Adapting to Unknown Disturbance Autocorrelation in Regression with Long Memory

Abstract

We show that it is possible to adapt to nonparametric disturbance autocorrelation in time series regression in the presence of long memory in both regressors and disturbances by using a smoothed nonparametric spectrum estimate in frequency-domain generalized least squares. When the collective memory in regressors and disturbances is sufficiently strong, ordinary least squares is not only asymptotically inefficient but asymptotically non-normal and has a slow rate of convergence, whereas generalized least squares is asymptotically normal and Gauss-Markov efficient with standard convergence rate. Despite the anomalous behaviour of nonparametric spectrum estimates near a spectral pole, we are able to justify a standard construction of frequency-domain generalized least squares, earlier considered in case of short memory disturbances. A small Monte Carlo study of finite sample performance is included.

Semiparametric Sieve-Type Generalized Least Squares Inference

“…Robinson (1991)). In the presence of long-range-dependence, a related approach was investigated by Hidalgo and Robinson (2002), who demonstrated that the unknown spectral density of the errors may be replaced by a suitable smoothed nonparametric estimator without any effect on the first-order asymptotic distribution of their (approximate) GLS estimator.…”

Developments in the Analysis of Spatial Data

“…To some degree the extension of theory for the time series case d = 1 to d > 1 is straightforward but particular features cause difficulty: for multilateral models, least squares (LS) tends to be inconsistent and use of a likelihood approximation is important, as first noted by Whittle (1954); the "edge-effect" is a source of bias in the central limit theorem when d ≥ 2, and methods of overcoming it are discussed in Guyon (1982), Dahlhaus and Künsch (1987), Robinson and Vidal Sanz (2006). Under long range dependence, limit distributional behaviour may be nonstandard, without weighting of a type used more generally to achieve efficiency (see e.g., Fox and Taqqu (1986), Hidalgo and Robinson (2002)). …”

Testing a sub-hypothesis in linear regression models with long memory covariates and errors

“…Regression models with long memory in design and errors are useful when the long memory in design variables may be not enough to explain the long memory in the response process, cf. [13]. Such models are found useful in economics and finance when observing high frequency data where spot returns are regressed on forward premiums.…”