Adaptive Estimators and Tests of Stationary and Nonstationary Short- and Long-Memory ARFIMA–GARCH Models

Abstract

This article considers the fractionally autoregressive integrated moving average [ARFIMA(p,d,q)] models with GARCH errors. The process generated by this model is short memory, long memory, stationary, and nonstationary, respectively, when d ∈ (-1/2, 0), d ∈ (0, 1/2), d ∈ (-1/2, 1/2), and d ∈ (1/2, ∞). Using a unified approach, the local asymptotic normality of the model is established for d ∈ ∪∞j = 0(J - 1/2, J + 1/2). The adaptivity and efficiency of the estimating parameters are discussed. In a class of loss functions, the asymptotic minimax bound of the estimators for the model is given when the density f of rescaled residuals is unknown. An adaptive estimator is constructed for the parameters in the ARFIMA part when f is symmetric, and a general form of the efficient estimator is also constructed for all the parameters in the ARFIMA and GARCH parts. When the density f is unknown, Wald tests are constructed for testing the unit root + 1 against the class of fractional unit roots. It is shown that these tests asymptotically follow the chi-squared distribution and are locally most powerful. These results are new contributions to the literature, even for the ARFIMA model with iid errors, except for the adaptive estimator in this case with d ∈ (0, 1/2). The performance of the asymptotic results in finite samples is examined through Monte Carlo experiments. An application to the U.S. Consumer Price Index inflation series is given, and a clear conclusion from this is that the series is neither an I(0) nor an I(1), but rather than an I(d) process with d ≈ 0.288.