Knowledge that Transforms

THE RESEARCH on the economic theory of the cost-of-living index has not paid much attention to the proposals made by Frisch and Wald in the thirties. In Frisch [3,4] the was developed and tested on certain examples. Wald [8] succeeded in deriving a new for the index of cost of living in an article containing a curious editorial footnote by Frisch on the comparative advantages of both methods. Banerjee [1] presented a simplification of the derivation of Wald's new formula. In an article commemorating Frisch, Samuelson [7] asked for a study of the relative merits of Frisch's and Wald's proposals to which he added a variant of his own. Recently Banerjee [2] succeeded in showing that Wald's new formula is the true cost-of-living index for a general quadratic utility function. In this paper I give an alternative, mathematically equivalent but economically more meaningful, derivation which highlights the resemblance of Wald's index to the true cost-of-living index corresponding to the familiar Klein-Rubin-Stone-Geary utility function (Section 2). This derivation provides a convenient framework for discussing Frisch's double-expenditure method (Section 3) and Samuelson's proposal (Section 4) and for assessing the relative merits of them. I hope this note is a (partial) answer to Samuelson's question.

The asymptotic distribution of prediction is derived for the general simultaneous equation model with lagged endogenous variables and vector autoregressive errors. The results turn out to be particularly simple when no lagged endogenous variables are present. errors. It seems worthwhile to extend Schmidt's results, since the estimation of dynamic simultaneous equation models with autoregressive errors is now fairly commonplace in the literature. Also, as noted by Schmidt (5), the effects of parameter estimation can produce a significant effect on forecast confidence intervals. In fact, the results we obtain are computationally feasible making it possible to calculate the term due to parameter estimation in a forecast confi- dence interval. For the seemingly unrelated regression model with vector autore- gressive errors, but with no lagged dependent variables, the results turn out to be particularly straightforward generalizations of those obtained by Baillie (1), who

A. Ronald Gallant, Alberto Holly, Statistical Inference in an Implicit, Nonlinear, Simultaneous Equation Model in the Context of Maximum Likelihood Estimation, Econometrica, Vol. 49, No. 2 (Mar., 1981), p. 554

The exact finite-sample density function of the likelihood ratio (LR) test of overidentifying restrictions in linear equations with any number of included endogenous and exogenous variables is presented, along with its derivation. Properties of the density function and the LR test are presented and discussed. A new approximation for the LR test critical region is proposed and studied. 1. INTRODUCTION AND SUMMARY THE CRITICAL NATURE of identification in simultaneous equation models is well recognized. The identification status of equations within a simultaneous equation system helps determine which estimator to use and dictates some properties of the alternative estimators. It represents a statistical hypothesis that is logically prior to many other standard estimation and testing procedures carried out for model construction and evaluation. The major purposes of this paper are (i) analytical exploration of the likelihood ratio identifiability test statistic for equations with any number of endogenous and exogenous variables and (ii) proposal of a new approximate critical region for testing overidentifying restrictions in such equations. Anderson and Rubin [2] proposed a likelihood ratio test of overidentifying restrictions imposed on the structural parameter space along with their derivation of LIML estimates. Basmann [5] proposed a similar statistic based on the GCL estimation method. McDonald [20] derived the exact finite sample density function for the LR test statistic for an equation containing exactly two endogenous variables. Richardson [26] derived the finite sample density function for the two endogenous variable GCL version of the identifiability test statistic. Both McDonald and Richardson showed that the moments of the exact functions converge to moments of the F distribution as the concentration parameter increases without bound. Each gave necessary conditions for existence of exact moments. McDonald showed that the GCL test statistic will not possess more moments than the LIML statistic in the two-equation case. In further analysis, Kadane [13, 14] showed that both tests are consistent and that both the LR and GCL identifiability test statistics converge to the F distribution as the structural disturbances grow small. Some parallel developments have occurred in multivariate statistical analysis. These are not reviewed here because they are mainly tangential or else are cited at appropriate places in this paper.

Alice Nakamura, Masao Nakamura, A Comparison of the Labor Force Behavior of Married Women in the United States and Canada, with Special Attention to the Impact of Income Taxes, Econometrica, Vol. 49, No. 2 (Mar., 1981), pp. 451-489

[The effects of an improvement in information on the efficiency of risk-bearing are studied under various systems of incomplete markets. With sequential futures markets for uncontingent delivery, the welfare effects are indeterminate in sign, except under special circumstances. In the presence of options markets, however, an improved information structure is almost surely beneficial.]

[A discrete-choice probability model can be estimated from a sample stratified on the choice variable by maximizing the "pseudo-likelihood," a quantity closely related to the log likelihood for a random sample. We investigate the asymptotic properties of the estimator, and show that it is consistent, asymptotically normally distributed, and satisfies a commonly used criterion for asymptotic efficiency.]