Testing for fundamental vector moving average representations

We propose a test for invertibility or fundamentalness of structural vector autoregressive moving average models generated by non-Gaussian independent and identically distributed structural shocks. We prove that in these models and under some regularity conditions the Wold innovations are a martingale difference sequence (mds) if and only if the structural shocks are fundamental. This simple but powerful characterization suggests an empirical strategy to assess invertibility. We propose a test based on a generalized spectral density to check for the mds property of the Wold innovations. This approach does not require the specification and estimation of the economic agent's information flows or the identification and estimation of the structural parameters and the noninvertible roots. Moreover, the proposed test statistic uses all lags in the sample and it has a convenient asymptotic N(0, 1) distribution under the null hypothesis of invertibility, and hence, it is straightforward to implement. In case of rejection, the test can be further used to check if a given set of additional variables provides sufficient informational content to restore invertibility. A Monte Carlo study is conducted to examine the finite-sample performance of our test. Finally, the proposed test is applied to two widely cited works on the effects of fiscal shocks by Blanchard and Perotti (2002) and Ramey (2011).