Chapter 5. Structural Vector Autoregression

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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 average 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 structural 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 exogenous 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. Roots of C(z) that j j are inside the unit circle pose a much more di¢ cult problem, since models with roots inside the unit circle have the same second moment properties as models with roots outside the unit circle. The simplest example of this is the univariate MA(l) model

yt = (1 cL)"t;

2 where "t is i:i:d:(0; "). The same second moment of yt obtains for the model

yt = (1 cL~ )~"t; where 1 c~ = c 2 2 ~"t i:i:d:(0; c ): " This is important because it can lead to large speci…cation errors in structural VAR models that cannot be detected from the data. For example, suppose that the true structural model is

yt = (1 cL)"t with c > 1: j j A researcher using the invertible model would not recover the true structural shocks, but rather

1 (1 cL~ ) yt 1 = (1 cL~ ) (1 cL)"t 1 i = "t (~c c) c~ "t 1 i: i=1 X

6 A general discussion of this subject is contained in Hannan (1970) and Rozanov (1967). Implications of these results for the interpretation of structural VARs are discussed in Hansen and Sargent (1991) and Lippi and Reichlin (1993). For related discussion see Quah (1986). Hansen and Sargent (1991) provide a speci…c economic model in which non invertible structural moving average processes arise. In the model, one set of economic variables, say Xt, are generated by an invertible moving average process. Another set of economic variables, say Yt, are expectational variables, formed as discounted sums of expected future Xt0s. Hansen and Sargent then consider what would happen if only the Yt data were available to the econometrician. They show that the implied moving average process of Yt, written in terms of the structural shocks driving Xt, is not invertible. A simple version of the model is as follows: suppose Yt and Xt are two scalar time series, with Xt generated by the MA(1) process

Xt = "t "t 1:

Suppose Yt is related to Xt by the expectational equation

1 i Yt = Et Xt+i = Xt + EtXt+1 = (1 ) "t "t 1 = C(L)"t: i=0 X The root of C(z) is (1 ) =; which may be less than 1 even when the root of (1 z) is greater than 1. For example, if = = 0:8: The Hansen-Sargent example provides an important and constructive lesson for researchers using structural VARs: it is important to include variables that are directly related to the exogenous shocks under consideration. If the only variables used in the model are indirect indicators with important expectational elements, severe misspeci…- cation may result.

Assumption 4. Te determinantal polynomial C(z) has all of its roots outside j j the unit circle.

7 4. Identi…cation of the structural VAR

We assume Assumptions 1, 3 and 4 hold. Also assume that the lag polynomial of A(L) is of …nite order p, then the structural VAR can be written as

A0Yt = A1Yt 1 + A2Yt 2 + ::: + ApYt p + "t: (3)

In this section, we ask the following question: under what conditions we can uniquely recover A0;A1; :::; Ap and " if we are granted with a sample of in…nite size generated from the above model.

Since A0 is not restricted to be diagonal, equation (3) is a dynamic simultaneous equations model. It di¤ers from standard representations of the simultaneous equations model because observable exogenous variables are not included in the equations.

However, since exogenous and predetermined variables - lagged values of Yt are treated identically for purposes of identi…cation and estimation, equation (3) can be analyzed using techniques developed for simultaneous equations. The reduced form of (3) is

Yt = 1Yt 1 + 2Yt 2 + ::: + pYt p + et; where

1 i = A0 Ai (i=1,...,p) (4) 1 1 e = A0 "A0 :

With an in…nite sample, i and e can be uniquely determined without error. There- fore, the identi…cation question becomes: under what conditions we can uniquely deter- mine

A0;A1; :::; Ap; " from

1; :::; p; e using (4).

8 4.1. The order condition

This is the standard identi…cation question in the simultaneous equations literature. A necessary condition is the well known “order”condition. We have:

2 (Known) There are pn elements in 1; :::; p and n(n+1)=2 elements in e:These [n2p + n(n + 1)=2] parameters completely characterize the second moment of the data.

2 (Unknown) In the structural model (3) there are (p+l)n elements in A0;A1; :::; Ap and n(n + 1)=2 elements in ":

Since #Unknown #Known = n2 > 0: We need at least n2 restrictions for identi…cation.

Setting the diagonal elements of A0 equal to 1 gives the …rst n restrictions. This leaves n(n 1) restrictions that must be deduced from economic considerations.

4.2. What shouldn’tbe done

The identifying restrictions must be dictated by the economic model under consideration. It makes little sense to discuss the restrictions without reference to a speci…c economic system. An identi…cation strategy that makes sense for one model can be nonsense for another. This is the …rst "what shouldn’tbe done". Here, some general remarks on identi…cation are made in the context of a simple bivariate model explaining output and money; a more detailed discussion of identi…cation in structural VAR models is presented in Giannini (1991). Let the …rst element of Yt, say y1t; denote the rate of growth of real output, and the second element of Yt, say y2t, denote the rate of growth of money. Writing the typical element of Ak as aij;k;

9 Writing the typical element of A as Aij;k equation (3) becomes

p p y1;t = a12;0y2;t + a11;iy1;t i + a12;iy2;t i + "1;t; (5) i=1 i=1 Xp Xp y2;t = a21;0y1;t + a21;iy1;t i + a22;iy2;t i + "2;t; (6) i=1 i=1 X X Equation (5) is interpreted as an output or “aggregate supply” equation, with "1;t interpreted as an aggregate supply or productivity shock. Equation (6) is interpreted as a money supply “reaction function ” showing how the money supply responds to contemporaneous output, lagged variables, and a money supply shock "2;t. Identi…cation requires n(n - 1) = 2 restrictions on the parameters of these two equations. In the standard analysis of simultaneous equation models, identi…cation is achieved by imposing zero restrictions on the coe¢ cients for the predetermined variables. For example, the order condition is satis…ed if imposing the two constraints

a11;2 = a21;1 = 0:

In this case, y1;t 1 shifts the output equation but not the money equation, while y2;t 1 shifts the money equation but not the output equation. Of course, this is a very odd restriction in the context of the output-money model, since the lags in the equations capture expectational e¤ects, technological and institutional inertia arising from pro- duction lags and sticky prices, information lags, etc..There is little basis for identifying the model with the restriction a11;2 = a21;1 = 0: Indeed there is little basis for identifying the model with any zero restrictions on lag coe¢ cients. Sims (1980) persuasively makes this argument in a more general context, and this has led structural VAR modelers to avoid imposing zero restrictions on lag coe¢ cients. Instead, structural VARs have been identi…ed using restrictions on the covariance matrix of structural shocks, the matrix of contemporaneous coe¢ cients A0, and the matrix of long-run multipliers A(l).

10 4.3. A common normalization that provides n(n - 1)/2 restrictions

Restrictions on " have generally taken the form that ", is diagonal, so that the structural shocks are assumed to be uncorrelated. In the example above, this means that the underlying productivity shocks and money supply shocks are uncorrelated, so that any contemporaneous cross equation impacts arise through nonzero values of a12;0 and a21;0. Some researchers have found this a natural assumption to make, since it follows from a modeling strategy in which unobserved structural shocks are viewed as distinct phenomena which give rise to comovement in observed variables only through the speci…c economic interactions studied in the model. The restriction that " is diagonal imposes n(n 1)=2 restrictions on the model, leaving only n(n 1)=2 additional necessary restrictions. These additional restrictions can come from a priori knowledge about the A0 or the A(1) matrices.

4.4. Identi…cation through short run restrictions

In (5) and (6), n(n 1)=2 = 1, so a priori knowledge of a12;0 or a21;0 will serve to identify the model. For example, if it was assumed that the money shocks a¤ect output only with a lag, then a12;0 and this restriction identi…es the model. The generalization of this restriction in the n-variable model produces the Wold causal chain [see Wold (1954) and Malinvaud (1980, pp. 605-608)], in which @Y i;t = 0 for i

This restriction leads to a recursive model with A0 lower triangular, yielding the required n(n - 1)/2 identifying restrictions. This restriction was used in Sims (1980), and has become the "default" identifying restriction implemented automatically in com- mercial econometric software. Like any identifying restriction, it should never be used automatically. In the context of the output-money example, it is appropriate under the maintained assumption that exogenous money supply shocks, and the resulting change in interest rates, have no contemporaneous e¤ect on output. This may be a reasonable assumption for data sampled at high frequencies, but loses its appeal as the sampling interval increases.

11 Other restrictions on A0, can also be used to identify the model. Blanchard and Watson (1986) Bernanke (1986) and Sims (1986) present empirical models that are identi…ed by zero restrictions on A0, that don’tyield a lower triangular matrix. Keating (1990) uses a related set of restrictions. Of course, nonzero equality restrictions can also be used; see Blanchard and Watson (1986) and King and Watson (1993) for examples.

4.5. Identi…cation through long run restrictions

An alternative set of identifying restrictions relies on long-run relationships. In the context of structural VARs these restrictions were used in papers by Blanchard and Quah (1989) and King et al. (1991). These papers relied on restrictions on

p A(1) = A0 Ai i=1 X for identi…cation. Since 1 C(1) = A(1) ; these can alternatively be viewed as restrictions on the sum of impulse responses. To motivate these restrictions, consider the output-money example above. Using

(1), the long run cumulative e¤ect of the j-th element in "t on the i-th element in Yt is

1 cij;k; k=0 X which is the (i; j)-th element of C(1). Now, suppose that money is neutral in the long run, in the sense that shocks to money have no permanent e¤ect on the level of output. 1 This means that C(1) is a lower triangular matrix. Since A(1) = C(1) ; this implies A(1) is also lower triangular, and this yields the single extra identifying restriction that is required to identify the bivariate model. The analogous restriction in the general n-variable VAR, is the long-run Wold causal chain in which "i;t has no long-run e¤ect on Yj;t, for j < i. This restriction implies that A(1) is lower triangular yielding the necessary n(n 1)=2 identifying restrictions.

12 5. Estimation and inference

5.1. Estimating exactly identi…ed models

The most commonly used method is indirect least squares. It consists of two steps.

Step 1. Estimate the reduced form model equation by equation using OLS. 1 Step 2. Estimate the structural parameters using i = A Ai (i=1,...,p), e = 0 1 1 A0 "A0 and additional appropriate identi…cation restrictions.

5.2. Estimating overly identi…ed models

It can be estimated using GMM or quasi-maximum likelihood assuming normal errors.

GMM. Let

0 Zt = (Yt 1; :::; Yt0 p)0 denote the vector of predetermined variables in the model, and let denote the vector of unknown parameters in A0;A1; :::; Ap and " (imposing identi…cation restrictions to reduce the parameter space). The population moment conditions that implicitly de…ne the structural parameters are

E "tZt0 = 0

E "t"t0 = ": GMM estimators are formed by choosing ^ such that the above two equations are satis…ed, or satis…ed as closely as possible, with sample moments used in place of the population moments. This is a fairly standard GMM estimation problem. The theory and inferential procedure discussed earlier applies. We omit the details.

MLE It is the same as FIML in standard simultaneous models, we omit the details.

13 5.3. Inference

Under the additional assumption that the variables involved are weakly stationary, conventionally asymptotic theory can be used for inference. In many structural VAR exercises, the impulse response functions and variance decompositions de…ned are of more interest than the parameters of the structural VAR. 1 Since C(L) = A(L) , the moving average parameters/impulse responses and the variance decompositions are di¤erentiable functions of the structural VAR parameters. The continuous mapping theorem directly yields the asymptotic distribution of these parameters from the distribution of the structural VAR parameters. Formulae for the resulting covariance matrix can be determined by delta method calculations. Conve- nient formulae for these covariance matrices can be found in Hamilton (1994). Many applied researchers have instead relied on Bootstrap methods for estimating standard errors of estimated impulse responses and variance decompositions. The same procedure that we discussed earlier can be applied. A drawback of this standard procedure is that it does not recognize that we are dealing with an VAR process. Kilian (1998) provided a bootstrap procedure that is tailored to the dynamic feature of the model.

6. Structural VAR versus fully speci…ed economic models

We ask a very basic question: does it even make sense to talk about impulse without specifying an economic model? Of course, we need to state this question more precisely. How informative are unrestricted VARs about how particular economic models re- spond to preference, technology, and information shocks? Is it possible to check for whether a theoretical model has the property in population that it is possible to infer economic shocks and impulse responses to them from the innovations and the impulse responses associated with a vector autoregression (VAR). The answer is yes for the simplest possible case. Suppose the equilibrium of an economic model or an approximation to it has the

14 following representation

Xt = AXt 1 + Bt; (7) Yt = CXt 1 + Dt; where

where Xt is an nX by 1 vector of possibly unobserved state variables,

Yt is an nY -by-1 vector of variables observed by an econometrician (the same as those in the VAR)

is an n-by-1 vector of economic shocks impinging on the states and observables t (to compare with "t in the VAR).

Equilibrium representations of the form (7) are obtained in one of two widely used procedures. The …rst is to compute a linear or log linear approximation of a nonlinear model as exposited, for example, in Harald Uhlig (1999). It is straightforward to collect the linear or log linear approximations to the equilibrium decision rules and to arrange them into the state space form (7). A second way is to derive (7) directly as a representation of a member of a class of dynamic stochastic general equilibrium models with linear transition laws and quadratic preferences. The question is: under what conditions do the economic shocks in the state-space system (7) match up with the shocks associated with a VAR? That is, under what conditions is

t = "t; where is a matrix of constants depending on the structural parameters in (7), can potentially be uncovered by “structural” VAR analysis? When (3) holds, impulse responses from the SVAR match the impulse responses from the economic model (7).

Assume the dimensions of "t and t are the same. Assume D is nonsingular. This gives the simplest case. Then, the second equation in (7) implies

1 t = D (Yt CXt 1) : 15 Substituting this into the …rst equation and rearranging it gives

1 1 I A BD C L Xt = BD Yt: It will turn out that the answer will depend on the eigenvalues of A BD 1C:

Case 1: The eigenvalues of A BD 1C are strictly less than one in modulus. Then, we have 1 1 1 Xt = I A BD C L BD Yt: Again apply the second equation in (7),

1 1 1 Yt = I A BD C L BD Yt 1 + Dt: n o 1 1 1 This equation de…nes a VAR for Yt because I A BD C L BD is well de- …ned and t is orthogonal to Yt 1: Therefore, in this case, the answer is positive: the structural VAR can successfully uncover the economic shocks and impulse responses implied by the economic model.

Case 1: Some eigenvalues of A BD 1C are greater than one in modulus The most important di¤erence from the previous case is that I A BD 1C L is not invertible, therefore Xt can not be expressed as a linear function of Yt and its lagged values. In other words, it is impossible to uniquely pin down the state variable Xt after observing the history Yt;Yt 1; :::: . Therefore, Yt is not a Markov process conditioning f g on its own past.

= "t t 6 for any : The detailed argument can be found in Fernandez-Villaverde, Rubio-Ramirez, Sargent and Watson (2007). Fernandez-Villaverde, Rubio-Ramirez, Sargent and Watson (2007) hesitate to draw sweeping conclusions about VARs. Some applications of VARs are informative about the shapes of impulse responses to some economic shocks that theories should attempt to match, while others are not.

16 It is easy to reiterate the recommendation to estimate the deep parameters of a com- plete and fully trusted model by likelihood-function based methods. If you trust your model, you should accept that recommendation. The enterprise of identifying economic shocks and their impulse-response functions from VAR innovations aims, however, to coax interesting patterns from the data that will prevail across a set of incompletely speci…ed and not fully trusted models. Despite pitfalls, it is easy to sympathize with the enterprise of identifying economic shocks from VAR innovations if one is not dogmatic in favor of a particular fully speci…ed model.

17 References

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