Date of Award
Doctor of Philosophy
The least squares estimator of the autoregressive parameter, LS((gamma)), in a first-order stochastic difference equation with independent, identically distributed random innovations is known to be asymptotically unbiased, efficient and consistent (as T (--->) (INFIN) or (sigma) (--->) 0) under the proper model specification. Further, LS((gamma)) has a limiting normal distribution around the true parameter, (gamma), if the random innovations are drawn from a normal population. These properties are not observed, however, in sample sizes that are typical of economic time series.;This thesis analytically derives the exact bias and mean squared error (MSE) for LS((gamma)) and provides detailed numerical calculations under various specifications of the nuisance parameter space. The analysis is applied to cases where the regression function is properly specified and the errors are normally distributed, nonnormally distributed and finally under conditions of regression function misspecification characterized by the exclusion of relevant or inclusion of irrelevant exogenous regressors. Additionally, small disturbance asymptotics is used to develop higher-order asymptotic approximations to the exact bias and MSE as well as some asymptotic results as (sigma) (--->) 0. Detailed numerical calculations of the analytic formulae are also provided.;The results suggest that both exact bias and MSE are significantly different from their asymptotic values in small samples and are very sensitive to the parameter space and exogenous data driving the process. Misspecification has a large effect on bias and MSE in finite samples, although, more so in cases where relevant information has been ignored. It was found that bias is relatively robust to Gram-Charlier nonnormality but MSE can be significantly affected. As a further result, the exact bias and MSE tended to be smaller in absolute value for models that possessed a unit or explosive root in all cases considered. The higher-order approximations were able to provide an improved measure of exact bias and MSE over the asymptotic ((sigma) (--->) 0) values in cases where the signal to noise ratio was relatively high. The approximations performed consistently well in cases where the model was suspected of having a unit or explosive root.
Peters, Thomas Armstrong, "The Finite Sample Analysis Of Least Squares Estimators In Dynamic Econometric Models" (1986). Digitized Theses. 1533.