Abstract
Summary Tests for identification through heteroskedasticity in structural vector autoregressive analysis are developed for models with two volatility states where the time point of volatility change is known. The tests are Wald-type tests for which only the unrestricted model, including the covariance matrices of the two volatility states, has to be estimated. The residuals of the model are assumed to be from the class of elliptical distributions, which includes Gaussian models. The asymptotic null distributions of the test statistics are derived, and simulations are used to explore their small-sample properties. Two empirical examples illustrate the usefulness of the tests in applied work.
Highlights
Identification by heteroskedasticity of structural shocks has become a standard tool in structural vector autoregressive (VAR) analysis
We focus on the simpler case to make the problem tractable and leave more general models for future research.) For developing our tests, we assume that the distribution of the residuals is elliptically symmetric, which covers the case of Gaussian VAR processes and models where the residuals have t distributions or mixtures of normal distributions
This is accomplished by obtaining the structural shocks from the reduced-form errors as εt = B−1ut, such that B is the matrix of impact effects of the shocks, and the covariance matrices of the structural errors are given by
Summary
Identification by heteroskedasticity of structural shocks has become a standard tool in structural vector autoregressive (VAR) analysis (see, e.g., Kilian and Lutkepohl, 2017, Chapter 14). If competing identifying restrictions are available, which are just-identifying in a conventional setting, identifying information from heteroskedasticity can be used as overidentifying restrictions, which opens up the possibility of formally testing identifying restrictions that are otherwise not testable Such tests are not considered in the present paper but are discussed in, for example, Lanne and Lutkepohl (2008), Netsunajev (2013), and Lutkepohl and Netsunajev (2017). We will develop formal frequentist tests that can help in assessing identification through heteroskedasticity for the special case of stable VAR models with two volatility regimes of the residuals Such simple models for the change in volatility have been considered by, for example, Rigobon (2003), Lanne and Lutkepohl (2008, 2014), and Lutkepohl and Schlaak (2018). The proofs of the asymptotic results for the test statistics are provided in Appendix A
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