To export this article to Microsoft Word, please log in or subscribe.
Have an account? Please log in
Not a subscriber? Sign up today
Sarte, Pierre-Daniel G.. "On the identification of structural vector autoregressions." Economic Quarterly. Federal Reserve Bank of Richmond. 1997. HighBeam Research. 24 Apr. 2018 <https://www.highbeam.com>.
Sarte, Pierre-Daniel G.. "On the identification of structural vector autoregressions." Economic Quarterly. 1997. HighBeam Research. (April 24, 2018). https://www.highbeam.com/doc/1G1-20515443.html
Sarte, Pierre-Daniel G.. "On the identification of structural vector autoregressions." Economic Quarterly. Federal Reserve Bank of Richmond. 1997. Retrieved April 24, 2018 from HighBeam Research: https://www.highbeam.com/doc/1G1-20515443.html
Following seminal work by Sims (1980a, 1980b), the economics profession has become increasingly concerned with studying sources of economic fluctuations. Sims's use of vector autoregressions (VARs) made it possible to address both the relative importance and the dynamic effect of various shocks on macroeconomic variables. This type of empirical analysis has had at least two important consequences. First, by deepening policymakers' understanding of how economic variables respond to demand versus supply shocks, it has enabled them to better respond to a constantly changing environment. Second, VARs have become especially useful in guiding macroeconomists towards building structural models that are more consistent with the data.
According to Sims (1980b), VARs simply represented an atheoretical technique for describing how a set of historical data was generated by random innovations in the variables of interest. This reduced-form interpretation of VARs, however, was strongly criticized by Cooley and Leroy (1985), as well as by Bernanke (1986). At the heart of the critique lies the observation that VAR results cannot be interpreted independently of a moire structural macroeconomic model. Recovering the structural parameters from an estimation procedure requires that some restrictions be imposed. These are known as identifying restrictions. Implicitly, the choice of variable ordering in a reduced-form VAR constitutes such an identifying restriction.
As a result of the Cooley-Leroy/Bernanke critique, economists began to focus more precisely upon the issue of identifying restrictions. The extent to which specific innovations were allowed to affect some subset of variables, either in the short run or in the long run, began to be derived explicitly from structural macroeconomic models. Consequently, what were previously considered random surprises could be interpreted in terms of specific shocks, such as technology or fiscal policy shocks. This more refined use of VARs, known as structural vector autoregressions; (SVARs), has become a popular tool for evaluating economic models, particularly in the macroeconomics literature.
The fact that nontrivial restrictions must be imposed for SVARs to be identified suggests, at least in principle, that estimation results may be contingent on the choice of restrictions. To take a concrete and recent example, in estimating a system containing employment and productivity variables, Gali (1996) achieves identification by assuming that aggregate demand shocks do not affect productivity in the long run. Using postwar U.S. data, he is then able to show that, surprisingly, employment responds negatively to a positive technology shock. One may wonder, however, whether his results would change significantly under alternative restrictions. This article consequently investigates how the use of different identifying restrictions affects empirical evidence about business fluctuations. Two important conclusions emerge from the analysis.
First, by thinking of SVARs within the framework of instrumental variables estimation, it will become clear that the method is inappropriate for certain identifying restrictions. This finding occurs because SVARs use the estimated residual from a previous equation in the system as an instrument in the current equation. Since estimation of this residual depends on some prior identifying restriction, the identification scheme necessarily determines the strength of the instrument. By drawing from the literature on estimation with weak instruments, this article points out that in some cases, SVARs will not yield meaningful parameter estimates.
The second finding of interest suggests that even in cases where SVAR parameters can be properly estimated, different identification choices can lead to contradictory results. For example, in Gali (1996) the restriction that aggregate demand shocks not affect productivity in the long run also implies that employment responds negatively to a positive technology shock. But the opposite result emerges when aggregate demand shocks are allowed to have a small negative effect on productivity in the long run. This latter restriction is appropriate if demand shocks are interpreted as fiscal policy shocks in a real business cycle model. More importantly, this observation suggests that sensitivity analysis should form an integral part of deciding what constitutes a stylized fact within the confines of SVAR estimation.
This article is organized as follows. We first provide a brief description of reduced-form VARs as well as the basic idea underlying the Cooley-Leroy/Bernanke critique. In doing so, the important assumptions underlying the use of VARs are laid out explicitly for the nonspecialist reader. We then introduce the mechanics of SVARs--that is, the details of how SVARs are usually estimated--and link the issue of identification to the estimation procedure.(1) The next section draws from the literature on instrumental variables in order to show the conditions in which the SVAR methodology fails to yield meaningful parameter estimates. We then describe the type of interpretational ambiguities that may arise when the same SVAR is estimated using alternative identifying restrictions. Finally, we offer a brief summary and some conclusions.
1. REDUCED-FORM VARs AND THE COOLEY-LEROY/BERNANKE CRITIQUE
In this section, we briefly describe the VAR approach first advocated by Sims (1980a, 1980b). In doing so, we will show that the issue of identification already emerges in interpreting estimated dynamic responses for a given set of variables. To make matters more concrete, the analysis in both this and the next section is framed within the context of a generic bivariate system. However, the basic issues under consideration are invariant with respect to the size of the system. Thus, consider the joint time series behavior of the vector ([Delta][y.sub.t], [Delta][x.sub.t]), which we summarize as
(1) B(L)[Y.sub.t] = [e.sub.t], with b(0) = [B.sub.0] = I,
where [Y.sub.t] = ([Delta][y.sub.t], [Delta][x.sub.t])', and B(L) denotes a matrix polynomial in the lag operator L. B(L) is thus defined as [B.sub.0] + [B.sub.1]L + ... + [B.sub.k][L.sup.k] + ..., where [L.sup.k][Y.sub.t] = [Y.sub.t-k]. Since B(0) = I, equation (1) is an unrestricted VAR representation of the joint dynamic behavior of the vector [Y.sub.t]. In Sims's (1980a) original notation, the vector [e.sub.t] = ([e.sub.yt], [e.sub.xt])' would carry the meaning of "surprises" or innovations in [Delta][y.sub.t] and [Delta][x.sub.t] respectively.
In its simplest interpretation, the reduced form in (1) is a model that describes how the historical data contained in [Y.sub.t] was generated by some random mechanism. As such, few would question its usefulness as a forecasting tool. However, in the analysis of the variables' dynamic responses to the various innovations, the implications of the unrestricted VAR are not unambiguous. Specifically, let us rewrite (1) as a moving average representation,
(2) [Y.sub.t] = B[(L).sup.-1][e.sub.t] = C(L)[e.sub.t],
where C(L) is defined to be equal to [B(L).sup.-1], with C(L) = [C.sub.0] + [C.sub.1]L + ... + [C.sub.K][L.sup.K] + ..., and [C.sub.0] = C(O) = B[(0).sup.-1] = I. To obtain the comparative dynamic responses of [Delta][y.sub.t] and [Delta][x.sub.t], Sims (1980a) first suggested orthogonalizing the vector of innovations [e.sub.t] by defining [f.sub.t] = [Ae.sub.t], such that A is a lower triangular matrix with 1s on its diagonal and [f.sub.t] has a normalized diagonal covariance matrix. …
Business Review (Federal Reserve Bank of Philadelphia); September 22, 2011
Brookings Papers on Economic Activity; September 22, 2007
Federal Reserve Bank of Minneapolis Quarterly Review; July 1, 2010
Browse back issues from our extensive library of more than 6,500 trusted publications.
HighBeam Research is operated by Cengage Learning. © Copyright 2018. All rights reserved.
The HighBeam advertising network includes: womensforum.com GlamFamily