Consistency of the LSE in Linear regression with stationary noise

Tom 100 / 2004

Guy Cohen, Michael Lin, Arkady Tempelman Colloquium Mathematicum 100 (2004), 29-71 MSC: Primary 37A30, 37A50, 62J05; Secondary 28D05, 47A35. DOI: 10.4064/cm100-1-5


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.


  • Guy CohenDepartment of Mathematics
    Ben-Gurion University
    Beer-Sheva, Israel
  • Michael LinDepartment of Mathematics
    Ben-Gurion University of the Negev
    Beer-Sheva, Israel
  • Arkady TempelmanDepartment of Statistics
    Penn State University
    University Park, PA 16802, U.S.A.

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