Consistency of the LSE in Linear regression with stationary noise

Guy Cohen¹ ; Michael Lin² ; Arkady Tempelman³

¹ Department of Mathematics Ben-Gurion University Beer-Sheva, Israel
² Department of Mathematics Ben-Gurion University of the Negev Beer-Sheva, Israel
³ Department of Statistics Penn State University University Park, PA 16802, U.S.A.

Colloquium Mathematicum, Tome 100 (2004) no. 1, pp. 29-71

Résumé

We obtain conditions for $L_2$ and strong consistency of the least square estimators of the coefficients in a multi-linear regression model with a stationary random noise. For given non-random regressors, we obtain conditions which ensure $L_2$-consistency for all wide sense stationary noise sequences with spectral measure in a given class. The condition for the class of all noises with continuous (i.e., atomless) spectral measures yields also $L_p$-consistency when the noise is strict sense stationary with continuous spectrum and finite absolute $p$th moment, $p\geq 1$ (even without finite variance). When the spectral measure of the noise is not continuous, we assume that the non-random regressors are Hartman almost periodic, and obtain a spectral condition for $L_2$-consistency. An additional assumption on the regressors yields strong consistency for strictly stationary noise sequences. We also treat the case when the regressors are random sequences, with trends having some good averaging properties and with additive stationary ergodic random fluctuations independent of the noise. When the noise and the fluctuations have disjoint point spectra and the noise is strict sense stationary, we obtain strong consistency of the LSE. The results are applied to amplitude estimation in sums of harmonic signals with known frequencies.

Détail

DOI : 10.4064/cm100-1-5

Keywords: obtain conditions strong consistency least square estimators coefficients multi linear regression model stationary random noise given non random regressors obtain conditions which ensure consistency wide sense stationary noise sequences spectral measure given class condition class noises continuous atomless spectral measures yields p consistency noise strict sense stationary continuous spectrum finite absolute pth moment geq even without finite variance spectral measure noise continuous assume non random regressors hartman almost periodic obtain spectral condition consistency additional assumption regressors yields strong consistency strictly stationary noise sequences treat regressors random sequences trends having averaging properties additive stationary ergodic random fluctuations independent noise noise fluctuations have disjoint point spectra noise strict sense stationary obtain strong consistency lse results applied amplitude estimation sums harmonic signals known frequencies