁	2.490	0.247	10.064	0.995051	0.747 (q₄₁)	0.755 (q₁₁, q₄₁)
q₃₁	1.555	0.038	40.815	0.999700	0.951 (q₄₁)	0.960 (q₁₁, q₄₁)
q₄₁	2.290	0.654	3.504	0.958405	0.951 (q₃₁)	0.958 (q₂₁, q₃₁)
q₂₂	0.716	0.045	15.827	0.998002	0.622 (q₄₂)	0.632 (τlss, q₄₂)
q₃₂	10.978	0.388	28.313	0.999376	0.951 (q₄₂)	0.951 (τlss, q₄₂)
q₄₂	1.894	0.062	30.769	0.999472	0.951 (q₃₂)	0.955 (q₂₂, q₃₂)
q₃₃	1.543	0.024	63.950	0.999878	0.909 (q₄₃)	0.910 (ρτx, q₄₃)
q₄₃	0.319	0.027	12.003	0.996524	0.909 (q₃₃)	0.909 (z_ss, q₃₃)
q₄₄	0.986	0.058	16.985	0.998265	0.113 (σ)	0.389 (δ, σ)
δ	0.129	0.013	9.575	0.994531	0.934 (g_ss)	0.966 (z_ss, α)
σ	0.440	0.011	40.138	0.999690	0.940 (α)	0.980 (τxss,ρg,τl)
α	1.200	0.002	504.842	0.999998	0.940 (σ)	0.983 (τxss,ρg,τl)

B. Appendix – Model Derivations

B.1. Representative Consumer

B.1.1. Optimization Problem of the Household

Suppose households own the capital stock and rent it out at rate r_t. They also work for wages at rate w_t per unit of labour input and pay takes on labour, investment and bond holdings. Then, the optimization problem for the household looks as follows:

Objective function:

max{ct(st),lt(st),kt+1(st),bt(st)}Σt=0∞ΣstβtUct(st),1−l(st)Nt(st)︸=(1+gn)tsettingN0=1,

where ct(st),xt(st)≥0,∀st,t.

Let xt(st) be any random variable. The expectation over the discounted sum of future possible realizations can be written as:

E0Σt=0∞βtxt(st)≡Σt=0∞Σstβtxt(st)μt(st)

The second sum expresses that the expectation is a probability weighted average of the different possible realizations of the variable. We can thus rewrite the objective function as follows:

max{ct(st),lt(st),kt+1(st),bt(st)}E0Σt=0∞βtUct(st),1−l(st)(1+gn)t,

Budget constraint:

ct(st)+[1+τx(st)]xt(st)+[1+τb(st)](1+gn)bt(st)[1+Rt(st)]pt(st)−bt−1(st−1)pt(st)=[1−τl(st)]wt(st)lt(st)+rt(st)kt(st−1)+Tt(st)

Capital accumulation law:

(B-1)Nt+1(st+1)kt+1(st)=[(1−δ)kt(st)+xt(st)]Nt(st)⇔(1+gn)kt+1(st)=(1−δ)kt(st)+xt(st)

B.1.2. Lagrangian Function

Given that there is one budget constraint for each realization of s^t in each time period t one can write the Lagrangian function in the following way:

L=E0Σt=0∞βtUct(st),1−l(st)Nt(st)−Σt=0∞Σstλ˜t{ct(st)+[1+τx(st)]xt(st)+[1+τb(st)](1+gn)bt(st)[1+Rt(st)]pt(st)−bt−1(st−1)pt(st)−[1−τl(st)]wt(st)lt(st)−rt(st)kt(st−1)−Tt(st)}

Making use of expression (B-1) for investment and recalling that the expectation is a probability weighted sum of the possible realizations, one can integrate the budget constraints into the first part of the function one obtains:

L=E0{Σt=0∞βt[Uct(st),1−l(st)Nt(st)−λt[ct(st)+[1+τx(st)][(1+gn)kt+1(st)−(1−δ)kt(st)]+[1+τb(st)][(1+gn)bt(st)[1+Rt(st)]pt(st)−bt−1(st−1)pt(st)]−[1−τl(st)]wt(st)lt(st)−rt(st)kt(st−1)−Tt(st)]]}

with λt=λ˜tβtμt(st).

B.1.3. First-order Necessary Conditions

Differentiating the Lagrangian with respect to the choice variables of the household, one obtains the following first-order necessary conditions which hold ∀st,t:

For each state of the world, s^t, and each point of time t it holds that:

(B-2)∂L∂ct(st)=βtμt(st)[Uc,t(st)(1+gn)t−λt]=0⇔ λt=Uc,t(st)(1+gn)t

(B-3)∂L∂nt(st)=βtμt(st)[Ul,t(st)(1+gn)t+λt[1−τl,t(st)]wt(st)]=0⇔ λt=−Ul,t(st)(1+gn)t[1−τl,t(st)]wt(st)

The intratemporal optimality condition is then given by:

(B-4)−Ul,t(st)Uc,t(st)=[1−τl,t(st)]wt(st).

When differentiating the Lagrangian with respect to capital kt+1(st), one has to note that at time t + 1 the net capital stock of the previous period, kt(st−1), becomes kt+1(st), which leads to an additional component in the derivative.

(B-5)∂L∂kt+1(st)=βtμt(st)[−λt[1+τx,t(st)](1+gn)]+Σst+1>stβt+1μt+1(st+1)[λt+1[1+τx,t+1(st+1)](1−δ)+rt+1(st+1)]=0⇔1=β(1+gn)Σst+1>stμt+1(st+1)μt(st)λt+1λt[1+τx,t+1(st+1)](1−δ)+rt+1(st+1)1+τx,t(st)⇔1=β(1+gn)Σst+1>stμt+1(st+1|st)Uc,t+1(st+1)(1+gn)t+1Uc,t(st)(1+gn)t×[[1+τx,t+1(st+1)](1−δ)+rt+1(st+1)1+τx,t(st)]⇔1=βΣst+1>stμt+1(st+1|st)Uc,t+1(st+1)Uc,t(st)[[1+τx,t+1(st+1)](1−δ)+rt+1(st+1)1+τx,t(st)]⇔1=βEtUc,t+1(st+1)Uc,t(st)[1+τx,t+1(st+1)](1−δ)+rt+1(st+1)1+τx,t(st),

where in the third step we used μt+1(st+1)μt(st)≡μt+1(st+1|st) and equation (B-2).

When differentiating the Lagrangian with respect to bonds, one has to note that at time t + 1 the bonds of the previous period, bt−1(st−1), become bt(st), which leads to an additional component in the derivative.

(B-6)∂L∂bt(st)=βtμt(st)−λt[1+τb,t(st)](1+gn)[1+Rt(st)]pt(st)+Σst+1>stβt+1μt+1(st+1)λt+1[1+τb,t+1(st+1)]1pt+1(st+1)=0⇔1=β(1+gn)Σst+1>stμt+1(st+1)μt(st)λt+1λt1+τb,t+1(st+1)1+τb,t(st)pt(st)pt+1(st+1)[1+Rt(st)]⇔1=β(1+gn)Σst+1>stμt+1(st+1)μt(st)Uc,t+1(st+1)(1+gn)t+1Uc,t(st)(1+gn)t1+τb,t+1(st+1)1+τb,t(st)pt(st)pt+1(st+1)[1+Rt(st)]⇔1=βΣst+1>stμt+1(st+1|st)Uc,t+1(st+1)Uc,t(st)1+τb,t+1(st+1)1+τb,t(st)pt(st)pt+1(st+1)[1+Rt(st)]⇔1=βEtUc,t+1(st+1)Uc,t(st)1+τb,t+1(st+1)1+τb,t(st)pt(st)pt+1(st+1)[1+Rt(st)]⇔1=βEtUc,t+1(st+1)Uc,t(st)1+τb,t+1(st+1)1+τb,t(st)pt(st)pt+1(st+1)[1+Rt(st)]

where in the fourth step we used μt+1(st+1)μt(st)≡μt+1(st+1|st) and equation (B-2).

B.2. Representative Producer

The representative producer operates an aggregate CRS production function:

(B-7)yt(st)=Fkt(st−1),Zt(st)lt(st),

where F(.,.) has the standard properties and Zt(st)=zt(st)1+gzt.

B.2.1. Optimization Problem of the Firm

The producer maximizes profits:

(B-8)yt(st)−wt(st)lt(st)−rt(st)kt(st−1)

by setting:

(B-9)wt(st)=Fl,tkt(st−1),zt(st)(1+gz)tlt(st)

(B-10)•••

(B-11)•••

B.3. Additional Model Equations

The aggregate resource constraint is given by:

(B-12)yt(st)=ct(st)+gt(st)+xt(st).

Monetary policy is assumed to set the interest rate according to a Taylor rule of the following type:

(B-13)Rt(st)=(1−ρR)R+ωy(lnyt(st)−lny)+ωπ(πt(st)−π)+ρRRt−1(st−1)+R˜t(st),

where ρR∈[0,1),πt(st)≡lnpt(st)−lnpt−1(st−1) is the inflation rate and a variable’s symbol without a time subscript denotes the variable’s steady-state (or balanced growth path) value. In addition, it is assumed that ωπ>1, thus eliminating explosive paths.

B.4. Functional Forms and Auxiliary Assumptions

From here on, we make the following functional form assumptions:

and F(k,Zl)=kα(Zl)1−α and U(c,1−l)=c1−lψ1−σ1−σ

It is assumed that the state s^t stfollows a Markov process of the form μ(st|st−1) and that the wedges in period t can be used to uncover the event s^t uniquely, in the sense that the mapping from the event s^t to the wedges zt,1−τl,t,11+τl,t,gt,11+τb,t,Rt˜ is one-to-one and onto. Given this assumption, without loss of generality, let the underlying event st=(szt,slt,sxt,sgt,sbt,sR˜t), and let logzt(st)=szt,τl,t(st)=slt,τx,t(st)=sxt and loggt(st)=sgt. Given the unique mapping between s_t and the wedges we make following auxiliary choices:

logzt=logz(st), logg^t=logg^t(st),τl,t=τl(st),τx,t=τx(st),τb,t=τb(st),R˜t=R˜(st)

Note that we have effectively assumed that agents use only past wedges to forecast future wedges and that the wedges in period t are sufficient statistics for the event in period t. More precisely, the VAR representation of the underlying state s_t is modelled as follows

st+1=P0+Pst+Qεs,t+1,

where εs,t+1∼N(0,I).

B.5. Operational Model

In the operational model, we consider quantities which are not only expressed in per-capita terms but also detrended. To highlight the differences between this model’s and the previous model’s variables we introduce the notation v^≡vt(1+gz)t≡VtNt(1+gz)t. The model is then given by:

CRS production function:
(B-14)y^t(st)=k^t(st−1)αztlt(st)1−α
Aggregate resource constraint:
(B-15)y^t(st)=c^t(st)+g^t+x^t(st)
Capital accumulation law:
(B-16)(1+gn)(1+gz)t+1k^t+1(zt)=(1−δ)(1+gz)tk^t(zt−1)+(1+gz)tx^t(zt)⇔(1+gn)(1+gz)k^t+1(zt)=(1−δ)k^t(zt−1)+x^t(zt)
Taylor rule:
(B-17)Rt(st)=(1−ρR)R+ωy(lny^t(st)−lny^)+ωπ(πt(st)−π)+ρRRt−1(st−1)+R˜t,
F.O.C. labour:
(B-18)ψc^t(st)1−lt(st)=(1−τl,t)(1−α)k^t(st−1)αzt1−αlt(st)−α
F.O.C. capital:
(B-19)1=βEt{c^t+1(1+gz)t+1−σ(1−lt+1)ψ(1−σ)c^t(1+gz)t−σ(1−lt)ψ(1−σ)× [(1+τx,t+1)(1−δ)+αk^t+1(st)α−1zt+1lt+1(st+1)1−α1+τx,t]}=β˜Et{c^t+1c^t−σ1−lt+11−ltψ(1−σ)[(1+τx,t+1)(1−δ)+αk^t+1(st)α−1zt+1lt+1(st+1)1−α1+τx,t]},

where β˜=β/(1+gz)−σ.
F.O.C. bonds:
(B-20)1=βEt{c^t(st)(1+gz)tc^t+1(st+1)(1+gz)t+11+τb,t+11+τb,tpt(st)pt+1(st+1)[1+Rt(st)]}⇔1=β˜Et{c^t(st)c^t+1(st+1)1+τb,t+11+τb,tpt(st)pt+1(st+1)[1+Rt(st)]}

where β˜=β/(1+gz).

B.6. Steady State

The model in steady state is given by:

CRS production function:
(B-21)y^=k^αzl1−α
Aggregate resource constraint:
(B-22)y^=c^+g^+x^
Capital accumulation law:
(B-23)(1+gn)(1+gz)k^=(1−δ)k^+x^
Taylor rule:
(B-24)π=−R˜(1−ρR)ωπ+πSS,
F.O.C. labour:
(B-25)ψc^1−l=(1−τl)(1−α)k^αz1−αl−α
F.O.C. capital:
(B-26)1=β˜(1+τx)(1−δ)+αk^α−1zl1−α1+τx
F.O.C. bonds:
(B-27)1=β˜exp(−π)(1+R)⇒R=1β˜exp(−π)−1,

where we used that πt=logptpt−1.

To solve for the steady state of real variables, we start by solving (B-26) w.r.t. k^:

(B-28)k^=[αβ˜(1+τx)[1−β˜(1−δ)]]11−αzl≡Λzl

Plugging this expression for k^ in equation (B-25) yields:

(B-29)ψc^1−l=(1−τl)(1−α)(Λzl)αz1−αl−α⇔ψc^1−l=(1−τl)(1−α)Λαz⇔c^=1ψ(1−τl)(1−α)Λαz(1−l)

Inserting (B-28), (B-29) and (B-23) into the aggregate resource constraint leads to:

(B-30)k^α(zl)1−α=c^+g^+(1+gn)(1+gz)k^−(1−δ)k^Λzlα(zl)1−α=1ψ(1−τl)(1−α)Λαz(1−l)+g^+(1+gn)(1+gz)−(1−δ)ΛzlΛαzl=1ψ(1−τl)(1−α)Λαz−1ψ(1−τl)(1−α)Λαzl+g^+(1+gn)(1+gz)−(1−δ)Λzl

(B-31)Λαzl+1ψ(1−τl)(1−α)Λαzl−(1+gn)(1+gz)−(1−δ)Λzl=g^+1ψ(1−τl)(1−α)Λαzl=g^+1ψ(1−τl)(1−α)ΛαzΛαz1+1ψ(1−τl)(1−α)−Λz(1+gn)(1+gz)−(1−δ)

Using (B-31) in (B-28) one obtains:

(B-32)k^=Λzl=Λzg^+1ψ(1−τl)(1−α)ΛαzΛαz1+1ψ(1−τl)(1−α)−Λz(1+gn)(1+gz)−(1−δ)=g^+1ψ(1−τl)(1−α)ΛαzΛα−11+1ψ(1−τl)(1−α)−(1+gn)(1+gz)−(1−δ)

B.7. Definitions

Below we define the notation used for the model’s variables and parameters.

B.7.1. Variables

Lower-case variables define quantities in per-capita terms as follows:

(a)
Z: Labour-augmenting technical change Z=z(1+gz)t.
(b)
b: One period, nominal risk-free bonds; purchased in period t, pay off in period t + 1.
(c)
c: Consumption.
(d)
g: Government consumption.
(e)
k: Net capital stock.
(f)
l: Labour.
(g)
N: Population Nt=N0(1+gn)t.
(h)
r: Rental rate on capital.
(i)
R: Nominal interest rate.
(j)
t: Time period.
(k)
w: Wage rate.
(l)
x: Investment.
(m)
y: Output.
(n)
s: State of the world at time t.
(o)
π: Inflation rate.
(p)
τ_l: Labour tax.
(q)
τ_x: Tax on investment.
(r)
τ_b: Tax on bond holdings.

B.7.2. Parameters

The following parameters appear in the model:

(a)
g_n: Population growth rate of labour-augmenting technological process.
(b)
g_z: Growth rate of labour-augmenting technological process.
(c)
α: Parameter which determines the share of (and weight on) net capital stock in the Cobb-Douglas CRS production function.
(d)
β: Subjective discount factor, reflecting the time preference of the household.
(e)
δ: Depreciation rate of net capital stock.
(f)
ψ: Frisch elasticity of labour supply.
(g)
ρ_R: Weight on lagges nominal interest rate in Taylor rule (extent of ‘interest rate smoothing’).
(h)
ωπ: Coefficient on deviations of inflation from its steady-state value in Taylor’s rule.
(i)
ω_y: Coefficient on deviations of output from its steady-state value in Taylor’s rule.

C. Appendix – Gensys State Space

C.1. Log-linearized Equilibrium Conditions

We start by writing the system of equations in terms of k and s. This is done by replacing r, w, c^ and x^ in the first-order conditions with functions of the states. Thus, we start with:

(C-1)c^t+g^t+(1+gz)(1+gn)k^t+1−(1−δ)k^t=y^t=k^tα(ztlt)1−α

(C-2)ψc^t1−lt=(1−τlt)(1−α)k^tαlt−αzt1−α

(C-3)(1+τxt)c^t−σ(1−lt)ψ(1−σ) =β^Etc^t+1−σ(1−lt+1)ψ(1−σ)[αk^t+1α−1(zt+1lt+1)1−α+(1−δ)(1+τxt+1)],

where ψ=λ/(1−λ) and which can be reduced to the following:

ψ[k^tα(ztlt)1−α−(1+gn)(1+gz)k^t+1+(1−δ)k^t−g^t] =(1−τlt)(1−α)k^tαlt−αzt1−α(1−lt)(1+τxt)[k^tα(ztlt)1−α−(1+gn)(1+gz)k^t+1+(1−δ)k^t−g^t]−σ(1−lt)ψ(1−σ) =β^Et[k^t+1α(zt+1lt+1)1−α−(1+gn)(1+gz)k^t+2 +(1−δ)k^t+1−g^t+1]−σ(1−lt+1)ψ(1−σ) [αk^t+1α−1(zt+1lt+1)1−α+(1−δ)(1+τxt+1)].

Next, we compute the steady state of the system for constant values for z, the taxes and government spending:

k^/l=(1+τx)(1−β^(1−δ))β^αz1−α1/(α−1)c^=(k^/l)α−1z1−α−(1+gz)(1+gn)+1−δk^−g^=ξ1k^−g^c^=(1−τl)(1−α)(k^/l)αz1−α/ψ(1−1/(k^/l)k^)=ξ2−ξ3k^,

where the last two equations imply k^=(ξ2+g^)/(ξ1+ξ3),c^=ξ1k^−g^, l=(1/(k^/l))k^.

The log-linearization is done around these steady-state values. Detrended consumption is obtained via (C-1) and given approximately by:

c^t≈c^logc^t≈k^α(zl)1−α[αlogk^t+(1−α)(logzt+loglt)] −(1+gz)(1+gn)k^logk^t+1+(1−δ)k^logk^t−g^logg^t.

The labour input is then derived from the static first-order condition (C-2):

0≈ψ{k^α(zl)1−α[αlogk^t+(1−α)(logzt+loglt)]−(1+gz)(1+gn)k^logk^t+1+(1−δ)k^logk^t−g^logg^t}+(1−α)(1−τl)k^αl−αz1−α(1−l){1/(1−τl)τlt−αlogk^t+αloglt−(1−α)logzt+l/(1−l)loglt},

which we write succinctly as

loglt=φlklogk^t+φlzlogzt+φllτlt+φlglogg^t+φlk′logk^t+1.

Using this equation for logl, we use the other static first-order conditions to write logy^, logx^ and logc^ as follows:

logy^t=φyklogk^t+φyzlogzt+φylτlt+φyglogg^t+φyk′logk^t+1=(α+(1−α)φlk)logk^t+(1−α)(1+φlz)logzt+(1−α)[φllτlt+φlk′logk^t+1]logx^t=(1+gz)(1+gn)k^/x^logk^t+1−(1−δ)k^/x^logk^tlogc^t=φcklogk^t+φczlogzt+φclτlt+φcglogg^t+φck′logk^t+1=[y^logyt−x^logxt−g^logg^t]/c^,

where the φ ’s are known functions of the parameters.

Capital is derived from the dynamic first-order condition (C-4):

0≈(1+τx)c^−σ(1−l)ψ(1−σ)−ψ(1−σ)l/(1−l)loglt−σlogc^t+c^−σ(1−l)ψ(1−σ)τxt−β^Et{[αk^α−1(zl)1−α+(1−δ)(1+τx)]·[c^−σ(1−l)ψ(1−σ)−ψ(1−σ)l/(1−l)loglt+1−σlogc^t+1]+c^−σ(1−l)ψ(1−σ)[αk^α−1(zl)1−α(1−α)·(loglt+1+logzt+1−logk^t+1)+(1−δ)τxt+1]},

which simplifies to:

0≈(1+τx)−ψ(1−σ)l/(1−l)loglt−σlogc^t+τxt −Et{(1+τx)−ψ(1−σ)l/(1−l)loglt+1−σlogc^t+1 +β^[r(1−α)(loglt+1+logzt+1−logk^t+1)+(1−δ)τxt+1]},

where r=αy^/k^ and can be rewritten as:

(C-4)0≈φklloglt+φkclogc^t+τxt+φkElEtloglt+1+φkEcEtlogc^t+1+φkEzEtlogzt+1+φkEkEtlogk^t+1+φkEtxEtτxt+1.

C.2. Allowing for Adjustment Costs

To do log-linear computation (as in the baseline economy) in the case with adjustment costs and τct=τkt=0, we start with:

c^t+g^t+(1+gz)(1+gn)k^t+1−(1−δ)k^t+φ(x^t/k^t)k^t=y^t=k^tα(ztlt)1−αψc^t1−lt=(1−τlt)(1−α)k^tαlt−αzt1−α(1+τxt)c^t−σ(1−lt)ψ(1−σ)/(1−φ′(x^t/k^t)) =β^Etc^t+1−σ(1−lt+1)ψ(1−σ)[αk^t+1α−1(zt+1lt+1)1−α+(1−δ −φ(x^t+1/k^t+1)+φ′(x^t+1/k^t+1)x^t+1/k^t+1) (1+τxt+1)/1−φ′(x^t+1/k^t+1)],

where,

φ(x/k)=a2xk−b2.

and b is set equal to the investment-capital trend rate (i.e. b=(1+gz)(1+gn)−1+δ). To allow for different intensities of adjustment costs, a is raised from 0 (no adjustment costs) to 12.88 (the level used by Bernanke, Gertler, and Gilchrist (1999), the normal adjustment costs BGG level) to 4*12.88 (extreme adjustment costs).

Assuming φ(x^/k^)=φ′(x^/k^)=0, the log-linearization of these equations yields the same results as in the benchmark with the exception of the intertemporal condition:

0≈(1+τx){−ψ(1−σ)l/(1−l)loglt−σlogc^t+η(logx^t−logk^t)}+τxt −Et{(1+τx)−ψ(1−σ)l/(1−l)loglt+1−σlogc^t+1 +β^[r(1−α)(loglt+1+logzt+1−logk^t+1) +(1+τx)(1+gz)(1+gn)η(logx^t+1−logk^t+1) +(1−δ)τxt+1]},

where r=αy^/k^,η=φ′′(x^/k^)(x^/k^)=ab and the term in red is what is added to the baseline log-linearized dynamic equilibrium condition due to the presence of adjustment costs. This system can be rewritten as

(C-5)0≈φklloglt+φkclogc^t+τxt+φkxlogx^t+φkklogk^t+φkElEtloglt+1+φkEcEtlogc^t+1+φkEzEtlogzt+1+φkEkadjEtlogk^t+1+φkEtxEtτxt+1+φkExadjEtxt+1.

and differs from its baseline counterpart (C-4) by the terms in red.

C.3. Extension to MBCA – Šustek (2011)

The MBCA model features also bonds and prices. We thus have two additional equations which describe the dynamics of the nominal interest rate and inflation. The Taylor rule takes the form:

(C-6)Rt=(1−ρR)R+ωy(logy^t−logy^)+ωπ(πt−π)+ρRRt−1+R˜t⇔Rt=φπ0+φπylogy^t(st)+φππt+φπRRt−1+R˜t.

whereas the first-order condition for bonds is given by

(C-7)(1+τbt)c^t−σ(1−lt)ψ(1−σ)=β^Etc^t+1−σ(1−lt+1)ψ(1−σ)[(1+τbt+1)exp(−πt+1)(1+Rt)],

We follow the lines of Chari, Kehoe and McGrattan (2007) for the log-linearization of the F.O.C. for bonds and obtain

0≈(1+τb)c^−σ(1−l)ψ(1−σ)−ψ(1−σ)l/(1−l)loglt−σlogc^t +c^−σ(1−l)ψ(1−σ)τbt −β^Et{[(1+τb)exp(−π)(1+R)] ·[c^−σ(1−l)ψ(1−σ)−ψ(1−σ)l/(1−l)loglt+1−σlogc^t+1] +c^−σ(1−l)ψ(1−σ)[(1+τb)exp(−π)(1+R) ·((1+τb)−1τbt+1−πt+1+Rt)]},

which simplifies to:

0≈−ψ(1−σ)l/(1−l)loglt−σlogc^t+(1+τb)−1τbt −Et{[−ψ(1−σ)l/(1−l)loglt+1−σlogc^t+1] +[(1+τb)−1τbt+1−πt+1+Rt]},

and can be rewritten as:

0≈φRlloglt+φRclogc^t+φRτbτbt+φRElEtloglt+1+φREcEtlogc^t+1+φREτbEtτbt+1+Etπt+1−Rt.

C.4. Gensys State Space Representation

C.4.1. BCA – Chari et al. (2007)

The models we are interested in can be cast in the form:

(C-8)Γ0yt=Γ1yt−1+C+Ψϵt+Πηt

t=1,…,T, where C is a vector of constants, ∈_t is an exogenously evolving, possibly serially correlated, random disturbance and η_t is an expectational error, satisfying Etηt+1=0, all t. The η_t terms are not given exogenously, but instead are treated as determined as part of the model solution.

Within the context of the BCA model, the matrices Γ0yt,Γ1yt−1, C, Ψϵt,Πηt are given by:

Γ0yt=φkEk0010000φkl0φkcφkElφkEcφkEzφkEτx010000000000000001000000000000000100000000000000010000000000000001000000000φyk′φyzφyτl0φyg0−100000000φxk′000000−10000000φlk′φlzφlτl0φlg000−10000000000−100001000000000φcg0φcyφcx00−10000000000001000000000000000010000010000000000000000100000000000logk^t+1logztτltτxtlogg^t1logy^tlogx^tlogltlogg^tobslogc^tEt{loglt+1}Et{logc^t+1}Et{logzt+1}Et{τx,t+1}

Γ1yt−1=0000000000000000ρzρz,τlρz,τxρz,gz¯0000000000ρτl,zρτlρτl,τxρτl,gτ¯l0000000000ρτx,zρτx,τlρτxρτx,gτ¯x0000000000ρg,zρg,τlρg,τxρgg¯000000000000001000000000−φyk00000000000000−φxk00000000000000−φlk00000000000000000000000000000000000000000000000000000001000000000000000100000000000000010000000000000001logk^tlogzt−1τlt−1τxt−1logg^t−11logy^t−1logx^t−1loglt−1logg^t−1obslogc^t−1Et−1{loglt}Et−1{logc^t}Et−1{logzt}Et−1{τx,t}

C=000000000000000′

Ψϵt=0000q11000q21q2200q31q32q330q41q42q43q440000000000000000000000000000000000000000εz,tετl,tετx,tεg,t

Πηt=000000000000000000000000000000000000000000001000010000100001ηEl,tηEc,tηEz,tηEτx,t

C.4.2. MBCA – Šustek (2011)

Within the context of the MBCA model, the matrices Γ0yt,Γ1yt−1, C, Ψϵt,Πηt are given by:

Γ0yt=φkEk001000000φkl000φkcφkElφkEcφkEzφkEτx00010000000000000000000001000000000000000000000100000000000000000000010000000000000000000001000000000000000000000100000000000000000000010000000000000φyk′φyzφyτl0φyg000−1000000000000φxk′00000000−100000000000φlk′φlzφlτl0φlg00000−100000000000000−100000010000000000000001φπ0φπy000φπ−1000000000000φRτb0000φRl00−1φRcφRElφREc00φREτb10000φcg000φcyφcx0000−1000000000000000010000000000000000000000001000000010000000000000000000000100000000000000000000001000000000000000000000000000100000000logk^t+1logztτltτxtlogg^tτbtR˜t1logy^tlogx^tlogltlogg^tobsπtRtlogc^tEt{loglt+1}Et{logc^t+1}Et{logzt+1}Et{τx,t+1}Et{τb,t+1}Et{πt+1}

Γ1yt−1=0000000000000000000000ρzρz,τlρz,τxρz,gρz,τbρz,R˜z¯00000000000000ρτl,zρτlρτl,τxρτl,gρτl,τbρτl,R˜τ¯l00000000000000ρτx,zρτx,τlρτxρτx,gρτx,τbρτx,R˜τ¯x00000000000000ρg,zρg,τlρg,τxρgρg,τbρg,R˜g¯00000000000000ρτb,zρτb,τlρτb,τxρτb,gρτbρτb,R˜τ¯b00000000000000ρR˜,zρR˜,τlρR˜,τxρR˜,gρR˜,τbρR˜R˜¯0000000000000000000010000000000000−φyk00000000000000000000−φxk00000000000000000000−φlk000000000000000000000000000000000000000000000000000000−φπR0000000000000000000000000000000000000000000000000000000000000000100000000000000000000010000000000000000000001000000000000000000000100000000000000000000010000000000000000000001logk^tlogzt−1τlt−1τxt−1logg^t−1τbt−1R˜t−11logy^t−1logx^t−1loglt−1logg^t−1obsπt−1Rt−1logc^t−1Et−1{loglt}Et−1{logc^t}Et−1{logzt}Et−1{τx,t}Et−1{τb,t}Et−1{πt}

C=000000000000000000000′

Ψϵt=000000q1100000q21q220000q31q32q33000q41q42q43q4400q51q52q53q54q550q61q62q63q64q65q66000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000εz,tετl,tετx,tεg,tετb,tεR˜,t

Πηt=000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000010000001000000100000010000001ηEl,tηEc,tηEz,tηEτx,tηEτb,tηEπ,t

C.4.3. BCA – Chari et al. (2007) With Adjustment Costs

Allowing for adjustment costs in the standard BCA model requires modifying the state vector y(t) and the matrices Γ0yt,Γ1yt−1, C, Ψϵt,Πηt in the following way:

Γ0yt=φkEkadj001000φkxadjφkl0φkcφkElφkEcφkEzφkEτxφkExadj01000000000000000010000000000000000100000000000000001000000000000000010000000000φyk′φyzφyτl0φyg0−1000000000φxk′000000−100000000φlk′φlzφlτl0φlg000−100000000000−1000010000000000φcg0φcyφcx00−10000000000000100000000000000000100000010000000000000000010000000000000000000100000000logk^t+1logztτltτxtlogg^t1logy^tlogx^tlogltlogg^tobslogc^tEt{loglt+1}Et{logc^t+1}Et{logzt+1}Et{τx,t+1}Et{logxt+1}

Γ1yt−1=−φkkadj0000000000000000ρzρz,τlρz,τxρz,gz¯00000000000ρτl,zρτlρτl,τxρτl,gτ¯l00000000000ρτx,zρτx,τlρτxρτx,gτ¯x00000000000ρg,zρg,τlρg,τxρgg¯00000000000000010000000000−φyk000000000000000−φxk000000000000000−φlk0000000000000000000000000000000000000000000000000000000000100000000000000001000000000000000010000000000000000100000000000000001logk^tlogzt−1τlt−1τxt−1logg^t−11logy^t−1logx^t−1loglt−1logg^t−1obslogc^t−1Et−1{loglt}Et−1{logc^t}Et−1{logzt}Et−1{τx,t}Et−1{logxt}

C=0000000000000000′

Ψϵt=0000q11000q21q2200q31q32q330q41q42q43q44000000000000000000000000000000000000000000000000εz,tετl,tετx,tεg,t

Πηt=00000000000000000000000000000000000000000000000000000001000001000001000001000001ηEl,tηEc,tηEz,tηEτx,tηExt

C.4.4. MBCA – Šustek (2011) with Adjustment Costs

Allowing for adjustment costs in the standard BCA model requires modifying the state vector y(t) and the matrices Γ0yt,Γ1yt−1, C, Ψϵt,Πηt in the following way:

Γ0yt=φkEkadj00100000φkxadjφkl000φkcφkElφkEcφkEzφkEτx00φkEτxadj0100000000000000000000001000000000000000000000010000000000000000000000100000000000000000000001000000000000000000000010000000000000000000000100000000000000φyk′φyzφyτl0φyg000−10000000000000φxk′00000000−1000000000000φlk′φlzφlτl0φlg00000−1000000000000000−1000000100000000000000001φπ0φπy000φπ−10000000000000φRτb0000φRl00−1φRcφRElφREc00φREτb100000φcg000φcyφcx0000−100000000000000000100000000000000000000000001000000001000000000000000000000001000000000000000000000001000000000000000000000000000010000000000000000001000000000000logk^t+1logztτltτxtlogg^tτbtR˜t1logy^tlogx^tlogltlogg^tobsπtRtlogc^tEt{loglt+1}Et{logc^t+1}Et{logzt+1}Et{τx,t+1}Et{τb,t+1}Et{πt+1}Et{logxt+1}

Γ1yt−1=−φkkadj0000000000000000000000ρzρz,τlρz,τxρz,gρz,τbρz,R˜z¯000000000000000ρτl,zρτlρτl,τxρτl,gρτl,τbρτl,R˜τ¯l000000000000000ρτx,zρτx,τlρτxρτx,gρτx,τbρτx,R˜τ¯x000000000000000ρg,zρg,τlρg,τxρgρg,τbρg,R˜g¯000000000000000ρτb,zρτb,τlρτb,τxρτb,gρτbρτb,R˜τ¯b000000000000000ρR˜,zρR˜,τlρR˜,τxρR˜,gρR˜,τbρR˜R˜¯000000000000000000000100000000000000−φyk000000000000000000000−φxk000000000000000000000−φlk00000000000000000000000000000000000000000000000000000000−φπR00000000000000000000000000000000000000000000000000000000000000000001000000000000000000000010000000000000000000000100000000000000000000001000000000000000000000010000000000000000000000100000000000000000000001logk^tlogzt−1τlt−1τxt−1logg^t−1τbt−1R˜t−11logy^t−1logx^t−1loglt−1logg^t−1obsπt−1Rt−1logc^t−1Et−1{loglt}Et−1{logc^t}Et−1{logzt}Et−1{τx,t}Et−1{τb,t}Et−1{πt}Et−1{logxt}

C=0000000000000000000000′

Ψϵt=000000q1100000q21q220000q31q32q33000q41q42q43q4400q51q52q53q54q550q61q62q63q64q65q66000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000εz,tετl,tετx,tεg,tετb,tεR˜,t

Πηt=0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000010000000100000001000000010000000100000001ηEl,tηEc,tηEz,tηEτx,tηEτb,tηEπ,tηEx,t

D. Appendix – Derivatives with Alternative Stepsize

We here report the results for the case where the stepsize used to compute the numerical derivatives is not set to 1e−3 like in Komunjer and Ng (2011) but rather automatically selected by Matlab using the function nuderst. The returned step size is the maximum of 1e−4 times the absolute value of the current parameter and 1e−7.

The main results of the previous analysis hold through with two notable exceptions. Fist, as becomes evident from Tables D-1 and D-2, both the baseline BCA and MBCA model are strictly identifiable already at a tolerance level of 1e−9 (vs. 1e−11). Second, the BCA model is strictly identifiable even when the deep parameters are estimated at a tolerance level of 1e−10, as reported in Table D-3. This is not the case for the MBCA model (see Table D-5). A few things:

Table D-1.

Komunjer and Ng Test Results BCA Model.

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	29	25	15	51	40	62	0
e−03	29	25	16	54	45	69	0
e−04	29	25	16	54	45	69	0
e−05	29	25	16	54	45	69	0
e−06	29	25	16	54	45	69	0
e−07	30	25	16	54	46	69	0
e−08	30	25	16	54	46	69	0
e−09	30	25	16	55	46	71	1
e−10	30	25	16	55	46	71	1
e−11	30	25	16	55	46	71	1
Default = 2.756906e−12	30	25	16	55	46	71	1
Required	30	25	16	55	46	71	1

Summary: nθ=30,nX=5,nε=4.

Order condition: nθ=30,nδ=50.

Table D-2.

Komunjer and Ng Test Results BCA Model (Deep Parameters Estimated).

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	33	25	15	52	44	64	0
e−03	34	25	16	57	50	71	0
e−04	34	25	16	57	50	71	0
e−05	35	25	16	57	51	71	0
e−06	36	25	16	57	52	71	0
e−07	36	25	16	59	52	72	0
e−08	37	25	16	59	53	73	0
e−09	37	25	16	60	53	74	0
e−10	37	25	16	62	53	76	0
e−11	37	25	16	62	53	78	1
Default = 5.513812e−12	37	25	16	62	53	78	1
Required	37	25	16	62	53	78	1

Summary: nθ=37,nX=5,nε=4.

Order condition: nθ=37,nδ=50.

Table D-3.

Komunjer and Ng Test Results MBCA Model.

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	60	49	33	101	89	130	0
e−03	60	49	36	108	96	143	0
e−04	60	49	36	109	96	144	0
e−05	60	49	36	109	96	144	0
e−06	60	49	36	109	96	144	0
e−07	61	49	36	109	97	145	0
e−08	61	49	36	110	97	145	0
e−09	61	49	36	110	97	146	1
e−10	61	49	36	110	97	146	1
Default = 1.165290e−11	61	49	36	110	97	146	1
e−11	61	49	36	110	97	146	1
Required	61	49	36	110	97	146	1

Summary: nθ=61,nX=7,nε=6.

Order condition: nθ=61,nδ=105.

Table D-4 (D-6) reports results for when τbss and R˜ss (and deep parameters) are estimated.

Table D-4.

Komunjer and Ng Test Results MBCA Model (τbss and R˜ss Estimated).

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	60	49	33	101	89	130	0
e−03	60	49	36	108	96	143	0
e−04	60	49	36	109	96	144	0
e−05	60	49	36	109	96	144	0
e−06	60	49	36	109	96	144	0
e−07	61	49	36	109	97	145	0
Default = 1.165290e−11	61	49	36	110	97	146	0
e−08	61	49	36	110	97	146	0
e−09	61	49	36	110	97	146	0
e−10	61	49	36	110	97	146	0
e−11	61	49	36	110	97	146	0
Required	63	49	36	112	99	148	1

Summary: nθ=63,nX=7,nε=6.

Order condition: nθ=63,nδ=105.

Problematic parameters at Tol = 1e−3: τbss, R˜ss.

Problematic parameters at Tol = 1.000000e−11: τlss, τxss, g_ss, τbss, R˜ss, ρ_z, ρτl,z, ρτx,z, ρg,z, ρτb,z, ρR˜,z, ρz,τl, ρτl, ρτx,τl, ρg,τl, ρτb,τl, ρR˜,τl, ρz,τx, ρτl,τx, ρτx, ρg,τx, ρτb,τx, ρR˜,τx, ρz,g, ρτl,g, ρτx,g, ρ_g, ρτb,g, ρR˜,g, ρz,τb, ρτl,τb, ρτx,τb, ρg,τb, ρτb, ρz,R˜, ρτl,R˜, ρτx,R˜, ρg,R˜, ρτb,R˜, ρR˜, q₂₁, q₃₁, q₅₁, q₂₂, q₃₂, q₄₂, q₅₂, q₃₃, q₅₃, q₆₃, q₄₄, q₅₄.

Table D-5.

Komunjer and Ng Test Results MBCA Model (Deep Parameters Estimated).

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	67	49	35	104	97	133	0
e−03	68	49	36	113	104	146	0
e−04	69	49	36	115	105	148	0
e−05	70	49	36	115	106	148	0
e−06	70	49	36	116	106	149	0
e−07	71	49	36	118	107	149	0
e−08	72	49	36	118	108	151	0
e−09	72	49	36	120	108	154	0
e−10	72	49	36	121	108	156	0
Default = 2.330580e−11	72	49	36	121	108	156	0
e−11	72	49	36	121	108	157	1
Required	72	49	36	121	108	157	1

Summary: nθ=72,nX=7,nε=6.

Order condition: nθ=72,nδ=105.

Problematic parameters at Tol = 1e−3: z_ss, τlss, τxss, g_ss, ρτl,z, ρz,τl, ρτx,τl, ρg,τl, ρτb,τl, ρR˜,τl, ρτl,g, ρτl,τb, ρτl,R˜, ρτx,R˜, q₂₁, q₂₂, ψ.

Problematic parameters at Tol = 2.330580e−11: z_ss, τlss, τxss, g_ss, ρ_z, ρτl,z, ρτx,z, ρg,z, ρτb,z, ρR˜,z, ρz,τl, ρτl, ρτx,τl, ρg,τl, ρτb,τl, ρR˜,τl, ρz,τx, ρτl,τx, ρτx, ρg,τx, ρτb,τx, ρR˜,τx, ρz,g, ρτl,g, ρτx,g, ρ_g, ρτb,g, ρR˜,g, ρz,τb, ρτl,τb, ρτx,τb, ρg,τb, ρτb, ρR˜,τb, ρz,R˜, ρτl,R˜, ρτx,R˜, ρg,R˜, ρτb,R˜, ρR˜, q₁₁, q₂₁, q₃₁, q₄₁, q₅₁, q₆₁, q₂₂, q₃₂, q₄₂, q₅₂, q₆₂, q₃₃, q₄₃, q₅₃, q₆₃, q₄₄, q₅₄, q₆₄, q₅₅, q₆₅, q₆₆, g_n, g_z, β, δ, ψ, σ, α, ρ_R, ωπ, ω_y, π_ss.

Table D-6.

Komunjer and Ng Test Results MBCA Model (τbss,R˜ss and Deep Parameters Estimated).

Tol	ΔΛS	ΔTS	ΔUS	ΔΛTS	ΔΛUS	ΔS	Pass
e−02	67	49	35	104	97	133	0
e−03	68	49	36	113	104	146	0
e−04	69	49	36	115	105	148	0
e−05	69	49	36	115	105	148	0
e−06	70	49	36	116	106	148	0
e−07	71	49	36	118	107	150	0
Default = 2.330580e−11	72	49	36	121	108	157	0
e−08	72	49	36	119	108	152	0
e−09	72	49	36	121	108	154	0
e−10	72	49	36	121	108	156	0
e−11	72	49	36	121	108	157	0
Required	74	49	36	123	110	159	1

Summary: nθ=74,nX=7,nε=6.

Order condition: nθ=74,nδ=105.

Problematic parameters at Tol = 1e−3: z_ss, τlss, τxss, g_ss, τbss, R˜ss, ρτl,z, ρz,τl, ρτx,τl, ρg,τl, ρτb,τl, ρR˜,τl, ρτl,g, ρτl,τb, ρz,R˜, ρτl,R˜, ρτx,R˜, ρτb,R˜, q₂₁, q₂₂, ψ, σ, ωπ, π_ss.

Problematic parameters at Tol = 1.000000e−11: z_ss, τlss, τxss, g_ss, τbss, R˜ss, ρ_z, ρτl,z, ρτx,z, ρg,z, ρτb,z, ρR˜,z, ρz,τl, ρτl, ρτx,τl, ρg,τl, ρτb,τl, ρR˜,τl, ρz,τx, ρτl,τx, ρτx, ρg,τx, ρτb,τx, ρR˜,τx, ρz,g, ρτl,g, ρτx,g, ρ_g, ρτb,g, ρR˜,g, ρz,τb, ρτl,τb, ρτx,τb, ρg,τb, ρτb, ρR˜,τb, ρz,R˜, ρτl,R˜, ρτx,R˜, ρg,R˜, ρτb,R˜, ρR˜, q₁₁, q₂₁, q₃₁, q₄₁, q₅₁, q₆₁, q₂₂, q₃₂, q₄₂, q₅₂, q₆₂, q₃₃, q₄₃, q₅₃, q₆₃, q₄₄, q₅₄, q₆₄, q₅₅, q₆₅, q₆₆, g_n, g_z, β, δ, ψ, σ, α, ρ_R, ωπ, ω_y, π_ss.

Notes

1

Global identification would extend the uniqueness requirement to the whole parameter space.

2

A point is regular if it belongs to an open neighborhood where the rank of the matrix does not vary.

3

Please refer to Iskrev (2010) for further details.

4

This detrending method is used in Brinca et al. (2016) and is found to make the ML estimation procedure very robust across a large set of OECD countries.

5

We use Chari et al. (2007) actual data for the study of the 1982 recession.

6

Testing that the f_i statistics are not significantly different from each other would be the most rigorous way of answering this question.

7

In cases where the upper bound and the lower bound only overlapped at a given number (e.g. f_i statistic of wedge 1 UB = 0.02 and f_i statistic of wedge 2 LB = 0.02), we still counted them as the f_i statistic overlapping and so the number would not be reported in bold.

8

As pointed out by Qu and Tkachenko (2012) it is usually the case that δ(θ¯0) is unknown and, thus, so is the domain of the curve. This is why in constructing the non-identification curve first a wide support is considered, the model solved and the spectrum computed. The resulting curve is then truncated so as to exclude points which (i) are associated with indeterminacy, (ii) violate the natural bounds of the parameters and (iii) yield fθ¯(ω) different from fθ¯0(ω).

9

There is no need to consider ω∈[−π,0] because fθ¯(ω) is equal to the conjugate of fθ¯(−ω).

References

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Canova, F., & Sala, L. (2009, May). Back to square one: Identification issues in DSGE models. Journal of Monetary Economics, 56(4), 431–449.

Chari, V. V., Kehoe, P. J., & McGrattan, E. R. (2007). Business cycle accounting. Econometrica, 75(3), 781–836.

Cociuba, S. E., & Ueberfedt, A. A. (2010). Trends in U.S. hours and the labor wedge. Bank of Canada Working Paper Series No. 28. https://ideas.repec.org/p/bca/bocawp/10-28.html

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Acknowledgements

The views expressed are those of the authors and do not necessarily reflect those of the Bank of Portugal, the Federal Reserve Board, the Federal Reserve System or their staff. We thank Stéphane Bonhomme, Fabio Canova, Ellen McGrattan, Ricardo Reis, Barbara Rossi, Roman Šustek and Harald Uhlig for stimulating comments. We are also grateful to seminar participants at Norges Bank Brown Bag seminar, 2nd HenU/INFER Workshop on Applied Macroeconomics, The University of Chicago Econometrics Reading Group, 3rd Annual Conference of the International Association for Applied Econometrics, 22nd International Conference of the Society for Computational Economics on Computing in Economics and Finance, 10th Annual Meeting of the Portuguese Economic Journal and 31st Annual Congress of the European Economic Association. Pedro Brinca acknowledges funding from Fundação para a Ciência e a Tecnologia (UID/ECO/00124/2013, UID/ECO/00124/2019, PTDC/EGE-ECO/7620/2020; and Social Sciences DataLab, LISBOA-01-0145-FEDER-022209), POR Lisboa (LISBOA-01-0145-FEDER-007722 and LISBOA-01-0145-FEDER-022209), POR Norte (LISBOA-01-0145-FEDER-022209) and CEECIND/02747/2018.

Book Chapters

Prelims

Introduction

Chapter 1: Real-Time Real Economic Activity: Entering and Exiting the Pandemic Recession of 2020

Chapter 2: State Correlation and Forecasting: A Bayesian Approach Using Unobserved Components Models

Chapter 3: On Identification Issues in Business Cycle Accounting Models

Chapter 4: The Effect of News Shocks and Monetary Policy

Chapter 5: Statistical Identification of Economic Shocks by Signs in Structural Vector Autoregression

Chapter 6: Skewed SVARs: Tracking the Structural Sources of Macroeconomic Tail Risks

Index

Abstract

Keywords

Citation

Publisher

1 Introduction

2 The Prototype (M)BCA Economy

2.1 Description of the Economy

2.2 Equilibrium Conditions

2.3 Operational Model

3 Methodology

3.1 State Space Form

3.2 Estimated Parameters

3.3 Komunjer and Ng (2011) Test for Strict Identification

3.4 Iskrev (2010) Test for Strict and Weak Identification

3.4.1 Preliminaries

3.4.2 General Principles of Identification Analysis

3.4.3 Identification Strength

4 Results

4.1 Komunjer and Ng (2011)

4.1.1 Chari et al. (2007) BCA Model

4.1.2 Šustek (2011) MBCA Model

4.2 Iskrev (2010)

4.2.1 Chari et al. (2007) BCA Model

Fig. 1.

Fig. 2.

4.2.2 Šustek (2011) MBCA Model

Fig. 3.

Fig. 4.

5 Economic Relevance

5.1 Chari et al. (2007) BCA Model

5.2 Brinca et al. (2016) Multi-country BCA Analysis

6 Statistics for Practitioners

6.1 Empirical Distance Measures

7 Conclusion

A. Appendix – BCA and MBCA Model with Investment Adjustment Costs

A.1. Komunjer and Ng (2011)

A.1.1 Chari et al. (2007) BCA Model

A.1.2 Šustek (2011) MBCA Model

A.2 Iskrev (2010)

A.2.1 Chari et al. (2007) BCA Model

B. Appendix – Model Derivations

B.1. Representative Consumer

B.1.1. Optimization Problem of the Household

B.1.2. Lagrangian Function

B.1.3. First-order Necessary Conditions

B.2. Representative Producer

B.2.1. Optimization Problem of the Firm

B.3. Additional Model Equations

B.4. Functional Forms and Auxiliary Assumptions

B.5. Operational Model

B.6. Steady State

B.7. Definitions

B.7.1. Variables

B.7.2. Parameters

C. Appendix – Gensys State Space

C.1. Log-linearized Equilibrium Conditions

C.2. Allowing for Adjustment Costs

C.3. Extension to MBCA – Šustek (2011)

C.4. Gensys State Space Representation

C.4.1. BCA – Chari et al. (2007)

C.4.2. MBCA – Šustek (2011)

C.4.3. BCA – Chari et al. (2007) With Adjustment Costs

C.4.4. MBCA – Šustek (2011) with Adjustment Costs

D. Appendix – Derivatives with Alternative Stepsize

Notes

References

Acknowledgements

Book Chapters

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