Heteroscedastic Proxy Vector Autoregressions

Abstract

In proxy vector autoregressive models, the structural shocks of interest are identified by an instrument. Although heteroscedasticity is occasionally allowed for in inference, it is typically taken for granted that the impact effects of the structural shocks are time-invariant despite the change in their variances. We develop a test for this implicit assumption and present evidence that the assumption of time-invariant impact effects may be violated in previously used empirical models.

Keywords:

• Identification through heteroscedasticity
• Proxy VAR
• Structural vector autoregression

Supplementary Materials

The online supplement contains additional theoretical derivations of the testing procedure, additional material related to the simulation study, as well as supplementary results for the empirical example. Moreover, the code for all computations is available via the online supplement.

Acknowledgments

The authors thank Lutz Kilian, Ambrogio Cesa-Bianchi and three anonymous reviewers for useful comments on an earlier version of the article. We thank Diego Känzig for making his dataset on oil market determinants available to us. The article was presented in the Vienna Joint Economics Seminar in November 2020 and at the Granger Centre Seminar of the University of Nottingham in February 2021. The authors are grateful for the comments of the participants. The authors thank the high-performance computing service at Freie Universität Berlin for providing the computational resources required for this work (available at DOI: 10.17169/refubium-26754).

Notes

1 The underlying matrices ${\Sigma }_b(m)$ were computed by simulation.

2 The correlation in Känzig’s (2019) study is even slightly lower. For our simulations, we have chosen correlations similar to Cesa-Bianchi, Thwaites, and Vicondoa (2020) which is considered as an example in Section 5 (see Table S.2 of the online supplement for details).

3 The dataset is available as supplementary material to the article on the web page of the European Economic Review.

4 An LM test for heteroscedasticity as described in Lütkepohl (2005, pp. 600–601) yields a test statistic of 138.33 which clearly rejects the null hypothesis at the 1% significance level and, hence, provides strong evidence for a change in variance.

5 We have also computed correlations between the proxy and the estimated monetary policy shock and show them in Table S.2 of the online supplement. They are in the lower range of what was used in the simulations. Thus, the proxy is not a particularly strong instrument.

6 We have also computed confidence intervals for the impulse responses using a wild bootstrap and present them in Figure S.4 of the online supplement. Although they provide somewhat smaller intervals for some of the impulse responses they are also problematic as Jentsch and Lunsford (2019) showed that they are invalid asymptotically. Apart from that they convey a similar picture as Figure 2. The same is true for Hall confidence intervals that could be used instead of the Efron intervals in Figure 2. We show them in Figure S.3 of the online supplement. We also emphasize that we show 68% confidence intervals in Figure 2 and the other figures. Using a larger confidence level, such as 90%, the intervals would, of course, be even wider than those in the figures.