Second Order Behaviour of M-Estimators in Regression with Long Memory Errors
by
Hira L. Koul
Michigan State University

A second order discrete time stochastic process is said to have long memory if its auto- correlations decrease hyperbolically to zero in the lag parameter, as the lag tends to infinity. The importance of the long memory in econometrics, hydrology, finance and numerous other physical sciences is abundantly demonstrated in Beran (1992, 1994), Baillie (1996), and references therein.

A surprising fact for these processes is as follows. Consider, for example, the problem of estimating the common mean based on n observations from a long memory Gaussian process. Since the marginal distribution is symmetric there are at least two estimators of this parameter, namely, the sample median and the sample mean. The surprising fact is that under the long memory Gaussianity, these two estimators are asymptotically equivalent in the first order, i.e., the difference between the sample median and the sample mean divided by the standard dviation of the sample mean tends to zero in probability, as the as the sample size n tends to infinity. This is in complete contrast to the i.i.d. situation where it is known that this sequence of r.v.'s converges in distribution to a normal r.v., with mean 0 and some positive variance. This phenomenon occurs more generally in terms of the other estimators of the mean, for long memory moving averages, not necessarily Gaussian, and in general regression models.

This talk will begin by discussing this first order equivalence of a class of M-estimators of the regression parameter vector to the least square estimator in linear regression models with the long memory moving average errors. Then a higher order asymptotic expansion of a class of these M-estimators is given. It is observed that a suitably standardized difference between an M-estimator and the least square estimator has a limiting distribution. The nature of the limiting distribution depends on the range of the dependence parameter d. If, for example, 1/3 < d < 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 < d < 1/3 then the corresponding limiting distribution is non-normal.

This talk is based on some joint work with D. Surgailis and Liudas Giraitis.

Some References

Baillie, R.T. and T. Bollerslev, (1994), The long memory of the forward premium. J. of International Money and Finance, 13, 309-324.

Beran, J. (1992). Statistical methods for data with long-range dependence. (With discussion). Statist. Sci. 7, 404-427.

Beran, J. (1994). Statistics for Long-Memory Processes. Monographs on Statistics and Applied Probab., 61. Chapman and Hall. N.Y.

Giraitis, L., Koul, H.L. and Surgailis, D. (1996). Asymptotic normality of regression estimators with long memory errors. Statist. and Probab. Letters, 29, 317-335.

Koul, H. L. and Surgailis, D. (1997). Asymptotic expansion of M-estimators with long memory errors. Ann. Statist. 25, 818-850.

Koul, H. L. and Surgailis, D. (1999). Second order behavior of M-estimators in linear regression with long memory parameter. To appear in J. Statist. Planning & Inference. (2000).

Date received: October 12, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafr-37.