## Date of Award

1990

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Department

Economics

## First Advisor

R. Carter Hill

## Abstract

The dissertation will address small sample properties of estimators and test statistics in a nonlinear regression model. The Box-Cox transformation is attractive to economists because a family of functional forms can be compared simultaneously within the framework of classical statistical inference. Usually, maximum likelihood (ML) methods are used to estimate the Box-Cox model. In the present study, nonlinear two stage and iterative generalized least squares (IGLS) method are considered. The accuracy of probability statements concerning nonlinear models is often questionable in small samples. Therefore, the finite sample distribution of the asymptotic t-statistic in the Box-Cox model is derived using an Edgeworth expansion. Bootstrapping, the more practical method for obtaining small sample distributions, is also discussed. ML estimation of Box-Cox transformation suffers from a violation of the usual regularity conditions since the likelihood function of the Box-Cox model is not a proper density function. Since it is required that $y\sb t$ $>$ 0 in order for the Box-Cox transformation to be well-defined, the dependent variable is assumed to have a truncated normal distribution. The asymptotically equivalent covariance matrix estimators and test statistics--Lagrange multiplier, likelihood ratio and Wald--are compared in small samples. The risk superiority of the Stein-rule estimator to the ML estimator is known in the context of the linear model. The usefulness of Stein-like estimation in the nonlinear Box-Cox model is investigated by considering the finite sample risk properties of ML and Stein-like estimators. It is expected that this dissertation will make four major contributions to the current econometric literature. First, we introduce IGLS estimation of the Box-Cox model, and thus make liner statistical inference applicable to the nonlinear model. Second, the exact distribution of asymptotic t-ratios is derived and the bootstrap inversion of Edgeworth expansion is used. Third, the small sample distribution and power properties of three asymptotically equivalent test statistics are investigated. Fourth, shrinkage estimation is used in the determination of functional form.

## Recommended Citation

Kim, Minbo, "Small Sample Properties of Estimators and Test Statistics in Nonlinear Regression: The Box-Cox Transformation." (1990). *LSU Historical Dissertations and Theses*. 5066.

https://repository.lsu.edu/gradschool_disstheses/5066

## Pages

209

## DOI

10.31390/gradschool_disstheses.5066