Open Access
May, 1979 Strong Consistency of Least Squares Estimates in Dynamic Models
T. W. Anderson, John B. Taylor
Ann. Statist. 7(3): 484-489 (May, 1979). DOI: 10.1214/aos/1176344670


The least squares estimate of the parameter matrix $\mathbf{B}$ in the model $\mathbf{y}_t = \mathbf{B'x}_t + \mathbf{u}_t$, where $\mathbf{u}_t$ is an $m$-component vector of unobservable disturbances and $x_t$ is a $p$-component vector, converges to $\mathbf{B}$ with probability one under certain conditions on the behavior of $x_t$ and $\mathbf{u}_t$. When $\mathbf{x}_t$ is stochastic and the conditional expectation of $\mathbf{u}_t$ given $\mathbf{x}_s$ for $s \leqslant t$ and $\mathbf{u}_t$ for $s < t$ is zero, then the least squares estimates are strongly consistent if the inverse of $\mathbf{A}_T = \sigma^T_{t=1} \mathbf{x}_t\mathbf{x}'_t$, where $T$ is the sample size, converges to the zero matrix and if the ratio of the largest to the smallest characteristic root of $\mathbf{A}_T$ is bounded with probability one.


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T. W. Anderson. John B. Taylor. "Strong Consistency of Least Squares Estimates in Dynamic Models." Ann. Statist. 7 (3) 484 - 489, May, 1979.


Published: May, 1979
First available in Project Euclid: 12 April 2007

zbMATH: 0407.62040
MathSciNet: MR527484
Digital Object Identifier: 10.1214/aos/1176344670

Primary: 62J05
Secondary: 60F15

Keywords: dynamic models , least squares , Linear regression , strong consistency

Rights: Copyright © 1979 Institute of Mathematical Statistics

Vol.7 • No. 3 • May, 1979
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