*Econometric Theory, 7, 1991, 307-340.*

**ADAPTIVE RULES FOR SEMINONPARAMETRIC ESTIMATORS THAT ACHIEVE ASYMPTOTIC NORMALITY**

BRIAN J. EASTWOOD

*Dalhousie University*

A. RONALD GALLANT

*North Carolina State University*

In econometrics, seminonparametric (SNP) estimators originated in the consumer demand literature. The Fourier flexible form is a well-known example. The idea is to replace the consumer's indirect utility function with a truncated series expansion and then use a parametric procedure, such as nonlinear multivariate regression, to set a confidence interval on an elasticity. More recently, SNP estimators have been used in nonlinear time series analysis. A truncated Hermite expansion with an ARCH leading term is used as the conditional density of the process. The method of maximum likelihood is used to fit it to data.

One problem, though, is selecting the truncation point so that a confidence interval set on an evaluation functional such as an elasticity is accurate. We examine this problem in a simple situation: a univariate Fourier series expansion over its natural domain [0,2) fitted by least squares to equally space data. The true data-generating process is assumed to have a periodic, smooth response function and additive i.i.d. errors.

We first consider deterministic truncation procedures. The asymptotic normality and consistency of estimates of evaluation functionals are established. We find the rate of growth in the truncation point that minimizes a sharp upper bound on the Kolmogorov-Smirnov distance from the standard normal distribution. We examine the coverage of confidence intervals in a two-factor Monte Carlo experiment. The two factors are the variance of the errors and the rough ness of the response function of the data-generating process. Both theory and the Monte Carlo experiment suggest that deterministic truncation procedures have undesirable properties and do not give accurate confidence intervals.

We next consider adaptive procedures. We establish the asymptotic normality and consistency for a class of adaptive procedures which includes an upwards F-testing procedure. We re-examine confidence interval coverage by repeating the Monte Carlo experiment. It is much improved, and the undesirable properties of deterministic procedures are overcome. The results give much encouragement to use of SNP procedures in more complex estimation problems.