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 study introduces ordinally additive, ordinally linear, and ordinally Cobb-Douglas utility functions for the analysis of risky decisions when the uncertainty affects several attributes. Practical algorithms for the determination of utility functions with these forms are provided. Further, the study offers several risk invariance axioms on choice behavior under multidimensional risk. These axioms, for the first time, extend to the multidimensional context the heuristic correspondence between risk aversion and subjective wealth, heretofore familiar in only one dimension. In addition, the consequences of these new risk invariance axioms for utility functional forms in the multi-dimensional context are investigated. The result is a sequence of theorems which show that ordinally linear, ordinally Cobb-Douglas, and ordinally additive von Neumann-Morgenstern utility functions are characterized by the risk invariance axioms. IN THE STUDY OF DECISION MAKING under uncertainty, it is well known that plausible sets of axioms imply that the decision maker acts as if he maximizes his expected von Neumann-Morgenstern utility. (See [1], for example.) If the uncertain outcomes are multidimensional, then the appropriate utility concept is a function of many variables. This is the case, for example, for a firm choosing marketing policies which will affect sales and profits, for an individual faced with investment choices which will affect consumption during several years, and for a government deciding among projects which differ in their costs, outputs, and environmental impacts. In grappling with such problems, decision analysts have found it impossibly difficult to proceed with utility measured by a general function of the outcome variables. Instead, they have used multi-attribute utility functions with special forms, and found that their conclusions are sensitive to the particular form utilized. (See [13], for example.) Thus, it falls to theorists to develop testable hypotheses about risky choice which are equivalent to special (and, hopefully, convenient) functional forms for multiattribute utility. Fishburn [2, 3, and 4], Keeney [5 and 6], and Pollak [9, 10, and 11] have made contributions in this vein. This study introduces ordinally additive von Neumann-Morgenstern utility functions (i.e., those which are a monotonic transformation of a sum of functions, each of one variable) to the literature. I show that they should be well suited to practical decision analysis by presenting algorithms for their use. I propose several risk invariance axioms which plausibly extend to the multidimensional context the intuitive link between risk aversion and wealth in one dimension. These axioms are shown to characterize (in the presence of some other assumptions) ordinally additive, ordinally linear, and ordinally Cobb-Douglas von NeumannMorgenstern utility functions.

[A kernel of a set of alternative actions over which there is a partial order is defined in terms of optimality properties. It is shown to be the same as the generalized efficient set. A variety of theorems such as uniqueness, existence, and composition in terms of other sets are established. Related sets, such as quasi kernels and weak kernels, are also considered.]

[A certain set of weak rationality conditions is shown to be necessary and sufficient for a social decision function to be a cooperative game according to the formulation of von Neumann and Morgenstern. In exhibiting this broad connection between game theory and the theory of social choice, attention is focused on the critical role played by the blocking coalitions in such games.]

This paper analyzes three quarterly investment models for the detection of certain specifi- cation errors. The models are those of Anderson (1 and 2), Eisner (4), and Meyer-Glauber (10). The models are applied to thirteen manufacturing industries. A set of specification error tests developed by Ramsey (12, 13, and 14) are applied to the above models so as to detect the specification errors of omission of variables, incorrect functional form, simul- taneous equation problems, and heteroskedasticity. The models are ranked in order of the number of times they failed to be rejected by the specification error tests and the rank scheme is compared to that found in a previous study by Jorgenson, Hunter, and Nadiri (6), where more conventional criteria are used for ranking the models. industries, making use of both quarterly and annual data. Accelerator models and their variations (flexible accelerator models) as well as models considering internal and external finance are common in the estimation of the investment function. The lag structure between investment and its determinants and the manner in which replacement or the depreciation of capital is accounted for has also evoked the interest of researchers.2 From a perusal of the literature it is apparent that we face almost as many possible models for investment behavior as there are researchers. The problem at hand then is to come closer to a single general investment model from the numerous possibilities suggested. To do this we must investigate models of investment which differ both in terms of the determinants of investment as well as their lag structure so as to cover the broad range of specifications suggested. In a recent study, Jorgenson, Hunter, and Nadiri (6) (hereafter JHN) investi- gated various investment functions for several manufacturing industries using deflated, seasonally adjusted, quarterly data. JHN chose the best model based on the following criteria: (i) comparison of a given investment function with an auto- regressive scheme with regard to goodness to fit; (ii) comparison of a given invest- ment function with a model regressing investment on past anticipated investment expenditures; (iii) R2; (iv) estimates of the standard error of the fitted regression residuals corrected for degrees of freedom; and (v) Durbin-Watson ratio. The last three criteria mentioned above (and especially the third and fourth) are often the standard techniques employed by researchers in selecting a model specifica- tion.3

ITS DISAGREEABLE IMPLICATIONS about social choices have convinced many people that Arrow's impossibility theorem rests on unacceptably strong conditions. Dissatisfaction has centered on (but is not limited to) the conditions that the social ordering should be a weak ordering (WO) and that it should be independent of irrelevant alternatives (IIA). A long line of research, culminating in the results of Mas-Colell and Sonnenschein [18] and Fishburn [7], has shown, however, that the impossibility theorem is robust against reasonable relaxations of WO. (More accurately, if WO is weakened, say by waiving completeness or transitivity of the social ordering, and the remaining conditions are correspondingly strengthened to keep the problem interesting, the impossibility remains.) It seems that this line

A. D. Owen, A Proof that Both the Bias and the Mean Square Error of the Two-Stage Least Squares Estimator are Monotonically Non-Increasing Functions of Sample Size, Econometrica, Vol. 44, No. 2 (Mar., 1976), pp. 409-411

DISCUSSIONS OF IDENTIFICATION of parameters in simultaneous equation econometric models almost invariably assume that data are in the form of aggregative time series, i.e., only one measurement of each variable is available in each time period. This note shows that parameters in a model which is underidentified by the usual rank and order criteria at the aggregative level may be identified when disaggregated data aie available. The argument is presented in terms of a traditional textbook example of an underidentified model which consists of a demand and a supply function for a single commodity that are linear in price, and a market clearing equilibrium equation.