Are Structural VARs with Long-Run Restrictions Useful in Developing Business Cycle Theory?

The central finding of the recent structural vector autoregression (SVAR) literature with a differenced specification of hours is that technology shocks lead to a fall in hours. Researchers have used this finding to argue that real business cycle models are unpromising. We subject this SVAR specification to a natural economic test and show that when applied to data from a multiple-shock business cycle model, the procedure incorrectly concludes that the model could not have generated the data as long as demand shocks play a nontrivial role. We also test another popular specification, which uses the level of hours, and show that with nontrivial demand shocks, it cannot distinguish between real business cycle models and sticky price models. The crux of the problem for both SVAR specifications is that available data require a VAR with a small number of lags and such a VAR is a poor approximation to the model’s VAR. ∗The authors thank the National Science Foundation for financial support. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. The growing interest in structural vector autoregressions (SVARs) with long-run restrictions stems largely from the recent finding of researchers using this procedure that a technology shock leads to a fall in hours. Since a technology shock leads to a rise in hours in most real business cycle models, the researchers argue that their SVAR analyses doom existing real business cycle models and point to other types of models, such as sticky price models, as promising. (See Galí 1999, Francis and Ramey 2005a, and Galí and Rabanal 2005.) For example, Francis and Ramey write that “the original technology-driven real business cycle hypothesis does appear to be dead” (2005a, p. 1380) and that the recent SVAR results are “potential paradigm-shifters” (2005a, p. 1380). Similarly, Galí and Rabanal state that “the bulk of the evidence” they report “raises serious doubts about the importance of changes in aggregate technology as a significant (or, even more, a dominant) force behind business cycles” (2005, p. 274). We argue that these researchers’ conclusions–and the usefulness of their procedure–are suspect when the procedure is closely examined. In general, using SVARs to evaluate alternative economic models is an attempt to develop business cycle theory using a simple time series technique and minimal economic theory. In the common approach to this sort of analysis, researchers run VARs on the actual data, impose some identifying assumptions on the VARs in order to back out empirical impulse responses to various shocks, and then compare those empirical SVAR impulse responses to theoretical responses that have been generated by the economic model being evaluated. Models that generate theoretical responses that come close to the SVAR responses are thought to be promising, whereas others are not. Here we focus on the SVAR literature that uses a version of this common approach with long-run restrictions in order to identify the effects of technology shocks on economic aggregates. The main claim of this literature is that its particular SVAR procedure can confidently distinguish between promising and unpromising classes of models without the researchers having to take a stand on the details of nontechnology shocks, other than minimal assumptions like orthogonality. We evaluate this claim by subjecting the SVAR procedure to a natural economic test. We treat a multiple-shock business cycle model as the data-generating mechanism, apply the SVAR procedure to the model’s data, and see if the procedure can do what is claimed for it. We find that, in principle, the SVAR claim of not needing to specify the details of nontechnology shocks is correct if the researcher has extremely long time series to work with. Regardless of the magnitude and persistence of other shocks, a researcher who applies the SVAR procedure to extremely long time series drawn from our model will conclude that the data are generated from our model and will be able to confidently distinguish whether the data are generated by our model or by a very different model. With series of the length available in practice, however, the SVAR claim is incorrect. Our test shows that the impulse responses to technology shocks identified by the SVAR procedure vary significantly as the magnitude and persistence properties of other shocks vary, even though, obviously, the theoretical impulse responses do not. In particular, depending on the specification of the VAR, when other shocks play a nontrivial role in output fluctuations, a researcher who applies the SVAR procedure to data from our model either will conclude that the data are not generated from our model or will not be able to confidently distinguish whether the data are generated by our model or by a very different model. If, however, other shocks play only a trivial role in output fluctuations, then the SVAR impulse responses are close to the theoretical ones, and researchers can use the impulse responses to confidently distinguish between our model and very different models. We obtain intuition for our findings from two propositions–an infinite-order representation result and a first-order representation result. The infinite-order representation result shows that when a VAR has the same number of variables as shocks, the variables in the VAR have an infinite-order autoregressive representation in which the autoregressive coefficients decay at a constant rate. Since we use a two-variable VAR and our model has two shocks, this result implies that the VAR has an infinite-order representation. With our parameter values, the coefficients in this representation decay very slowly. Even so, if very long time series are available, the empirical impulse responses are precisely estimated and close to the theoretical impulse responses. With series of the length available in practice, however, the estimated impulse responses are not close to the theoretical impulse responses when the nontechnology shock is not trivial. A deconstruction of the SVAR’s poor performance reveals that its problem is that the small number of lags in the estimated VAR dictated by available data lengths makes the estimated VAR a poor approximation to the infinite-order VAR of the observables from the model. That is, the VAR suffers from lag-truncation bias. Our other proposition shows that, when the VAR has sufficiently many variables rel-