THE CHOW TEST is a widely used procedure for testing for the equality of sets of coefficients in two linear regression models. However, when the number of observations in one of the models is less than the number of regression coefficients, the Chow test is incapable of testing the hypothesis of equality against that of inequality. It can never be concluded from the Chow test itself that the two sets are equal, although at times it may be possible to conclude that they are unequal. This point is implicit in [1], but has not been specifically discussed heretofore. The indeterminacy of the Chow test results from the insufficient number of observations. The two linear regression models, each of which is assumed to satisfy the conditions of the standard normal linear regression model, can be written as

In this paper an explicit and computationally convenient expansion of the exact finite sample distribution function of a quasi-maximum likelihood spectral estimator is given. In the majority of practical situations it will be necessary to estimate certain nuisance parameters of the distribution. Therefore, a method of evaluating these parameters is suggested and some Monte-Carlo evidence concerning the practical implementation of the results is given. 1 INTRODUCTfON AN EXTENSIVE SET of asymptotic results relating complex statistical analysis to the problems that arise in estimating spectra from the discrete Fourier transforms of time series data has been well established; see, for example, Brillinger [3] and Goodman [6]; and Hannan [11] has recently extended these results to cover the modified Fourier coefficients, proposed by Bingham, Godfrey, and Tukey [1] and defined in (2.3) below. Unfortunately it seems likely that the sample sizes required for one to approach the asymptotic position will not be available when considering the analysis of many economic time series. In a recent article, however, Hatanaka [12] has shown that the elimination of leakage produced by the modified Fourier coefficients is effective for small finite realizations and that it may be possible to recover the loss of degrees of freedom associated with the familiar estimator obtained by averaging over the modified periodogram.2 The purpose of the present paper is to extend these results by using complex statistical analysis to derive expressions for both the form and exact finite sample distribution of a spectral estimator obtained from the modified Fourier coefficients. Thus in the following section a brief exposition of some basic theory is given and a quasi-maximum likelihood estimation procedure is suggested. In Section 3 an exact expression for the finite sample distribution of the proposed estimator is given and shown to incorporate an established distributional result as a particular special case. In the majority of practical situations it will be necessary to estimate certain nuisance parameters of this distribution if it is to be employed and a method of evaluating these parameters is also suggested. It is well known, however, that while the replacement of nuisance parameters by consistent estimates will generally leave asymptotic theory intact the consequences of estimating nuisance parameters are unlikely to be negligible in finite sample theory. Since it is not possible to determine analytically the effect that the estimation of these nuisance parameters will have, the results of some simple Monte-Carlo

This paper provides necessary and sufficient conditions for it to be optimal to base decisions on estimates of the parameters that characterize a decision problem (e.g., profit maximization with an estimated price elasticity of demand). We show that the separation of parameter estimation from decision making generally yields lower utility than an integrated approach which takes account of estimation uncertainty. We evaluate the decision in the parameter estimation method and show that the resulting utility loss can be substantial. MANY ACTUAL DECISIONS are based on statistical estimates of parameters that help to characterize the decision environment. For example, a firm maximizing the expected utility of profit might find that its input and output decisions depend on unknown parameters of its demand function. Econometric estimates of such parameters might then be derived and utilized in making these decisions. The first purpose of this paper is to rigorously investigate whether it is correct to make decisions in this manner; in general, it is not. The second purpose is to investigate the decis'ion bias in decisions based on commonly employed parameter estimates. We will determine, for example, whether a price setting monopolist is mistakenly setting prices too high or too low when he bases his pricing decision on the maximum likelihood estimate of his demand equation. Finally, we provide a detailed numerical example to show that basing decisions on conventional parameter estimates can lead to large losses of utility. In Section 2, we introduce all notation, explain the procedure commonly used when basing decisions on values of unknown underlying parameters, and exhibit the decision-theoretic correct alternative procedure. When the optimal decisions under these two procedures are identical, we call the proper. We use the term summary value to refer not only to standard parameter estimates but to any single substituted for an unknown parameter in order to make decisions. This generalized concept is necessary because a that is appropriate for making decisions, in a sense defined below, need not have any of the properties of conventional parameter estimators. In Section 3, we derive under general assumptions necessary and sufficient conditions for the existence of proper values that are independent of the decision maker's utility function, U( ). This independence restriction is