#### Degree Type

Dissertation

#### Date of Award

1990

#### Degree Name

Doctor of Philosophy

#### Department

Statistics

#### First Advisor

Wayne A. Fuller

#### Abstract

We consider estimation for the p-th order autoregressive time series Y[subscript]t = [alpha][subscript]1 Y[subscript]t-1 + ·s + [alpha][subscript]pY[subscript]t-p + e[subscript]t, where Y[subscript]-p + 1, ·s, Y[subscript]0 are initial conditions and \e[subscript]t is a sequence of martingale differences with finite 2 + [nu] moments for [nu] > 0. We investigate estimation of ([alpha][subscript]1,[alpha][subscript]2,·s,[alpha][subscript]p) = [alpha] under two assumptions. In the first, all roots of the characteristic equation, m[superscript]p - [alpha][subscript]1[subscript] m[superscript]p-1 - ·s - [alpha][subscript]p = 0, are less than one in absolute value. In the second, one characteristic root is equal to one and the others are less than one in absolute value. The vector [alpha], the sum [theta][subscript]1 = [sigma][subscript]spi = 1p [alpha][subscript]i, and the largest root of the characteristic equation are parameters of interest;By considering the p-th order autoregressive time series in the forward direction and in the backward direction, we obtain an estimation method called symmetric least squares. We investigate an estimator of the largest characteristic root based on the symmetric least squares method, an estimator of the largest root based on the maximum entropy method, and an estimator based upon standardized variables. The pivotal statistic for the ordinary least squares estimator and the pivotal statistic for the symmetric least squares estimator converge in distribution to different distributions. The standardized least squares estimator, the symmetric least squares estimator, and the maximum entropy estimator converge to the same distribution;The simulation results indicate that the ordinary least squares one-sided test for a unit root is the most powerful test in the model without an intercept. The ordinary least squares test is less powerful than the other tests in the model with an intercept. In the stationary case, the ordinary least squares estimator of [sigma][subscript]spi = 1p [alpha][subscript]i is less efficient than the symmetric least squares estimator in both the model with intercept and the model without intercept;We also consider the first order multivariate time series, Y[subscript]t = A[subscript]1 Y[subscript]t - 1 + e[subscript]t where Y[subscript]0 is the initial condition and \ e[subscript]t is a sequence of independent identically distributed random vectors with finite 2 + [nu] moments for [nu] > 0. Tests for a unit root of the characteristic equation, ǁ [lambda] I[subscript]k - A[subscript]1 ǁ = 0, based on the ordinary least squares estimator and the standardized least squares estimator are studied. The standardized least squares test for a unit root is more powerful than the ordinary least squares test for the multivariate model.

#### DOI

https://doi.org/10.31274/rtd-180813-12400

#### Publisher

Digital Repository @ Iowa State University, http://lib.dr.iastate.edu/

#### Copyright Owner

Heon Jin Park

#### Copyright Date

1990

#### Language

en

#### Proquest ID

AAI9110553

#### File Format

application/pdf

#### File Size

198 pages

#### Recommended Citation

Park, Heon Jin, "Alternative estimators of the parameters of the autoregressive process " (1990). *Retrospective Theses and Dissertations*. 9880.

https://lib.dr.iastate.edu/rtd/9880