Estimation of stochastic difference equations with nonlinear restrictions

Abstract

<p>Let the parameters of the stochastic difference equation;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);(UNFORMATTED TABLE FOLLOWS);satisfy f((eta)) = 0, (2)(TABLE ENDS);where f((eta)) is a continuous, twice differentiable function of (eta) that takes values in r-dimensional Euclidean space r (LESSTHEQ) q + p, and (eta) = ((alpha)(,1), ...,(alpha)(,q), (gamma)(,1), (gamma)(,2), ..., (gamma)(,p))'. It is assumed that (psi)(,ti), i = 1, 2, ..., q are independent of the error process e(,t) . The model described by (1) and (2) contains a wide class of regression models used in practice. One example is the ordinary regression model with errors that follow a stationary or a nonstationary autoregression;Asymptotic properties of the least squares estimator of (eta) con- structed subject to the restrictions (2) are derived. In the derivation, the sum of squares for different (psi)(,ti) are permitted to increase at different rates as the sample size increases. The asymptotic results justify use of the usual regression statistics for inference about the parameter vector (eta). Models where the theory is applicable are discussed;The results of a Monte Carlo experiment comparing several esti- mation procedures are reported. The model for the experiment is a regression model with two regressors, one a random walk and one a sequence of normal independent random variables. The error proc- ess is a first order autoregressive process. The empirical variance of the nonlinear least squares estimator of (eta) and the empirical vari- ance of the generalized least squares estimator of (eta) based on the true value of the first order autocorrelation coefficient are compared. The ratio of the empirical variance of the nonlinear least squares estimator to that of the generalized least squares estimator varied from 0.98 to 1.48 for samples of size 25 and from 0.98 to 1.26 for samples of size 100. The empirical percentiles of the "t-statistics" were compared to the percentiles of the corresponding limiting distributions. The study indicated that if the parameter of the error process is very close to the stationarity boundary, a larger sample size is required for the approximations to work well than if the parameter is close to zero.</p>