John S. Chipman

On Least Squares with Insufficient Observations

Abstract The optimality of the method of least squares is reconsidered when multicollinearity is present. An analysis is presented of the relationship between estimability and identifiability. The criterion of best linear minimum bias is developed, and shown to be equivalent to that of best linear conditionally unbiased estimation subject to complementary (non-estimable) linear restrictions. Imposition of erroneous estimable linear restrictions is shown to lower variances of estimators if and only if it biases them. All these results rely heavily on the use of the generalized inverse of a matrix, for which a new proof of existence and uniqueness is presented from the viewpoint of duality in linear spaces. Finally, estimation by minimum mean square error is proposed, and this is shown to reduce to the least squares method when either (a) regression coefficients have infinite prior variances, or (b) least squares estimators have small sampling variances.

Factor Price Equalization and the Stolper-Samuelson Theorem

It is important at the outset to remove an ambiguity in the statement just made. It is one thing to say that, given any initial equilibrium position, there exists a one-to-one association between commodities and factors such that a change in any commodity price will lead to a more than proportionate change (in the same direction) in the corresponding factor price. It is quite another thing to state that it is possible to find a one-to-one association between goods and factors in advance such that, starting from any equilibrium, a change in any commodity price will lead to a more than proportionate change in the price of the already specified factor. The first may be called the local version of the Stolper-Samuelson theorem, and the second the global version. Another distinction must be made. In the case of two factors and two commodities (the only case treated rigorously by Stolper and Samuelson), it turns out that if, as a result of an increase in the price of a good, one of the factor prices rises more than proportionately, then the other factor price must actually fall. In generalizing the theory to more than two commodities and two factors, it no longer holds that a more than proportionate increase in one factor price entails a fall in all the remaining factor prices. The case in which this does occur will be referred to as the strong form of the StolperSamuelson theorem, whereas the more general case will be called the weak form. We shall explore the various ways in which the theory first set forth by Stolper and Samuelson may be extended to n goods and factors. The main conclusions can be summarized as follows (the wording is necessarily vague, inasmuch as it constitutes a translation of mathematical conditions): (1) The Stolper-Samuelson theorem (strong, as well as weak form) is true locally (almost everywhere) for n 2, and globally whenever reversal of factor intensity is ruled out, as is, by now, quite well known. However, it is no longer true for n > 2, even under conditions which guarantee full factor price equalization. (2) Under certain special conditions, the weak form of

EFFICIENCY OF LEAST-SQUARES ESTIMATION OF LINEAR TREND WHEN RESIDUALS ARE AUTOCORRELATED

first-order stationary Markoff process with zero mean and autocorrelation coefficient p, - 1 < p < 1, the greatest lower bound for the efficiency of the least-squares estimator of 8 (relative to the Gauss-Markoff estimator) over the interval 0;p < 1 is .753763. This compares with a greatest lower bound of .535898 for the relative efficiency of the Cochrane-Orcutt estimator of 38.

An empirical implication of Auspitz-Lieben-Edgeworth-Pareto complementarity

Abstract It is shown that if a differentiable demand function satisfying the budget identity is generated by a strongly concave twice-differentiable utility function such that all commodities are complements according to the Auspitz-Lieben-Edgeworth-Pareto definition, then all goods are normal and the demand for each commodity is a decreasing function of its own price.

Estimation and Aggregation in Econometrics: An Application of the Theory of Generalized Inverses

Publisher Summary This chapter discusses a linear regression model in which X is the n × k matrix of n observations on k independent variables and V is the n × n sample covariance matrix of the residual errors in the regression. Multicollinearity refers to circumstances in which X has rank less than k; approximate multicollinearity is very common and is generally expected except in cases, where variables have been either omitted or aggregated expressly to avoid it. Singularity of V could result from certain types of sampling covariance patterns and sometimes is inherent in the economic structure of the problem. The k × k matrix U is interpreted in the chapter as the prior variance of the k × 1 vector β of regression coefficients, the latter being considered as a random variable in the Bayesian sense, with a subjective probability distribution. The formulation of the problems of best approximate aggregation and disaggregation presented in the chapter differs from W. D. Fishers in a number of essential respects.

Social utility and the gains from trade

The proposition that a country’s inhabitants will, or at least could, gain from participating in free international trade as opposed to remaining in a state of autarky, has long been believed, although only sketches of rigorous proofs have been supplied, notably those of Samuelson (1939, 1962), Kemp (1962), and Kenen (1957). 1 In this paper we shall pursue the question by developing the concept of a social utility function first formulated by Samuelson (1956). Full generality will be sacrificed in the interest of simplicity. We shall limit ourselves to the general framework of the Heckscher-OhlinSamuelson model, in which the factors of production are in fixed aggregate supply within countries, and are perfectly mobile among industries and without occupational preferences (cf. Samuelson (1953)). The production possibility set for the n final commodities which are assumed freely tradable internationally *,t zero transport costs isI assumed to be convex, and production is assumed to be carried out efficiently. Consumer demand is assumed generated by continuous, strictly monotone, and strictly convex preference orderings, which are represented by continuous concave utility functions Ui which attach zero utilities to zero bundles. We consider a class # of social welfare

THE TRANSFER PROBLEM ONCE AGAIN

Publisher Summary This chapter discusses the transfer problem. As a transfer would result in a larger volume of goods being shipped from the paying to the receiving country, it would raise freight rates from the paying to the receiving country but lower them for goods moving in the opposite direction, on account of the additional ballast traffic. The result would be a rise in the price spread between the two countries with respect to commodities moving from the paying to the receiving country and a fall in the price spread for commodities moving in the opposite direction. According to Wicksell, this would entail a rise in the prices of both import and export goods in the receiving country and a fall of both of these prices in the paying country, with no presumption as to the effects on the terms of trade. The approach adopted here is the classical one in which full employment is assumed to be maintained in both countries, rather than the approach of post-Keynesian employment theory which was fully and definitively treated by Metzler.

Projections, generalized inverses, and quadratic forms☆

The purpose of this paper is to present a unified treatment of the apparently unrelated topics, namely projections and generalized inverses, with some applications to the distribution theory of quadratic forms in Gaussian random variables. Other applications will also be mentioned. Only finite dimensional euclidean spaces are considered, but some of the theory can be extended to the infinite dimensional case using, for instance, the methods of [l]. However, the treatment then will no longer be elementary. It is customary to consider linear operators in finite dimensional spaces as matrices, and we shall do so here. For this we choose a fixed but arbitrary basis, and all considerations are related to it. Thus “projection” is synonymous with “idempotent matrix”. The properties of such matrices play a key role in what follows. The importance of projections has been noticed in the past (e.g., refs. 2-4, to name a few). However, they limited themselves to the cases of orthogonal projections, or symmetric idempotent matrices. Unless otherwise stated, the idempotent matrices are not symmetric in this paper. In Section I, several properties of such matrices are proved. It turns out that, by our methods, the results of the above authors can be simplified. In Section II, we deduce the existence (and uniqueness) of the so-called generalized inverse of a matrix [5], a computational formula [6], as well as a “spectral theorem” for such matrices using the results of Section I. Finally in Section III, we apply this theory in deriving the distributions of some quadratic forms in

the theory of exploitative trade and investment policies: a reformulation and synthesis

Publisher Summary This chapter discusses the theory of exploitative trade and investment policies. It is recognized in the classical doctrine that achievement of an international optimum would require not only a consensus as to the optimal distribution of output but also a willingness to implement that consensus by undertaking lump-sum redistributions of income. If the home country is a borrower and the foreign country specializes, it will be the home countrys advantage to encourage the foreign country to withdraw some of its capital, as the resulting excess of capital in the lending country will lower its rate of return there, thus lowering the opportunity cost of borrowing capital. The rationale for taxing income from home capital invested abroad, or from foreign capital invested at home, is quite a different one when the foreign country diversifies. The cases in which the tariff rate and tax rate on income from foreign investment have opposite sign are termed as paradoxical.

Intra-Industry Trade, Factor Proportions, and Aggregation

In the literature on the empirical explanation of trade flows, it appears to have become a universally accepted dictum that the existence of so-called “intra-industry trade,” i.e., trade between countries of products within the same industrial category, is prima facie evidence of the existence of economies of scale or monopolistic competition, or both, and in particular is incompatible with the “factor-proportions theory,” by which is meant the theory developed by Heckscher and Ohlin according to which trade flows among countries are explained by differences in their relative factor endowments, it being assumed that production functions among countries are the same.