Knowledge that Transforms

Some decision rules for discriminating among alternative regression models are proposed and mutually compared. They are essentially based on the Akaike Information Criterion as well as the Kullback-Leibler Information Criterion (KLIC) : namely, the distance between a postulated model and the true unknown structure is measured by the KLIC. The proposed criteria combine the parsimony of parameters with the goodness of fit. Their relationships with conventional criteria are discussed in terms of a new concept of unbiasedness .

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

[Very often, an index number used in an economic model has been constructed in two or more stages. If the two stage procedure gives the same answer as a single stage procedure, then Vartia calls the index number formula "consistent in aggregation." Paasche and Laspeyres indexes have this consistency in aggregation property, but these index number formulae are consistent only with very restrictive functional forms for the underlying aggregator (i.e., utility or production) function. The present paper shows that the class of superlative index number formulae has an approximate consistency in aggregation property, where a superlative index number formula is one which is consistent with a flexible functional form for the underlying aggregator function. The paper also contains some empirical examples which both illustrate the main theorem and also indicate that the chain principle for constructing index numbers is preferable to the fixed base method. Finally, the paper proves some theorems about the class of pseudo-superlative index numbers.]

central objective of this paper is to present a series expansion of nonlinear estimators in order to facilitate an analysis of the distributions of such estimators. Where the estimator under consideration is a maximum likelihood estimator, the method provides somewhat more information, as well as higher order approximations to the distributions of the nonlinear estimators than does the usual theory which demonstrates asymptotic normality. The method is also useful for a wide class of estimators including those defined only implicitly by the estimating procedure. Approximations to the distributions of the nonlinear estimators can be obtained in many cases even when the moments do not exist. In any event, it is to be hoped that the analytic procedures discussed in this paper will simplify the analysis of specific cases and will shed more light on the general formulation of nonlinear estimation problems. The remainder of this paper is in four sections. The first section presents the basic theory and analyzes the asymptotic distributions of nonlinear estimators in correctly specified models. This is followed in the second section by a brief discussion of a number of interesting examples. The third section compares the approach outlined in this paper with the traditional maximum likelihood and general nonlinear series expansions. In the fourth section the approximate asymptotic distribution of the regression residuals is derived. The general statement of the model to be considered in the following sections is given by:

DESPITE THE INTENSE ACTIVITY in the area of estimation of limited dependent variable (LDV) models (see, for example, the Fall 1976 issue of the Annals of Economic and Social Measurement), the interpretation of regression coefficients in truncated regression models has been largely ignored. This issue should not be taken lightly since the obvious interpretation which equates regression coefficients to partial derivatives of the conditional mean of the dependent variable is unfortunately incorrect. One specific implication of the results presented here is that whenever the dependent variable y in the usual classical linear normal regression model is truncated above and/or below, then the effect of the fth regressor on the conditional mean of y is proportional to, but not equal to, the fth regression coefficient. However, more generally this note presents a simple expression for the effect of the fth regressor on any conditional moment of y in terms of a generalized LDV model introduced by Poirier [3]. Poirier introduced a general LDV model which permitted skewness in a pre-truncated variable by transforming it within the class of transformations suggested by Box and Cox

[A social choice function is exactly and strongly consistent if for each profile of true preferences of individuals it possesses a strong equilibrium point which yields the same social choice as that corresponding to the profile of true preferences. We construct exactly and strongly consistent anonymous social choice functions, and investigate various additional desirable properties of our construction.]

We consider dynamic choice behavior under conditions of uncertainty, with emphasis on the timing of the resolution of uncertainty.Choice behavior in which an individual distinguishes between lotteries based on the times at which their uncertainty resolves is axiomatized and represented, thus the result is choice behavior which cannot be represented by a single cardinal utility function on the vector of payoffs.Both descriptive and normative treatments of the problem are given and are shown to be equivalent.Various specializations are provided, including an extension of "separable" utility and representation by a single cardinal utility function.

NUMEROUS STUDIES OF OPTIMAL MODELS in economic growth theory conducted with the aid of Pontryagin's maximum principle [3] led to important qualitive conclusions about the optimal development of economic systems over a finite or even infinite horizon (the latter is the more natural statement of the problem). At the same time almost all authors have been limited by consideration of production functions of only a narrow class, as a rule the class of concave functions. Concave production functions are known to be a good approximation of economic reality when the economy is in a high state of economic development (for instance, when the ratio of capital K to labor force L is great). However, accurate analysis of growth in certain less developed countries leads one to the conclusion that economic description by a concave function is not always applicable and that it is necessary to expand the class of production functions under consideration for a more adequate description of an economic system. Of special interest in this respect are functions which possess increasing returns to scale at an early stage of economic development and diminishing returns at a later stage. In turn, introduction of such functions generates a number of difficulties of a mathematical character (for example, Mangasarian's theorem on the sufficiency of the Pontryagin's conditions is not valid in this case). At present we do not know works where this problem has been studied definitively even in the one-dimensional case. Meanwhile, in our opinion, it is of considerable interest. In the present paper we consider a one-sector dynamic model of an economy with a convex-concave production function. The study is based on application of a maximum principle in Arrow's form [1] which is extremely useful for the analysis of the economic processes, since it allows taking phase constraints into consideration. Arrow's proposition has not been strictly proved; however, to my knowledge, there does not exist any contradictory examples.

In this paper we are concerned with the following question: in any economy with several public goods, what are the conditions under which the conventional optimality rule of equality between the sum of marginal rates of substitution and the marginal rate of transformation still holds even in the presence of distortionary taxation?Two cases are considered.In the first case, the taxes may be arbitrary.In the second case, the taxes are optimally chosen.