FEDERAL RESERVE BANK of ATLANTA

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1 FEDERAL RESERVE BANK of ALANA Information Criteria for Impulse Response Function Matching Estimation of DSGE Models Alastair Hall, Atsushi Inoue, James M. Nason, and Barbara Rossi Working Paper b October 2009 WORKING PAPER SERIES

2 FEDERAL RESERVE BANK of ALANA WORKING PAPER SERIES Information Criteria for Impulse Response Function Matching Estimation of DSGE Models Alastair Hall, Atsushi Inoue, James M. Nason, and Barbara Rossi Working Paper b October 2009 Abstract: We propose new information criteria for impulse response function matching estimators (IRFMEs). hese estimators yield sampling distributions of the structural parameters of dynamic stochastic general equilibrium (DSGE) models by minimizing the distance between sample and theoretical impulse responses. First, we propose an information criterion to select only the responses that produce consistent estimates of the true but unknown structural parameters: the valid impulse response selection criterion (VIRSC). he criterion is especially useful for mis-specified models. Second, we propose a criterion to select the impulse responses that are most informative about DSGE model parameters: the relevant impulse response selection criterion (RIRSC). hese criteria can be used in combination to select the subset of valid impulse response functions with minimal dimension that yields asymptotically efficient estimators. he criteria are general enough to apply to impulse responses estimated by VARs, local projections, and simulation methods. We show that the use of our criteria significantly affects estimates and inference about key parameters of two well-known new Keynesian DSGE models. Monte Carlo evidence indicates that the criteria yield gains in terms of finite sample bias as well as offering tests statistics whose behavior is better approximated by first order asymptotic theory. hus, our criteria improve on existing methods used to implement IRFMEs. JEL classification: C32, E47, C52, C53 Key words: impulse response function, matching estimator, redundant selection criterion he authors thank the editors, two anonymous referees, J. Boivin, C. Burnside, Jeff Campbell, P. Hansen, Ò. Jordà, J. Lindé, S. Mavroedis, F. Schorfheide and seminar participants at the 2006 EC2, 2007 SED, EMSG at Duke U., the UBC Macro Lunch, the 2007 Conference in Waterloo, the 2007 ESEM, Kyoto University, Osaka University, Brown University, ilburg University, University of Amsterdam, and University of Cincinnati for comments, and they thank C. Burnside and L. Christiano for sharing their codes. Atsushi Inoue acknowledges financial support by the SSHRC under project he views expressed here are the authors and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the authors responsibility. Please address questions regarding content to Alastair Hall, Manchester University, Economics Department, School of Social Sciences, Manchester, United Kingdom M3 9PL, alastair.hall@manchester.ac.uk; Atsushi Inoue, Department of Agricultural and Resource Economics, North Carolina State University, Campus Box 809, Raleigh, NC , , atsushi@unity.ncsu.edu; James M. Nason, Federal Reserve Bank of Atlanta, Research Department, 000 Peachtree Street, N.E., Atlanta, GA 30309, jim.nason@atl.frb.org; or Barbara Rossi, Duke University, Department of Economics, 204 Social Science Building, Durham, NC 27708, brossi@econ.duke.edu. Federal Reserve Bank of Atlanta working papers, including revised versions, are available on the Atlanta Fed s Web site at Click Publications and then Working Papers. Use the WebScriber Service (at to receive notifications about new papers.

3 Introduction Since the seminal work of Rotemberg and Woodford (997), there has been increasing use of impulse response function matching to estimate parameters of dynamic stochastic general equilibrium (DSGE) models. Impulse response function matching estimation (IRFME) is a limited information approach that minimizes the distance between sample and DSGE model generated impulse responses. hose applying this estimator to DSGE models include, among others, Christiano, Eichenbaum and Evans (2005, CEE hereafter), Altig, Christiano, Eichenbaum, and Lindé (2005, ACEL hereafter), Iacoviello (2005), Jordà and Kozicki (2007), DiCecio (2005), Boivin and Giannoni (2006), Uribe and Yue (2006) DiCecio and Nelson (2007), and Dupor, Han, and sai (2007). Despite the widespread use of impulse response function (IRF) matching, only ad hoc methods have been used to choose which IRFs and how many lags to match. In a recent paper, Dridi et al (2007) present a comprehensive statistical framework for analyzing econometric estimation of DSGE in which the parameters of a structural economic model are estimated by matching moments using a binding function obtained from an instrumental model. Using this terminology in our context, the DSGE is the structural model, the VAR is the instrumental model and the impulse response function is the binding function. heir framework allows the model to be mis-speci ed and distinguishes the parameters into three categories: parameters of interest, estimated nuisance parameters and calibrated nuisance parameters. Since the structural model may be mis-speci ed, an important issue is whether the structural model encompasses - or partially encompasses - the instrumental model in the sense that estimation based on the binding function nevertheless yields consistent estimators of the parameters of interest. As well as introducing this conceptual framework, Dridi et al (2007) derive rst order asymptotic properties of (Partial) Indirect Inference estimators within this set-up, and propose a statistic for testing whether the structural model (potentially) encompasses the instrumental model. Dridi et al s (2007) discussion of encompassing highlights the importance of the choice of binding function within this type of estimation. As noted by Dridi et al, the choice of the binding function is analogous to the choice of moment function in moment based estimation. In the literature on moment based estimation, it is recognized that two aspects of the choice of moment function are important: (i) valid moment conditions are needed to obtain consistent estimation: (ii) not all valid moment conditions are informative and that it may be desirable in terms of nite sample

4 2 properties to base the estimation on a subset of valid moment conditions. Exploiting the analogy between moment conditions and binding functions, it can be seen that Dridi et al s encompassing test addresses (i) above; however, their analysis does not address the sequential testing issues that arise if the statistic is used repeatedly as part of a speci cation search. o our knowledge, (ii) above has not been considered in the context of IRFME, where it is common to use IRF up to a relatively large, pre-speci ed number of lags. In this paper, we address the issue of which impulse response functions to use in impulse response function matching estimation. Working within the framework provided by Dridi et al (2007), we propose two new criteria: the Valid Impulse Response Selection Criterion (VIRSC) and the Relevant Impulse Response Selection Criterion (RIRSC). he VIRSC is designed to determine which impulse response functions to include in order to obtain consistent estimators of the parameters of interest. he RIRSC is designed to select from the set of valid impulse responses those which are informative about the parameters of interest by excluding those that are redundant. We provide conditions under which the two criteria are weakly and strongly consistent, and report simulation evidence that shows the sensitivity of IRFME to the choice of impulse responses and also the e cacy of our criteria. he RIRSC criterion is applied to the DSGE models of CEE and ACEL. We often obtain point estimates that are little changed from those CEE and ACEL report. Nonetheless, the RIRSC yields economically important changes in inference regarding several key parameters that lead to strikingly di erent conclusions than those of CEE and ACEL. We conjecture that the parameter estimates in CEE and ACEL may be subject to small sample biases, and we investigate this issue in Monte Carlo experiments. he Monte Carlo exercises indicate that, in general, the small sample bias of IRFMEs is mitigated by RIRSC compared to using a relatively large xed lag length, and that the VIRSC works well in small samples. hus, the criteria that we propose should be attractive to analysts at central banks and other institutions conducting policy evaluation with DSGE models, as well as academic researchers testing newly developed DSGE models. As mentioned above, the framework considered in this paper is inspired by the work of Dridi et al (2007), and we believe that our results both complement and extend their analysis in the following ways. We propose information criteria for the selection of both valid and relevant responses, whereas they focus more on hypothesis testing and model selection. We focus on commonly used

5 3 methodologies for IRFME, including but not limited to simulation-based estimators, whereas they focus on general simulation-based estimators. Our criteria can be used not only for IRFMEs but also for general classical minimum distance and Indirect Inference Estimators. Relative to this literature, and in particular relative to Dridi et al. (2007), we add useful information criteria to select valid as well as relevant restrictions, thus signi cantly extending the scope of their analysis. Our approach is also connected to the literature on moment selection, with VIRSC and RIRSC being extensions to Classical Minimum Distance estimators of criteria proposed by Andrews (999) and Hall et al (2007). he criteria that we propose are also connected to several strands of the literature that estimate DSGE models. Rotemberg and Woodford (996), CEE (2005), and ACEL (2005) employ IRFMEs that minimize the di erence between sample and theoretical IRFs using a non-optimal weighting matrix. Jordà and Kozicki (2007) show that our RIRSC meshes with an IRFME estimator based on local projections and an optimal weighting matrix. Note that our criterion is applicable whether the weighting matrix is e cient or not. Finally, we show that our criterion can be an element of the Sims (989) and Cogley and Nason (995) simulation estimator. he paper is organized as follows. Section 2 presents our new criterion for the IRFME in the leading VAR case and discusses the assumptions that guarantee its validity. In section 3, we provide a clarifying example. he projection and simulation-based estimators are studied in section 4. Sections 5 and 6 present the empirical results and Monte Carlo analyses. Section 7 concludes. All technical proofs and assumptions are collected in the Appendix. 2 he VAR-based IRF Matching Estimator In this section, we consider the leading case in which the researcher is interested in estimating the parameters of a DSGE model by using a VAR-based IRFME. his estimator is obtained by minimizing the distance between the sample IRFs obtained by tting a VAR to the actual data and the theoretical IRFs generated by the DSGE model. he sample and the theoretical IRFs are identi ed by restrictions implied by the DSGE model. his requires we assume that the DSGE model admits a structural VAR representation, so that the sample IRFs are informative for the DSGE model parameters. We are interested in the VAR: Y t = 0 + Y t + 2Y t 2 + ::: + p 0 Y t p0 + " t ; ()

6 4 where Y t = [Y ;t ; Y 2;t ; :::; Y ny ;t] 0 is n Y, t = ; 2; :::;, and " t = [" ;t ; " 2;t ; :::; " ny ;t] 0 is white noise with zero mean and variance ". he population VAR lag order is p 0. For () to have an in nite order Vector Moving Average (VMA) representation and IRFs, we make the following standard assumption: Assumption (A0). In eq., (L) = I ny L 2L 2 p 0 L p 0 is invertible, where L is the lag operator and I ny is the (n Y n Y ) identity matrix and p 0 is nite. he model s parameters are (; ; ): is a p vector of structural parameters of interest, is a p vector of estimated nuisance parameters and is a p vector of calibrated nuisance parameters. 2, and belong to < p, A < p and < p. Let denote a vector of population IRFs with maximum horizon H estimated from data Y [Y 0 ; :::; Y 0 ]0. In particular, we will use i;j; to denote IRFs of each variable Y i;t+ to a structural shock " j;t at horizon, where i; j = ; :::; n Y, = ; :::; H. 3 Let be a n 2 Y vector that collects the population IRFs at a particular horizon : i= z } { = ( ;; ; ;2; ; :::; ;ny ; ; i=2 z } { 2;; ; 2;2; ; :::; 2;nY ; ; :::; i=n Y z } { ny ;; ; :::; ny ;n Y ; ) 0 he population IRFs at horizons = ; 2; :::; H can be further collected in the 0 so that = 0 ; 0 2 H ; :::; 0 : We will use 0 to denote the true value of. n 2 Y H vector Note that even if the true theoretical model has a VAR() representation, our results still apply for the Indirect Inference estimator that we discuss in Section 4. 2 Our, and correspond to, 2 and 22 of Dridi et al. (2007). 3 For simplicity we assume that the dimension of y t, n y; and that of " t, n ", are equal. However, n y can be greater than n ". For example, suppose that a tri-variate VAR(2) with two shocks is tted to the actual data in order to estimate eight DSGE model parameters using an optimal weighting matrix. When H = 2, suppose the 88 asymptotic covariance matrix of all possible IRFs is singular with rank of 2. Suppose the Moore-Penrose generalized inverse of the asymptotic covariance matrix is used as the weighting matrix and that the 88 Jacobian matrix of the theoretical IRF has rank of 8, which is implicit in assumption (). In this case, the eight DSGE model parameters will be identi ed. If instead the tri-variate VAR(2) is driven only by one shock, the asymptotic covariance matrix has rank six. As a result, the inverse of the asymptotic covariance matrix of the IRFME is singular and the DSGE model parameters will not be identi ed. he dimension of shocks matters for identi cation but not necessarily relative to the dimension of y t. Provided rank conditions are satis ed, adding a redundant vector of variables to the VAR system, while holding the number of shocks xed, will not violate the identi cation condition. However, the nite-sample performance of the IRFME estimator can deteriorate.

7 5 Finally, let g(; ; ) be a n 2 Y H vector of impulse responses obtained by a DSGE model. hese impulse responses are the binding function between the structural parameters ; ; and the impulse responses : he structural parameters and are estimated from where is calibrated. = g(; ; ) he question that we address in this paper is which subset of and g should be used. address this question, we introduce selection vectors that choose subsets of impulse responses. o De nition of c. Let c be a n 2 Y H selection vector that indicates which elements of the candidate impulse responses are included in estimation. We use c to index functions of impulse responses, that is, (c) and g(; c). If c j = then the j th element of is included in (c), and c j = 0 implies this element is excluded. For example, if only impulse responses up to horizons h < H are selected, c = [ n 2 Y h 0 n 2 Y (H h)]0 where mn and 0 mn denote m n matrices of ones and zeros, respectively. If the impulse responses of the second element of Y t with respect to the rst element of " t are used, c = H [0 ny 0 (ny ) 0 (ny 2)n Y ] 0. Note that jcj = c 0 c equals the number of elements in (c). he set of all possible selection vectors is denoted by C H, that is n o C H = c 2 < n2 Y H ; c j = 0; ; for j = ; 2; : : : n 2 Y H; and c = (c ; : : : c n 2 Y H )0 ; jcj : he methodologies proposed in this paper are valid for general minimum distance estimators as well as indirect inference estimators. In particular, a special case on which we focus is the IRF Matching Estimator (IRFME). he IRFME is a classical minimum distance estimator such that: ^ ( ; c) A b ( ; = arg min b (c) g ; ; ; c 0 b (c) b (c) g ; ; ; c ; (2) 2;2A c) where ^ (c) is an estimate of (c) and b (c) is a weighting matrix. b (c) could be the inverse of the covariance matrix of the IRFs ^ (c) or, as often found in practice, a restricted version of this matrix that has zeros everywhere except along its diagonal, which displays the variances of the IRFs. In general, b (c) can be readily obtained from standard package procedures that compute IRF standard error bands. 4 4 In this paper, we focus on optimal weighting matrix estimators because the VIRSC is based on the overidentifying restrictions test statistic, although both methods can be extended to non-optimal weighting matrices.

8 6 In order to implement the IRFME in practice, the researcher has to choose which impulse responses to use in (2). Our contribution to the existing literature is to provide statistical criteria to choose c. he criterion that we propose allows the researcher to avoid using the IRFs that are mis-speci ed and/or that contain only redundant information; at the same time, we enable the researcher to identify the relevant horizon of the IRFs. Since IRFs that do not contain additional information only add noise to the estimation of the deep parameters, these IRFs should be eliminated. he following de nitions formalize these concepts: We make the following Assumptions, which are slight modi cations of the assumptions in Dridi et al. (2007): Assumption (A). (a) is non-empty and compact in < p < p. (b) he parameter value [ 0 0; 0 ] 0 belongs to the interior of. 0 is the true parameter value whereas is the pseudo-true parameter value. (c) here is a function Q (Y ; (c); c) that is twice continuously di erentiable in such that b (c) = 0 2 Q (Y ; 0 (c); (Y ; 0 (c); 0 + o p (): Assumption (A2). he true parameter value 0 2 is a unique solution to max 2 max c0;h 2C 0;H (; ) jc 0;Hj where C 0;H (; ) = fc 2 C H : (c) = g(; ; ) for some 2 Ag: (3) In addition, we de ne C 0;H ( ) C 0;H ( 0 ; ): 5 Assumption (A3). Let P 0 denote the true probability distribution. (Y ; 0 (c) ; c) is asymptotically normally distributed with zero mean and a nite p.d. asymptotic covariance matrix: I 0 (c) = P 0 lim V (Y ; 0 (c); c) (b) here is a jcj jcj nite matrix J 0 (c) such that: J 0 (c) = P (c) (Y ; 0 (c); c): (5) 5 his assumption ensures that the true parameter value 0 is the unique parameter value at which there are most valid IRFs. For just-identi ed cases, there may be other parameter values at which the restriction holds, but none of other parameter values give as many valid IRFs as the true parameter value does. (4)

9 7 Assumption (A4). g (; ; ) is continuously di erentiable in (; ) 0;(c); 0 ; 0 ] 0 column rank (p + p ) : has full Finally, let: 0 (c) J 0 (c)i 0(c)J 0 (c) (c) 0 (c) W ( 0 ( 0 ; (c) ; ; c) 0 ; 0 ] 0 0 (c)@g( 0; (c) ; ; 0 ; 0 ] 0 ; (6) and let W (;) ( ; c) denote the (p p ) upper-left diagonal sub-matrix of the (p + p ) (p + p ) matrix W ( ; c). Using results from Dridi et al. (2007), Lemma 9 in the Appendix shows that the estimates of and are asymptotically normal, centered around their true and pseudo-true parameter values, respectively, with joint asymptotic covariance matrix W ( ; c). 2. he Valid IRF Selection Criterion Our rst goal is to identify the largest subset of IRFs that guarantee consistent estimation of the parameters of interest, that is the set of valid IRFs. We de ne the Valid Impulse Responses Selection Criterion by V IRSC ( ; c) = b ( ; c) h(jcj) ; (7) where b ( ; c) = [b (c) g(^ ( ; c); ^ ( ; c); ; c)] 0 ^ (c) [b (c) g(^ ( ; c); ^ ( ; c); ; c)]; ^ (c) is a consistent estimator of (c); h(jcj) is a deterministic penalty that is an increasing function of the number of impulse responses. For example, the SIC-type penalty term imposes h(jcj) = jcj and = ln ( ) and it is acceptable for the VIRSC under the restrictions of Assumption B3 listed and discussed below. We select impulse response functions by minimizing the criterion (7): ^c V IRSC; = arg min c2c H V IRSC ( ; c): (8) In this section, our main result shows that bc V IRSC; converges in probability to the (unique) selection vector, c 0, that chooses only valid restrictions. his selection vector provides consistent

10 8 estimates of the DSGE model parameters (which satisfy almost sure convergence in Section 5). We de ne the following sets of selection vectors for valid restrictions: C max;h ( ) = fc 2 C 0;H (; ) : jcj jc 0;H j for all c 0;H 2 C 0;H ( 0 ; )g; (9) C 0;H (; ) was de ned in (3) and it is the set of selection vectors in which the remaining restrictions are valid. C max;h ( ) is the set of selection vectors in C 0;H (; ) that maximizes the number of selected vectors in C 0;H ;. Also, in addition to Assumptions (A0) (A4), we consider: Assumption (B). C max;h ( ) = fc 0 g. Assumption (B2). ^ (c)! p (c); where (c) is positive de nite. Assumption (B3). h() is strictly increasing and! as! with = o( ). Note that Assumption B ensures that the set of valid IRFs is uniquely identi ed. 6 Assumption B2 ensures that there are consistent estimators for the asymptotic variances that enter our formulae. Such consistent estimators of 0 (c) can be found in Lütkepohl (990) and Lütkepohl and Poskitt (99). Assumption B3 imposes appropriate assumptions on the penalty terms that guarantee the validity of the proposed information criterion. Also, note that the SIC-type penalty term (h(jcj) = jcj and = ln ( )) and the Hannan-Quinn-type penalty term (h(jcj) = jcj and = ln [ln ( )]) satisfy Assumption B3, but the AIC-type penalty term, for which h(jcj) = jcj and = 2; does not. heorem (Valid IRF Selection Criterion (VAR case)) Suppose that Assumptions A0, A, A3 A4 hold for c 2 C H, that Assumption A2 holds for c 2 C 0;H hold. Let ^c V IRSC; be de ned in (8). hen ^c V IRSC; p! c0. ; and that Assumptions B B3 2.2 he Relevant IRF Selection Criterion Our second goal is to identify the fewest number of IRFs that guarantee that, asymptotically, the covariance matrix of the IRFME is as small as possible. 7 We de ne the Relevant Impulse Responses 6 Suppose that there are two elements, c and c 2, and that they are distinct. hen de ne c 3 as the maximum of c and c 2. hen c 3 includes more IRFs than c and c 2 and c 3 consists of valid IRFs, a contradiction. So it has to be unique. 7 For two covariance matrices, A and B, we say that A is smaller than B if B A is positive semi-de nite.

11 9 Selection Criterion by where RIRSC ( ; c) = ln(j ^W (;) ( ; c)j) + k(jcj)m ; (0) (;) ^W ( ; c) is a consistent estimator of W (;) ( ; c), and k(jcj)m is a deterministic penalty that is an increasing function of the number of impulse responses. For example, the SIC-based criterion selects k(jcj) = jcj and m = ln(p ) p (see Assumption C3 and the remarks thereafter for more discussion on the choice of these parameters). We select impulse response functions by minimizing the criterion (0): ^c RIRSC; = arg min RIRSC ( ; c): () c2c 0;H ( ) where C 0;H ( ) is the set of selection vectors in which the restrictions are valid, de ned below (3). In this section, the main result shows that ^c RIRSC; converges in probability to the (unique) selection vector c r. he vector c r chooses, among the valid IRFs, those: (i) with the smallest asymptotic variance and (ii) if a relevant IRF is dropped, the asymptotic variance is larger. 8 It is useful to de ne selection vectors that pick IRFs yielding e cient estimators: C E;H ( ) = fc 2 C 0;H ( ) : W (;) ( ; c) = W (;) ( ; c 0 )g; C min;h ( ) = fc 2 C E;H ( ) : jcj jc E;H j for all c E;H 2 C E;H ( )g: C E;H ( ) is the set of selection vectors in which selected restrictions yield e cient estimators. C min;h ( ) is the set of selection vectors that pulls out non-redundant IRFs from the valid IRFs. We sequence the selection vectors rst to nd an element in C E;H () and second to acquire an element in C min;h (). In addition to Assumptions (A0) (A4), we consider: Assumption (C). C min;h ( ) = fc r g. Assumption (C2). ^W (;) ( ; c) = W (;) ( ; c) + O p ( =2 ) for c 2 C I;H ( ); where C I;H ( ) fc 2 C 0;H (; ) : Assumptions A(0) A(4) holdg; 8 he reason why we require (ii) is that even if one adds redundant IRFs to the relevant IRFs one still obtains the same smallest asymptotic variance.

12 0 and for c 2 C 0;H ( ) \ C c I;H ( ). j ^W ( ; c)j p! : Assumption (C3). k() is strictly increasing and m satis es m! 0 and =2 m! as!. Remarks. C I;H ( ) is the set of selection vectors that maximizes the number of selected valid responses. Assumption C ensures that the set of relevant IRFs is uniquely identi ed. Although the validity of the assumption depends on the model and has to be checked on a case by case basis, we can show that Assumption C is satis ed in the simpli ed practical method discussed in Section 2.3 below as well as in Example and in the AR() case discussed respectively in Sections 3 and 7. below. Even when Assumption C is not satis ed, we expect our criterion will select elements in the set C min;h ( ), and therefore the estimator remains consistent. 9 Note that we allow for some selection vectors for which the parameters are not identi ed provided Assumption C holds. Assumption C2 ensures that there are consistent estimators for the asymptotic variances that enter our formulas. Assumption C3 imposes appropriate assumptions on the penalty terms that guarantee the validity of the proposed information criterion. Furthermore, note that the SICtype penalty term, k(jcj) = jcj and m = ln(p ) p satis es Assumption C3 whereas the AIC-type penalty term, for which k(jcj) = jcj and m = p 2 does not. We show that our criterion is weakly consistent in the following theorem: heorem 2 (Relevant IRFs Selection Criterion (VAR case)) Suppose that Assumptions (A0), (A) (A4) and (C) (C3) hold. Let ^c RIRSC; = arg min c2c0;h ( ) RIRSC ( ; c). hen ^c RIRSC; p! c r. 2.3 Suggestions for Practical Implementation heorems and 2 describe the asymptotic behavior of the two criteria, the VIRSC and the RIRSC, that consider all possible combinations of IRFs. However, implementing these criteria in practice 9 Suppose C min;h( ) = fc r; c 2rg. We expect that bc 2 fc r; c 2rg with probability approaching one. hen b = ;r b ; c r + ( ;r) b ; c 2r + op (), where ;r equals if c r is selected and equals zero if c 2r is selected. Since b ; c r and b ; c 2r converge in probability to 0, then b p! 0.

13 might become very computationally intensive, given the large number of variables included in typical VARs and the fact that H can be large (for example, CEE chose H = 2). Applied researchers, however, often impose an ad hoc maximum lag length to all the IRFs a DSGE model is asked to match. For practical reasons, we suggest the following. First, select the valid IRFs among the (n Y n Y ) set of all IRFs given a pre-speci ed choice for H. Second, among the valid IRFs, select the relevant IRFs across horizons h, for h 2 f; 2; : : : ; Hg. Our VIRSC and RIRSC can be easily tailored to this special case. his ad hoc approach selects the valid IRFs using a set of selection vectors that consider all IRFs up to the maximum horizon H. Let where bc V IRSC; = arg min c2c H V IRSC ( ; c): (2) C H = fc 2 C H : c = H d for some d = [d ; d 2 ; :::; d n 2 Y ] 0 ; d j 2 f0; g; j = ; 2; :::; n 2 Y g: De ne C 0;H by C 0;H as in (3), with C H replaced by C H. Also de ne C max;h by C max;h as in (9) with C 0;H replaced by C 0;H. We have: Corollary 3 (Simpli ed VIRSC Criterion (VAR case)) Let the structural model have a VAR representation (), and the estimator of the parameters be de ned as (2), where c is chosen by (2). Suppose that Assumptions A-A4, B-B3 hold with C H, C min;h and c 0 replacing C H, C min;h and p c 0, respectively. hen bc V IRSC;! c0. Relevant IRFs are selected among the set of possible IRFs ; 2 ; :::; H that satisfy the validity criterion. Let c Y denote the n 2 Y n 2 Y component of bc V IRSC;. Let c Y h = [c 0 Y ; c 0 Y ; :::c 0 Y {z } n 2 Y h C H = vector that selects the valid IRFs for h = H; that is the last 0 n2 Y (H h)]0 and n o c Y h = [ n 2 Y h 0 n 2 Y (H h)]0 ; for h = ; 2; :::; H : De ne C E;H, C min;h and c r by C E;H, C min;h and c r, respectively, with C H replaced by C H. Note that, in this case, Cmin;H directly satis es Assumption C. 0 0 In fact, if the order of the VAR is nite, then elements of C min;h are all nite. Suppose that there are two

14 2 Using these de nitions, we implement the RIRSC by selecting the maximum horizon of IRFs that minimizes the RIRSC across horizons for the given choice of IRF shocks and variables obtained by the VIRSC: ^h = arg min ( ; c Y h): h2f;2;:::;hg (3) Let h r denote the corresponding IRF lag length implied by c r : c r = [ n 2 Y h r 0 n 2 Y (H h r) ]0 : It follows immediately from heorems and 2 that ^h is a consistent estimate of h r : Corollary 4 (Simpli ed RIRSC Criterion (VAR case)) Let the structural model have a VAR representation (), and the estimator of the parameters be de ned as (2), where c is chosen by (2). Suppose that Assumptions A-A4, C-C3 hold with C H, C min;h and c r replaced by C H, C min;h and c r, respectively. hen ^h p! hr. 3 Interpretation of the Criteria his section provides examples that clarify the identi cation problem for DSGE model parameters estimated by IRF matching. he examples also make concrete the de nitions of redundant and relevant IRFs under the RIRSC, as well as the usefulness of the VIRSC in the presence of model mis-speci cation. 3. Interpretation of the VIRSC Example (A Simple New Keynesian Model) Consider the following simpli ed New Keynesian model (cfr. Canova and Sala, 2009): y t = k + a E t y t+ + a 2 (i t E t t+ ) + e t (4) t = k 2 + a 3 E t t+ + a 4 y t + e 2t (5) i t = k 3 + a 5 E t t+ + e 3t (6) where y t is the output gap, t is the in ation rate, i t is the nominal interest rate, and e t ; e 2t ; e 3t are i.i.d.(0,) contemporaneously uncorrelated shocks. he rst equation is the IS curve, the second elements in the set C min;h, and denote these elements by h r; and h r;2, and let h r; and h r;2 be di erent, with h r; < h r;2. By de nition of C min;h; they achieve the same asymptotic variance but since h r; < h r;2 then h r;2 cannot be element of C min;h, thus inducing a contradiction. herefore, the set C min;h must be unique.

15 3 is a Phillips curve and the third characterizes monetary policy. o consider the issue of mis-speci cation, assume that the researcher imposes a 4 = a 4 in his/her model, whereas a 4 is di erent from zero in the true data generating process (5). In addition, suppose the true values of the parameters a ; a 3 are known and equal to 0:5 and k = 0: hen the solution of the model is: i where B y A = i t y t t i 0 0 C A = B y A + B 2 0 C B a 2 a 4 a 4 a 5 a 4 0:25a 5 0:25 + a 2 a 4 0:5 ( a 5 ) a 2 :5a 2 a 4 0:5 a 2 a 4 0 C B k k 2 k 3 e ;t e 2;t e 3;t C A C A and = 0:25+(a 5 )a 2 a 4 : (7) Suppose the researcher is mainly interested in estimating the slope of the IS equation (a 2 ). Using the notation of the previous sections, = fk ; a ; a 3; ag ; = fa 2 g ; and = fk 2 ; k 3 ; a 5 g is the empty set. He has two options to estimate the parameter of interest: MLE or IRFME. Recall that the researcher works with a mis-speci ed model that assumes that a 4 = a 4, where we let a 4 =. MLE will recover consistent estimates of the mean parameters (y, and i). Let the researcher recover the parameter of interest from (7) assuming a 4 = using information on the variances and covariances: 0 i t y t t 00 0 i a 2 a 2 C A N BB y A ; B 2 a a2 2 + CC AA a 2 a a For example, the researcher may estimate ba 2 = cov( c t ; i t ); while in reality the distribution implied (8) by (7) is 0 i t y t t 00 C A N i y 0 C A ; a 2 a 2 a 4 a 2 a a 4a a 4 a 2 a 4 a 4 a a 4 a 2 2 a2 4 + a2 4 + CC AA ; (9) then ba 2! a 2 cov( t ; i t ). Note that a 2 = a 2 a 4 6= a 2, which in general will not recover the true value of a 2 unless the true parameter value is a 4 =. It is important to note that in this example

16 4 the response of y t to e ;t is the valid and relevant response among the three IRFs at impact, whereas the responses of t to e ;t ; e 2;t are not valid. Consider now the IRFME based on the response of y t to e ;t from (7). Since that IRF is not a ected by assuming a 4 = or not, then the researcher will correctly estimate a 2 whether or not the model is correctly speci ed or not. 3.2 Interpretation of the RIRSC Example 2 (Labor Productivity and Hours) his example follows Watson (2006). Watson (2006) derives the VAR representation for a simple RBC model outlined in Christiano, Eichenbaum and Vigfusson (2006, Section 2), in which a technology shock is the only disturbance that a ects labor productivity in the long run. Watson (2006) shows that the RBC model can be written as a bivariate VAR(3) model in labor productivity (y t =l t ) and employment (l t ): y 5 4 ln yt l t 5 = 4 0 y 5 4 ln yt l t ln(l t ) ln(l t ) + l y y l ln yt 3 l t 3 ln(l t 3 ) 3 2 t 0 0 l y ( + l ) ln yt 2 l t 2 ln(l t 2 ) t 5 (20) where is the capital share in production; l is the serial correlation coe cient in the tax rate on labor income process; ~a z and a z are, respectively, the parameters associated with the lagged state of technology in the policy rule for labor in the standard and recursive versions of the model; = (~a z a z l )=( y ); 2 = ~a z l =( y ); t = ( y ) z " z t, v t = a l l " l t; and " l t, " z t are i.i.d. zero mean and unit variance shocks (cfr. Watson, 2006, eqs. 3 and 4). he structural parameters of interest in this example are y ; and l. For identi cation purposes, we impose the short-run restriction that a z = 0, which yields the short-run restriction, = ~a z =( y ): hus is informative for the structural parameter ~a z, given the estimate of y. 3 5 In the notation of the previous sections, the structural parameters of interest are = f y ; l g, the estimated nuisance parameter is = f g, and the calibrated nuisance parameters are = fa z ; a l ; l ; z g. here are two trivial examples of redundant impulse responses. One is the response of ln yt l t to " z t, which is always zero see (20). Another is restrictions on the impulse responses j for j > 3: since the model is a VAR(3) model, restrictions for j > 3 are nonlinear transformation of

17 5 the rst three impulse responses, and thus are rst-order equivalent to some linear combinations of the above restrictions. herefore, adding these restrictions will not reduce the asymptotic variance. However, even if an impulse response depends on the parameters of interest and its horizon is less than or equal to p, the impulse response may be redundant. 4 Alternative IRF Matching Estimators Although the VAR-based IRFME is the most widely used IRFME, alternative IRFME have been proposed in the literature. Jordà and Kozicki (2007) proposed IRFME based on local projections. In addition, researchers have been interested in simulation-based methods to approximate theoretical impulse responses. his section extends the RIRSC to these IRF matching estimators, and describes how our criterion is implemented in these contexts. 4. he IRF Matching Projection Estimator Consider rst the local projections method advocated by Jordà (2005) and used in Jordà and Kozicki (2007). he simplest version of his estimator for the th step impulse response is ^B ; D, where ^B ; is directly estimated from Y t+ = B 0; + B ;+ Y t + B 2;+ Y t B p;+ Y t p + u t+ for = ; :::; H, and D is a matrix derived from the identi cation procedure. Let the vector of structural impulse responses estimated by local projections be denoted by b J; (c). Jordà s local projection impulse response estimator is: ^ J; ( ; c) b J; ( ; c) A = arg min (b J; (c) g(; ; ; c)) 0 b (c)(b J; (c) g(; ; ; c)) (2) 2;2A where g(; ; ; c) is the vector of the model s theoretical impulse responses given structural parameter, and b J; (c) is the inverse of a consistent estimator of the asymptotic covariance matrix of ^ J; (c). Our main result for the local projection estimator is:

18 6 heorem 5 (Consistent IRF Selection (Local Projections case)) Suppose that Assumptions A-A4, B-B3, and C-C3 hold with c W ( ; c) replaced by c W J; ( ; c) which is a consistent estimator of (6) constructed using b J; (c) and b J; (c), and W c(;) J; (; c) the (p p ) upper-left diagonal sub-matrix of c W J; (; c): Let the estimator of be (2), where c is chosen such that: ^c V IRSC; = arg min c2c H V IRSC J; ( ; c); where V IRSC J; ( ; c) = b J; ( ; c) h(jcj) ; b J; ( ; c) = (^ J; (c) g(^ J; ( ; c); ^ J; ( ; c); ; c)) 0 ^J; (c)(^ J; (c) g(^ J; ( ; c); ^ J; ( ; c); ; c)) and: ^c RIRSC; = arg hen ^c V IRSC; p! c0 and ^c RIRSC; p! cr. min c2c 0;H ( ) RIRSC J; ( ; c); and RIRSC J; ( ; c) = log(j c W (;) J; (; c)j) + k(jcj)m : 4.2 Indirect Inference Estimators he third estimator that we consider is the simulation-based estimator. he simulation-based estimator is an indirect-inference (II) estimator with a sequence of nite-order VAR models used as an auxiliary model (see Smith (993) and Dridi, Guay and Renault (2007), and Gourieroux, Monfort and Renault (993) for examples of indirect inference applied to DSGE models and nancial models, respectively). In the macroeconomics literature, the application of simulation-based estimators to IRFMEs is referred to as the Sims-Cogley-Nason estimator. he II estimator is implemented as follows. First, t a VAR(p) to the actual data to obtain sample impulse responses b (c). 2 Note that all the results in this section still hold if the VAR is of in nite order provided that H is nite. Next, simulate synthetic data of length from the DSGE model with parameter vector (; ; ). Let the s th simulated synthetic data obtained using initial condition Y0 s be denoted by Y s (; ; ; Y0 s ), and repeat this process for s = ; ::; S, where S is the total number of simulation replications. Estimate the VAR(p) on the synthetic he Sims-Cogley-Nason estimator was popularized by Kehoe (2006). 2 All the subsequent estimated parameters should also be function of p, the estimated VAR lag length. However, in order to simplify notation, we suppress this dependence in the notation.

19 7 samples s = ; ::; S to obtain a matrix of simulated theoretical IRFs. Let eg (s) (; ; ) denote the vector of simulated impulse responses from the s-th synthetic sample. Finally, de ne eg S (; ; ) to be the average across the ensemble of simulated IRFs, which we refer to as the (approximate) theoretical impulse responses, eg S (; ; ) =(=S) P S s= eg(s) (; ; ). As the paper did earlier, c is used to index subsets of IRFs, eg S (; ; ; c). Note that the matrix of shock innovations is drawn only once and held xed as is adjusted to move eg S (; ; ; c) closer to b (c). he II estimator of minimizes the distance between the average simulated theoretical impulse responses and the sample impulse responses: ^ II; (S; ; c) A b II; (S; ; = arg min (b (c) eg S (; ; ; c)) 0 b (c)(b (c) eg S (; ; ; c)); (22) 2 c) where b (c) is a weighting matrix. 3 Next, consider the problem of selecting the impulse responses for the IRFME. We impose additional assumptions: Assumption (A ). Let P denote the probability measure of the simulated data. For the simulated theoretical impulse responses eg S (; ; ; c), Q (Y ; (c); c) satis es eg S (; ; ; c) = S SX s= eg (s) (; ; ; c) = 0 (c) eg (s) (; ; ; 2 Q (Y s (; ; ; Y0 s); 0(c); 0 (Y s (; ; ; Y0 s); 0(c); c) + o p Assumption (A3 ). (Y s ( 0; (c) ; ; Y s 0 ); 0(c); c) is asymptotically normally distributed with zero mean and asymptotic covariance matrix I0( ; c) = P lim V (Y s ( 0 ; (c) ; ; Y0 s ); 0 (c); c) and independent of the initial values Y s 0, s = ; 2; :::; S. (b) here is a jcj jcj matrix J 0 ( ; c) such that J0 ( ; c) = P 2 0 (c) (Y s ( 0 ; (c) ; ; Y0 s ); 0 (c); c): 3 he Appendix shows that, under quite mild conditions, ^ II; (S; ; c) is consistent and asymptotically normal.

20 8 Assumption (A4 ). eg 0 ;(c); 0 ; 0 ] 0. (; ; ; c) is continuously di erentiable in (; ; ) and lim (s)! Assumption (A5). Let Cov denote covariance under P. here are jcj jcj matrices K 0 ( ; c) and K 0 ( ; c) such that lim! (Y ; 0 (c); (Y s ( 0 ; (c) ; ; Y0 s ); e 0 ( 0 ; (c) ; ; c); c) ( 0;(c); ;c) 0 ; 0 ] 0 = K 0 ( ; c) lim (Y s ( 0 ; (c) ; ; Y 0); 0 (c); c); (Y l ( 0 ; (c) ; ; Y l 0);e 0 ( 0 ; (c) ; ; c); c) = K 0( ; c) independent of the initial values Y0 s and Y 0 l, s 6= l, s; l = ; 2; :::; S. Let the following de nitions hold: 0 (S; ; c) = J0 (c)i 0(c)J0 (c) + S J 0 ( ; c)i0( ; c)j0 ( ; c) + J0 ( ; S c)k0( ; c)j0 ( ; c) J0 (c)k 0( ; c)j0 ( ; c) J0 ( ; c)k0( 0 ; c)j0 (c) (S; ; c) = 0 (S; ; c) W (S; 0 ( 0 ; (c) ; c) c) 0 ; 0 ] 0 ( 0 (S; ; 0; (c) ; 0 ; 0 ] 0 (23) and W (;) (S; ; c) denotes the (p p ) upper-left diagonal sub-matrix of the (p + p ) (p + p ) matrices W (S; ; c). Lemma 0 in the Appendix shows that the simulation-based estimators ^II; ( ; c); ^ II; ( ; c) are asymptotically normal, centered around their true and pseudo-true parameter values, respectively, with an asymptotic covariance matrix equal to W (S; ; c): Finally, let cw II; (S; ; c) be a consistent estimator of W (S; ; c) and b II; (S; ; c) be de ned as b II; (S; ; c) = (b (c) eg S ( b II; S; ; c ; b II; S; ; c ; ; c)) 0 b (c)(b (c) eg S ( b II; S; ; c ; b II; S; ; c ; ; c)): Finally, let W c(;) II; (S; ; c) be the (p p ) upper-left diagonal sub-matrix of the (n 2 Y Hn2 Y H) matrix cw II; (S; ; c): Note that Gourieroux, Monfort and Renault (993, S2 S3) propose consistent estimators of 0 (S; ; c) for some special cases which can be used to construct c W II; (S; ; c) and cw II; (S; ; c) can be computed using (7) in the Appendix. heorem 6 describes the IRF selection criteria we propose for the II estimator:

21 9 heorem 6 (Consistent IRF Selection (Simulation-based case)) Let Assumptions A, A, A2, A3, A4, A5 hold. Let c be chosen such that: and ^c V IRSC; = arg min c2c H V IRSC II; ( ; c); where V IRSC II; ( ; c) = b II; (S; ; c) h(jcj) ; ^c RIRSC; = arg min RIRSC II; ( ; c); and c2c 0;H ( ) RIRSC II; ( ; c) = log(j c W (;) II; (S; ; c)j) + k(jcj)m : hen, under Assumptions B-B3, ^c V IRSC; p! c0 and, under the additional Assumptions C-C3, ^c RIRSC; p! cr. 5 Strongly Consistent IRFs Selection his section derives additional results that guarantee almost sure convergence of the VIRSC and RIRSC. his analysis will provide more guidance on the choice of the penalty term than the weak consistency results of Sections 2. and 2.2 (in fact, the weak consistency results only require Assumption C3 which is valid for many choices for (jcj) and m ), although at the cost of imposing additional assumptions on the data. For simplicity, we will do so only in the simulation-based estimator: the results for the VAR-based and the Projection-based estimators follow directly as special cases. We impose additional assumptions: Assumption (D). (a) here is a unique 0 ( ; c) and 0 ( ; c) such that [ 0 ( ; c) 0 ; 0 ( ; c) 0 ] 0 = arg min 2;2A [ 0 (c) g(( ; c); ( ; c); ; c)] 0 (S; ; c) [ 0 (c) g(( ; c); ( ; c); ; c)]; and [^ II; (S; ; c) 0 ; ^ II; (S; ; c) 0 ] 0 a:s:! [ 0 ( ; c) 0 ; 0 ( ; c) 0 ] 0 for all c 2 C H : (b) here is a sequence of positive semi-de nite matrices f^ (S; ; c)g such that ^ (S; ; c) a:s:! (S; ; c) where (S; ; c) is positive de nite for all c 2 C 0;H. (c) he Law of the Iterated Logarithm (LIL) holds: b lim sup 0! (2 lnln ) =2 =2 (S; ; c) ^ (c) g( 0 ( ; c); 0 ( ; c); c) = ; a:s:

22 20 for all b 2 < jcj such that jbj= (where b is a generic vector) and c 2 C 0;H where lnln denotes ln(ln( )). (d) sup 2;2A kd~g S (; ; ; c) Dg(; ; ; c)k = o as (). he following results holds: heorem 7 (Strong Consistency of VIRSC ) Suppose that Assumptions A4, B, B3 and D hold. hen ^c V IRSC; a:s:! c 0. heorem 7 shows that the results of heorem can be strengthened to almost sure convergence. We can derive similar results for the RIRSC under the following Assumption. Let Y s t (; ; ; Y s 0 ) denote the time t component of the vector Y s (; ; ; Y s 0 ): Assumption (E). (a) Let ; ; c)=@[ 0 ; 0 ] 0 j =0 ;=(c); G(c) = (@=@ 0 ; ; c)=@[ 0 ; 0 ] 0 let the following approximations hold: j =0 ;=(c); b G(c) be a consistent estimator of G(c), and =2 (^ ( ; c) 0 (c)) = G(c) 0 (S; ; c)g(c) G(c) 0 (S; ; c) =2 [^ (c) eg S ( 0 ; ; ; c)] + o() a:s: (24) =2 vecf ^G (c) G(c)g = G(c) =2 (^ ( ; c) 0 ) + o() a:s: (25) Also,! v (Y t ; Y s t (; ; ; Y s 0 ); 0; c) and! (Y t ; Y s t (; ; ; Y s 0 ); 0; c) exist such that: =2 vechf b (S; ; c) 0 (S; ; c)g = =2 X =2 (^ (c) ~g S ( 0 ; (c) ; ; c)) = =2 for some 0 < m <. t=+m X t=+m! v (Y t ; Y s t ( 0 ; (c) ; ; Y s 0); 0 ; c) + o() a:s: (26)! (Y t ; Y s t( 0 ; (c) ; ; Y s 0); 0 ; c) + o() a:s: (b) Let!(Y t ; Y s t (; ; ; Y s 0 ); 0; c) = [! (Y t ; Y s t (; ; ; Y s 0 ); 0; c) 0! v (Y t ; Y s t (; ; ; Y s 0 ); 0; c) 0 ] 0. De ne! (S; ; c) = lim! V ar[ =2 P t=!(y t; Y s t (; ; ; Y s 0 ); 0; c)]. P t=!(y t; Y s t (; ; ; Y s 0 ); 0; c) satis es the LIL in the sense that for all b 2 < dim(!) with kbk =, ( ) lim sup! (2 lnln ) =2 b0! (S; ; X c) =2!(Y t ; Yt s (; ; ; Y0 s ); 0 ; c) = ; a:s: for all c 2 C. t=

23 2 Assumption (F). Let the penalty function be k(jcj)m where k() is strictly increasing and m = o() and either: (i) lim inf! =2 m =(lnln ) =2 = where z < < and z is a positive constant that is de ned in Appendix A; or (ii) lim inf! =2 m =(lnln ) =2 = +. he following results holds: heorem 8 (Strong Consistency of RIRSC ) Suppose that Assumptions A4, C, C3, E and F hold. hen ^c RIRSC; a:s:! c r. Remarks. heorem 8 establishes conditions under which ^c RIRSC; is strongly consistent for c r. It can be seen that the conditions on the penalty term are necessarily satis ed if (jcj; p; ) = (jcj p)ln[ =2 ]= =2, which is the penalty term associated with the Schwarz information criterion. However, the conditions are not necessarily satis ed if (jcj; p; ) = (jcj p)ln[ln =2 ]= =2, which is the penalty term associated with the Hannan and Quinn information criterion. In the latter case, if selection is over all possibilities then strong consistency requires that z = 2 =2 (! (c r ) +! (c)) for all c 2 C E;H where! (c) is de ned in Appendix A. Notice that if this condition fails for some c 2 C E;H, then one of two scenarios unfolds: if = z then the assignment is random between c r and c; if < z then ^c RIRSC; a:s:! c and so more moments are included than is necessary to achieve the minimum variance. he data dependence of the condition governing these outcomes makes this choice of penalty term unattractive. It is interesting to contrast the conditions on the penalty term for the case considered here in which the order of convergence of W c(;) (S; ; c) to W (;) (S; ; c) is =2. For weak consistency, it is only necessary that m! and m = o( =2 ). Given the discussion above, the strong consistency results suggest the use of a penalty term for which lim inf! =2 m =(lnln ) =2 diverges. heorem 8 therefore provides more guidance on the choice of penalty term than the corresponding weak consistency result. heorem 8 relies crucially on Assumptions E and F, which are high level assumptions that guarantee approximations by the law of iterated logarithms. For simplicity, heorem 8 also relies on the use of the optimal weighting matrix, which however is not crucial. heorem 8 would still hold with any positive de nite weighting matrix.

24 22 6 Empirical Analysis of wo Representative DSGE Models his section applies a VAR-based IRFME and the information criteria to the new Keynesian DSGE models of CEE and ACEL. he goal is to assess the impact of the VIRSC and the RIRSC on the estimated parameters of these DSGE models. Since the VIRSC con rms that all IRFs are valid, we focus on the RIRSC. We estimate the CEE and ACEL models xing the maximum number of impulse response lags at 20 (excluding those that are zero by assumption) and employing the RIRSC. In either case, the IRFME is implemented with a diagonal weighting matrix. 4 he CEE and ACEL DSGE models use di erent schemes to identify IRFs. he CEE model is estimated by matching the responses of nine aggregate variables only to an identi ed monetary policy shock. he identi cation relies on an impact restriction that orthogonalizes the monetary policy shock with respect to the nine aggregate series. We use this identi cation to estimate nine parameters of the CEE DSGE model. Long-run neutrality restrictions identify the IRFs engaged to estimate the parameters of the ACEL DSGE model. he ACEL DSGE model is constructed such that: (i) neutral and capital embodied shocks are the only shocks that a ect productivity in the long run; (ii) the capital embodied shock is the only shock that a ects the price of investment goods; and (iii) monetary policy shocks do not contemporaneously a ect aggregate quantities and prices. hese restrictions identify IRFs for ten aggregate variables with respect to neutral technology, capital embodied and monetary policy shocks. he ACEL DSGE model presents 8 parameters to estimate. able (a) reports the results for the ACEL DSGE model. From the left to right of the table, the columns list parameters, parameter estimates and standard errors under RIRSC, and parameter estimates and standard errors given a xed IRF lag length of 20. We implement the RIRSC by matching the IRFs with respect to the three shocks and progressively reduce the lags in all three IRFs one by one. Next, the RIRSC criterion (0) is applied as the number of lags in each IRF ranges from two to 20, which gives a total of number of IRF points (h) ranging between 6 and 60. he RIRSC selects h = 3 for the three IRFs, which makes it possible for the 8 ACEL DSGE model parameters to be identi ed. he RIRSC has one important e ect on ACEL DSGE model parameter estimates. Across the 4 ACEL remark that the diagonal weighting matrix ensures that the estimated DSGE model parameters are such that theoretical IRFs lie as much as possible within con dence bands of estimated IRFs.