Nonstationary Nonlinear Heteroskedasticity in Regression

This paper considers the regression with errors having nonstationary nonlinear heteroskedasticity. For both the usual stationary regression and the nonstationary cointegrating regression, we develop the asymptotic theories for theleast squares methods in the presence of conditional heterogeneity given as a nonlinear function of an integrated process. The conditional heteroskedasticity generated by an integrated process has more fundamental effects on the regression asymptotics than the one generated by a stationary process. In particular, the nonstationarity of volatility in the regression errors may induce spuriousness of the underlying regression. This is true for both the usual stationary regression and the nonstationary cointegrating regression, if excessive nonstationary volatility is present in the errors. Mild nonstationary volatilities do not render the underlying regression spurious. However, their presence makes the least squares estimator asymptotically biased and inefficient and the usual chi-square test invalid. We provide some illustrations to demonstrate the empirical relevancy of the model and theory developed in the paper. For this purpose, examined are US consumption function, EURO/USD forward-spot spreads and capital-asset pricing models for some major NYSE stocks.