Nonstationary Volatility Regressions

A popular approach to forecast variance is to use the fitted value of a simple OLS autoregression of realized variance measures. However, many financial returns are known to have highly persistent and possibly nonstationary volatilities. Under the nonstationarity, the asymptotic behaviors of the OLS estimators are unclear. We consider the autoregressions with spot, integrated, and realized variance measures when the spot variance process is nonstationary, and derive the asymptotic properties of the OLS estimators of the autoregressions. In particular, the asymptotic biases of the OLS estimators for the regressions with the integrated and realized variances are obtained. We then consider a feasible instrumental variable (IV) approach to reduce the bias of the OLS estimator, where the instrument equals the lagged value of the variable of interest, and show that the feasible IV estimator obtained from the realized variance is asymptotically equivalent to the infeasible OLS estimator obtained from the regression with the spot variance. Simulation results corroborate the theoretical findings of the paper.