Testing persistence in the context of conditional heteroscedasticity errors

Abstract

This article examines the power of two well-known procedures of fractional integration in the context of conditional heteroskedasticity in the variance. One of the methods is parametric while the other is semiparametric. Several Monte Carlo experiments conducted in this article show that both methods perform well to detect the order of integration of the series under the assumption that the underlying disturbances follow Generalized Autoregressive Conditional Heteroscedasticity (GARCH)-type errors. The methods are applied to 10 European stock market indices. The results indicate that the orders of integration of the series are close to 1 in all cases, being strictly higher than 1 in four countries. Moreover, taking the d-differenced processes, and estimating Fractionally Integrated GARCH (FI-GARCH) models on the squared residuals, fractional degrees of integration are obtained in the majority of the series.

Acknowledgements

The author gratefully acknowledges the financial support from the Ministerio de Ciencia y Tecnología (ECO2008-03035 ECON Y FINANZAS, Spain) and from a PIUNA Project at the University of Navarra. Comments of an anonymous referee are also gratefully acknowledged.

Notes

¹ The statistical properties of the GARCH models can also be found in the surveys by Gourieroux (1997) and Palm (1996).

² That means that if the test is directed against local alternatives of form: H _a : d = d _o  + δT ^−1/2, with δ ≠ 0, the limit distribution is normal, with unit variance and mean that cannot be exceeded in absolute value by any rival regular statistic under Gaussianity of u_t .

³ In spite of the infinite unconditional variance of the FIGARCH models, Baillie et al. (1996) suggest that the IGARCH result of Nelson (1990) and Bougerol and Picard (1992) can be used to show strict stationarity and ergodicity for 0 ≤ d ≤ 1.

⁴ Some methods to calculate the optimal bandwidth numbers in semiparametric contexts have been examined in Delgado and Robinson (1996) and Robinson and Henry (1996). However, in the case of the Whittle estimator of Robinson (1995a), the use of optimal values has not been theoretically justified. Other authors, such as Lobato and Savin (1998) use an interval of values for m, but we have preferred to report the simulations for the whole range of values of m.

⁵ The inclusion of a linear trend is relevant for some purposes. Thus, for example, suppose that we cannot reject the null with d_o  = 1 and a linear trend. In such a case, y_t becomes, for t > 1, a random walk model with a drift.

⁶ Allowing for weakly autocorrelated disturbances the results were substantially the same as those reported here for the white noise case.