Adaptive Inference in Heteroscedastic Fractional Time Series Models

Abstract

We consider estimation and inference in fractionally integrated time series models driven by shocks which can display conditional and unconditional heteroscedasticity of unknown form. Although the standard conditional sum-of-squares (CSS) estimator remains consistent and asymptotically normal in such cases, unconditional heteroscedasticity inflates its variance matrix by a scalar quantity, λ > 1, thereby inducing a loss in efficiency relative to the unconditionally homoscedastic case, λ = 1. We propose an adaptive version of the CSS estimator, based on nonparametric kernel-based estimation of the unconditional volatility process. We show that adaptive estimation eliminates the factor λ from the variance matrix, thereby delivering the same asymptotic efficiency as that attained by the standard CSS estimator in the unconditionally homoscedastic case and, hence, asymptotic efficiency under Gaussianity. Importantly, the asymptotic analysis is based on a novel proof strategy, which does not require consistent estimation (in the sup norm) of the volatility process. Consequently, we are able to work under a weaker set of assumptions than those employed in the extant literature. The asymptotic variance matrices of both the standard and adaptive CSS (ACSS) estimators depend on any weak parametric autocorrelation present in the fractional model and any conditional heteroscedasticity in the shocks. Consequently, asymptotically pivotal inference can be achieved through the development of confidence regions or hypothesis tests using either heteroscedasticity-robust standard errors and/or a wild bootstrap. Monte Carlo simulations and empirical applications illustrate the practical usefulness of the methods proposed.