## The Annals of Statistics

- Ann. Statist.
- Volume 9, Number 3 (1981), 531-543.

### Estimation of the Parameters of Stochastic Difference Equations

Wayne A. Fuller, David P. Hasza, and J. Jeffery Goebel

#### Abstract

Let $Y_t$ satisfy the stochastic difference equation $Y_t = \sum^q_{i=1} \psi_{ti} \alpha_i + \sum^p_{j=1} \gamma_j Y_{t-j} + e_t,$ where the $\{\psi_{ti}\}$ are fixed sequences and (or) weakly stationary time series and the $e_t$ are independent random variables, each with mean zero and variance $\sigma^2$. The form of the limiting distributions of the least squares estimators of $\alpha_i$ and $\gamma_j$ depend upon the absolute value of the largest root of the characteristic equation, $m^p - \sum^p_{j=1} \gamma_jm^{p-j} = 0$. Limiting distributions of the least squares estimators are established for the situations where the largest root is less than one, equal to one, and greater than one in absolute value. In all three situations the regression $t$-type statistic is of order one in probability under mild assumptions. Conditions are given under which the limiting distribution of the $t$-type statistic is standard normal.

#### Article information

**Source**

Ann. Statist., Volume 9, Number 3 (1981), 531-543.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176345457

**Digital Object Identifier**

doi:10.1214/aos/1176345457

**Mathematical Reviews number (MathSciNet)**

MR615429

**Zentralblatt MATH identifier**

0499.62082

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Secondary: 62J05: Linear regression

**Keywords**

Stochastic difference equation time series regression for time series autoregressive process

#### Citation

Fuller, Wayne A.; Hasza, David P.; Goebel, J. Jeffery. Estimation of the Parameters of Stochastic Difference Equations. Ann. Statist. 9 (1981), no. 3, 531--543. doi:10.1214/aos/1176345457. https://projecteuclid.org/euclid.aos/1176345457