Long Memory Errors Research Articles

Nonparametric regression with long-memory errors

The fixed design regression model with long-memory errors is considered. The finite-dimensional asymptotic distributions of the properly normalised kernel estimators of the regression function are shown to be normal when the errors are a linear process.

Asymptotic expansion of M-estimators with long-memory errors

This paper obtains a higher-order asymptotic expansion of a class of M-estimators of the one-sample location parameter when the errors form a long-memory moving average. A suitably standardized difference between an M-estimator and the sample mean is shown to have a limiting distribution. The nature of the limiting distribution depends on the range of the dependence parameter $\theta$. If, for example, $1/3 < \theta < 1$, then a suitably standardized difference between the sample median and the sample mean converges weakly to a normal distribution provided the common error distribution is symmetric. If $0 < \theta < 1/3$, then the corresponding limiting distribution is nonnormal. This paper thus goes beyond that of Beran who observed, in the case of long-memory Gaussian errors, that M-estimators $T_n$ of the one-sample location parameter are asymptotically equivalent to the sample mean in the sense that $\Var(T_n)/\Var(\bar{X}_n) \to 1$ and $T_n = \bar{X}_n + o_P(\sqrt{\Var(\bar{X}_n)}).$

Testing for structural change in a long-memory environment

Long-memory time-series analysis is apt to be applied to economic time series which extend over many years, in which circumstances the possibility of structural breaks is likely to be entertained. Tests for a change in parameter values at a given time point are proposed in linear regression models with long-memory errors. Existing tests based on the assumption of serially independent or weakly dependent errors will typically be invalid in such an environment. The tests are derived in case of certain nonstochastic and stochastic regressors, and are given large-sample justification. A small Monte Carlo study of finite-sample behaviour is included.

Semiparametric Analysis of Long-Memory Time Series

We study problems of semiparametric statistical inference connected with long-memory covariance stationary time series, having spectrum which varies regularly at the origin: There is an unknown self-similarity parameter, but elsewhere the spectrum satisfies no parametric or smoothness conditions, it need not be in $L_p$, for any $p > 1$, and in some circumstances the slowly varying factor can be of unknown form. The basic statistic of interest is the discretely averaged periodogram, based on a degenerating band of frequencies around the origin. We establish some consistency properties under mild conditions. These are applied to show consistency of new estimates of the self-similarity parameter and scale factor. We also indicate applications of our results to standard errors of least squares estimates of polynomial regression with long-memory errors, to generalized least squares estimates of this model and to estimates of a cointegrating relationship between long-memory time series.