Knowledge that Transforms

A WELL DEVELOPED EXACT STATISTICAL THEORY exists for hypothesis testing in the normal linear regression model when the errors are independent and homoscedastic. In the more general case where the error covariance matrix is nonscalar and depends on a set of unknown parameters, exact analysis is difficult and reliance is usually placed on asymptotic approximations for large sample size n. In this paper, higher-order asymptotic expansions are developed for comparing the size and power of some common procedures for testing linear hypotheses on the regression coefficients in a class of generalized normal linear models. The class investigated is essentially the same as in Breusch [4] and Magnus [6] and includes many of the examples of heteroscedasticity and autocorrelation discussed in the literature. We assume simply that the regressors are nonrandom and that the error covariance matrix is a smooth function of a few parameters that can be efficiently estimated by maximum likelihood. Tests based on the Wald, likelihood ratio, and Lagrange multiplier principles are considered. These principles lead to three tests which, though distinct in finite samples, are locally asymptotically equivalent and share certain asymptotic optimality properties. Of course, there are infinitely many other tests that are asymptotically equivalent to the ones examined here. Although the techniques of this paper can be applied to any of them, our results concern only the tests arising from the three traditional principles. We show that, to a second order of approximation under local alternatives, the likelihood ratio test statistic is a simple average of the Wald statistic and the Lagrange multiplier statistic. When the null hypothesis is one dimensional, the three tests are, to second order, equally powerful; that is, after the critical regions are adjusted so that the tests have (to order n - i) the same size, the local power functions differ by terms of smaller order than n -. When the null hypothesis contains more than one

[This paper investigates the exact sampling distribution of the least squares estimator of β in the model y"t = @m + @by"t"-"1 + @u"t where the @u"t are independently N(0, @s extasciicircum2). The distribution is calculated for the case where y"0 is a known constant and where y"0 is a random variable. Given y"0 is a constant we prove a small @s asymptotic result and compute the exact powers of nonsimilar tests of the random-walk hypothesis β = 1 and of the stability hypothesis β = 0.9. The exact powers of a test of the stability hypothesis are calculated for the case where y"0 is random. The accuracy of the standard normal approximation is examined for both start-up regimes.]

[I examine the effects of price discrimination on the equilibrium prices, number of firms, and level of total surplus in a monopolistically competitive market. The main finding is that uniform pricing is more (less) efficient than is price discrimination when the purchases made by the consumers who are discriminated against constitute a small (large) proportion of the total purchases.]

[This paper investigates two-person bargaining under incomplete information where one player has strictly better information about the potential value of the transaction than the other. The implications of informational barriers to trade are explored, and optimal bargaining mechanisms are characterized.]

An approximation to the inconsistency introduced by imposing an incorrect restriction on a parametric model is given. The approximation can be applied to estimators generated by optimizing any objective function satisfying certain regularity conditions. Examples given include analysis of misspecification in discrete choice and time-series models estimated by maximum likelihood, and in a nonlinear regression model. SPECIFICATION ERROR ANALYSIS in the linear regression model has been studied by Theil [1], who gives formulas for, e.g., the effect of leaving out relevant variables on the expected values of the estimators of the coefficients of the included variables. In this paper we suggest analogous formulas for estimators obtained by optimizing an objective function subject to restrictions. We have in mind maximizing (1/n) x loglikelihood and will usually use this terminology. We consider the effect on the limit of the restricted estimator of a small violation of the restrictions. In the linear regression case our formula coincides with that given by Theil. In order to keep our results widely applicable and to avoid a mass of unnecessary detail we make assumptions on the asymptotic behavior of the loglikelihood function itself, rather than on the data-generating process per se. Many alternative sets of assumptions on the data densities can lead to the behavior we require of the loglikelihood functions. These will not be pursued here. The interested reader is referred to, e.g., White [12] for the case of independent observations and Kohn [8] for the time-series case. 1. GENERAL FORMULAS The general approach we take is based on a linear approximation to the likelihood function at the maximum likelihood estimator. It is in this sense that our analysis is local. For some models the local and global specification error results coincide; a well known case is the effect of omitted regressors in the linear regression model. Essentially the only cases involve linearity, although often there is agreement regarding the signs of the inconsistency. We show below that the local and global results even fail to coincide in the case of misspecified AR processes. Generally however, the global results are unknown.2 Taylor expansions are typically used together with assumptions on the data generating process to obtain the asymptotic distribution of the maximum likelihood estimator (Cramer [3]). In this paper we will not concern ourselves with asymptotic distributions of Vn -normed MLE's since these have been worked

A game form is acceptable if for every preference profile, a Nash equilibrium exists and the outcomes corresponding to Nash equilibria are Pareto efficient. A game form is strongly consistent if the set of strong Nash equilibria is always nonempty. The paper shows that no game form can be both acceptable and strongly consistent. The set of game forms which are both acceptable and dominance-solvable is also characterized in terms of the effectivity functions of game forms.

How potential entrants to an open-access fishery form their expectations determines the fishery's adjustment path to a steady state but not the steady state values themselves. It is well known that, in the standard model with myopic expectations (those based on current values), boats enter the fishery only when the fish stock is greater than its steady state stock. We show that, with rational expectations (perfect foresight), however, boats may enter when the fish stock is much lower than its steady state value if the boat fleet is sufficiently small. This paper contrasts myopic and rational expectations within a general dynamic model of an open-access fishery.

Estimators obtained by maximizing a likelihood function are studied in the case where the true p.d.f. does not necessarily belong to the family chosen for the likelihood function. When such a procedure is applied to the estimation of the parameters of the first order moments, it is possible to prove a necessary and sufficient condition for its consistency. Asymptotic normality is shown as well as the existence of a lower bound for the asymptotic covariance matrix. It is also seen that this bound can be reached if consistent estimates are available for the parameters of the second order moments. Finally, a necessary and sufficient condition for the consistency if the pseudo maximum likelihood estimation of the first and second moments is given.

This paper considers price competition among firms when there are capacity constraints and buyers have limited ability to visit firms. A natural method of allocating buyers among firms arises in the equilibrium of the buyers' search game. Sufficient conditions are given under which the buyers' equilibrium varies continuously with the prices charged by firms. Capacity constraints are used to guarantee that this ensures existence of (mixed strategy) equilibria for the pricing game played by sellers. We show that natural pure strategy equilibria arise when the game is made large in appropriate ways. SINCE BERTRAND'S CRITICISM OF COURNOT, it has been known that models where sellers compete with prices may have undesirable properties when the commodities traded are homogenous. In the simplest models with constant marginal costs, price may be driven down to the competitive level, even when there are only two firms. More problematic, if sellers are subject to capacity constraints, Nash (in prices) equilibrium may not exist at all. These problems all arise because of a discontinuity in firms' profit functions that arises when the prices of two or more firms are equal. There have been many attempts to resolve this problem. The simplest is to constrain sellers' prices to be equal, and to force them to compete in quantities. This is the basis of most of the literature on monopolistic competi