The Slow Convergence of Ordinary Least Squares Estimators of α, β and Portfolio Weights under Long‐Memory Stochastic Volatility

Abstract

We consider inference for the market model coefficients based on simple linear regression under a long memory stochastic volatility generating mechanism for the returns. We obtain limit theorems for the ordinary least squares (OLS) estimators of α and β in this framework. These theorems imply that the convergence rate of the OLS estimators is typically slower than if both the regressor and the predictor have long memory in volatility, where T is the sample size. The traditional standard errors of the OLS‐estimated intercept () and slope (), which disregard long memory in volatility, are typically too optimistic, and therefore the traditional t‐statistic for testing, say, α = 0 or β = 1, will diverge under the null hypothesis. We also obtain limit theorems (which imply slow convergence) for the estimated weights of the minimum variance portfolio and the optimal portfolio in the same framework. In addition, we propose and study the performance of a subsampling‐based approach to hypothesis testing for α and β. We conclude by noting that analogous results hold under more general conditions on long‐memory volatility models and state these general conditions which cover certain fractionally integrated exponential generalized autoregressive conditional heteroskedasticity (EGARCH) models.