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Working Paper Series Approximating time varying structural models with time invariant structures WP 15-10 Fabio Canova BI Norwegian Business School and CEPR Filippo Ferroni Banque de France and University of Surrey Christian Matthes Federal Reserve Bank of Richmond This paper can be downloaded without charge from: http://www.richmondfed.org/publications/ Approximating time varying structural models with time invariant structures Fabio Canova, BI Norwegian Business School and CEPR Filippo Ferroni Banque de France and University of Surrey Christian Matthes Federal Reserve Bank of Richmond September 8, 2015 Working Paper No. 15-10 Abstract The paper studies how parameter variation a¤ects the decision rules of a DSGE model and structural inference. We provide diagnostics to detect parameter variations and to ascertain whether they are exogenous or endogenous. Identi…cation and inferential distortions when a constant parameter model is incorrectly assumed are examined. Likelihood and VAR-based estimates of the structural dynamics when parameter variations are neglected are compared. Time variations in the …nancial frictions of Gertler and Karadi’ (2010) model are studied. s Keywords: Structural model, time varying coe¢ cients, endogenous variations, misspeci…cation. JEL Classi… cation: C10, E27, E32. We thank Michele Lenza, Marco del Negro, Tao Zha, Ferre de Graeve, James Hamilton, Frank Schorfheide, and the participants to seminars at Goethe University, University of Milan, Bank of England, Carlos III Madrid, Humboldt University Berlin, Federal Reserve Board, and the conferences ESSIM 2015; Identi…cation in Macroeconomics, National Bank of Poland; Econometric Methods for business cycle analysis, forecasting and policy simulations, Norges Bank; the NBER Summer Institute group on Dynamic equilibrium models for comments and suggestions. The views presented in this paper do not re‡ those of ect the Banque de France, the Federal Reserve Bank of Richmond, or the Federal Reserve system. 1 1 INTRODUCTION 1 Introduction In macroeconomics it is standard to study models that are structural in the sense of Hurwicz (1962); that is, models where the parameters characterizing the preference and the constraints of the agents and the technologies to produce goods and services are invariant to changes in the parameters describing government policies. Such a requirement is crucial to distinguish structural from reduced form models, and to conduct correctly designed policy counterfactuals in dynamic stochastic general equilibrium (DSGE) models. Recently, Dueker et al. (2007), Fernandez Villaverde and Rubio Ramirez (2007), Canova (2009), Rios Rull and Santaeularia Llopis (2010), Liu et al. (2011), Galvao, et al. (2014), Vavra (2014), Seoane (2014), and Meier and Sprengler (2015) have shown that DSGE parameters are not time invariant and that variations display small but persistent patterns. Parameter variations can not be taken as direct evidence that DSGE models are not structural. For example, Cogley and Yagihashi (2010), and Chang et al. (2013) showed that parameter variations may result from the misspeci…cation of a time invariant model, while Schmitt Grohe and Uribe (2003) indicated that parameter variations may be needed in certain small open economy models to ensure the existence of a stationary equilibrium. The approach the DSGE literature has taken to model parameter variations follows the VAR literature (see Cogley and Sargent, 2005, and Primiceri, 2005): they are assumed to be exogenously drifting as independent random walks. Many economic questions, however, hint at the possibility that parameter variations may instead be endogenous. For example, is it reasonable to assume that a central bank reacts to in‡ ation in the same way in an expansion or in a contraction? Davig and Leeper (2006) analyze state-dependent monetary policy rules and describe how this feature a¤ects structural dynamics. Does the propagation of shocks depend on the state of private and government debt? Do …scal multipliers depend on inequality, see e.g. Brinca et al. (2014)? Are households as risk averse or as impatient when they are wealthy as when they are poor? Questions of this type are potentially numerous. Clearly, policy analyses conducted assuming time invariant parameters or an inappropriate form of time variations may be misleading; comparisons of the welfare costs of business cycles biased; and growth prescriptions invalid. This paper has three main goals. First, we want to characterize the decision rules of a DSGE when parameter variations are either exogenous or endogenous, and in the latter case, when agents internalize or not the e¤ects that their decisions may have on parameter variations. Second, we wish to provide diagnostics to detect misspeci…cations due to neglected parameter variations. Third, we want to study the consequences in terms of identi…cation, estimation, and inference of using time invariant models when the DGP features parameter variations and compare likelihood-based and SVAR-based estimates of the structural dynamics when parameter variations are neglected. The existing literature is generally silent on these issues. Seoane (2014) uses 2 1 INTRODUCTION parameter variations as a respeci…cation tool. Kulish and Pagan (2014) characterize the decision rules of a DSGE model when predictable structural breaks occur. Magnusson and Mavroedis (2014) and Huang (2014) examine how variations in the certain parameters may a¤ect the identi…cation of other structural parameters and the asymptotic theory of maximum likelihood estimators. Fernandez Villaverde et al. (2013) investigate to what extent variations in shock volatility matter for real variables. Ireland (2007) assumes that trend in‡ ation in a standard New Keynesian model is driven by structural shocks; Ascari and Sbordone (2014) highlight that it may be a function of policy decisions. The next section characterizes the decision rules in a general setup where both exogenous and endogenous variations in the parameters regulating preferences, technologies, and constraints are possible. We consider both …rst order and higher order perturbed approximations. We present a simple RBC example to provide intuition for the results we obtain. We show that if parameter variations are exogenous, structural dynamics are the same as in a model with no parameter variations. Thus, if one correctly identi…es structural disturbances, she would make no mistakes in characterizing structural impulse responses, even if she employes a constant coe¢ cient model. Clearly, variance and historical decompositions exercises will be distorted, since some sources of disturbances will be omitted. If parameter variations are instead endogenous, structural dynamics may be di¤erent from those of a constant coe¢ cient model. Di¤erences exist because the income and substitution e¤ects present in the constant coe¢ cient model are altered. These conclusions do not necessarily hold when higher order approximations are used. Section 3 provides diagnostics to detect misspeci…cation induced by neglecting parameter variations and to distinguish exogenous vs. endogenous parameter variations. In the context of a Monte Carlo exercise, we show that they are able to detect the true DGP with high probability In section 4 we are interested in measuring the identi…cation repercussions that neglected time variations may have for time invariant parameters. Since the likelihood is constructed using forecast errors, which are generally misspeci…ed when parameter variations are neglected, one expects the likelihood shape to be both ‡ attened and distorted. In the context of the RBC example, we show that indeed both pathologies occur; we also show that weakly identi…ed (time invariant) parameters do not become better identi…ed when time variations in other parameters exist. Section 5 considers structural estimation of a time invariant model when the data is generated by models with time varying parameters. We expect distortions because the dynamics assumed by the constant coe¢ cient model are generally incorrect and because shock misaggregation is present. Indeed, important biases in parameter estimates are present, occur primarily in parameters controlling income and substitution e¤ects, and do not die away as sample size increase. Estimated impulse responses di¤er from the true ones both in quantitative and qualitative sense. Section 6 studies whether a less structural time invariant SVARs model can capture the dynamics induced by structural shocks. We show that the performance 3 2 THE SETUP 4 is comparable if not superior to the one of structural models. The performance of SVARs worsens when shocks to the parameters account for a considerable portion of the variability of the endogenous variables but the deterioration is not as large as with likelihood -based approaches. Section 7 estimates the parameters of Gertler and Karadi’ (2010) model of uns conventional monetary policy, applies the diagnostics to detect parameter variations, and estimates versions of the model where the bank’ moral hazard parameter is als lowed to vary over time. We …nd that a …xed coe¢ cient model is misspeci…ed, that making parameter variations endogenous function of net worth is preferable, and that the dynamic e¤ects of capital quality shocks on the spread and on bank net worth can be more persistent than previously thought. Section 8 concludes. 2 The setup The optimality conditions of a DSGE model can be represented as: Et [f (Xt+1 ; Xt ; Xt 1 ; Zt+1 ; Zt ; t+1 ; t )] =0 (1) where Xt is an nx 1 vector of endogenous variables, Zt is an nz 1 vector of strictly exogenous variables, t = [ 1t ; 2t ]; vector of possibly time varying structural parameters, where 2t is a n 1 nx1 1 vector, nx nx1 ; appearing in the case agents internalize the e¤ects that their decisions have on the parameters and 1t is an n 1 1 vector, while f is a continuous function, assumed to be di¤erentiable up to order q, mapping onto a Rnx space. Since the distinction between variables and parameters is blurred when we allow for parameter variations, we use the convention that parameters are those variables that typically assumed to be constant by economists. The law of motion of the exogenous variables is: Zt+1 = (Zt ; z t+1 ) (2) where is a continuous function, assumed to be di¤erentiable up to order q, mapping onto a Rnz space; z is a ne 1 vector of i.i.d. structural disturbances with mean t+1 zero and identity covariance matrix; nz ne ; 0 is an auxiliary scalar; is a known ne ne matrix. The law of motion of the structural parameters is: t+1 = ( ; Xt ; Ut+1 ) (3) where is a continuous function, assumed to be di¤erentiable up to order q, mapping onto the Rn space; Ut is a nu 1 vector of exogenous disturbances, n = n 1 (1+nx1 ) nu ; is a vector of constants: The law of motion of Ut+1 is: Ut+1 = (Ut ; u u t+1 ) (4) where is continuous and di¤erentiable up to order q, mapping onto the Rnu space; u is a n 1 vector of i.i.d. disturbances, with mean zero and identity covariance u t matrix, uncorrelated with the z ; and u is a known nu nu matrix. t+1 2 THE SETUP 5 The decision rule is assumed to be of the form: Xt = h(Xt 1 ; Wt ; t+1 ; ) (5) where h is a continuous function, assumed to be di¤erentiable up to order q, and 0 mapping onto a Rnx space, t+1 = [ z0 ; u0 ]0 ; = diag[ z ; u ]; Wt = [Zt ; Ut0 ]0 : t+1 t+1 Few features of the setup need some discussion. First, t will be serially correlated if Ut or Xt or both are serially correlated. Second, the vector of structural disturbances z may be smaller than the vector of exogenous variables t+1 and the dimension of u may be smaller than the dimension of the structural t+1 parameters. Thus, there may be common patterns of variations in Zt+1 and Ut+1 : Third, we allow for time variations in the parameters regulating preferences, technologies, and constraints, but we do not consider variations in the auxiliary parameters regulating the law of motion of Zt and Ut ; as we are not interested in stochastic volatility, GARCH, or rare events phenomena (as in e.g. Andreasan, 2012), nor in time variations driven by evolving persistence of the exogenous processes. Fourth, (5) makes no distinction between states and controls. Thus, it has the format of a …nal form (endogenous variables as a function of the exogenous variables and the parameters) rather than of a state space form (control variables as a function of the states and of the parameters). 2.1 First order approximate decision rule We start by studying the implications of structural parameters variation for the optimal decision rule when a …rst order approximate solution is considered. Taking a linear expansion of (1) around the steady states leads to 0 = Et [F xt+1 + Gxt + Hxt 1 + Lzt+1 + M zt + N t+1 + O t] (6) where F = @f =@Xt+1 , G = @f =@Xt , H = @f =@Xt 1 , L = @f =@Zt+1 , M = @f =@Zt , N = @f =@ t+1 O = @f =@ t ; all evaluated at the steady states values of (Xt ; Zt ; t ) and lower case letters indicate deviations from the steady states. Linear expanding (5) leads to: xt = P xt 1 + Qzt + Rut (7) where P = @h=@Xt 1, Q = @h=@Zt , R = @h=@Ut ; all evaluated at steady state values. Proposition 2.1. The matrices P, Q, R satisfy: P solves F P 2 + (G + N Given P , Q solves V Q = Given P , R solves W R = G + N x) x )P + (H + O vec(L vec(N z +M ) x) = 0. and V = u!u + O u) 0 z F +Inz where W = !0 u (F P +G+N F + In x ). (F P + where u = @ =@Ut+1 ; x = @ =@Xt ; z = @ =@Zt ; ! u = @ =@Ut , vec denotes the columnwise vectorization, and where we assume that all the eigenvalues of z and of ! u are strictly less than one in absolute value. 2 THE SETUP 6 Proof. The proof is straightforward. Substituting (7) into (6), we obtain 0 =[F P 2 + (G + N x )P + [(F P + G + N + (H + O x )R x )]xt 1 + F R! u + N + [(F P + G + N u!u +O x )Q + FQ z +L z + M ]zt u ]ut Since the solution must hold for every realization of xt their coe¢ cient to zero and the result obtains. 1, zt , ut , we need to equate Corollary 2.2. If x = 0, the dynamics in response to the structural shocks zt are identical to those obtained when parameters are time invariant. Variations in the j-th parameter have instantaneous impact on the endogenous variables xt , if and only if the j th column of N u ! u + O u 6= 0. Corollary 2.3. If u = 0 and the matrices N x and O tions have no e¤ ects on the endogenous variables xt . x are zero, parameter varia- Proposition 2.1 indicates that the …rst order approximate decision rule will, as in a constant coe¢ cient setup, be a VARMA(1,1) but with an additional set of disturbances. Corollaries 2.2 and 2.3 give conditions under which parameter variations alter the dynamics induced by structural disturbances. If parameter variations are purely exogenous, x = 0, the P and Q matrices are identical to those of a constant coe¢ cient model. Thus, parameter variation adds variability to the endogenous variables without altering the dynamics produced by structural disturbances. In other words, suppose an economy is perturbed by technology shocks. Then, the dynamics induced by these shocks do not depend on whether the discount factor is constant or time varying, provided technological innovations are exogenous and unrelated to the innovations in the discount factor. This result implies that if one is able to identify the structural disturbances z from t a time invariant version of the model, she would make no mistakes in characterizing structural dynamics. Clearly, variance or historical decomposition exercises will be distorted, since certain sources of variations (the u disturbances) are omitted. One t interesting question is whether standard procedures allow a researcher employing a time invariant model to recover z from the data when the DGP features time varying t structural parameters. If not, one would like to know which structural disturbance absorbs the missing shocks. Sections 5 and 6 study these issues in a practical example. On the other hand, if parameter variations are purely endogenous, u = 0, the dynamics in response to structural shocks may be altered. To know if distortions are present; one needs to check whether the columns of the matrices N x and O x are equal to zero. If they are not, a researcher employing a time invariant model is likely to incorrectly characterize both the structural dynamics and the relative importance of di¤erent sources of disturbances for the variability of the endogenous variables. The equilibrium dynamics, as encoded in the P matrix, can thus help us to distinguish between models with endogenous time variation featuring di¤erent laws of motion for the parameters (i.e. the x matrix). Distinguishing between models with 2 THE SETUP 7 exogenous time variation that di¤er in how the parameters respond to exogenous disturbances (i.e. the u matrix) is possible if the cross equation restrictions present in R are di¤erent across models. 2.2 Higher order approximate decision rule Are the conclusions maintained when higher order approximations are considered? In the second order approximation, the …rst order terms are the same as in the linear approximation. To examine whether quadratic terms will be a¤ected by the presence of time variations insert (5) in the optimality conditions so that (1) is 0 = Et [F (Xt ; Wt ; t+1 ; )] (8) The second order approximation of (8) is Et [(F x xt 1 + F w wt + F ) + 0:5(F xx (xt xt 1 F xw (xt 1) + F ww (wt wt ) + F x x t 1 wt ) + F 2 ) + + F w wt ] = (9) 0 1 Note that F ; F x xt 1 ; F w wt are all zero, see Schmitt Grohe and Uribe (2004). The second order expansion of (5) is xt = hx xt 1 + hw wt + 0:5(hxx (xt + hxw (xt 1 wt ) + hx xt 1 xt 1 1) + hww (wt wt ) + h + hw wt 2 ) (10) It is hard to make general statements about the properties of second order solutions of models with time varying coe¢ cients. As long as Fxx , Fww , Fwx are not a¤ected by parameter variations, as is the case when variations are exogenous, second order approximations in time varying coe¢ cient and in …xed coe¢ cient models will be the same. However, when these expressions are a¤ected, the approximations will be di¤erent. As an example of this latter case, consider the model Et yt+1 = 0:95 t xt xt = 0:8 xt t = (2 (11) 1 + 0:2 x + ut 0:5 (exp( (12) 1 (xt 1 x) + exp( 2 (xt 1 x)) + vt (13) where both vt and ut are i.i.d. and x Ext = 1. It is easy to verify that when 1 = 2 ; the …rst order solution (including only the terms concerning structural dynamics) is yt = 0:76xt 1 + 0:95ut (14) and it is the same as in the constant coe¢ cient model ( 1 = 2 = 0; vt = 0; 8t), since N x and O x are both zero. However, the second order solution (including only terms concerning structural dynamics) is yt = 0:76xt 1 + 0:95ut 0:01565x2 t 1 0:2375u2 + 0:038xt t 1 ut (15) 2 THE SETUP 8 while the second order solution of the constant coe¢ cient model is yt = 0:76xt 1 + 0:95ut 0:01520x2 t 1 0:2375u2 t 0:038xt 1 ut (16) The hxx matrix di¤ers in the two cases because, in general, vt may a¤ect yt+1 and thus alter higher order derivatives. For higher order approximate solutions, the dynamics induced by structural shocks in constant coe¢ cient and time varying coe¢ cient models will generally di¤er, even with exogenous time variations. For example, in a third order approximation, the optimality conditions will feature terms in F x and F w ; which require a correction of the linear terms to account for uncertainty. Since in the constant coe¢ cient model some shocks are omitted, one should expect the correction terms to di¤er in constant and time varying coe¢ cient models. 2.3 Discussion The results we derived require parameter variations to be continuous and smooth. This is in line with the evidence produced by Stock and Watson (1996) and with the standard practice employed in time varying coe¢ cient VAR. Our framework is ‡ exible and can accommodate once-and-for-all breaks (at a known date), as long as the transition between states is smooth. For example, a smooth threshold exogenously switching speci…cation can be approximated with t+1 = (1 ) + t + a exp(t T0 )=(b + exp(t T0 )), t = 1; : : : ; T0 1; T0 ; T0 + 1; : : : T , where a and b are vectors, while t+1 = (1 ) + t + a exp( (Xt X))=(b + exp( (Xt X)); where X is the steady state value of Xt ; can approximate smooth threshold endogenously switching speci…cations. What the framework does not allow for are Markov switching variations, occurring at unknown dates, as in Liu, et al. (2011), or abrupt changes, as in Davig and Leeper (2006), since the smoothness conditions on the f function may be violated. Note, however, that our model becomes a close approximation to a Markov switching setup when the number of states is large. It is important to emphasize that the (linear) solution we derive is a standard VAR with …xed coe¢ cients and additional shocks. Thus, DSGE models with time varying coe¢ cients do not generate new issues for aggregation or non-fundamentalness relative to a …xed coe¢ cient DSGE model. More importantly, it is incorrect to consider time varying coe¢ cient VAR as the reduced form counterpart of continuously varying coe¢ cient DSGE models. One can show that there exists a state space representation of the solution where the (exogenously) time varying coe¢ cients play the role of additional states of the model. What Proposition 2.1 shows is that the state space representation can be solved out to produce a standard VAR representation for the endogenous variables. Moreover, the proposition indicates that the matrices P and Q will be time varying only if x is itself time varying. Thus, to match the time varying coe¢ cient VAR evidence, it is necessary to consider variations in DSGE auxiliary parameters rather than variations in DSGE structural parameters. Kulish and Pagan (2014) have developed solution and estimation procedures for models with abrupt breaks and learning between the states. Their solution for 2 THE SETUP 9 the pre-break and post-break period is a constant coe¢ cient VAR, while for the learning period is a time varying coe¢ cient VAR. Thus, a few words distinguishing the two approaches are needed. First, they are interested in characterizing the solution during the learning period, when the structure is unchanged, while we are interested in the decision rule when parameters are continuously varying. Second, their modelling of time variations is abrupt and the solution is designed to deal with that situation. Third, in our setup expectations are varying with the structure; in Kulish and Pagan they vary only in anticipation of a (foreseeable) break. An alternative way of modelling time variations in (3) would be to make parameters functions of the exogenous rather than the endogenous variables, ( ; Zt ; Ut+1 ) as, for example, in Ireland (2007). While the equations the t+1 = coe¢ cients of the decision rule solve are di¤erent, the conclusions we have derived are unchanged by this modi…cation. For, example, in the …rst order approximation, P now solves F P 2 +GP +H = 0; given P, Q solves V Q = vec(L z +M +N z z +O z ) and V = Inz (F P + G + F z ); and given P, R solves W R = vec(N u ! u + O u ), where W = Inz (F P + G + F ! u ). While there are obvious economic di¤erences between exogenous vs. endogenous coe¢ cient variations, an alternative (statistical) way to think about the two speci…cations is that in the former each parameter evolves independently and covariations, if they exist, can be modelled by selecting the matrix u to be of reduced rank. With endogenous variations, instead a set of observable factors (the X’ drives coms) mon parameter variations. Thus, u is diagonal and full rank, unless some parameter variations are purely endogenous. As (7) makes clear, it is hard to distinguish models with time varying coe¢ cients from time invariant models with an additional set of shocks. In fact, models with n1 structural shocks and n2 time varying parameters, models with n = n1 + n2 structural shocks and models with n1 structural shocks and n2 measurement errors are observationally equivalent: xt = P xt 1 + Qzt + Rut (17) = P xt 1 + Q zt (18) = P xt 1 + Qzt + vt (19) 0 where Q = [Q; R]; zt = [zt ; u0 ]0 , vt = Rut . Thus, when designing time variation t diagnostics, one must rule out a-priori all these potentially observational equivalent structures. In applications, procedures like the one described in section 3 or the one of Seoane (2014) can be used to select the interpretation of the additional shocks. Finally, it is useful to compare the (linear) solution we derive with the solution obtained when coe¢ cients are constant but the volatility of the shocks is stochastic. Neglecting second order terms, the solution in this latter case is xt = P xt 1 + Qzt + A 2 : Thus, in empirical applications, it is crucial to allow for t stochastic volatility to avoid to misrepresent volatility changes for parameter variations - a point made earlier by Sims (2001). 2 THE SETUP 2.4 10 An example To convey some intuition into the mechanics of corollaries 2.2-2.3, we use a simple, closed economy, RBC model. The representative agent maximizes max E0 1 X t( t=1 1 Ct 1 A Nt1+ ) 1+ (20) subject to the sequence of constraints Yt (1 gt ) = Ct + Kt (1 t )Kt 1 1 t Kt 1 Nt Yt = where Yt is output, Ct consumption, Kt the stock of capital, Nt is hours worked, and G gt = Y t is the share of government expenditure in output. The system is perturbed t by two exogenous structural disturbances: one to the technology Zt; and one to the government spending share, gt , both assumed to follow time invariant AR(1) processes ln t = (1 ) ln + ln gt = (1 g ) ln g + ln g t 1 ln gt 1 + et + eg t (21) where variables without time subscript denote steady state quantities. There are 12 parameters in the model: 6 structural ones ( is the capital share, the risk aversion coe¢ cient, the inverse of the Frisch elasticity of labor supply, A the constant in front of labor in utility, t the time discount factor, and t the depreciation rate), and 6 auxiliary ones (the steady state values of the government expenditure share and of TFP, ( ; g); their autoregressive parameters, ( ; g ); and their standard deviations ( ; g )). We assume that all parameters but t and t are time invariant. Dueker et al. (2007), Liu et al (2011), and Meier and Sprenger (2015) provide evidence that these parameters are indeed evolving over time. The …rst order approximation to the law of motion of ( t , t ) is described below. The optimality conditions of the problem are: ACt Nt t Ct (1 gt )Yt = (1 gt )Yt =Nt (1 gt+1 )Yt+1 = Et +1 t+1 Ct+1 ( Kt+1 @ t+1 @ t+1 + Et u(Ct+1 ; Nt+1 ) Kt ) @Kt @Kt = Ct + Kt (1 t )Kt 1 Yt = )(1 1 t Kt 1 Nt (22) t+1 (23) (24) (25) Time variations in t and t a¤ect optimal choices in two ways. There is a direct e¤ect in the Euler equation and in the resource constraint when t and t are time varying; and if agents take into account that their decisions may a¤ect parameter 2 THE SETUP 11 variations, there will be a second (endogenous) e¤ect due variations in the derivatives of t+1 and t+1 with respect to the endogenous states - see equation (23). Note that varying parameters can not be considered wedges in the sense of Chari et al. (2007), because there are cross-equation restrictions that need to be satis…ed. Furthermore, while the rank of the covariance matrix of the wedges is full, this is not necessarily the case in our setup. We specialize this setup to consider various possibilities. 2.4.1 Model A: Constant coe¢ cients. As a benchmark, we let 0 t t = and t = . The optimality conditions are Et [f (Xt+1 ; Xt ; Xt 1 ; Zt+1 ; Zt ; B C Et B t @ )] = 1 +1 ACt Nt (1 )(1 gt )Yt Et Ct+1 ( (1 gt+1 )Yt+1 =Kt + 1 (1 gt )Yt Ct + Kt (1 )Kt 1 Yt Kt 1 Nt1 t ) C C=0 A (26) Xt = (Kt ; Yt ; Ct ; Nt )0 , Zt = ( t ; gt )0 : In the steady state, we have: K = Y (1 g) C ; =1 1 + 1= Y 2.4.2 " Model B: Exogenous parameter variations Set dt = postulate t+1 = t . We let K Y g N ; = Y Y ud;t+1 = u Since t+1 is exogenous, @ 0 B 1 Et B @ 1 (dt+1 t+1 ;t+1 t+1 =@Kt = =@ K Y 1 (1 d ud;t u ;t 1 ; Y = ) ; t+1 (1 A )(1 (1 C g) Y (27) ) )0 = Ut+1 and + ed;t+1 (28) +e (29) t+1 =@Kt Et [f (Xt+1 ; Xt ; Xt ;t+1 = 0 and the f function becomes 1 ; Zt+1 ; Zt ; t+1 ; +1 ACt Nt (1 )(1 gt )Yt dt Ct+1 =Ct ( (1 gt+1 )Yt+1 =Kt + 1 (1 gt )Yt Ct Kt + (1 t )Kt 1 Yt Kt 1 Nt1 t t+1 ) 1 t )] C C=0 A = (30) where Xt = (Kt ; Yt ; Ct ; Nt )0 , Zt = ( t ; gt )0 and t = 1t : C With the selected parameterization the steady state values of ( K ; Y ; N ; Y ) coinY Y cide with those of the constant coe¢ cient model. In addition, since x = 0, variations N Y 1+ # 1 + : 2 THE SETUP 12 in (dt+1 ; t+1 ) leave the decision rule matrices P and Q as in model A. Thus, as far as structural dynamics are concerned, models A and B are observationally equivalent. To examine whether variations in t have an instantaneous impact on Xt , we need to check the columns of N u ! u + O u . 0 1 0 0 B 1= = C C 6= 0 N u u+O u =B (31) @ 0 K A 0 0 Note that if dt were a fast moving variable, the impact e¤ect on Xt would depend on the persistence of shocks to the growth rate of the discount factor. For example, if d = 0, shocks to the growth rate of the time discount factor have no e¤ects on Xt . Thus, if only the discount factor is time varying and variations in its growth rate are i.i.d., models A and B have identical decision rules. 2.4.3 Model C: Endogenous parameter variations, no internalization Assume that the time variations in the growth rate of the discount factor and in the depreciation rate are driven by the aggregate capital stock. We specify t+1 =[ u ( u l )e a (Kt K) ]+[ u ( u l )e b (Kt K) ]+U ;t+1 (32) where a ; b ; u ; l are vectors of parameters and U ;t+1 is a zero mean, i.i.d. vector of shocks. This speci…cation is ‡ exible and depending on the choice of 0 s;we can accommodate linear or quadratic relationships, which are symmetric or asymmetric. To ensure that models C and A have the same steady states, we set l = ( =2, =2). We assume that agents treat the capital stock appearing in (32) as an aggregate variable. This assumption is similar to the ’ small k -big k’ situation encountered in standard rational expectations models or to the distinction between internal and external habit formation. Thus, agents’…rst order conditions do not take into account the fact that their optimal capital choice changes dt and t so @ t+1 =@Kt = @ t+1 =@Kt = 0 and the equilibrium conditions are then as in (30).Since the f function is the same as in model B, the matrices N and O are unchanged. To examine whether parameter variations a¤ect the matrices regulating structural dynamics note that 0 1 0 0 B 0 1= C (du =2)( 11 21 ) 0 0 0 C N x = B (33) @ 0 0 A ( u =2)( 12 22 ) 0 0 0 0 0 0 1 0 0 B 1= 0 C (du =2)( 11 21 ) 0 0 0 C O x = B (34) @ 0 k A ( u =2)( 12 22 ) 0 0 0 0 0 2 THE SETUP 13 Endogenous variations in dt ; t leave P and Q una¤ected, unless 1 6= 2 and/or 3 6= 4 , i.e. unless there are asymmetries in the law of motion of (dt ; t ). To verify whether parameter variations impact on Xt , check the columns of N u ! u + O u . We have: 0 1 0 0 B 1= (du C =2)( 1 + 2 ) 0 C 6= 0 (35) N u!u + O u = B @ 0 K( u =2)( 3 + 4 ) A 0 0 if 1 6= 2, 2.4.4 or 3 6= 4 and regardless of persistence of the shocks to the parameters. Model D: Endogenous parameter variations, internalization. We still assume that time variations in the discount factor and in the depreciation rate are driven by the aggregate capital stock and by an exogenous shock, as in equation (32). Contrary to case C, we assume that agents internalize the e¤ects their capital decisions have on parameter variations. The relevant derivatives are d0 t+1 @dt+1 =@Kt = 0 t+1 @ t+1 =@Kt = ( ( u u =2)[ =2)[ 1e 3e 1 (Kt K) 3 (Kt K) + + 2e 4e K) 2 (Kt 4 (Kt K) ] (36) ] (37) In order for the steady states of model D to equal to those of model A, we restrict 1 = 2 = 1 , 3 = 4 = 3 . The optimality conditions are: 0 = Et [f (Xt+1 ; Xt ; Xt 0 1 ; Zt+1 ; Zt ; t+1 ; t )] +1 B 1 Et B @ d0 u(Ct+1 ; Nt+1 )=Ct t ACt Nt (1 )(1 gt )Yt dt Ct+1 =Ct ( (1 gt+1 )Yt+1 =Kt+1 + 1 (1 gt )Yt Ct Kt + (1 t )Kt 1 Kt 1 Nt1 Yt t t+1 + 0 t+1 Kt ) where as before Xt = (Kt ; Yt ; Ct ; Nt )0 , Zt = ( t ; gt )0 but now t = (dt ; t ; d0 ; 0 )0 and t t 0 1 1 0 1 (Kt K) + e 1 (Kt K) ] + U 2du (du =2)[e dt+1 ;t+1 B t+1 C B =2)[e 3 (Kt K) + e 3 (Kt K) ] + U ;t+1 C B C C = ( ; Kt ; Ut+1 ) = B 2 u ( u @ d0 A A @ (du =2) [ e 1 (Kt K) + e 1 (Kt K) ] t+1 0 (Kt K) + e 3 (Kt K) ] ( u =2) [ e 3 t+1 (39) The relevant matrices of derivatives evaluated at the steady states are ! u = 02 2 , N= x @f @ 0 B =B @ t+1 0 0 0 B 0 1= =B @ 0 0 0 0 0 0 2( 2( u u =2) =2) 2 1 2 3 0 0 0 0 0 u(C; N )=C 0 0 1 0 0 0 0 C C; 0 0 A 0 0 u 1 0 0 B K C C ; O = @f = B 0 A @ t @ 0 0 0 B 0 =B @ 2( u =2) 2 1 0 2( 0 1= 0 0 0 0 K 0 0 0 0 0 1 0 0 0 u =2) 2 3 C C: A 1 0 0 C C 0 A 0 1 = C C (38) A 2 THE SETUP 14 Clearly, N x 6= 0, and N u ! u + O u = 0. Thus, a shock to the law of motion of the parameters alters the dynamics produced by structural shocks, even when the relationship between parameters and states is symmetric: In sum, parameter variations matter for the structural dynamics either if the relationship between parameters and the states is asymmetric or if agents internalize the consequences their decisions have on parameter variations, or both. 2.4.5 Impulse responses Why are structural dynamics in models C and D di¤erent from those in model A? To understand what drives economic di¤erences, we compute impulse responses. For the parameters common to all models, we choose = 0:30, = 0:99, = 0:025, = 2, = 2, A = 4:50, =1; = 0:90, = 0:00712, g = 0:18; g = 0:50 and g = 0:01. For the other parameters, we choose: Model B: = 0:985; Model C : 1 = 0:01; 0:999; u = 0:025. Model D : u = 0:999; 1 u = 0:0001; = 0:025. = 0:95 and = 0:03; 2 2 = 0:002 = 0:2; 1 = 0:016; 1 = 0:2; = 0:07. = 0:1, 2 2 = 0:1, d = = 0:5, d = 0.0001; u = = 0:1, Figure 1 reports the responses of hours, capital, consumption, and output to the two structural shocks in the four models. The …rst column has the responses to technology shocks; the second has the responses to government expenditure shocks 1 . Note …rst, that the sign of the responses is unchanged by the presence of parameter variations. The responses of models C and D di¤er from those of model A in the shape and the persistence of consumption and capital responses. Di¤erences occur because income and substitution e¤ects are di¤erent. For instance, in response to technology shocks, agents work and save less and consume more in models C and D than in the constant coe¢ cients model, while in response to government expenditure shocks, consumption falls more and capital falls less relative to the constant coe¢ cients case. Thus, parameter variations play the same role as uncertainty variations and make agents desire to smooth less transitory structural shocks. 1 Since the responses of hours and output to government expenditure shocks are di¤erent from what the conventional wisdom indicates, a few words of explanation are needed. In a standard RBC in response to government expenditure shocks, hours and output typically increase because of a wealth e¤ect. However, here the shock a¤ects the share of government expenditure in GDP. Thus, the positive wealth e¤ect on labor supply is absent because government expenditure increase in exactly the same proportion as output, thus disincentivizing agents to try to increase private output. 3 CHARACTERIZING TIME VARYING MISSPECIFICATION Te c h o lo g y s h o c k s -3 x 10 15 G ex pendit ure s hoc k s -3 x 10 Hours 0 5 -1 0 -2 -5 5 10 15 20 25 30 35 40 5 Capital Consumption 20 25 30 35 40 5 10 15 20 25 30 35 40 -3 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 -0.005 -0.01 -0.015 -0.02 -0.025 0.2 5 10 15 20 25 30 35 40 x 10 0.025 0.02 0.015 0.01 0.005 Output 15 -4 0.4 10 -5 -10 -15 5 10 15 20 25 30 35 40 x 10 0 0.1 0.08 0.06 0.04 0.02 -2 -4 5 10 15 20 25 30 35 40 B C D A Figure 1: Impulse responses, …rst order approximation 3 Characterizing time varying misspeci…cation Because the decision rules of constant coe¢ cient models are generally misspeci…ed when the data generating process (DGP) features parameter variations, it is important to diagnose potential time varying problems. This section describes two diagnostics useful for the purpose: one based on ” wedges”and one based on forecast errors. Consider the optimality conditions of a constant coe¢ cient model Et F (Xt 1 ; Wt ; z t+1 ; ) =0 (40) obtained substituting for Xt the decision rule: Xt = h(Xt 1 ; Wt ; z t+1 ; ) (41) When Xt 1 has been generated by the constant coe¢ cient model, F is a martingale di¤erence. When instead Xt 1 has been generated by a time varying coe¢ cient model Xt = h (Xt 1 ; Wt ; (42) t+1 ; ) E[F (Xt 1 ; Wt ; F (Xt 1 ; Wt ; z ; t+1 z ; ) t+1 z )] 6= 0; since t+1 and h6= h : Furthermore, t+1 6= will be predictable using past values Xt 1 : To see why, 3 CHARACTERIZING TIME VARYING MISSPECIFICATION 16 consider the …rst order approximate optimality conditions. In this system of equations, the wedge is (F (P (F (Q Q) z + G(Q P )2 + G(P Q) + F (P (F (P P )(G P ))xt 1 + G))zt + P )R + GR + F R ! u )ut (43) When P = P; Q = Q; as in the exogenously varying model, the wedge reduces to (GR + F R ! u )ut (44) which di¤ers from zero if R 6= 0 and will be predictable using xt j ; j 1; if ! u 6= 0: When, as in the endogenously varying model, P 6= P; Q 6= Q; the wedge will di¤er from zero, even when R = 0, and will be predictable using past xt 1 ; even when ! u = 0. Hence, to detect time varying misspeci…cation, one can compute wedges and regress them on the lags of the observables. If they are signi…cant, the martingale di¤erence condition is violated, and there is evidence of time varying parameters. Note that the diagnostic uses the assumption that the model is correctly speci…ed up to parameter variations. If the model is incorrect, lags of the observables may be signi…cant, even without time varying coe¢ cients. Inohue, Kuo, and Rossi (2015) apply this idea to detect generic model misspeci…cation. The logic of the forecast error diagnostic is similar. The linearized decision rule in a constant coe¢ cients model is xt = P xt 1 +Qzt , while in a time varying coe¢ cient model it is xt = P xt 1 + Q zt + R ut . Let vt be the forecast error in predicting xt using the decision rules of the constant coe¢ cient model and the data generated from the time varying coe¢ cient model: The forecast error can be decomposed as vt = xt j P xt 1 = Q zt + R ut + (P P )xt 1 (45) Thus, forecast errors are functions of the lags of the observables xt 1 when P 6= P: However, even if P = P , forecasts error linearly depend on the lags of the observables if ut is serially correlated. Hence, an alternative way to check for parameter variations involves regressing the forecast errors vt on lagged values of the observables and checking the signi…cance of the regression coe¢ cients. We apply the two diagnostics to 1,000 samples constructed using the RBC model previously considered. Table 1 reports the rejection rate of an F-statistic for the null hypothesis of no time variations at the 0.05 percent con…dence level. The Euler wedge diagnostic has very good size properties (does not reject the hypothesis of no time variations) when the model has …xed coe¢ cients; when it has …xed coe¢ cients but it is locally misspeci…ed - capacity utilization is neglected; and when the exogenous time variations are i.i.d.. It is somewhat conservative in detecting time variations when exogenous parameter variations are persistent and has excellent power properties when variations are endogenous. The forecast error diagnostic has good size properties when the DGP has no time variation and no misspeci…cation is present but tends to overreject the null 3 CHARACTERIZING TIME VARYING MISSPECIFICATION 17 DGP Euler wedge Forecast errors output F-test ct 1 ; rt 1 = 0 F-test ct 1 ; nt 1 ; yt 1 = 0 T=1000 T=150 T=1000 T=150 Fixed coe¢ cients 0.00 0.00 0.00 0.00 Fixed coe¤ and capacity utilization 0.001 0.003 1.00 0.98 Exogenous TVC no serial correlation 0.07 0.001 0.91 0.24 Exogenous TVC 0.53 0.40 1.00 0.90 Endogenous TVC 1.00 0.93 1.00 0.99 Table 1: Percentage of rejections at the 0.05 con…dence level of the null of no time variations in 1000 experiments. The dependent variable is either the Euler wedge or the forecast error in the output equation. The regressors are lagged consumption and interest rates for the Euler wedge; lagged output, consumption and hours for the forecast error. if misspeci…cation is present or exogenous time variations are i.i.d.. On the other hand, it has good power properties when time variations are present. Because of the di¤erences they display, it seems wise to use both diagnostics in empirical applications. 3.1 Exogenous vs. endogenous parameter variations If the diagnostics of the previous subsection indicate the presence of parameter variations, one may interested in knowing whether they are of exogenous or endogenous type. One way to distinguish the two options is to use the DGSE-VAR methodology of Del Negro and Schorfheide (2004). In a DSGE-VAR, one uses the DSGE model as a prior for the VAR of the observable data and employs the marginal likelihood to measure the value of the additional information the DSGE provides. If the additional observations come from the DGP, the quality of the estimates improves (standard errors are reduced), and the marginal likelihood increases. On the other hand, if the additional observations come from a DGP di¤erent from the one generating the data, biases may be introduced, noise added, and the precision of the estimates and the …t of the model reduced. Formally, let L( jy) be the likelihood of the VAR model for data y and let gj ( j j ; Mj ) be the prior induced by the DSGE model Mj using parameters j on the VAR paR rameters :The marginal likelihood is hj (yj j ; Mj ) = L( jy)gj ( j j ; Mj )d ; which, for given y; is a function of Mj . Since L( jy) is …xed, hj (yj j ; Mj ) re‡ ects the plausibility of gj ( j j ; Mj ) in the data. Thus, if g1 and g2 are two DSGE-based priors and h1 (yj 1 ; M1 ) > h2 (yj 2 ; M2 ), there is better support for in the data for g1 . Thus, for a given data set, a researcher comparing the marginal likelihood produced by adding data from the exogenous and the endogenous speci…cations should detect whether the observable sample is more likely to be generated by one of the two models. We prefer to use the DSGE-VAR device rather 3 CHARACTERIZING TIME VARYING MISSPECIFICATION 18 than comparing the marginal likelihood of di¤erent models directly because small samples may led to distortions in marginal likelihood comparisons, distortions that will be reduced in our DSGE-VAR setup. T1 =150 T1 =750 DGP Model B Model C Model D Model B Model C Model D Simulated from B 1.00 0.00 0.00 0.99 0.00 0.00 Simulated from C 0.01 0.99 0.00 0.00 0.98 0.00 Simulated from D 0.00 0.00 1.00 0.00 0.00 0.99 Table 2: Probability that Bayes factor exceeds 3.0 in a sample of 1,000 experiments. Marginal likelihoods are obtained using T=150 data points produced by the models listed in the …rst row and T1 simulated data from the model listed in the …rst column. When rows do not sum to one, the Bayes factor is inconclusive (below 3.0). Table 2 reports results using this technology in the RBC example. The sample size is T = 150 and Bayes factors computed when T1 = 150; 750 simulated data from the DSGE listed in the …rst row are added to the actual data and 1,000 experiments are run. The statistic is powerful since marginal likelihood di¤erences are quite large, even when T1 = 150. 3.2 Some practical suggestions Given that, in practice, we do not know if a model is misspeci…ed or not, we suggest users the following checklist as a way to approach the diagnostic problem: i) Take a conventional model that has been used and tested in the literature and estimate its structural parameters, potentially allowing for time variations in the variance of the shocks. ii) Run the time variation diagnostics and, if time variations are found to be present, check whether endogenous vs. exogenous variations are more appropriate. When the model is of large scale, running regressions on all potential endogenous variables leads to overparameterization and muticollinearity. Thus, it is important to select the relevant variables to make the test powerful. We recommend users to employ the states of the model, as they determine the endogenous variables. Similarly, when performing the exogenous vs. endogenous check, having the proper state variables for the endogenous speci…cation is important to make the comparison fair. One way do this is to estimate a model with exogenous time variation, take the smoothed residuals and run auxiliary regressions of the smoothed residuals on potential determinants of time variations. To avoid overparameterization, we also suggest users to a-priori shrink the coe¢ cients of the auxiliary diagnostic regressions toward zero. Rejection of the null of no time variations in this case provides stronger con…dence that parameter variations are indeed present. 4 PARAMETER IDENTIFICATION When the diagnostics detect time variations, one needs to specify which parameter may be time varying for the next stage of the analysis. In theory, one could specify time variations in all the structural parameters of interest, but this may lead again to an overparametrized model, which is di¢ cult to estimate. We suggest two approaches here: either introduce time variations in parameters which have been documented in the literature to be unstable or in parameters a researcher suspects variations to be present. Alternatively, one could look at the smoothed residuals of the time invariant model, equation by equation, and restrict time variations to the parameters appearing of the equations whose residuals show the largest evidence of serial correlation. 4 Parameter identi…cation Since forecast errors are used to construct the likelihood function via the Kalman …lter, one should expect the misspeci…cation present in the forecast errors to spread to the likelihood function. In this section we examine whether time invariant parameters can be identi…ed from a potentially misspeci…ed likelihood function. Canova and Sala (2009) have shown that standard DSGE models feature several population identi…cation problems, intrinsic to the models and to the solution method employed. The issue we are concerned with here is whether parameters that could be identi…ed if the correct likelihood is employed became poorly identi…ed when the wrong likelihood is used. In other words, we ask whether identi…cation problems in time invariant parameters may emerge as a byproduct of neglecting variations in other parameters. Magnusson and Mavroedis (2014) have shown that when GMM is used, time variations in certain parameters help the identi…cation of time invariant parameters. Huang (2014) quali…es the result by showing that time variations in weakly identi…ed parameters have no e¤ect on the asymptotic distribution of strongly identi…ed parameters. Figures 2 and 3 plot the likelihood function of the RBC model in the risk aversion coe¢ cient and the share parameter ; and in the labor share and the autoregressive parameter of the technology ; when the forecast errors of the correct model (top row) and of the constant coe¢ cient model (bottom row) are used to construct the likelihood function. The …rst column considers data generated by the model B, the second and the third data generated by models C and D. While the likelihood curvature in the correct model is not large, it is easy to verify that the maximum occurs at = 2; = 2; = 0:30; = 0:9 for all three speci…cations. When the decision rules of the constant coe¢ cients model are used to construct the likelihood function and the true DGP is model B, the likelihood is ‡ attened and the risk aversion coe¢ cient become very weakly identi…ed. When the true model features endogenous time variations, distortions are larger. The likelihood function becomes locally convex in ; and become weakly identi…ed, and the maximum in the is shifted away from the true value. 19 4 PARAMETER IDENTIFICATION 20 True R BC B - Estim ated with RBC B True R BC C - Estim ated with R BC C True R BC D - Estim ated with R BC D log likeli 2900 3250 3220 2880 3210 3200 2860 3200 2840 2.5 2.5 2 γ 3150 2.5 2 2 1.5 1.5 2.5 γ η 3190 2.5 1.5 1.5 2.5 2 2 γ η 2 1.5 1.5 η True R BC B - Estim ated with RBC A True R BC C - Estim ated with R BC A True R BC D - Estim ated with R BC A log likeli 1000 2000 2800 1800 500 2600 1600 0 2.5 2.5 2 γ 1400 2.5 2 2 1.5 1.5 2.5 γ η 2400 2.5 1.5 1.5 2.5 2 2 γ η 2 1.5 1.5 η Figure 2: Likelihood surfaces True R BC B - Estim ated with R BC B True R BC C - Estim ated with R BC C True R BC D - Estim ated with R BC D log likeli 2900 3400 3500 3200 2850 3000 3000 2800 0.35 1 0.3 α 2800 0.35 0.3 0.9 0.25 0.8 ρ 1 α ζ 2500 0.35 0.25 0.8 ρ 1 0.3 0.9 α ζ 0.9 0.25 0.8 ρ ζ True R BC B - Estim ated with R BC A True R BC C - Estim ated with R BC A True R BC D - Estim ated with R BC A log likeli 2000 2500 3000 2000 0 2500 1500 -2000 0.35 1 0.3 α 0.9 0.25 0.8 ρ ζ 1000 0.35 1 0.3 α 0.9 0.25 0.8 Figure 3: Likelihood surfaces ρ ζ 2000 0.35 1 0.3 α 0.9 0.25 0.8 ρ ζ 4 PARAMETER IDENTIFICATION 21 These observations are con…rmed by the Koop et al. (2013) statistic, see table 4. Koop et al. show that asymptotically the precision matrix grows at the rate T for identi…ed parameters and at rate less than T for underidenti…ed parameters. Thus, the precision of the estimates, scaled by the sample size, converges to a constant for identi…ed parameters and to zero for underidenti…ed parameters. Furthermore, the magnitude of the constant measures identi…cation strength: a large value indicates a strongly identi…ed parameter; a small value a weakly identi…ed one. Parameter T=150 T=300 T=500 T=750 T=1000 T=1500 T=2500 DGP Model B, Estimated model A 15.9 17.8 17.2 18.8 18.4 19.3 17.9 28.5 45.7 108.4 81.4 93.6 104.2 90.17 1.8e+4 2.6e+4 4.2e+4 4.2e+4 4.5e+4 4.9e+4 4.37e+4 z 209.2 655.5 2741 2190 2860 3417 2802 g 927.3 973.8 1.7e+4 1.7e+4 2.4e+4 2.3e+4 2.5e+4 140.2 156.2 264.2 215.5 239.1 252.1 229.3 A 28.42 30.67 7.99 10.99 9.15 7.83 9.83 DGP Model C, Estimated model A 822 1033 743 785 759 746 752 2261 3147 2682 2809 2720 2579 2566 3073 2673 2952 2909 2799 2806 2877 z 1.74 2.23 2.44 2.96 3.17 2.82 2.90 g 4.6e+5 4.4e+5 4.3e+5 4.0e+5 3.8e+5 4.4e+5 4.3e+5 1.8e+4 1.1e+4 1.4e+4 1.2e+4 1.1e+4 1.6e+4 1.5e+4 A 351 493 441 505 500 449 444 DGP Model D, Estimated model A 550 575 592 610 545 542 494 3577 2442 2660 2870 2564 2711 2430 1613 1243 1120 1162 1068 1189 1074 z 1.22 1.28 1.44 1.53 1.60 1.62 1.67 g 5.2e+5 6.7e+5 6.5e+5 6.0e+5 5.7e+5 5.8e+5 5.7e+5 1.1e+4 2.5e+4 2.4e+4 1.9e+4 2.1e+4 2.0e+4 2.1e+4 A 488 276 340 382 349 395 334 Table 3: Koop, Pesaran, and Smith diagnostic. Reported are the diagonal elements of the precision matrix scaled by the sample size When the DGP is model B and a …xed coe¢ cients model is considered, all parameters are identi…ed, even though A and are only weakly identi…ed. When the DGP are models C and D, all parameters but g seem identi…able. Interestingly, in models C and D, g is weakly identi…ed, even when the correct likelihood is used. Thus, time variations in t and t do not help in the identi…cation of g , in line with Huang (2014). 5 STRUCTURAL ESTIMATION WITH A MISSPECIFIED MODEL 5 Structural estimation with a misspeci…ed model To study the properties of likelihood -based estimates of a misspeci…ed constant coe¢ cients model, we conduct a Monte Carlo exercise. We generate 150 or 1,000 data points from versions B, C, D of the RBC model previously considered, estimate the structural parameters using the likelihood function constructed with the decision rules of the time invariant model A, and repeat the exercise 150 times using di¤erent shock realizations. We also estimate the structural parameters using the likelihood constructed with the correct decision rules (i.e. model B rules if the data is generated with model B, etc.) for benchmarking estimation distortions. We consider two setups: one where parameter variations are small (2-5 percent of the variance of output is explained by shocks to the parameters; henceforth, DGP1) and one where parameter variations are substantial (around 20 percent of the variance of output is explained by shocks to the parameters; henceforth, DGP2). Table 4 has the results for DGP1: it reports the …xed parameters used to generate the data (column 1), the mean posterior estimate (across replications) obtained when the likelihood uses the correct decision rules (column 2), and the mean posterior estimate, the 5th and the 95th percentile of the distribution of estimates obtained when the likelihood function uses the decision rules of the time invariant model, when T=150 (columns 3-5) and when T=1000 (columns 6-8). Table A1 in the appendix has the results for DGP2. Figures A1 and A2 in the appendix plot the distributions of estimates for the two DGPs. The vertical line represents the true parameter value; solid black lines represent distributions obtained with the correct model; solid blue (red) lines represent the distributions obtained with the incorrect constant coe¢ cient model when T=150 (T=1000). When the model is correctly speci…ed, the distribution of estimates should collapse around the true value. Thus, if the mean is away from the true parameter value and/or the spread of the distribution is large, likelihood -based methods have di¢ culties in recovering the constant parameters of the data generating process. Figure 4 presents the impulse responses for DGP1: the …rst two columns have the responses to technology shocks and government expenditure shocks in model B, the next two the responses in model C, and the last two the responses in model D. In each box we report the response obtained using mean value of the correct distribution of estimates, and the 16th and 84th percentiles of the distribution of responses obtained using the estimated distribution of parameters produced by the time invariant model. Figure A3 in the appendix has the same information for DGP2. Table 5 presents the long run variance decomposition for DGP1 (table A2 has the information for DGP2) when T=150 and the mean posterior estimate is used in the computations. In the …rst two columns we have the contribution of technology and government spending shocks in the correct model; the last two columns have the contribution when the constant coe¢ cient model is used. For the two time varying parameters, we set dt = t+1 = t ; and assume that in model B, t+1 (dt+1 (1 ) ; t+1 (1 ) )0 = Ut+1 , where = 0:99; = 0 are independent AR(1) process with 0:025 the components of Ut+1 = (ud;t+1 ; u ;t+1 ) 22 5 STRUCTURAL ESTIMATION WITH A MISSPECIFIED MODEL persistence d = 0:9; = 0:8; and standard deviations d = 0:002; = 0:07: For models C and D, the law of motion of the time varying parameters is t+1 = [ u (Kt K) ] + [ (Kt K) ] + U 0 = (0:9999; 0:03); ( u ( u u t+1 ; where l )e a l )e b u 0 0 a = (0:03; 0:2); b = (0:031; 0:1); Ut+1 is i.i.d. with u =diag(0.03,0.008). True value Estimated Correct Estimated Time invariant Estimated Time invariant Mean Mean 5 percentile 95 percentile Mean 5 percentile 95 percentile T=150 T=150 T=1000 DGP Model B = 2:0 2.00 2.03 1.47 2.88 2.32 1.55 3.37 = 2:0 2.02 1.23 -0.14 2.07 0.96 -0.38 2.04 0.97 0.99 0.97 1.00 0.99 0.96 1.00 z = 0:98 = 0:5 0.47 0.74 0.60 0.96 0.87 0.77 0.98 g = 0:025 0.03 0.01 0.01 0.02 0.01 0.01 0.05 = 0:3 0.30 0.19 0.11 0.28 0.23 0.15 0.40 A = 4:5 4.55 2.79 1.33 4.12 2.68 1.23 4.06 DGP Model C = 2:0 2.00 2.42 1.63 3.85 2.85 1.73 6.14 = 2:0 2.00 0.64 -0.26 1.77 0.60 -0.50 1.79 0.98 0.99 0.97 1.00 0.97 0.85 1.00 z = 0:98 = 0:5 0.48 0.43 -0.10 0.96 0.65 0.27 0.98 g = 0:025 0.03 0.01 0.01 0.02 0.02 0.01 0.09 = 0:3 0.30 0.22 0.13 0.34 0.29 0.18 0.47 A = 4:5 4.49 2.14 1.18 3.47 2.37 1.18 3.66 DGP Model D = 2:0 2.00 2.58 1.69 3.34 2.40 1.74 3.26 = 2:0 2.01 0.29 -0.28 1.54 1.09 -0.30 1.99 = 0:97 0.96 0.99 0.94 1.00 0.96 0.91 1.00 z 0.48 0.51 -0.26 0.96 0.66 0.39 0.98 g = 0:5 = 0:025 0.02 0.01 0.01 0.03 0.01 0.01 0.02 = 0:3 0.30 0.22 0.14 0.35 0.22 0.15 0.30 A = 4:5 4.52 2.32 1.42 3.68 3.45 1.37 4.51 Table 4: Distributions of estimates, DGP1. A few features of the results are worth discussing. First, when the correct model is employed, estimation is successful even when T=150, regardless of the DGP and of whether time variations are exogenous or endogenous. Thus, numerical distortions seem minor. Second, with DGP1, a number of distortions occur when a time invariant model is used in estimation. For example, when exogenous variations are present, the persistence of government spending shock is poorly estimated (mean persistence is about 50 percent larger than the true one), while estimates of ; and A are severely 23 5 STRUCTURAL ESTIMATION WITH A MISSPECIFIED MODEL 24 biased downward. The distortions are smaller when the time variations are endogenous (models C and D). Nevertheless, signi…cant downward biases exist in the inverse of the Frisch elasticity , in and . Third, the performance of the time invariant model is roughly independent of whether the data features external or internal endogenous time variations and does not improve when the sample size increases. When parameter variations explain a signi…cant portion of output variability, all features become more striking. For example, when parameter variations are exogenous, estimating a time invariant model leads to an overestimation of the persistence of the structural shocks. In fact, the only way a time invariant model can accommodate the additional dynamics and variability present in the endogenous variables is by increasing the persistence of both shocks. In models C and D the distortions become considerably larger and, for example, the mean posterior estimate of inverse of the Frisch elasticity is now negative. In addition, the distribution of estimates is typically skewed and multimodal. Thus, neglected parameter variations are more detrimental when they account for a signi…cant portion of the variability of the endogenous variables. -3 x 10 Te c h n o lo g y -3 x 10 Hours 0 G .e x p e n d itu r e -3 x 10 Te c h n o lo g y -3 x 10 0 0 -0.5 G .e x p e n d itu r e -3 x 10 -3 x 10 0 -2 -2 -4 -2 -2 -4 -2 -6 -3 -2 10 20 30 -3 -6 -3 -2.5 G .e x p e n d itu r e 0 -1 -1 -1 -1 -1.5 Te c h n o lo g y 1 0 -8 -4 40 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 30 40 -3 x 10 0.06 0.06 Capital 0.05 -5 0.06 0.05 0.04 0.03 -10 0.04 -0.02 0.03 0.02 0.02 -15 0.01 10 20 30 10 -3 20 30 40 10 -3 x 10 20 30 40 10 -3 x 10 20 30 -0.03 40 10 -3 x 10 -0.02 0.02 -0.03 0.01 40 -0.01 -0.01 0.04 20 30 40 -3 x 10 -3 x 10 x 10 10 Output 0 8 0 8 0 -4 -2 6 10 -2 8 -1 6 -5 6 -6 -3 4 -8 -10 4 4 -10 -4 10 20 30 40 10 -3 20 30 40 10 -4 x 10 20 30 10 -3 x 10 2 40 20 30 40 10 -4 x 10 20 30 40 -3 x 10 -4 x 10 x 10 Consumption 5 -1 4.5 4.5 4 -2 2.5 -6 -2 3 -4 2 4 -2 -4 3.5 3.5 3 -3 3 -6 2.5 2.5 -8 10 20 30 Mo d e l B- A 40 10 A84 20 A16 30 40 B50 10 20 30 Mo d e l C - A 40 10 A84 20 30 A16 40 C50 10 20 30 Mo d e l D - A 40 10 20 A84 Figure 4: Impulse responses, DGP1 Impulse responses are in line with these conclusions. When parameter variations explain a small fraction of the variability of output, responses to technology shocks are o¤ in terms of impact magnitude, in particular for output; and the response produced with estimates of the true model tend to be on the upper limit of the estimated 68 percent band produced with estimates of the incorrect model. Interestingly, output responses are those more poorly characterized and, consistent with previous …ndings, the misspeci…cation is larger when the true model features exogenous time variations. A16 D50 5 STRUCTURAL ESTIMATION WITH A MISSPECIFIED MODEL The responses to government expenditure shocks obtained with a time invariant model are di¤erent from those obtained estimating the correct model in terms of magnitude, shape and persistence. Since the signal that government expenditure shocks produce is weak, it is not surprising that it is obscured by the presence of parameter variations. The dynamic distortions obtained when parameter variations matter for the variance of output are generally larger. For example, the persistence of the responses to technology shocks is poorly estimated. While true responses tend to zero, the bands obtained estimating a time invariant model do not include zero even after 10 years. Variable Technology Government Technology Government DGP: Model B Estimated: Time invariant Y 94.100 0.300 0.997 0.004 C 89.500 0.200 0.999 0.001 N 60.200 0.500 0.986 0.014 K 70.200 0.400 0.995 0.006 DGP: Model C Estimated: Time invariant Y 97.200 0.300 0.988 0.016 C 88.100 0.300 0.999 0.001 N 44.600 0.600 0.990 0.012 K 84.400 0.200 0.990 0.014 DGP: Model D Estimated: Time invariant Y 98.000 0.100 0.993 0.015 C 92.200 0.200 0.998 0.003 N 35.900 0.500 0.973 0.034 K 96.600 0.300 0.992 0.012 Table 5: Long run variance decomposition, DGP1. What is the contribution of structural shocks to the variability of the endogenous variables when the forecast errors of the time invariant model are used to construct the likelihood function? One should expect the structural shocks of the time invariant model to be a contaminated version of the structural shocks of the time varying DGP for two reasons. First, the wrong P matrix is used to compute forecast errors. Second, we are aggregating m (primitive and parameter) shocks into n < m (structural) shocks, thus generating VARMA decision rules where the n structural shocks are functions of the leads and lags of the original disturbances (see e.g. Canova and Paustian, 2011). Thus, even if the P matrix were correctly speci…ed, distortions should occur, unless the shocks to the parameters are unimportant and feature low persistence. When parameter variations explain a small portion of output, technology shocks in the time invariant model absorb the missing variability, regardless of the nature of parameter variations and the e¤ect seems strong for hours worked. When parameter variations explain a larger portion of the variance of output, technology shocks still absorb a large amount of the missing variability but, in some cases, spending shocks 25 6 STRUCTURAL DYNAMICS AND SVAR METHODS 26 also capture the missing variability see, e.g., the case of endogenous variations 2 In sum, for the DGP we consider and the parameterization employed, estimating a constant parameter model when the DGP features time varying parameters leads to distortions, regardless of the sample size, of whether variations are exogenous or endogenous, and of whether parameter variations matter for output variability or not. The parameters mostly a¤ected are those regulating the estimated persistence of the shocks and those controlling income and substitution e¤ects. 6 Structural dynamics and SVAR methods The previous section showed that if time variations are neglected, structural estimates are biased and structural responses distorted. Because of these problems, one may wonder whether less structural and computationally less demanding methods can be used if structural dynamics are all that matters to the investigator. Canova and Paustian (2011) showed that when the model misses features of the DGP, SVAR methods employing robust sign restrictions can be e¤ective in capturing qualitative features of structural dynamics. Here we ask if SVARs are good also when parameter variations are neglected. The exercise is as follows. Using the illustrative RBC model, we simulate data from the decision rules of models B, C, and D when parameter variations generate small output volatility (DGP1). We then estimate a VAR, compute residuals, and rotate them using an orthonormal matrix. We then keep the resulting impulse responses if (simultaneously) technology shocks generate a positive response of hours, capital, output, and consumption on impact and government expenditure shocks generate a negative response of hours, output, consumption and capital. These restrictions hold in the four model speci…cations we consider and are robust to variations of the (constant) structural parameters within a reasonable range. We repeat the exercise 150 times and collect the distribution of structural responses when the correct and the time invariant SVAR speci…cations are used. Figure 5 plots the median response in the correct model (red line) and the 16th and 84th percentiles of the distribution of responses obtained with the time invariant model. Overall, SVAR methods are competitive with structural methods when parameter variations are neglected. When the DGP is model B, the sign and the shape of the responses are correctly captured. Although the responses to technology shocks obtained with the correct model are on the upper bound of the band obtained with the time invariant model and the responses to government spending shocks 2 We have also performed a Monte Carlo exercise allowing the labor share to be time varying. Variations in the labor share have been documented in, e.g., Rios Rull and Santaeularia Llopis (2010), and there is evidence that they are strongly countercyclical. This is relevant for our exercise because all four optimatility conditions are a¤ected by time variations, altering the strength of the income and substitution e¤ect distortions. Indeed, we do …nd that distortions become quite large and it many cases it becomes di¢ cult to estimate the time invariant model regardless of the DGP (results available on request). 7 TIME VARYING FINANCIAL FRICTIONS? 27 obtained with the correct model tend to be on the lower bound of the bands obtained, no major distortions occur. The performance with the other two DGPs is similar. With model C, it is the magnitude of the response of consumption that is mainly misrepresented, while with model D it is primarily the persistence of certain responses that is underestimated. T4e c h n o lo g y s h o c k s G - 4e x p e n d itu r e s h o c k s x 10 x 10 T4e c h n o lo g y s h o c k s G - 4e x p e n d itu r e s h o c k s x 10 x 10 T4e c h n o lo g y s h o c k s Hours 0 0 -5 10 15 -5 -1 0 -5 -1 0 -5 -1 5 -1 0 5 0 0 -5 -5 -1 5 x 10 5 5 0 -1 0 G - 4e x p e n d itu r e s h o c k s x 10 5 0 5 10 15 -1 0 -1 0 -1 5 20 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 0 .0 5 -0 .0 1 -0 .0 1 0 .0 4 -0 .0 2 -0 .0 3 0 .0 3 -0 .0 3 -0 .0 4 0 .0 2 -0 .0 5 0 .0 1 -0 .0 1 0 .0 3 0 .0 5 -0 .0 2 0 .0 4 -0 .0 2 Capital 0 .0 4 0 .0 3 0 .0 2 -0 .0 3 0 .0 1 -0 .0 4 0 .0 2 5 10 15 20 5 -3 10 15 20 5 -3 x 10 10 15 -0 .0 6 20 5 -3 x 10 10 15 20 5 -3 x 10 10 15 20 -3 x 10 -3 x 10 x 10 -2 7 -2 -2 6 6 6 -4 Output -0 .0 4 -0 .0 5 0 .0 1 -4 5 -4 4 4 4 -6 -6 3 -6 2 -8 2 2 -8 -8 5 10 15 20 5 -3 15 20 5 15 20 5 10 15 20 5 -3 10 15 20 -3 x 10 x 10 -0 .5 3 -3 x 10 x 10 -0 .5 8 4 -2 -1 2 .5 10 -3 x 10 3 .5 Consumption 10 -3 x 10 -1 .5 6 -1 -1 .5 3 -4 2 -2 1 .5 -2 .5 -2 4 2 -2 .5 -6 2 1 -3 5 10 15 M o d e ls B - A 20 5 upper 84 10 15 l o we r 1 6 20 m e a n t ru e -3 1 5 10 15 M o d e ls C - A 20 5 upper 84 10 l o we r 1 6 15 20 m e a n t ru e 5 10 15 M o d e ls D - A 20 upper 84 l o we r 1 6 m e a n t ru e Figure 5: Impulse responses, DGP1, SVAR models. Recall that there are two sources of misspeci…cation in time invariant models: the P matrix is generally incorrect; aggregation problems are present. Our analysis indicates that, with the DGP we use, i) distortion in the P matrix are small; ii) the Q matrix is not very strongly a¤ected by time variations; iii) shock misaggregation is minor. Because parameter shocks are i.i.d., timing distortions are also small. 7 Time varying …nancial frictions? We apply the technology we developed to the unconventional monetary policy model of Gertler and Karadi’ (GK) (2010). Our contribution is three fold. We provide s likelihood estimates of the parameters speci…c to the model (the fraction of capital that can be diverted by banks , the proportional transfer to entering bankers !, and the survival probability of bankers ), which the authors have informally calibrated to match a steady state spread, a steady state leverage, and a notional length of bank activity; we use the diagnostics we developed to gauge the extent of parameter variations; we estimate a model with time variations in and compare its …t with the 7 TIME VARYING FINANCIAL FRICTIONS? 28 …t of the time invariant model augmented with an extra shock; and examine responses to capital quality shocks in the …xed coe¢ cient and the time varying coe¢ cient models. The equations of the GK model are summarized in appendix C. We use U.S. data from 1985Q2 to 2014Q3 on the growth rate of output, growth rate of consumption, growth rate of leverage, and growth intermediary demand for assets (credit) and the spread. The spread is measured by the di¤erence between BAA 10-year corporate bond yields and a 10-year treasury constant maturity, and it is from the FRED, as are real personal consumption expenditures and GDP data. Leverage is from Haver and measures Tier 1 (core) capital as a percent of average total assets. Credit is measured as total loans (from Haver), scaled by size of US population. While the data transformation is su¢ cient to eliminate volatility variations, the credit and the spread variable display a signi…cant structural break in the last few years of the sample. Thus, we present estimates obtained in the full sample and in the sample ending at 2007:4. Equation T-stat Creditt 1 Leveraget 1 Spreadt 1 1 Sample 1985:3-2014:3 Y 0.84 2.61 0.24 0.52 10.00 C -0.85 1.11 0.85 -0.65 0.33 Credit 1.06 2.61 1.65 -0.58 8.49 Leverage -1.11 -2.50 -1.63 0.63 -8.25 Spread -1.26 -3.06 -1.10 0.81 -8.46 Sample 1985:3-2007:4 Y -1.79 3.87 -2.23 -0.38 6.86 C -1.37 1.19 -0.26 0.38 1.40 Credit -1.18 3.53 -0.69 -0.08 7.02 Leverage -1.06 -3.46 0.75 0.09 -6.80 Spread 1.16 -3.84 -1.03 0.17 -6.86 Yt Ct F-stat 1 4.39 1.26 7.11 7.04 8.16 4.23 0.81 3.60 3.72 4.29 Table 6: Regression diganostic for time variation. The left-hand side of the regression is the forecast in the equation listed in the …rst column; the right-hand side the variables listed in the second to the …fth column. Critical levels of the F-stat(5,112)=2.56 and F(5,85)=2.90. Using a ‡ prior, the posterior mode estimates for the full sample are = 0:245, at = 0:464, ! = 0:012; the standard errors are tight (0.0182, 0.0098, 0.0008), making the estimates highly signi…cant. For the shorter 1985-2007 sample the modal estimates are = 0:178, = 0:399, ! = 0:008; and the standard errors are 0.0127, 0.0129, 0.0006. Thus, while estimates of the three crucial parameters are altered when data for the last …nancial crisis is used, di¤erences are small a-posteriori. For comparison, GK calibrated these three parameters to = 0:318, = 0:972, ! = 0:002. In the GK model regulates private leverage: our estimate implies a higher steady state leverage than the one implied by the authors (our estimate for the full sample is 3.32, GK is 7 TIME VARYING FINANCIAL FRICTIONS? 29 1.38), which is closer to the leverage found in the U.S. in corporate and non-corporate business sectors over the sample. Our estimates also suggest that bankers’ survival probability is lower than the one assumed by GK (about 10 years). With the parameter estimates obtained in the full sample, we perform our diagnostics. Table 6 indicates that the forecast errors of all equations except consumption are predictable and, typically, lagged consumption and lagged spread matter. The mean value of the Euler wedge is 0.02 with a standard error of 0.03; but both lagged consumption and lagged investment to output ratios signi…cantly explain its movements (coe¢ cients are respectively -0.10 and 0.72, with standard errors of 0.01 and 0.13). When we run our diagnostics on the shorter sample, we reach the same conclusions: all forecast errors but the one of the consumption equation are predictable, and lagged consumption and lagged spread matter; lagged consumption and lagged investment to output ratios predict the wedge (coe¢ cients are respectively -0.13 and 0.96, with standard errors of 0.02 and 0.23). Thus, the time variations we detect are not due to the crisis and to the potential break it generates. Parameter Time Invariant Time Invariant Exogenous TVC Endogenous TVC 6 shocks Function of net worth h 0.43 (0.006) 0.11 (0.02) 0.19 (0.03) 0.09 (0.02) 0.24 (0.01) 0.97 (0.01) 0.37 (0.03) 0.55 (0.03) ! 0.01 (0.008) 0.02 (0.001) 0.02 (0.002) 0.11 (0.008) 0.46 (0.009) 0.80 (0.01) 0.54 (0.01) 0.52 (0.02) 0.99 (0.004) 0.02 (0.002) 0.03 (0.003) 0.98 (0.008) u 0.02 (0.007) 1 0.15 (0.009) 2 Log ML -167.97 1098.32 1546.18 1628.69 Table 7: Parameter estimates, Gertler and Karadi model, standard errors in parenthesis. Armed with this preliminary evidence, we estimate the model allowing to be time varying. Since regulates leverage and drives movements in the credit and spread equations, whose smoothed residuals seem to be the most a¤ected by serial correlation, we opt to make this parameter time varying. We specify t = (1 t = (2 ) + u ( t 1 u + et; Exogenous variations ) (exp( 1 (Xt 2 Endogenous variations 1 X s )) + exp( (46) 2 (Xt 1 X s ))) + et; (47) where X is net bank wealth. We chose bank net wealth as the relevant state in the endogenous speci…cation because of its importance for the spread and the credit variable. Table 7 reports estimates of selected parameters. 7 TIME VARYING FINANCIAL FRICTIONS? 30 In the model with exogenously varying parameters, variations in t very persistent. Furthermore, estimates of ( ; !; ) are now larger making steady state leverage drop to about 2.9 and the lifetime of bankers to increase. With the endogenous speci…cation, estimates of and ! further increase, making steady state leverage fall to 1.9, but bankers’ survival probability is roughly unchanged. The data seem to require a very strong asymmetric speci…cation for time variations ( 1 < 2 ), implying a strong negative relationship between the fraction of funds that bankers can divert and their net worth. Finally, note that, in term of marginal likelihood, the endogenous speci…cation is superior to both the exogenous one and the …xed coe¢ cient speci…cation augmented by a shock to bankers’net worth accumulation equation. To investigate how inference di¤ers in the three estimated models, we plot in …gure 7 the responses of output, in‡ ation, investment, net worth, leverage, and the spread to a one percent capital quality shock. The constant coe¢ cient speci…cation closely replicates the dynamics presented by GK (see their …gure 3). There is a persistent decline in output and a temporary but strong decline in in‡ ation. Investment temporarily falls but it then increases because capital is below its steady state. Bankers’net worth falls and there is a sharp increase in the spread. -3 -3 x 10 x 10 1 0 Inflation Output -4 -6 -8 -1 -2 -3 -4 -10 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 -3 x 10 -0.05 4 Net worth Investments 6 2 0 -0.1 -0.15 -0.2 -2 5 10 15 20 25 30 35 40 -3 x 10 0.2 0.15 Spread Leverage 3 0.1 0.05 2 1 0 5 10 15 20 25 30 Cons tant 35 40 Ex ogenous Endogenous Figure 6: Dynamics in response to a capital quality shock When we allow to be exogenously varying, responses are qualitatively similar. Quantitatively, output falls more on impact but less in the short run; net worth falls less and the spread increases less in the short run. Thus, making exogenously time varying, reduces the model’ ability to capture recessionary e¤ects on impact. s 8 CONCLUSIONS When variations are endogenous, the model possesses an additional mechanism of propagation of shocks since lower net worth implies higher share of funds diverted by banks and generally stronger accelerator dynamics. Since the dynamic responses of net worth are highly persistent, the spread persistently increases making investment increase less and output fall more and more persistently relative to the other two cases. Thus, neglecting endogenous variations could impair our ability to correctly measure the e¤ects of capital quality shocks. While the economic interpretation of the relationship between and net worth is beyond the scope of the paper, attempts to endogenize crucial parameters in the Gertler and Karadi model exist (see e.g. Bi, Leeper and Leith, 2014; Ferrante, 2014). Their work could provide the microfundations for the evidence we uncover. 8 Conclusions This paper is interested in i) characterizing the decision rules of a DSGE when parameter variations are exogenous or endogenous, and in the latter case, when agents internalize or not the e¤ects their decisions may have on parameter variations; ii) providing diagnostics to detect misspeci…cation driven by parameter variations; and iii) studying the consequences of using time invariant models when the parameters are time varying in terms of identi…cation, estimation, and inference We show that if parameter variations are purely exogenous, the contemporaneous impact and the dynamics induced by structural shocks are the same as in a model with no parameter variations. However, if parameter variations are endogenous, the structural dynamics may be di¤erent and the extent of the di¤erence depends on the detail of the model. We provide diagnostics to detect the misspeci…cation due to neglected parameter variations and describe a marginal likelihood diagnostics to recognize whether exogenous or endogenous time variations should be used. We highlight that certain parameter identi…cation problems noted in the literature may be the result of misspeci…cation due to neglected time variations. Our Monte Carlo study indicates that parameter and impulse response distortions may be large even for modest time variations in the parameters. It also shows that, when parameter variations are neglected, SVAR methods are competitive with more structural likelihood-based methods, as far as the responses to structural disturbances are concerned. In the context of the Gertler and Karadi (2010) model, we show that the parameter regulating the amount of moral hazard is likely to be time varying. 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Time varying Phillips curve, NBER working paper 19790. 33 9 REFERENCES 34 Appendix A: Additional Monte Carlo …gures and tables η ρ γ ρ z δ g α A 1 Model B 0. 8 0. 8 0. 8 0. 8 0. 6 0. 6 0. 6 0. 6 0. 8 0. 8 0. 6 0. 6 0. 5 0. 4 0. 4 0. 4 0. 4 0. 4 0. 4 0. 2 0. 2 0. 2 0. 2 0. 2 0. 2 0 5 10 -1 0 1 2 3 0. 8 0. 9 1 0. 5 1 1 0. 02 0. 04 0. 2 0. 4 0. 6 0 0 Model C 5 10 1 0. 8 0. 8 0. 8 0. 8 0. 8 0. 6 0. 6 0. 6 0. 6 0. 6 0. 4 0. 4 0. 4 0. 4 0. 4 0. 2 0. 2 0. 2 0. 2 0. 2 0. 5 0. 5 0 0 5 10 -2 0 2 0. 8 0. 9 1 -0. 5 0 0. 5 1 1. 5 1 0. 02 0. 04 0. 2 0. 4 0. 6 1 0. 8 Model D 5 0. 8 0. 6 0. 6 0. 4 0. 4 0. 4 0. 2 0. 2 1 0. 8 0. 6 0. 8 0. 2 0. 6 0. 5 0. 5 0. 5 0. 4 0. 2 0 0 5 10 0 1 2 0. 8 0. 9 1 T rue T = 100 10 -1 0 1 I ncorrect T = 100 0 0. 02 0. 04 0. 2 0. 4 0. 6 I ncorrect T = 1000 Figure A1: Density of estimates; DGP1 (parameter variations explain 2-5 percent of output variance). 5 10 9 REFERENCES True 35 Estimated Correct Estimated Time invariant Estimated Time invariant Mean Mean 5 percentile 95 percentile Mean 5 percentile 95 percentile T=150 T=150 T=1000 DGP Model B = 2:0 2.00 2.29 1.53 3.87 2.45 1.61 3.09 = 2:0 2.01 1.11 -0.33 2.06 0.25 -0.27 1.95 0.94 0.99 0.96 1.00 0.99 0.97 1.00 z = 0:9 = 0:5 0.47 0.76 0.62 0.96 0.91 0.79 0.98 g = 0:025 0.03 0.01 0.01 0.03 0.01 0.01 0.01 = 0:3 0.30 0.19 0.11 0.41 0.21 0.10 0.34 A = 4:5 4.53 2.73 1.33 4.14 1.80 1.14 4.16 DGP Model C = 2:0 2.00 3.40 1.56 7.51 5.19 1.77 22.90 = 2:0 2.00 -0.08 -0.32 0.73 -0.19 -0.35 0.35 0.88 0.99 0.93 1.00 0.99 0.90 1.00 z = 0:9 = 0:5 0.48 0.56 0.08 0.97 0.91 0.59 0.98 g = 0:025 0.02 0.02 0.01 0.07 0.02 0.01 0.07 = 0:3 0.30 0.26 0.15 0.34 0.26 0.19 0.35 A = 4:5 4.50 1.71 1.25 2.77 2.27 1.24 8.17 DGP Model D = 2:0 2.00 3.05 1.68 4.59 2.40 1.98 4.81 = 2:0 2.00 -0.06 -0.28 0.54 1.63 -0.27 1.98 = 0:9 0.88 0.98 0.90 1.00 0.92 0.91 1.00 z 0.47 0.42 -0.46 0.96 0.50 0.32 0.97 g = 0:5 = 0:025 0.02 0.01 0.01 0.03 0.01 0.01 0.01 = 0:3 0.30 0.23 0.15 0.32 0.21 0.13 0.27 A = 4:5 4.49 1.91 1.45 3.57 4.10 1.65 4.51 Table A1: Distributions of estimates, DGP2 (parameter variations explain 20 percent of output variance). 9 REFERENCES 36 η ρ γ ρ z δ g α A 1 0 .8 0 .8 0 .8 0 .6 0 .6 0 .6 0 .6 0 .4 0 .4 0 .4 0 .4 0 .2 Model B 0 .8 0 .2 0 .2 1 0 .8 0 .2 0 .6 0 .5 0 .5 0 .4 0 .2 0 5 10 15 -1 0 1 2 3 0 .6 0 .8 1 - 0 .5 0 0 .5 1 1 0 .6 0 .6 0 .4 0 .4 0 .2 1 2 1 1 6 8 0 .2 - 0 .5 0 1 4 0 .5 0 0 .8 2 0 .4 0 .2 1 0 0 .6 0 .5 0 .4 0 0 .2 0 .4 0 .6 0 .8 0 .6 0 0 .6 0 1 0 .8 0 .5 5 10152025 0 .0 4 1 0 .8 0 .2 Model C 0 .8 0 0 .0 2 0 .5 1 1 0 0 .0 2 0 .0 4 0 .2 0 .4 0 .6 1 2 4 6 8 2 4 6 8 1 0 .8 Model D 0 .6 0 .5 0 .5 0 .5 0 .5 0 .5 0 .5 0 .4 0 .2 0 0 5 10 15 20 0 1 2 0 0 .6 0 0 .8 1 T rue T = 100 0 -1 0 1 0 0 .0 2 0 .0 4 0 .2 0 .4 0 .6 In c o r r e c t T = 1 0 0In c o r r e c t T = 1 0 0 0 Figure A2: Density of estimates, DGP2 (parameter variations explain 20 percent of output variance) 9 REFERENCES 37 -3 M o d e l B -A -3 M o d e l C -A x 10 -3 M o d e l D -A x 10 -3 M o d e l B -A x 10 -3 M o d e l C -A x 10 -3 M o d e l D -A x 10 x 10 2 0 Hours -1 -1 -2 -2 -4 10 20 30 10 20 30 10 20 30 0 .0 5 -5 -6 -8 -3 40 -1 0 40 10 20 30 40 10 20 30 40 10 -0 . 0 1 -0 . 0 3 40 20 30 40 20 30 40 20 30 40 -0 . 0 2 -0 . 0 1 5 30 -0 . 0 1 -0 . 0 2 20 10 -0 . 0 0 5 -0 . 0 1 0 .0 6 0 .0 4 0 .0 4 0 .0 4 0 .0 3 0 .0 2 0 .0 3 0 .0 2 0 .0 2 0 .0 1 0 .0 1 10 20 30 40 10 -3 20 30 40 10 -3 20 30 40 10 -3 x 10 8 6 40 6 4 4 10 20 30 40 20 30 40 30 40 -3 -3 x 10 x 10 0 -5 -5 -1 0 -1 0 -2 -4 2 10 20 0 4 2 -0 . 0 4 10 -3 8 6 30 0 10 10 8 20 x 10 12 -0 . 0 3 -0 . 0 4 -0 . 0 2 x 10 x 10 10 Output -4 -2 -4 40 0 .0 5 Capital -2 0 -2 0 0 0 0 10 20 30 40 -1 5 -1 5 10 20 30 40 10 20 30 40 10 -3 -3 -3 -4 -4 -4 x 10 x 10 x 10 x 10 x 10 x 10 4 4 3 Consumption -2 -2 -2 -4 -4 -4 -6 -6 3 3 2 2 2 1 -6 1 10 20 30 40 A 84 10 A 16 20 30 B 50 40 10 20 30 A 84 40 10 A 16 20 C 50 30 40 10 20 A 84 30 40 A 16 10 D 50 Figure A3: Impulse responses, DGP2 (parameter variations explain 20 percent of output variance) 9 REFERENCES small Variable Technology Government Technology Government DGP: Model B Estimated: Time invariant Y 81.300 0.100 0.998 0.006 C 55.300 0.100 0.998 0.002 N 15.600 0.400 0.978 0.025 K 40.600 0.100 0.994 0.008 DGP: Model C Estimated: Time invariant Y 81.900 0.100 0.927 0.082 C 26.500 0.100 0.999 0.001 N 5.400 0.400 0.966 0.039 K 37.400 0.100 0.974 0.030 DGP: Model D Estimated: Time invariant Y 82.200 0.100 0.936 0.072 C 32.800 0.100 0.996 0.008 N 10.200 0.500 0.928 0.079 K 60.000 0.400 0.979 0.028 Table A2: Variance decomposition, DGP2 (parameter variations explain 20 percent of output variance). 38 9 REFERENCES 39 Appendix B : The RBC model with variable capital utilization of section 3 The representative household maximizes the following stream of future utility max E0 1 X t j=0 k s ct + it = wt nt + rt kt it = kt s kt (1 = ut kt n1+ A t 1+ c1 t 1 s a(ut )kt )kt (48) Tt (49) (50) 1 (51) 1 where ct is consumption, it investment, kt the stock of capital, and nt is hours worked. Household chooses the utilization rate of capital, ut , and the amount of e¤ective capital s that she can rent to the …rm, i.e. kt . Household receives earnings from supplying labor k and capital services to the …rm, i.e. wt and rt respectively, subject to a cost of changing s . Finally, T are lump sum taxes levied by the government. capital utilization, a(ut )kt t The production function is s yt = zt (kt ) n1 t A fraction of output is consumed by the government and …nanced with lump-sum taxes. The government budget constrain is always balanced, i.e. gt yt = Tt The optimality conditions of the planner problem are (1 gt )yt = ct + kt Ant ct = (1 (1 )(1 a(ut )ut ) kt gt )yt =nt 1 = Et (ct =ct+1 ) (1 (1 gt )yt =kt 1 1 a(ut+1 )ut+1 + (1 gt+1 )yt+1 =kt ) 0 = ut (a (ut )ut + a(ut )) yt = zt (ut kt 1= + a(ut ) = 1) n1 t 1 e (ut 1) 1 The functional form for the adjustment cost of the capital utilization is in the last equation and it satis…es a(1) = 0, a0 (1) = 1= + 1, a00 (1) = (1= + 1). If c assume that, in the steady state u = 1, the steady states for ( k ; y ; n ; n) are the same y y as in the RBC model without variable capital utilization. 9 REFERENCES 40 Appendix C : The equations of Gertler and Karadi model exp(%t ) = (exp(Ct ) 1 = exp( t) = h exp(Ct exp(Rt ) exp( exp(%t ) exp(%t 1 ) 1 )) h(exp(Ct+1 ) h exp(Ct )) (52) t+1 ) (53) (54) exp( t ) = (1 exp(Yt ) exp(Lt ) ) exp( t+1 )(exp(Rk;t+1 ) exp(Rt )) + exp( t ) = (1 )+ exp(Lt )' = exp(%t ) exp(Pm;t )(1 exp( t ) = exp(zt ) = exp(xt ) = exp(Kt ) = exp(Nt ) = exp(N et ) = (55) ) exp( t+1 ) exp( t+1 ) exp(zt+1 ) exp( t+1 ) exp( t ) 1 (1 exp( t ) t) (exp(Rk;t ) exp(Rt 1 ))(1 t 1 ) exp( t 1 ) + exp(Rt exp( t )(1 t) exp(zt ) (exp( t 1 )(1 t 1 )) exp( t ) exp(Nt ) exp(Qt ) exp(N et ) + exp(N nt ) exp(zt ) exp(Nt exp(N nt ) = !(1 1 ) exp( eN e;t ) + exp(xt+1 ) exp( t+1 ) (56) (57) (58) 1) exp(Ym;t ) exp( )) + exp( t ) (exp(Qt ) exp(Kt 1 ) exp(Qt 1 ) exp(Ym;t ) = exp(at ) (exp( t ) exp(Ut ) exp(Kt 1 )) exp(Lt )1 (Int + I s ) (Int + I s ) (Int + I s ) 1)2 + i ( 1) exp(Qt ) = 1 + 0:5 i ( s s (Int 1 + I (Int 1 + I (Int 1 + I s s) (Int+1 + I s ) (Int+1 + I s ) 2 exp( t+1 ) i ( 1)( ) (Int + I s (Int + I s ) exp(Gt ) = G s 1) (61) (62) (64) exp(Rk;t ) = (exp(Pm;t ) exp(Kt ) = exp( t ) exp(Kt (60) (63) t t 1 ) exp(Qt ) exp( t ) exp(Kt 1 ) exp( ) = c + b=(1 + ) exp(Ut )1+ exp(Ym ) b exp(Ut ) exp( t ) exp(Kt 1 ) = exp(Ut ) exp(Pm;t ) Int = exp(It ) exp( ) exp( t ) exp(Kt (59) (65) (66) (67) (68) (69) 1) (70) + Int (71) exp(gt ) (Int + I s ) exp(Yt ) = exp(Ct ) + exp(Gt ) + exp(It ) + 0:5 i ( (Int 1 + I s exp(Ym;t ) = exp(Yt ) exp(Dt ) (72) 1)2 (Int + I s ) + exp(Kt ) (73) (74) 9 REFERENCES 41 exp(Dt ) = exp(Dt + (1 1) )((1 exp(inf lt exp(inf lt 1) 1) exp(inf lt ) P P (1 ) exp(inf lt ) 1 )=(1 )) =(1 ) exp(Xt ) = 1= exp(Pm;t ) (75) (76) exp(Ft ) = exp(Yt ) exp(Pm;t ) + exp( exp(F(77) t+1 ) exp(inf lt+1 ) (exp(inf lt )) t+1 ) 1 (1 ) exp(inf lt ) P exp(Zt+1 ) (78) t+1 ) exp(inf lt+1 ) P exp(Zt ) = exp(Yt ) + exp( exp(Ft ) exp(inf lt ) = exp(inf lt ) 1 exp(Zt ) (exp(inf lt ))1 = exp(inf lt 1) P (1 ) (79) )(exp(inf lt ))1 + (1 (80) exp(it ) = exp(Rt ) exp(inf lt+1 ) exp(it ) = exp(it t = at = t e ;t = t = i ( (Rk;t+1 a = gt = 1) at 1 gt e s Rk s (exp(Xt )=( =( +R )+e 1))) y ) i exp(ei;t ) ;t (82) (83) ea;t (84) e a 1 (85) ;t eg;t 1 ;t 1 t 1 exp(inf lt ) Rt t 1 g (81) 1 +e +e (86) ;t ;t Note: t = 0; 8t in the basic estimated model. It appears only in the augmented model of table 7. (87) (88)