Unbounded heteroscedasticity in autoregressive models

This paper develops the asymptotic theory for stable autoregressive models in which the noise variance grows in a polynomial-like fashion. It is shown that the asymptotic distribution of the OLS estimator of the coefficient vector is multivariate normal with a covariance matrix that depends on the order, k, of the variance growth. A consistent estimator of k is proposed, which delivers heteroscedasticity-robust test statistics. The case of “variance decline” is studied as well. It is demonstrated that by means of a simple data transformation producing the time reversed image of the original series, the problem of “variance decrease” can be reformulated in terms of that of polynomial-like variance growth. Simulation evidence suggests that the new procedures work quite well in small samples. Finally, the new methods are used in order to measure potential asymmetries in business cycles dynamics among several OECD countries.