Knowledge that Transforms

[The maximum likelihood approach is used to derive Hausman's specification test. Its asymptotic power is compared with the more conventional test procedures.]

Douglas W. Caves, Laurits R. Christensen, W. Erwin Diewert, The Economic Theory of Index Numbers and the Measurement of Input, Output, and Productivity, Econometrica, Vol. 50, No. 6 (Nov., 1982), pp. 1393-1414

MANY COMMODITIES can be viewed as bundles of individual attributes for which no explicit markets exist. It is often of interest to estimate structural demand and supply functions for these attributes, but the absence of directly observable attribute prices poses a problem for such estimation. In an influential paper published several years ago, Rosen [3] proposed an estimation procedure to surmount this problem. This procedure has since been used in a number of applications (see, for example, Harrison and Rubinfeld [2] or Witte, et al. [4]). The purpose of this note is to point out certain pitfalls in Rosen's procedure, which, if ignored, could lead to major identification problems. In Section 2 we summarize briefly the key aspects of Rosen's method as it has been applied in the literature. Section 3 discusses the potential problems inherent in this procedure and provides an example. Section 4 concludes with a few suggestions for future research.

For the Tobit model with independent observations, Amemiya [1] has established the strong consistency and asymptotic normality of a stationary point, 9, of the log-likelihood. The likelihood for dependent observations may be computationally intractable, so the behavior of 9 in the presence of serially correlated observations is of interest. Under a relaxation of Amemiya's assumption of independence, we prove that 9 is strongly consistent and asymptotically normal, and give an expression for the limiting covariance matrix.

A PROBLEM OF ESTIMATION that has long confronted many economists is the difficulty of estimating the parameters of equations with limited dependent variables on cross-section time-series (i.e., panel) data. While there are widely available packaged computer programs for estimating either (a) cross-section probit and Tobit models or (b) simple permanent-transitory, random-effects panel models with continuous dependent variables, there are no available computationally feasible methods of combining these two models. This is because the likelihood function that arises in such a combined model contains multivariate normal integrals whose evaluation is quite difficult, if not impossible, with conventional approximation methods. There is a widespread feeling among those working in the area that one possible method of evaluation, the use of quadrature techniques, is in principle possible but is in practice computationally too burdensome to consider (e.g., Albright et al. [2, p. 13]; Hausman and Wise [6, p. 12]). In this note we point out that this is true only of standard quadrature techniques such as trapezoidal integration or its improved variants; Gaussian quadrature, on the other hand, is extremely efficient and is well within the bounds of computational feasibility on modern computers. In what follows, we state the nature of the integrals that need to be evaluated, provide a brief exposition of Gaussian quadrature, and provide a numerical illustration of its use in

[In a portfolio problem with given asset returns, the portfolio efficient set is the set of portfolios chosen by any risk averse agent. Using an approach of Peleg and Yaari [13], we characterize the portfolio efficient set and derive some of its properties. In particular, we show that it may not be convex, proving that a central result of mean variance theory, the efficiency of the market portfolio, does not generalize. Finally, a characterization of the efficiency of several observations gives a version of revealed preference theory for incomplete markets.]

Noncooperative strategic behaviors are studied in the Malinvaud-Dreze-de la Vallee Poussin decentralized planning procedure. We depart from the assumption of myopic behavior by assuming that every agent takes into account the effect over a given period of time [0, T] of his answers to the Center. One shows that, for T large, every Nash equilibrium of the ensuing game in intertemporal strategies approaches: (i) a competitive equilibrium in an exchange economy, and (ii) a Lindahl equilibrium in an economy with public goods. Thus, the Center loses any significant influence on the income distribution.

Comments on a study by Conger and Campbell which postulated a 6 equation dynamic model including a fertility equation, estimated by 2 stage least squares on the basis of aggregate data for the U.S. The authors present a table with 3 sets of estimates for each of the 6 structural equations specified in the earlier study; 2 stage least squares estimates obtained for 1946-70 and 1946-76 and the estimates obtained by Conger and Campbell are indicated. The authors report that they were unable to obtain the results reported by the latter; their estimates were quite different and did not seem to support the conclusions offered in the earlier study. Estimates for the longer period were substantially different from those for the shorter. The authors believe that the Conger Campbell data also do not support the Conger-Campbell conclusions. Each point is discussed, with reference to the table. The equations are used to evaluate fertility, female participation, infant mortality, income, education, and medical expenditure.