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Chapter 5. Structural Vector Autoregression

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Chapter 5. Structural Vector Autoregression Chapter 5. Structural Vector Autoregression Contents 1 Introduction 1 2 The structural moving average model 1 2.1 The impulse response function . 2 2.2 Variance decomposition . 3 3 The structural VAR representation 4 3.1 Connection between VAR and VMA . 5 3.2 The invertibility requirement . 5 4 Identi…cation of the structural VAR 8 4.1 The order condition . 9 4.2 What shouldn’tbe done . 9 4.3 A common normalization that provides n(n - 1)/2 restrictions . 11 4.4 Identi…cation through short run restrictions . 11 4.5 Identi…cation through long run restrictions . 12 5 Estimation and inference 13 5.1 Estimating exactly identi…ed models . 13 5.2 Estimating overly identi…ed models . 13 5.3 Inference . 14 6 Structural VAR versus fully speci…ed economic models 14 1. Introduction Following the work of Sims (1980), vector autoregressions have been extensively used by economists for data description, forecasting and structural inference. The discussion here focuses on structural inference. The key idea, as put forward by Sims (1980), is to estimate a model with minimal parametric restrictions and then subsequently test economic hypothesis based on such a model. This idea has attracted a great deal of attention since it promises to deliver an alternative framework to testing economic theory without relying on elaborately parametrized dynamic general equilibrium models. The material in this chapter is based on Watson (1994) and Fernandez-Villaverde, Rubio-Ramirez, Sargent and Watson (2007). We begin the discussion by introducing the structural moving average model, and show that this model provides answers to the “impulse” and “propagation” questions often asked by macroeconomists. The relationship between the structural moving aver- age model and structural VAR is then discussed in Section 3. That section discusses the conditions under which the structural moving average polynomial can be inverted, so that the structural shocks can be recovered from a VAR. When this is possible, a struc- tural VAR obtains. Section 4 shows that the structural VAR can be interpreted as a dynamic simultaneous equations model, and discusses econometric identi…cation of the model’sparameters. Section 5 discusses issues of estimation and statistical inference. 2. The structural moving average model Consider Yt = C(L)"t; (1) where Yt is an nY -by-1 vector (e.g. it can quarterly GDP growth, in‡ation, interest rate). "t is an n"-by-1 vector of shocks to the economy. We allow the dimension of Yt and "t to be di¤erent. 1 C(L) is a matrix of lagged polynomials. It is allowed to be in…nite order. Equation (1) is called the structural moving average model, since the elements of "t, are given a structural economic interpretation. For example, one element of "t, might be interpreted as an exogenous shock to technology, another as an exogenous shock to in‡ation (say an unexpected change in oil price), another as an exogenous change in the quantity of money, and so forth. In the jargon developed for the analysis of dynamic simultaneous equations models, (1) is the …nal form of an economic model, in which the endogenous variables Yt are expressed as a distributed lag of the exogenous variables, given here by the elements of "t. It will be assumed that the endogenous variables Yt are observed, but that the ex- ogenous variables "t, are not directly observed. Rather, the elements of "t, are indirectly observed through their e¤ect on the elements of Yt. This assumption is made without loss of generality, since any observed exogenous variables can always be added to the Yt vector. The model (1) is considered to be general: if you log linearized a rational expectations model around its steady state, then the solution can almost always be represented as (1). Such a generality arises because the number of shocks is allowed to be di¤erent from the number of endogenous variables. 2.1. The impulse response function We are interested in how the system’s endogenous variables responded dynamically to exogenous shocks. Let 2 C(L) = C0 + C1L + C2L + ::::; where Ck is an nY n" 2 matrix. Then, clearly @Yt = C0; @"t0 @Yt = C1; @"t0 1 @Yt = C2; :::: @"t0 2 Therefore, C0 is the impact e¤ect of "t on the endogenous variable, C1 is the e¤ect of "t one period later, C2 is two periods later, and so on. Consider an arbitrary column, say the j th column in Ck. It measures the e¤ect of the j-th element of "t on Yt+k. Further, the (i; j)-th element in Ck measures the e¤ect of the j-th element of "t on i-th element of Yt+k. When this element is viewed as a function of k, it is called the impulse response function of "j;t for Yi;t. 2.2. Variance decomposition We are interested in which shocks are the primary causes of variability in the endogenous variables. For example, is technology shock or monetary shock the main contributor to the business cycle ‡uctuation? To answer this question, the probability structure of the model must be speci…ed. Assumption 1. The shocks are i:i:d:(0; "): The assumption implies that any serial correlation in the exogenous variables is captured in the lag polynomial C(L). The assumption of zero mean is inconsequential, since deterministic components such as constants and trends can always be added to (1). Viewed in this way, "t represents innovations or unanticipated shifts in the exogenous variables. The question concerning the relative importance of the shocks can be made more precise by casting it in terms of the h-step-ahead forecast errors of Yt. Let t h Yt=t h = E Yt "s s= j f g 1 3 denote the h-step ahead forecast of Yt, made at time t h, and let h 1 et=t h = Yt Yt=t h = Ck"t k k=0 X denote the the resulting forecast error. For small values of h, et=t h can be interpreted as “short-run” movements in Yt, while for large values of h, et=t h can be interpreted as “long-run” movements. In the limit as h , et=t h = Yt: The importance of a ! 1 speci…c shock can then be represented as the fraction of the variance in et=t h, that is explained by that shock; it can be calculated for short-run and long-run movements in Yt by varying h. When the shocks are mutually correlated there is no unique way to do this, since their covariance must somehow be distributed. However, when the shocks are uncorrelated the calculation is straightforward. 2 Assumption 2. " is diagonal with diagonal elements : j Under Assumptions 1 and 2, the variance of the i-th element of et=t h is n" h 1 2 2 j cij;k ; j=1 " k=0 # X X where cij;k is the (i,j)-th element of Ck. The faction of the h-step-ahead forecast error variance in Yi;t attributed to "j;t is given by 2 h 1 c2 2 j k=0 ij;k Rij;h = : n" 2 h 1 2 j=1 Pj k=0 cij;k P h P i It is called the variance decomposition of Yi;t at horizon h. 3. The structural VAR representation The structural moving average model (1) is useful but can not be estimated: it has an in…nite number of parameters. VAR is considered instead. 4 3.1. Connection between VAR and VMA If C(L) is invertible, then A(L)Yt = "t; (2) where 1 2 A(L) = C(L) = A0 A1L A2L + :::: a one-sided matrix of lag polynomial. (2) is called the structural VAR representation of (1). The structural VAR representation is useful for two reasons. First, when the model parameters are known, it can be used to construct the unobserved exogenous shocks as a function of current and lagged values of the observed variables Yt. Second, it provides a convenient framework for estimating the model parameters: with A(L) approximated by a …nite order polynomial, Equation (2) is a dynamic simultaneous equations model, and standard simultaneous methods can be used to estimate the parameters. 3.2. The invertibility requirement It is not always possible to invert C(L) and move from the structural moving average representation (1) to the VAR representation (2). Clearly, we need nY n":This requirement has a very important implication for structural analysis using VAR models: in general, small scale VARs can only be used for structural analysis when the endogenous variables can be explained by a small number of structural shocks. Thus, a bivariate VAR of macroeconomic variables is not useful for structural analysis if there are more than two important macroeconomic shocks a¤ecting the variables. In what follows we assume that nY = n". Assumption 3. nY = n" = n: This rules out the simple cause of non invertibility just discussed; it also assumes that any dynamic identities relating the elements of Yt, when nY > n" have been solved out of the model. 5 Under Assumption 3, C(L) is square and the general requirement for invertibility is that the determinantal polynomial C(z) has all of its roots outside the unit circle. j j Roots on the unit circle pose no special problems; they are evidence of over di¤erencing and can be handled by appropriately transforming the variables (e.g. accumulating the necessary linear combinations of the elements of Yt). In any event, unit roots can be detected, at least in large samples, by appropriate statistical tests.
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