Identification Analysis of DSGE models with DYNARE FP7 Funded, Project MONFISPOL Grant no.: 225149. Marco Ratto European Commission, Joint Research Centre with the contribution of Nikolai Iskrev Bank of Portugal MONFISPOL Final Conference, Frankfurt
Outline 1 Introduction 2 DSGE models Moments 3 Identification Local identification 4 DYNARE Implementation Identification strength Analyzing identification patterns Main features of the software DYNARE procedure Notes 5 Possible extensions 6 Examples
Intro Introduction In developing the identification software, we took into consideration the most recent developments in the computational tools for analyzing identification in DSGE models. A growing interest is being addressed to identification issues in economic modeling (Canova and Sala, 2009; Komunjer and Ng, 2011a,b; Iskrev, 2010b). The identification toolbox includes the new efficient method for derivatives computation presented in Ratto and Iskrev (2010a,b) and the identification tests proposed by Iskrev.
DSGE Models DSGE Models: Structural model and reduced form A DSGE model is summarized by a system g of m non-linear equations: ) E t (g(ẑ t,ẑ t+1,ẑ t 1,u t θ) = 0 (1) Most studies use linear approximations of the original models: Γ 0 (θ)z t = Γ 1 (θ)e t z t+1 +Γ 2 (θ)z t 1 +Γ 3 (θ)u t (2) where z t = ẑ t ẑ. The elements of the matrices Γ 0, Γ 1, Γ 2 and Γ 3 are functions of θ.
DSGE Models Assuming that a unique solution exists, it can be cast in the following form z t = A(θ)z t 1 +B(θ)u t (3) Some of the variables in z t are not observed, so the transition equation (3) is complemented by a measurement equation x t = Cz t +Du t +ν t (4) τ collects the non-constant elements of ẑ, A, and Ω, i.e. τ := [τ z, τ A, τ Ω ].
DSGE Models Moments Theoretical first and second moments The unconditional first and second moments of x t are given by Ex t := µ x = s (5) { cov(x t+i,x t ) := Σ CΣz (0)C if i = 0 x(i) = CA i Σ z (0)C (6) if i > 0 where Σ z (0) := Ez t z t solves the matrix equation Σ z (0) = AΣ z (0)A +Ω (7)
DSGE Models Moments Denote the observed data with X T := [x 1,...,x T ], and let Σ T be its covariance matrix, i.e. Σ T := EX T X T = Σ x (0), Σ x (1),..., Σ x (T 1) Σ x (1), Σ x (0),..., Σ x (T 2)............ Σ x (T 1), Σ x (T 2),..., Σ x (0) (8)
DSGE Models Moments We define m T := [µ,σ T ], where σ T := [vech(σ x (0)),vec(Σ x (1)),...,vec(Σ x (T 1)) ] m T is a function of θ. If either u t is Gaussian, or there are no distributional assumptions about the structural shocks, the model-implied restrictions on m T contain all information that can be used for the estimation of θ. The identifiability of θ depends on whether that information is sufficient or not.
Identification Global identification: the Gaussian case Theorem Suppose that the data X T is generated by the model (3)-(4) with parameter vector θ 0. Then θ 0 is globally identified if m T ( θ) = m T (θ 0 ) θ = θ 0 (9) for any θ Θ. If (9) is true only for values θ in an open neighborhood of θ 0, the identification of θ 0 is local.
Identification Local identification Local identification: The rank condition Theorem Suppose that m T is a continuously differentiable function of θ. Then θ 0 is locally identifiable if the Jacobian matrix J(q) := m q θ has a full column rank at θ 0 for q T. This condition is both necessary and sufficient when q = T if u t is normally distributed.
Identification Local identification Given the chain rule J(T) = m T τ τ θ (10) another necessary condition discussed in Iskrev (2010b): Corollary The point θ 0 is locally identifiable only if the rank of J 2 = τ θ at θ 0 is equal to k. The condition is necessary because the distribution of X T depends on θ only through τ, irrespectively of the distribution of u t. It is not sufficient since, unless all state variables are observed, τ may be unidentifiable.
DYNARE Implementation DYNARE implementation I The identifiability of each draw θ j is then established using the necessary and sufficient conditions discussed by Iskrev (2010b): Finding that matrix J 2 is rank deficient at θ j implies that this particular point in Θ is unidentifiable in the model. Finding that J 2 has full rank but J(T) does not, means that θ j cannot be identified given the set of observed variables and the number of observations. we also analyze the derivatives of the LRE form of the model (J Γ = γ θ ), to check for trivial non-identification problem, like two parameters always entering as a product in Γ i matrices; if θ is identified check J(q) with q < T. According to Theorem 2 this is sufficient for identification;
DYNARE Implementation Tracking singularities Whenever some of the matrices J 2, J(T) or J Γ is rank deficient, the code tries to diagnose the subset of parameters responsible for the rank deficiency: 1 if there are columns of zeros in the J( ) matrix, the associated parameter is printed on the MATLAB command window; 2 compute pairwise- and multi-correlation coefficients for each column of the J( ) matrix: if there are parameters with correlation coefficients equal to unity, these are printed on the MATLAB command window; 3 take the Singular Values Decomposition (SVD) of J( ) and track the eigenvectors associated to the zero singular values.
DYNARE Implementation Identification strength Identification strength A measure of identification strength is introduced, following the work of Iskrev (2010a) and Andrle (2010), based on the information matrix the strength of identification for parameter θ i : s i = θi 2/(I T(θ) 1 ) (i,i) (11) a sort of a priori t-test for θ i ; alternative normalization using the prior standard deviation σ(θ i ): s prior i = σ(θ i )/ prior i = σ(θ i ) (I T (θ) 1 ) (i,i) (12) I T (θ) (i,i) (13)
DYNARE Implementation Identification strength sensitivity and correlation components (Iskrev, 2010a), std(θ i ) {I T (θ) 1 } = 1 1 i 1 ρ 2 i i.e. weak identification may be due to i 0 or ρ i 1; (14) the sensitivity component is defined as i = θi 2 I T (θ) (i,i) (15) The identification toolbox shows, after the check of rank conditions, the plots of the strength of identification and of the sensitivity component for all estimated parameters.
DYNARE Implementation Analyzing identification patterns Analyzing identification patterns 1 Andrle (2010): the identification patterns are shown by taking the singular value decomposition of I T (θ) or of the J(q) matrix and displaying the eigenvectors corresponding to the smallest (or highest) singular values; 2 Iskrev (2010b): check which group of one, two or more parameters is most capable to mimic (replace) the effect of each parameter. A brute force search is done for each column of J(q) (j) to detect the group of columns J(q) (I j), having the highest explanatory power for J(q) (j) by a linear regression.
DYNARE Implementation Features Main features of the software The new DYNARE keyword identification triggers the routines developed at JRC. This option has two modes of operation; point identification check (default) and Monte Carlo mode.
DYNARE Implementation Features Point identification check With a list of estimated parameters: with prior definition: the program performs the local identification checks for the estimated parameters at the prior mean (prior mode, posterior mean and posterior mode are also alternative options); for ML estimation (no prior definition), local identification checks are performed for the estimated parameters at the actual or initial value declared for estimation (ML value is also possible); No list of estimated parameters: the program computes the local identification checks for all the model parameter values declared in the DYNARE model file.
DYNARE Implementation Features Monte Carlo exploration When information about prior distribution is provided, a full Monte Carlo analysis is also possible. For a number of parameter sets sampled from prior distributions, the local identification analysis is performed in turn. This provides a global prior exploration of local identification properties of DSGE models. This is a glocal procedure. This Monte Carlo mode can also be linked to the the global sensitivity analysis toolbox, also available in the official DYNARE as of version 4.3.
DYNARE Implementation Features The optional Monte Carlo implementation I a sample from Θ is made of many randomly drawn points from Θ, where Θ Θ discarding values of θ that do not imply a unique solution. The set Θ contains all values of θ that are theoretically plausible, and may be constructed by specifying a lower and an upper bound for each element of θ or by directly specifying prior distributions. After specifying a distribution for θ with support on Θ, one can obtain points from Θ by drawing from Θ and removing draws for which the model is either indetermined or does not have a solution. Conditions for existence and uniqueness are automatically checked by DYNARE.
DYNARE Implementation Features The optional Monte Carlo implementation II The checks for local identification are then performed for each parameter set in turn, obtaining a Monte Carlo sample of identification features for the model under analysis.
DYNARE Implementation Features The Advanced option (point) analysis of the LRE form and the reduced form; identification patterns (Iskrev, 2010b; Andrle, 2010).
DYNARE Implementation Features The Advanced option (Monte Carlo) analysis of the condition number of the Jacobians J(q), J 2 and J Γ and detection of the parameters that mostly drive large condition numbers (i.e. weaker identification); analysis of the identification patters across the Monte Carlo sample; detailed point-estimate (identification strength and collinearity analysis) of the parameters set having the smallest/largest condition number; when some singularity (rank condition failure) is detected for some elements of the Monte Carlo sample, detailed point-estimates are performed for such critical points.
DYNARE Implementation Features Test routines A library of test routines is also provided in the official DYNARE test folder. Such tests implement some of the examples described in the present document. Kim (2003) : the DYNARE routines for this example are placed in the folder dynare_root/tests/identification/kim; An and Schorfheide (2007) : the DYNARE routines for this example are placed in dynare_root/tests/identification/as2007;
DYNARE Implementation DYNARE procedure DYNARE procedure A new syntax is available in DYNARE. The simple keyword identification(<options>=<values>); triggers the point local identification checks, performed at the prior mean. Prior definitions and the list of observed values are needed, using the standard DYNARE syntax for setting-up an estimation.
DYNARE Implementation DYNARE procedure DYNARE procedure: Options I parameter set = prior mode prior mean posterior mode posterior mean posterior median. Specify the parameter set to use. Default: prior_mean. prior_mc = INTEGER sets the number of Monte Carlo draws (default = 1); prior_mc=1 triggers the default point identification analysis; prior_mc>1 triggers the Monte Carlo mode; prior_range = INTEGER triggers uniform sample within the range implied by the prior specifications (when prior_mc>1). Default: 0 load_ident_files = 0, triggers a new analysis, while load_ident_files = 1, loads and displays a previously performed analysis (default = 0);
DYNARE Implementation DYNARE procedure DYNARE procedure: Options II ar = <integer> (default = 3), triggers the value for q in computing J(q); useautocorr: this option triggers J(q) in the form of auto-covariances and cross-covariances (useautocorr = 0), or in the form of auto-correlations and cross-correlations (useautocorr = 1). The latter form normalizes all m q entries in [ 1,1] (default = 0). advanced = INTEGER triggers standard or advanced identification analysis (default = 0). max_dim_cova_group = INTEGER In the brute force search (performed when advanced=1) this option sets the maximum dimension of groups of parameters that best reproduce the behavior of each single model parameter. Default: 2
DYNARE Implementation DYNARE procedure DYNARE procedure: Options III periods = INTEGER triggers the length of the stochastic simulation to compute the analytic Hessian. Default: 300 periods = INTEGER When the analytic Hessian is not available (i.e. with missing values or diffuse Kalman filter or univariate Kalman filter), this triggers the length of stochastic simulation to compute Simulated Moments Uncertainty. Default: 300 replic = INTEGER When the analytic Hessian is not available, this triggers the number of replicas to compute Simulated Moments Uncertainty. Default: 100.
DYNARE Implementation DYNARE procedure DYNARE procedure: Options IV gsa_sample_file = INTEGER If equal to 0, do not use sample file. If equal to 1, triggers GSA prior sample. If equal to 2, triggers GSA Monte-Carlo sample (i.e. loads a sample corresponding to pprior=0 and ppost=0 in the dynare_sensitivity options). Default: 0 gsa_sample_file = FILENAME Uses the provided path to a specific user defined sample file. Default: 0
DYNARE Implementation Notes Notes I DYNARE symbolic preprocessor interprets and implements the model definitions as expressed in the DYNARE file; It will not reflect all parameter definitions which may be hidden in the <>_steadystate.m file. The # syntax should be used in the model block of the DYNARE file, instead.
Possible extensions Possible extensions I Komunjer and Ng (2011a): use structural properties of the canonical solution of DSGE models and restrictions implied by observational equivalence to derive rank conditions that do not require knowledge of infinite autocovariances or Markov parameters; like Iskrev, their conditions do not depend on data but also do not depend on the estimator (i.e. do not rely on Gaussian assumptions about shocks) nor on T (i.e. apply to T );
Possible extensions Possible extensions II useful framing in terms of controllability/observability and minimal state space representation (control system engineering); no big practical advance w.r.t. the present approach for identification purposes, but useful additional diagnostics to analyse DSGE models. use numerical derivatives.
Possible extensions Possible extensions I Koop et al. (2011) suggest two Bayesian identification indicators: is a subset of params is known to be identified, the marginal posterior is compared to the posterior expectation of the prior conditionasl on the identified parameters; considers the rate at which posterior precision gets updated as the sample size T is increased: for unidentified parameters, rate of update is smaller than T.
Examples Examples
Examples Kim (2003) Kim (2003) This paper demonstrated a functional equivalence between two types of adjustment cost specifications, coexisting in macroeconomic models with investment: intertemporal adjustment costs which involve a nonlinear substitution between capital and investment in capital accumulation, and multisectoral costs which are captured by a nonlinear transformation between consumption and investment. We reproduce results of Kim (2003), worked out analytically, applying the DYNARE procedure on the non-linear form of the model.
Examples Kim (2003) The representative agent maximizes β t logc t (16) t=0 subject to a national income identity and a capital accumulation equation: ( Ct (1 s) 1+θ ( It +s 1 s) s [ ( It δ δ K t+1 = where s = βδα, = 1 β +βδ. ) 1+θ = (At K α t )1+θ (17) ) 1 φ +(1 δ)k 1 φ t ] 1 1 φ (18)
Examples Kim (2003) φ( 0) is the inverse of the elasticity of substitution between I t and K t (represents the size of intertemporal adjustment costs) θ( 0) is the inverse of the elasticity of transformation between consumption and investment (called the multisectoral adjustment cost parameter). θ. the two adjustment cost parameter only enter through an overall adjustment cost parameter Φ = φ+θ 1+θ, thus implying that they cannot be identified separately.
Examples Kim (2003) On the effect of the number of states. Assume to have an additional equation for the Lagrange multiplier λ t = (1 s)θ, with λ t entering the Euler equation. (1+θ)C (1+θ) t We still assume that only C t and I t can be observed.
R t = ρ R R t 1 +(1 ρ R )ψ 1 π t +(1 ρ R )ψ 2 ( y t +z t )+ε R,t (21) Examples An Schorfheide (2007) An and Schorfheide (2007) y t = E t [y t+1 ]+g t E t [g t+1 ] 1/τ (R t E t [π t+1 ] E t [z t+1 ]) (19) π t = βe t [π t+1 ]+κ(y t g t ) (20) g t = ρ g g t 1 +ε g,t z t = ρ z z t 1 +ε z,t (22) (23) where y t is GDP in efficiency units, π t is inflation rate, R t is interest rate, g t is government consumption and z t is change in technology.
Examples An Schorfheide (2007) The model is completed with three observation equations for quarterly GDP growth rate (YGR t ), annualized quarterly inflation rates (INF t ) and annualized nominal interest rates (INT t ): where β = 1 1+r A /400. YGR t = γ Q +100 (y t y t 1 +z t ) (24) INFL t = π A +400π t (25) INT t = π A +r A +4γ Q +400R t (26)
Examples An Schorfheide (2007) Bibliography I Sungbae An and Frank Schorfheide. Bayesian analysis of DSGE models. Econometric Reviews, 26(2-4):113 172, 2007. DOI:10.1080/07474930701220071. Michal Andrle. A note on identification patterns in DSGE models (august 11, 2010). ECB Working Paper 1235, 2010. Available at SSRN: http://ssrn.com/abstract=1656963. Fabio Canova and Luca Sala. Back to square one: identification issues in DSGE models. Journal of Monetary Economics, 56(4), May 2009. Nikolay Iskrev. Evaluating the strenght of identification in DSGE models. an a priori approach. unpublished manuscript, 2010a. Nikolay Iskrev. Local identification in DSGE models. Journal of Monetary Economics, 57:189 202, 2010b.
Examples An Schorfheide (2007) Bibliography II Jinill Kim. Functional equivalence between intertemporal and multisectoral investment adjustment costs. Journal of Economic Dynamics and Control, 27(4):533 549, February 2003. URL http://ideas.repec.org/a/eee/dyncon/v27y2003i4p533-549.ht Ivana Komunjer and Serena Ng. Dynamic identification of DSGE models. Econometrica, 2011a. accepted. Ivana Komunjer and Serena Ng. Global identification in nonlinear models with moment restrictions. Econometric Theory, 2011b. accepted. Gary Koop, Hashem Pesaran, and Ron Smith. On identification of DSGE models. mimeo, 2011.
Examples An Schorfheide (2007) Bibliography III M. Ratto and N. Iskrev. Computational advances in analyzing identification of DSGE models. 6th DYNARE Conference, June 3-4, 2010, Gustavelund, Tuusula, Finland, 2010a. Bank of Finland, DSGE-net and Dynare Project at CEPREMAP. M. Ratto and N. Iskrev. Identification toolbox for DYNARE. 1st MONFISPOL conference, London, 4-5 November 2010, 2010b. The London Metropolitan University and the European Research project (FP7-SSH) MONFISPOL.