## The Annals of Statistics

- Ann. Statist.
- Volume 7, Number 3 (1979), 484-489.

### Strong Consistency of Least Squares Estimates in Dynamic Models

T. W. Anderson and John B. Taylor

#### Abstract

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.

#### Article information

**Source**

Ann. Statist., Volume 7, Number 3 (1979), 484-489.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176344670

**Digital Object Identifier**

doi:10.1214/aos/1176344670

**Mathematical Reviews number (MathSciNet)**

MR527484

**Zentralblatt MATH identifier**

0407.62040

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62J05: Linear regression

Secondary: 60F15: Strong theorems

**Keywords**

Least squares strong consistency linear regression dynamic models

#### Citation

Anderson, T. W.; Taylor, John B. Strong Consistency of Least Squares Estimates in Dynamic Models. Ann. Statist. 7 (1979), no. 3, 484--489. doi:10.1214/aos/1176344670. https://projecteuclid.org/euclid.aos/1176344670