Matching Theory and Data: Bayesian Vector Autoregression and Dynamic Stochastic General Equilibrium Models

Transcription

1 Matching Theory and Data: Bayesian Vector Autoregression and Dynamic Stochastic General Equilibrium Models Alexander Kriwoluzky University of Bonn July 25, 212 Abstract This paper shows how to identify the structural shocks of a Vector Autoregression (VAR) model while simultaneously estimating a dynamic stochastic general equilibrium (DSGE) model that is not assumed to replicate the data-generating process. It proposes a framework for estimating the parameters of the VAR model and the DSGE model jointly: the VAR model is identified by sign restrictions derived from the DSGE model; the DSGE model is estimated by matching the corresponding impulse response functions. JEL classification: C51. Keywords: Bayesian Model Estimation, Vector Autoregression, Identification. I am especially grateful to Wouter den Haan, Chris Sims and Harald Uhlig for numerous inspiring and fruitful discussions of this work. I would also like to thank Marek Jarocinski, Martin Kliem, Helmut Lütkepohl, Bartosz Mackowiak, Christian Pigorsch, Francesco Ravazzolo, Morten Ravn, Lenno Uusküla, and Felix Wellschmied for their comments and suggestions. The paper further benefited from discussions with participants at the Annual meeting of the European Econometric Society in Barcelona, the Dynare conference in Oslo and from seminar participants at Norges Bank. Part of the research was conducted while I was visiting Princeton University funded by the German Academic Exchange Service. Further grants from the DEKA Bank and the SFB 649 are gratefully acknowledged. University of Bonn, Department of Economics, Adenauerallee 24-42, Bonn, Germany, a.kriwoluzky@uni-bonn.de, tel:

2 1 Introduction How can we estimate the effects of an exogenous disturbance on the economy? One way is to estimate a Vector Autoregression (VAR) model. This is straightforward to do in the first place, but one needs additional restrictions to identify the structural effects of the disturbance. A popular way to identify the structural effects is to employ sign restrictions derived from a DSGE model. This method has been applied among others by Dedola and Neri (27),Pappa (29), Peersman and Straub (29), and Enders, Müller, and Scholl (211). However, sign restrictions are rarely unique across the DSGE model s potential parameter space. Another way to estimate the effects of an exogenous disturbance is to use the DSGE model directly. It contains structural shocks whose effects depend on the parameters of the DSGE model. Therefore, in order to estimate the effects of the shocks, the parameters of the DSGE model have to be estimated. This is typically done by full information likelihood based methods or by matching selected moments, for instance impulse response functions, of the DSGE model and the data. Estimation by full information likelihood based methods is plagued by at least two concerns: first, the DSGE model has to contain as many structural shocks as there are observable variables and second, it has to contain features and frictions which are not pertinent to any economic question but are necessary to match the data. Under the presumption that the DSGE model is too stylized to be thought of as the data-generating process, but is good enough to give a correct impulse response function, it can be estimated by minimizing the distance between the corresponding impulse response function of the VAR model and the DSGE model. Exemplary work include Christiano, Eichenbaum, and Evans (25) and Ravn, Schmitt-Grohé, Uribe, and Uuskula (21). To that end however, the VAR model has to be identified in the first place. In this paper I suggest an encompassing method which identifies the VAR model based on restrictions derived from the DSGE model and at the same time estimates the parameters of the DSGE model. The algorithm builds on the insight that the sign restrictions derived from the DSGE model are qualitative restrictions (positive or negative) and that not all signs of the impulse response functions are restricted. At the same time, when computing the probability density function of a DSGE model parameter vector by matching the corresponding impulse response functions of the DSGE model and the VAR model, all impulse response functions of the VAR model are considered and taken quantitatively into account. Thus, the identified impulse response functions contain additional information which can be employed to estimate the parameters of the DSGE model. I employ this additional information to update the prior distribution of the DSGE model s parameters. This way, the method yields posterior probability statements about the sets of different sign restrictions and the associated impulse response functions. The algorithm presented here is closely related to Canova and Paustian (211) and DelNegro and Schorfheide (24). My approach differs from the former in that it yields probability distributions about the parameters of the DSGE model and thus about the impulse response functions. In opposite to DelNegro and Schorfheide (24), I do not require the specification of as many structural shocks as there are observable variables. The remaining paper is organized in the following way. In Section 2 I use a simple model to sketch the main idea of the paper. In Section 3 I describe the general framework, 2

3 in Section 4 I apply the method to a medium-scale DSGE model in a Monte Carlo study. Section 5 concludes the paper. 2 A simple example To sketch out the idea of the paper, I consider the fiscal theory of the price level (FTPL) model developed by Leeper (1991) as the DSGE model. The FTPL model provides a traceable and simple environment to highlight the main idea of the paper. Section 2.1 sets out the DSGE model. In Section 2.2 I describe how the sign restrictions that identify the VAR model depend on the parameters of the DSGE model, and how the sign restrictions are determined. This setting is restricted to two parameter vectors of the DSGE model only. In the subsequent Section 2.3 I show how one can estimate the DSGE model given the information from the VAR model. 2.1 Brief description of the FTPL model The FTPL model can be reduced to two equations in inflation 1 (π F ) and real debt (b F ). There are two shocks, one monetary policy shock ɛ F,m and one fiscal policy shock ɛ F,f. The signs of the impulse response functions of inflation and real debt depend on the reaction coefficient of the nominal interest rate on inflation (α F ) and on the reaction coefficient of taxes on real debt (γ F ). I choose standard values for the remaining parameters of the DSGE model and solve for the recursive laws of motion of the endogenous variables. 2 The solution of the DSGE model takes the following form: yt F = Φ F 1 (θ F )yt 1 F + A F 1 (θ F )ɛ F t + A F 2 (θ F )ɛ F t 1, ɛ F N (, I) (1) where y t = [ ] [ ] πt F b F t denotes the vector of endogenous variables, ɛ F t = ɛ F,m t ɛ F,f t the vector of exogenous shock processes, and θ F = [ α F γ ] F collects the parameters of the DSGE model. The coefficient matrices Φ F 1, A F 1, and A F 2 contain the recursive laws of motion. The variances of the shock processes are normalized to one. The impulse response functions of the FTPL model depend on θ F. Denote the discount factor of the household by β F. In the case of α F β F > 1 and β F 1 γ F < 1, monetary policy is called active and fiscal policy passive. I refer to this combination as Regime I. In the case of α F β F < 1 and β F 1 γ F > 1, monetary policy is passive and fiscal policy is active. I refer to this combination as Regime II. Table 1 summarizes the signs of the impulse response functions of the endogenous variables on impact in the two regimes. Since the moving average part of the process in Equation (1) is invertible and the process is stationary, it can be approximated by a VAR model. The reduced form VAR model of Equation (1) is approximately given by 3 : y F t = B F 1 (θ F )y F t 1 + B F 2 (θ F )y F t 2 + u F t, u F N (, Σ F u (θ F ) ), (2) 1 All variables and parameters in this section exhibit an superscript F to distinguish them from the remaining parts of the paper. 2 For an overview of the calibration of the DSGE model is given in Appendix A.1.1. The analytical solution of the FTPL is provided in Appendix A The derivation of the VAR model is given in Appendix A

4 Regime I Regime II ɛ F,m ɛ F,f ɛ F,m ɛ F,f π F b F Table 1: Sign of impulse response functions of π F and b F to the shocks ɛ F,m and ɛ F,f on impact in each regime. where u F denotes the reduced form shock vector u F t = A F 1 (θ F )ɛ F t, Σ u (θ F ) = A F 1 (θ F )A F 1 (θ F ) its variance-covariance matrix, and B F (θ F ) = [B1 F (θ F ) B2 F (θ F )] denotes the reduced form coefficient matrix. The coefficients associated with further lags are close to zero. The identification issue arises, because it is not possible to determine A F 1 uniquely from knowing Σ F u (θ F ) only. Instead, the reduced form variance-covariance matrix Σ F u (θ F ) can be decomposed into the product of many impulse matrices ÃF (µ F, θ F ). The impulse matrices ÃF (µ F, θ F ) are rotations of the matrix A F 1 rotated by the rotation parameter µ F. Every impulse matrix ÃF (µ F, θ F ) can be decomposed into the lower Cholesky decomposition L F (θ F ) of Σ u and a rotation matrix R F (µ F ): Ã F (θ F, µ F ) = L F (θ F )R F (µ F ). The matrices L F (θ F ) and R F (µ F ) take the general form: [ ] L F σ = F π (θ F ) F sin(φ F (θ F ))σ F b (θ F ) cos(φ F (θ F ))σ F F b (θ F ) F and R F = [ cos(µ F ) sin(µ F ) sin(µ F ) cos(µ F ) where the standard deviation of inflation and of real debt, and their correlation is denoted by σ F π F (θ F ), σ F b F (θ F ), and sin(φ F (θ F )) respectively. In order to identify the impulse matrix A F I employ sign restrictions, i.e. I only allow for values of µ F which satisfy certain restrictions on the signs of some of the impulse response functions. 2.2 Identification of the VAR model In this section I use the simple example to show how the sign restrictions which identify the VAR model are derived from the DSGE model. Since sign restrictions are inequality restrictions, employing them as identifying restrictions yields a distribution of structural impulse response functions. In the example I want to investigate the effect of a monetary policy shock on inflation and therefore leave this response unrestricted. I restrict the signs of the impact response of real debt to a monetary policy shock, and the impact response of inflation and real debt to a fiscal policy shock. To further simplify the analysis I assume there exist only two possible realizations of the parameter vector θ F, one from Regime I ( θ F I = [1.31.1]), and one from Regime II 4 ],

5 ( θ F II = [.522.1] ). Both parameter vectors correspond to the mean values for each regime chosen by Davig and Leeper (25). Table 1 shows that each parameter vector is associated with a set of different sign restrictions. I conduct the following experiment: I assume that one parameter vector ( θ [θi F ; θf II ]) is the true parameter. That is, I solve the FTPL model parameterized with that parameter vector to derive Equation (1), and solve afterwards for B F (θ ) and L F (θ ). In the next step, I determine the range of the rotation parameter such that the impulse matrix Ã(µ, θ ) fulfills the sign restrictions associated with each regime on impact. The sign restrictions imposed in each regime are the following. In both regimes, real debt is supposed to decrease in response to a fiscal policy shock and to increase as response to a monetary policy shock. In Regime I inflation is restricted to be zero on impact after a fiscal policy shock, in Regime II inflation responds positively. In both regimes the response of inflation to a monetary policy shock is left unrestricted. Consider first the case of Regime I, i.e. assume that θ = θi F. Given BF (θ I ) and L F (θ I ), Regime I restricts µ F to be equal to π. The identifying restrictions associated with Regime II given θi F as the data generating parameter restrict µ F to be in the interval [1.711; π[. I redo the exercise assuming that θ = θii F and calculate BF (θii F ) and LF (θii F ). The interval of µ F that fulfills the sign restrictions associated with Regime II is [.244; 1.846]. Figure 1 plots the distribution of the identified impulse response function ( p(ϕ V,F ) ). If θii F is the data generating parameter there exists no µ F such that the restrictions associated with Regime I are fulfilled. 6 π F on ε F,m.4 b F on ε F,m π F on ε F,f b F on ε F,f Figure 1: Distribution of identified impulse response functions of the VAR model. θ F II as DGP, sign restrictions associated with Regime II. 98% probability bands. 2.3 Estimation of the DSGE model In this section I consider the following setup: the reduced form of the VAR model has been estimated 4, i.e. B F (θ F ) and L F (θ F ) are given. The sign restrictions associated with a parameter vector of the FTPL model yield a distribution of impulse response functions of the 4 The VAR model can be estimated by OLS. I will continue to employ the analytic solution derived in Appendix A

6 VAR model. I now study whether this distribution can be used to estimate the parameter vector of the DSGE model Two parameter vector example continued Consider again the case where there exist only two parameter vectors. The exercise is to recover the correct parameter vector (θ ) underlying B F (θ F ) and L F (θ F ) from the distribution of impulse response functions of the VAR model. To discriminate between the two parameter vectors θi F and θii F I do the following: each set of sign restrictions yields a distribution of impulse ( response functions of the VAR model which is thus conditional on the DSGE model p(ϕ V,F θ F ) ). For each parameter vector θ F I take one random realization out of p(ϕ V,F θ F ). Afterwards, I find the parameter vector of the DSGE model for which the implied impulse response functions are closest to that realization. In the two parameter vector example this can be conducted as an eyeballing exercise. Assume first that θ = θii F is the true parameter vector that determines the matrices of the VAR model ( B F (θii F ), LF (θii F )). In this case it is not possible to find a rotation parameter µ F such that ÃF fulfills the restrictions that correspond to θi F. Consequently, θf II is the estimated parameter vector. In the other case, when θi F is the true parameter vector underlying B F (θ F ) and L F (θ F ), there exist rotation parameters such that the sign restrictions of Regime I or Regime II are fulfilled. Consider first a realization out of p(ϕ V,F θi F ). There is only one rotation parameter such that the sign restrictions are fulfilled: µ F = π. Consequently, the DSGE model parameter vector closest to the impulse response functions of the VAR model is θi F. Consider next the distribution p(ϕ V,F θii F ). A possible random realization would be µf = 2.5. The corresponding impulse response function of the VAR model is plotted in Figure 2 as dotted lines. This figure also plots the impulse response function of the DSGE model for θi F and θii F. As the figure indicates, the impulse response function of the DSGE model closest to the impulse response function of the VAR model is the one implied by θi F. Thus, even if the identification of the impulse response function of the VAR model is conditioned on the sign restrictions associated with θii F, θf I is the estimated parameter vector of the DSGE model. This is possible for two reasons: first, the impulse response of inflation to a monetary policy shock has been left unrestricted in the identification step but is now included into the evaluation of the parameter vector. Second, the impulse response functions of the VAR model depend on all the entries of B F, a matrix dependent on θ F which has not been used in the identification step. This illustrates the point that the suggested method builds on the fact that the sign restrictions are qualitative restrictions only, but that the resulting identified impulse response functions contain additional information which can be employed to discriminate between parameter vectors of the DSGE model. The mechanism at work here is similar to the approach suggested by Canova and Paustian (211), who employ identifying restrictions on the one hand and evaluation restrictions on the other. In addition, the method in this paper allows to estimate the distribution of parameters and thus derive probability statements, while Canova and Paustian (211) discriminate between a set of sub-models about which they derive probabilistic statements. 6

7 1.5 π F on ε F,m θ I F θ II F µ F = b F on ε F,m θ I F θ II F µ F = π F on ε F,f θ I F θ II F µ F = b F on ε F,f θ I F θ II F µ F = Figure 2: Impulse response function of the DSGE model parameterized with θi F (solid line), and θii F (dashed line). Impulse response function of the VAR model with BF (θi F ), LF (θi F ) for µ F = More than two parameter vectors In this section I allow for a wider range of parameter vectors of the FTPL model. This necessitates a probability density function to discriminate between two different parameter vectors of the FTPL model. The probability distribution of the DSGE model parameter vector is conditional on the impulse response functions of the VAR model p(θ F ϕ V,F ). Under the assumption that the impulse response functions of the VAR model are asymptotically normal distributed with mean ϕ D,F and variance-covariance matrix Σ ω, the probability distribution of θ F can be described by employing a limited information likelihood function (L F ) as density function. The density function L F is defined as: L F (θ F ϕ V,F, Σ F ω ) = ( ) 1 Nϕ 2 ( (ϕ Σ F ω 1 2 exp V,F ϕ D,F (θ F ) ) Σ F 1 ( 2π ω ϕ V,F ϕ D,F (θ F ) )) (3) where ϕ D,F (θ F ) denotes the impulse response function of the DSGE model in dependence of its parameter vector, N ϕ the number of elements in ϕ D,F (θ F ), and Σ ω the variance-covariance matrix of the the error term ωt F = ϕ V,F ϕ D,F (θ F ). In this section, I assume Σ F ω to be known. The conditional distribution is evaluated using a Metropolis algorithm. For the parameter vector of the FTPL model I assume a uniform distribution which implies a unique and nonexplosive solution of the FTPL model. This distribution is labelled p(θ F ). At the i th step of the algorithm: 7

8 1. Take one draw θ i,f out of p(θ F ) and derive the sign restrictions. 2. Draw a µ i,f from a uniform distribution with bounds and π until the impulse response function ϕ F (µ F ) fulfills the sign restrictions associated with θ i,f. If no µ i,f with these characteristics can be found, start again at Given ϕ i,f (µ i,f ), maximize the density function (3) to find the parameter vector of the DSGE model θ i,f which is closest to the impulse response function of the VAR model, 4. Keep the pair θ i,f, ϕ V,i,F with probability min{ L Q(θ i,f ϕ V,i,F,Σ F ω ) L Q ; 1}. Return to 1. (θ i 1,F ϕ V,i 1,F,Σ F ω ) As an example, I choose θ = θii F to be the underlying data generating parameter vector and repeat the steps outlined above 1 times. The results are plotted in Figure 3. The first 2 draws have been discarded. The mean of the estimated 8 parameter vectors is θ F = [ ], i.e. it is very close to θii F = [.522.1, with a standard deviation of σ θf = [.1.1]. 1.5 π F on ε F,m.12 b F on ε F,m Quarter after shock Quarter after shock π F on ε F,f b F on ε F,f Quarter after shock Quarter after shock Figure 3: Distribution of identified impulse response functions of the VAR model (solid lines). 98% probability bands. Impulse response functions of the DSGE model at the mean estimate (dashed line). 8

9 3 The general framework In this section I set up the VAR model and the DSGE model in general. I show how the probability distributions of the impulse response functions of the VAR model conditional on the DSGE model and the parameters of the DSGE model conditional on the VAR model s impulse response functions are related and how the suggested sampling algorithm employs this relationship to evaluate both distributions. 3.1 The VAR model and the DSGE model The reduced form of the VAR model is defined as: y t = B 1 y t 1 + B 2 y t B l y t l + u t, t = 1,..., T (4) where y t is a m 1 vector, B 1... B l are coefficient matrices of size m m, u t the onestep ahead forecast error at time t. The error term is normally distributed with variancecovariance matrix Σ u. The structural error term is denoted by ɛ t. It is related to the reduced form error via u t = Aɛ t. The matrix A has the characteristic that Σ u = AA. The matrix A is identified by sign restrictions: the lower Cholesky decomposition L is rotated until the impulse response functions fulfill the sign restrictions. There might not be a unique matrix A fulfilling the sign restrictions. I denote any matrix fulfilling the sign restrictions by Ã: Ã = LR(µ) (5) The vector µ contains the rotation parameters. The coefficient matrices are stacked in a matrix B of dimension ml ml: [ ] B B = 1 B l I m(l 1) m(l 1),m The impulse response functions of the VAR model are denoted by ϕ V. They are defined as: ϕ V ij = B (i 1) A j i = 1, 2,... K, (6) where i denotes the impulse response period, K the impulse response horizon, and j the j th column of the m ml matrix A: A = [ Ã m,m(l 1) ] Given the vector of structural parameters θ, the fundamental solution of the DSGE model can be written as 5 : x t = T ( θ)x t 1 + R( θ)z t, (7) where T ( θ) and R( θ) are matrices one obtains after solving the DSGE model with standard solution techniques. The vector z collects the structural shocks of the DSGE model. z is assumed to be normally distributed with z N (, Σ DSGE ). The endogenous state variables of the DSGE model are related to the set of observable variables y via an observation equation: y t = Gx t (8) 5 x t denotes the percentage deviation of the generic variable X t from a deterministic steady state x chosen as approximation point. 9

10 where G denotes a matrix picking the corresponding endogenous states. The impulse response functions of the variables in y to a structural shock j at horizon i ϕ D ij are given by: ϕ D ij( θ) = GT ( θ) i R( θ)z j, i =, 2,...K. (9) I define the vector θ as comprising the structural parameters θ, the vectorized variancecovariance matrix Σ DSGE, as well as the vectorized variance-covariance matrix Σ ω. 3.2 The VAR model and the DSGE model connected The structural impulse response functions depend on sign restrictions, which I derive from a DSGE model. The sign restrictions rest on the parameter vector θ of the DSGE model. The distribution of ϕ V is therefore conditional on θ: p ( ϕ V θ ). The distribution of the DSGE model parameters is estimated conditional on the impulse response function of the VAR model. I denote this conditional distribution by p ( θ ϕ ) V. Both conditional distributions are linked by the equation: p ( ϕ ) V p ( θ ϕ ) V = p ( ϕ V θ ) p (θ) (1) Equation (1) shows that the Gibbs sampling algorithm can be applied to evaluate p ( θ ϕ ) V and p ( ϕ V θ ) in turn. The VAR model is estimated employing the dataset Y. Equation (1) is therefore rewritten as: p ( ϕ V Y ) p ( θ ϕ V, Y ) = p ( ϕ V θ, Y ) p (θ Y ) (11) In the following I will make two assumptions to apply the Gibbs sampling algorithm: first, since the DSGE model is estimated by matching the corresponding impulse response functions and not time series observations, the distribution of θ conditional on ϕ V and Y is equal to the distribution of θ conditional on ϕ V only 6. Second, Equation (8) can be augmented with a sufficient number of measurement errors such that p (θ Y ) is non-singular. Nevertheless, according to the first assumption the measurement errors are not needed to evaluate p ( θ ϕ ) V which is, besidesp ( ϕ V θ, Y ), the distribution of interest. 3.3 The conditional distribution p(θ ϕ V ) In order to evaluate the distribution p(θ ϕ V ) I employ a limited information Bayesian approach. This approach is based on Kim (22) and has recently been employed by Christiano, Trabandt, and Valentin (21). Following the latter authors I assume that the realization ϕ V is the data and that its asymptotic distribution is given by: 6 The following then holds: ϕ V a N ( ϕ D (θ), Σ ω ), (12) p(θ ϕ V, Y )p(ϕ V Y ) = p(ϕ V Y )p(θ ϕ V ). This argument is similar to the argument made by Smith (1993) and DelNegro and Schorfheide (24). 1

11 Conditional on an estimate of the variance-covariance matrix Σ ω, the approximate likelihood of the impulse response function ϕ D is then given by: L ( θ ϕ V, Σ ) ω = (2π) Nϕ 2 Σω 1 2 (13) ( 1 ( ) ) 1 ( exp ϕ V ϕ D ( θ) ( Σω ϕ V ϕ ( θ)) ) D. 2 Christiano, Trabandt, and Valentin (21) argue that 4 elements are already sufficient to validate the assumption concerning the asymptotic distribution. In the controlled Monte Carlo experiments in Section 4.2 I find that even for much fewer impulse response functions the density function (13) is an accurate description of the density. The variance-covariance matrix in Equation (13) is computed in the following way: the prior distribution of the DSGE model implies a wide range of impulse response functions of the DSGE model and corresponding sign restrictions. For every parameter vector of the DSGE model and the corresponding sign restrictions I generate draws ϕ V from the distribution of the impulse response functions of the VAR model (p(ϕ V θ, Y )). Given these realizations, the variance-covariance Σ ω matrix is be approximated. Again, this approach closely follows Christiano, Trabandt, and Valentin (21). 3.4 The conditional distribution p(ϕ V Y, θ) The second distribution of interest is the distribution of impulse response functions of the VAR model conditional on the restrictions derived from the DSGE model p(ϕ V Y, θ). The structural impulse response functions of a VAR model, defined in Equation (6), are the coefficients of the VMA representation of the VAR model: y t = ϕ V ɛ t + ϕ V 1 ɛ t 1 + ϕ V 2 ɛ t 2... ɛ (, I) (14) The posterior distribution p(ϕ V Y, θ) summarizes the knowledge about the structural VMA coefficients. Details about the specification and the derivation of the distribution can be found in Appendix B. Before turning to the VAR model, I briefly summarize the steps and the assumptions involved when deriving the posterior distribution for the coefficients in Equation (14). I adapt the specification laid out by Kociecki (25) and formulate the prior distribution for the structural VMA coefficients as a normal distribution. This formulation allows me to rewrite the joint distribution of the coefficients as the product of conditional normal distributions. In the next step these distributions are written in terms of the reduced form coefficients and the impulse matrix. While the distributions of the reduced form coefficients are again conditional normal distributions, the prior distribution of the impulse matrix is a non-standard distribution. This non-standard distribution cannot be evaluated by standard methods, but I assume that it can be approximated by the different impulse matrices which fulfill the sign restrictions. The likelihood of the reduced form VMA model is combined with the prior distribution to form the posterior. The sign restrictions put zero probability weights on the regions where the sign restrictions are not fulfilled. 11

12 In practical applications, whenever it is possible, VMA models are approximated by VAR models. The posterior distribution of the reduced form VAR model defined in Equation (4) is evaluated using the formulas provided inter alia by Uhlig (25). The prior distribution for B and Σ u is specified by choosing B, N, S, v : vec(b) Σ u N (vec(b ), Σ u N 1 ) Σ u IW(v S, v ). The maximum likelihood estimates of Σ u and B are Σ u = 1 T (Y X ˆB) (Y X ˆB) and ˆB = (X X) 1 X Y. The posterior is then given by: where N T = N + X X vec(b) Σ u N (vec(b T ), Σ u N 1 T ) (15) Σ u IW(v T S T, v T ), (16) B T = N 1 T (N B + X X ˆB) S T = v S + T Σu 1 (B v T v T v ˆB) N N 1 T X X(B ˆB) T v T = v + T. The distribution of the structural impulse response functions are obtained using sign restrictions based on Equation (5). 3.5 The sampling algorithm Section 3.2 illustrated that the Gibbs sampler can be used the evaluate the distributions p(ϕ V Y, θ) and p(θ ϕ V ). To discriminate between two parameter vectors of the DSGE I include a Metropolis step within the Gibbs sampler. Let superscript i denote the realizations of the i th iteration, the following steps are conducted at the i th iteration: 1. Draw from the prior distribution p(θ). Derive the corresponding sign restrictions. 2. Draw a reduced form coefficient matrix B i and a reduced variance-covariance matrix Σ i u from (15) and (16). 3. Given the identifying restrictions from Step 1 and the draws from Step 2, find a vector of rotation parameters such that à defined in Equation (5) and the associated structural impulse response functions ϕ V i defined in Equation (6) fulfill the sign restrictions. In the case that no rotation parameter vector can be found, return to Step Find θ i, which implies the closest fit of ϕ Di to ϕ V i by maximizing the posterior distribution p(θ i ϕ i ), combining Equation (13) with the prior density p(θ i ). 5. Accept the draw θ i with probability min{ p(θi ϕ V i ) p(θ i 1 ϕ V i 1 ), 1}. 12

13 6. Start again at Step 1. To test for convergence of the Markov chain I suggest to apply the Kolmogorov-Smirnov test. This tests whether two samples of the same chain have been drawn from the same distribution. In case the samples are drawn from the same distribution, it indicates convergence to a stationary distribution. The application of this test has the advantage that it can be applied to one chain only and yields a convenient stopping rule: stop the chain when the null hypothesis that the two samples are drawn from the same distribution cannot be rejected for all the parameters. Additionally, the evolution of the Kolmogorov-Smirnov statistic over the number of draws can be plotted. In the remaining section of the paper, I discuss the properties of the sampling algorithm in more detail using a Monte Carlo study. 4 Application to the deep habits model In this section I employ the deep habits model laid out by Ravn, Schmitt-Grohé, Uribe, and Uuskula (21) in a Monte Carlo exercise. I choose this particular DSGE model for two reasons: the underlying deep habits mechanism is well published (Ravn, Schmitt-Grohé, and Uribe, 26; Ravn, Schmitt-Grohé, Uribe, and Uuskula, 21; Ravn, Schmitt-Grohé, and Uribe, 212) and thus well known and explored. The model description can therefore be kept very brief. Second, when the authors take the deep habits model to the data, as in Ravn, Schmitt-Grohé, Uribe, and Uuskula (21), they explicitly state that the deep habits model is not thought of as a description of the DGP. Ravn, Schmitt-Grohé, Uribe, and Uuskula (21) address the question whether a simple model can account for the price puzzle, i.e. the increase of inflation after a contractionary monetary policy shock. The price puzzle occurs in the case the VAR model is identified using a Cholesky decomposition. Ravn, Schmitt-Grohé, Uribe, and Uuskula (21) develop a DSGE model which is able to generate a positive as well as a negative response of inflation after a monetary policy shock. I employ their model and the method of the paper to study in a controlled experiment whether the suggested method can recover the correct impulse response function of the VAR model and the corresponding correct parameter vector of the DSGE model. 4.1 Deep habits model The DSGE model consists of households, firms and a monetary authority. Households are identical and infinitely lived. Households act as monopolistically competitive labor unions in the labor market. The preferences are defined as deep habits: the utility derived from consuming a good of a certain variety is related to the past aggregate consumption of this variety. Furthermore, households own firms and receive dividends, and have access to a nominal risk-free bond. Firms are monopolistically competitive and face quadratic adjustment costs when optimizing their price. The main mechanism of the deep habits model works in the following way: firms have an incentive to lower prices today if they expect future demand to be high relative to current demand. Additionally, the firm increases its price elasticity of demand. 13

14 Parameter θh d ζ Hw ζ Hp ρ Hr α Hπ α Hy σ Hr description deep habit parameter wages adjustment costs parameter price adjustment costs parameter coefficient Taylor rule on lagged interest rate coefficient Taylor rule on inflation coefficient Taylor rule on output standard deviation monetary policy shock Table 2: Description of the estimated parameters of the deep habits model Monetary policy aims at stabilizing deviations in inflation and output from their steady state values. Further details of the deep habits model are described in Appendix A.2. The appendix contains a summary of the variables, the parameters, the steady-state in Appendix A.2.1, as well as the log-linearized equations of the deep habits model in Appendix A.2.2. All variables and parameters associated with the deep habits model are labeled with an H. 4.2 A Monte Carlo study In this section I employ a controlled experiment to study whether the suggested method can be applied to recover the correct impulse response function of the DSGE model and the corresponding parameter values. In the original paper, Ravn, Schmitt-Grohé, Uribe, and Uuskula (21) estimate only a subset of the parameters of the deep habits model, because not all parameters are identified. I follow the authors and calibrate some of the parameters. 7 Table 2 provides an overview of the parameters which are estimated. I choose a very wide prior distributions for the parameter vector of the DSGE model. Figure 4 displays the impulse response functions implied by the prior distribution. The response of inflation is centered around zero and associated with wide probability bands. In order to show the influence of the number of observations on the method I conduct two controlled experiments: one with 2, simulated observations and one with 2 simulated observations. Both cases use the same prior distribution. Since there are less shocks then observable variables I add small measurement errors to the data. I approximate the variance-covariance matrix Σ ω in two steps. First, I use only the prior distribution to compute the variance-covariance matrix as described in Section 3.3. After the first 5, draws, I use these draws to calculate the variance-covariance matrix again. Subsequently, I take 1, draws and discard the first 2, draws. The results, which are reported in Table 3, are based on the last 8, draws. As it is typically done in the literature that matches impulse response functions I employ only the diagonal of the variance-covariance matrix Σ ω in the estimation. The results, which are presented in Table 3, show that the method works very well for 2 as well as for 2, simulated observations. The estimated parameter values are close to 7 An overview of the calibrated values can be found in Table 5 in Appendix A

15 .5 Output 3.5 Interest rate.6 Inflation Quarters after shock Quarters after shock Quarters after shock 1 Consumption.5 Real wages Quarters after shock Quarters after shock Figure 4: Impulse response functions of the deep habits model implied by the prior distribution. 99% probability bands. the true parameter values. The deep habit parameter value, which is key for the response of inflation, is very precisely recovered. Figure 5 plots the prior distribution against the posterior distribution for the case with 2, observations. The posterior distribution is much smaller than the prior distribution. The true value is always part of the posterior distribution. The correct impulse response functions are recovered as well. Figure 6 displays 68% probability bands of the impulse response function of the VAR model and the DSGE model. Both models recover significantly the correct sign of the impulse response function of inflation. The algorithm converges for all parameters. Figure 7 plots of the corresponding Kolmogorov- Smirnov test statistic along the 8, draws. As Table 3 indicates, the estimation results for the case of only 2 observations are not very different. Here as well, the correct impulse response function is significantly estimated and the parameters, especially the key parameter θh d, estimated close to the true values. Correspondingly, the figures associated with 2 observations look similar to the case with 2, observations. They are therefore plotted in Appendix A

16 Prior distribution 2, observations 2 observations Parameter sim distribution mean std mean std mean std θh d.85 beta ζ Hw 4.9 normal ζ Hp 14.5 normal ρ Hr.7 beta α Hπ 1.46 gamma α Hy.1 gamma σ Hr 1 inv. gamma Table 3: Estimation results of the Monte Carlo experiments. The column denoted by sim contains the true parameter value. θ H d ζ Hw ζ Hp ρ Hr α Hπ α Hy σ Hr Figure 5: Prior (white) vs. posterior (black) distribution. Monte Carlo study with 2, observations. 16

17 .5 Output 1.2 Interest rate.1 Inflation Years after shock Years after shock Years after shock.1 Consumption.5 Real wages Years after shock Years after shock Figure 6: Impulse response functions of the deep habit model (dashed line) versus VAR model (solid lines). 68% probability bands. Monte Carlo study with 2, observations. θ H d ζ Hw ζ Hp ρ Hr α Hπ α Hy σ Hr Figure 7: Kolmogorov- Smirnov statistic along the number of draws. Monte Carlo study with 2, observations. 17

18 5 Conclusion In this paper I suggest an algorithm to estimate the effects of an exogenous disturbance on the economy. To this end, I identify a VAR model using sign restrictions from a DSGE model. Since these restrictions are usually not unique across the parameter space, I simultaneously estimate the distribution of the parameters of the DSGE model. This distribution is estimated by matching the identified impulse response functions of the VAR model and the impulse response functions of the DSGE model. By estimating the DSGE model that way, I circumvent strong assumptions such as that the DSGE model has to represent the data generating process. The suggested method combines two approaches, the identification of the VAR model on the one hand, and the estimation of the DSGE model on the other hand in one encompassing algorithm. The method utilizes the fact that sign restrictions are only qualitative restrictions and that typically not all signs of the impulse response functions are restricted. When estimating the DSGE model based on matching the identified impulse response functions and the impulse response functions of the DSGE model, all variables are considered and are quantitatively taken into account. The controlled Monte Carlo studies show that the method is capable of recovering the correct parameter vector of the DSGE model and correspondingly the correct impulse response function of the VAR model. 18

19 A Details to the DSGE models employed in the paper A.1 Calibration and solution of the FTPL model A.1.1 Calibration ḡ F =.25, c F = 1 ḡ F, b F =.4, π F = 1.343, R F = πf β F, β F =.99. A.1.2 Solution of the FTPL model The FTPL model can be reduced to: where: πt+1 F = β F α F πt F + β F ɛ F,m t (17) b F t + φ 1 πt F + φ 3 ɛ F,m t = φ 2 πt 1 F + φ 5 b F t 1 φ 4 ɛ F,m t 1 + ɛ F,f t (18) φ 1 = cf α F ( R F 1) + c F RF 2 ( R F 1) π F 2 + R F bf π F 2 [ ] φ 2 = αf c F π F ( R F 1) b F 2 c F φ 3 = ( R F 1) 2 φ 4 = φ 2 α The solution takes the form of Equation (1). For convenience I repeat the definitions: y F t = Φ F 1 (θ F )y F t 1 + A F 1 (θ F )ɛ F t + A F 2 (θ F )ɛ F t 1, ɛ F N (, I) with y t = [ ] [ ],, πt F b F t ɛ F t = ɛ F,m t ɛ F,f t θ F = [ α F γ ] F. The coefficient matrices B1 F, A F 1, and A F 2 contain the recursive laws of motion. [ Φ F ηπ 1 = F π F η ] π F b F η b F π F η b F b F The recursive laws of motion depend on the policy regime. For active monetary and passive fiscal policy the solution is given by: η b F b F = βf 1 γ F η π F b F = η π F π F = η b F π F = φ 2 19

20 For the case of passive monetary policy and active fiscal policy the solution is given by: η π F π F = ( βf α F + β F 1 γ F )φ 2 /φ1 φ 2 /φ 1 β F 1 + γ F η b F b F = βf α F η π F π F η π F b F = ( βf α F + β F 1 γ F ) + η π F π F φ 1 η b F π F = φ 2 φ 1 η π F π F The matrices A 1 and A 2 are defined in the following way: [ A F η1,π 1 = F ɛ F,m η 1,π F ɛ F,f η 1,b F ɛ F,m η 1,b F ɛ [ F,f A F η2,π 2 = F ɛ F,m ] η 2,b F ɛ F,m ] where: A.1.3 η 2,π F ɛ F,m = η π F b F φ 4 η π F π F η π F b F φ 1 β F α F η 2,b F ɛ F,m = φ 1η 2,π F ɛ F,m φ 4 η 1,π F ɛ F,m = βf + φ 3 η π F b F η 2,π F ɛ F,m η π F π F βf α F φ 1 η π F b F η 1,π F ɛ = η π F b F F,f η π F π F βf α F φ 1 η π F b F η 1,b F ɛ = φ 1η F,f 1,π F ɛ 1 F,f η 1,b F ɛ F,m = φ 1η 1,π F ɛ F,m φ 3 VAR representation of the FTPL model Rewrite Equation (1) using A 1 ɛ t = u t and the the lag operator L as: y t (Φ F 1 + A F 2 A F 1 1 )yt 1 ( ( ) A F 2 A F 1 1 Φ F 1 + (I Φ F 1 L)y t = (I + A F 2 A F 1 1 L)ut (I + A F 2 A F 1 1 L) 1 (I Φ F 1 L)y t = u t ( A F 2 A F 1 1 ) 2 ) y t 2... = u t (19) ( ) In both regimes is A F 2 A F close to zero. Lags of higher order than order two are therefore associated with zero coefficient matrices. 2

21 A.2 Solution of the deep habits model A.2.1 Overview of variables, parameters, steady states, and calibration Variable x H,t h H,t c H,t w H,t π H,t π Hw,t R H,t y H,t π H,t π Hw,t λ y H,t, λh H,t, λc H,t description composite consumption hours worked consumption real wages price inflation rate nominal wage growth nominal interest rate output indexation price growth indexation wage growth Lagrange multiplier Table 4: Description of the variables of the deep habits model Following Ravn, Schmitt-Grohé, Uribe, and Uuskula (21), calibrated values for the structural parameters are set as: Parameter or steady state value description calibration RH real interest rate 1.1 β H discount factor 1/RH φ H labor demand price elasticity 1 κ H inverse of the Frisch elasticity.5 σ H constant relative risk aversion 3 ν Hw wage indexation.96 πh Inflation steady state 1 πhw Nominal wage growth steady state 1 h H Hours worked steady state.3 Table 5: Calibration of the deep habits model 21

22 The remaining steady state values are computed the following way: x H = (1 θ d H) h H (2) c H = h H (21) λ c H = 1/((1 θh)η d H ) (22) λ y H = 1 + (θd Hβ H 1) λ c H (23) w H = λ y H (24) λ h H = w H /( x σ Hφ H ) (25) γ H = ( x σ H w H λ h H H )/ h κ H (26) A.2.2 Log-linearized equations x H ˆx H,t = c H ĉ H,t θ d H c H ĉ H,t 1 (27) γ H hκ H H κ H ĥ H,t = x σ H H w H ( σ H ˆx H,t + ŵ H,t ) λ H ˆλh H,t (28) φ H λh H hh x σ H(ˆλ h H,t + ĥh,t + σ H ˆx H,t ) = ζ Hw (ˆπ Hw,t ˆ π Hw,t ) + h H w H (ĥh,t + ŵ H,t ) + βζ Hw (ˆπ Hw,t+1 ˆ π Hw,t+1 ) (29) σ H ˆx H,t = R H ˆR H,t σ H ˆx H,t+1 ˆπ H,t+1 (3) ĉ H,t = ĥh,t (31) ˆλ y H,t = ŵ H,t (32) ĥ H,t = ŷ H,t (33) λ y H ˆλ y H,t + λ c H ˆλ c H,t = θ d Hβ H λc H ( σ H ˆx H,t+1 + σ H ˆx H,t + ˆλ c H,t+1) (34) η H λc H x H (ˆλ c H,t + ˆx H,t ) + ζ Hp (ˆπ H,t ˆ π H,t ) = c H ĉ H,t + β H ζ Hp (ˆπ H,t+1 ˆ π H,t+1 ) (35) ˆR H,t = ρ Hr ˆRH,t 1 + (1 ρ Hr )(α Hπˆπ H,t + α Hy ŷ H,t ) + ɛ H,t (36) ˆ π H,t = (1 ν Hp )ˆπ H,t 1 (37) ˆ π Hw,t = (1 ν Hw )ˆπ Hw,t 1 (38) w H ŵ H,t = w H ŵ H,t 1 + ˆπ Hw,t ˆπ H,t (39) 22

23 A.2.3 Additional results Monte Carlo study θ H d ζ Hw ζ Hp ρ Hr α Hπ α Hy σ Hr Figure 8: Prior (white) vs. posterior (black) distribution. Monte Carlo study with 2 observations. 23

24 θ H d ζ Hw ζ Hp ρ Hr α Hπ α Hy σ Hr Figure 9: Kolmogorov- Smirnov statistic along the number of draws. Monte Carlo study with 2 observations. Output.8 Interest rate.8 Inflation Years after shock Years after shock Years after shock Consumption Real wages Years after shock Years after shock Figure 1: Impulse response functions of the deep habit model (dashed line) versus VAR model (solid lines). 68% probability bands. Monte Carlo study with 2 observations. 24

25 B The distribution p(ϕ V Y, θ) B.1 Derivation of the prior distribution B.1.1 Prior distribution of the structural coefficients of the VMA model The structural VMA model is defined as: y t = ϕ V ɛ t + ϕ V 1 ɛ t 1 + ϕ V 2 ɛ t 2... ɛ (, I) (4) The vectorized prior distribution is assumed to be: vec(ϕ ) vec( ϕ ) vec(ϕ 1 ) vec( ϕ 1 ) vec(ϕ 2 ) N vec( ϕ 2 ),. vec(ϕ l ). vec( ϕ k ) V V1 V2 Vl V 1 V11 V12 V1k V 2 V21 V22 V2k. V k Vk1 Vk2 Vkk. (41) The probability distribution (41) can be decomposed into a marginal distribution of p(ϕ ) and succeeding conditional distributions: with p(ϕ, ϕ 1,..., ϕ k ) = p(ϕ k ϕ k 1 ϕ )p(ϕ k 1 ϕ k 2 ϕ ) p(ϕ 1 ϕ )p(ϕ ), (42) p(vec(ϕ )) = N (vec( ϕ ), V ) (43) p(vec(ϕ i vec(ϕ i 1 ) vec(ϕ )) = N (χ i, ii ), i = 1 k, (44) and χ i and ii abbreviate the usual definitions for conditional distributions: 1 χ i = vec( ϕ) + [ V Vi V ] Vi 1 vec(ϕ ϕ ) ii V i 1, Vi 1,i 1 vec(ϕ i 1 ϕ i 1 ) ii = V ii [ ] Vi Vii 1 V Vi V i 1, Vi 1,i 1 1 V i. V i 1,i. B.1.2 The Jacobian J ϕ V A, Φ The reduced form VMA model is given by: Y t = u t + Φ 1 u t 1 + Φ 2 u t Φ k u t k. (45) To write the prior distribution (42) in terms of the reduced-form coefficients, it is necessary to scale the probability distribution with the Jacobian J ϕ Φ : p(ϕ) = p(f(φ))j ϕ Φ. (46) 25

26 The relationship between structural and reduced-form moving average coefficients is given by: ϕ = A (47) ϕ i = Φ i A, i = 1 k. Φ is omitted as this matrix is normalized to an identity matrix by assumption. The Jacobian is calculated in the following way. Applying the vec-operator yields: 8 vec(ϕ i ) = (A I m m )vec(φ i ). (48) The Jacobian matrix is defined as: J ϕ Φ = det vec(ϕ 1 ) vec(φ ). vec(ϕ k ) vec(φ ) vec(ϕ 1 ) vec(φ 1 ).... vec(ϕ k ) vec(φ 1 ) vec(ϕ 1 ) vec(φ k ). vec(ϕ k ) vec(φ k ). (49) Due to the fact that vec(ϕ i) vec(φ j = for j > i, the matrix becomes a block triangular matrix, ) and the determinant is given by: B.1.3 J ϕ Φ = vec(ϕ ) vec(φ ) vec(ϕ 1) vec(φ 1 ) vec(ϕ k) vec(φ k ) J ϕ Φ = (A I m m ) k = A mk. (5) Prior distribution of the reduced form coefficients of the VMA model Given Equation (41), its decomposition defined in (42), and the Jacobian (5) a prior distribution for the reduced form coefficients conditional on ϕ = A is formulated as: where with p(a, Φ 1,..., Φ k θ) = p(φ k Φ k 1 A, θ) p(ϕ 1 A, θ)p(a θ)j ϕ Φ, (51) p(vec(a)) = N (vec( ϕ ), V ) (52) p(vec(φ i ) vec(φ i 1 ) vec(a)) = N ( Φ i, Vi i) (53) Φ i = (A I m m )χ i (54) V ii = (A 1 I m m ) ii (A 1 I m m ). (55) The prior distribution (51) consists of conditional normal distributions (53), which can be easily evaluated, and a non-standard distribution for the impulse matrix (52). The nonstandard distribution arises because the distribution in the impact period is scaled by the Jacobian (5). 8 I use the relationship vec(ab) = (I A)vec(B) = (B I)vec(A)). 26

27 B.2 The posterior distribution The likelihood of the process defined in Equation (45) is written in state space form: where F = ξ t+1 = F ξ t + U t+1 (56) y t = Hξ t, (57) ξ t = [ u t u t k ] m k 1 I m I m I m m k m k U t+1 = [ u t+1 ] m k 1 H = [ ] I m Φ 1 Φ k. m m k Given an initial condition for y and Σ u,, the likelihood can then be written as: p(y T,..., y Φ 1,..., Φ k, Σ u ) = p(y T y T 1... y, Φ 1,..., Φ k, Σ u ) p(y Φ 1,..., Φ k, Σ u ), (58) and evaluated using the Kalman filter. The posterior of the reduced form coefficients is derived by combining Equation (58) and Equation (51): p(φ 1, Φ k, A θ, Y ) = p(y T,..., y Φ 1,..., Φ k, Σ u )p(a, Φ 1,..., Φ k θ). (59) 27

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