Showing papers in "Econometrica in 1972"

Time Series Analysis Forecasting and Control

Abstract: From the Publisher: This is a complete revision of a classic, seminal, and authoritative book that has been the model for most books on the topic written since 1970. It focuses on practical techniques throughout, rather than a rigorous mathematical treatment of the subject. It explores the building of stochastic (statistical) models for time series and their use in important areas of application —forecasting, model specification, estimation, and checking, transfer function modeling of dynamic relationships, modeling the effects of intervention events, and process control. Features sections on: recently developed methods for model specification, such as canonical correlation analysis and the use of model selection criteria; results on testing for unit root nonstationarity in ARIMA processes; the state space representation of ARMA models and its use for likelihood estimation and forecasting; score test for model checking; and deterministic components and structural components in time series models and their estimation based on regression-time series model methods.

Methods of Estimation for Markets in Disequilibrium

Abstract: This paper is concerned with the econometric problems associated with estimating supply and demand schedules in disequilibrium markets. Tbe general problem is that in the absence of aa equilibrium condition the ex ante demand and supply quantities cannot in general be equated to the observed quantity traded in the market. Four methods of estimation, differing primarily in tbW use of information on price-setting behavior, are developed in this paper. The first method is a generalization of an earlier method developed by R. Quandt and is based upon the maximization of a likelihood function. The method does not require any specitic assnm,,tion about pricwettin~ behatior,.and it allows the sample separation (into demand and supply regimes) to be estimated along with the ce.ztlMent estimates. The second and third methods use the change in price as a qualirotiue proxy in determining the sample separation. The fourth method uses the change in price as a quantitative proxy for the amount of excess demand (supply) in the market. In the final section of the paper the four methods arc used to estimate a model of the housing attd mortgage market in an effort to gauge the potential usefulnesr of each of the methods.

The Exact Finite Sample Properties of the Estimators of Coefficients in the Error Components Regression Models

Abstract: Wallace and Hussain (1969) considered the use of an error components regression model in the analysis of time series of cross-sections and developed an Aitken estimator of the coefficient vector based on an estimated variance-covariance matrix of error terms. In this paper, we have shown that under the set of assumptions adopted by Wallace and Hussain there are an infinite number of estimators which have the same asymptotic variancecovariance matrix as the Wallace-Hussain estimator and also that it is not possible to choose an estimator on the basis of asymptotic efficiency. We have developed an alternative estimator of the variance-covariance matrix of error terms and have used this estimator in developing a feasible "Aitken" type estimator for the coefficient vector. We have derived some small sample properties of this estimator and have compared them with those of other estimators of the coefficient vector.

Social preference orderings and majority rule

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Weaker criteria and tests for linear restrictions in regression

Abstract: The standard F test for linear restrictions in regression is relevant as a criterion but fails to capture the notion of tradeoff between bias and variance. Average squared distance criteria yield operational tests that are more appropriate, depending upon objectives. In the present paper two alternative criteria are developed. The first allows testing of the hypothesis that the average squared distance of a restricted estimator from the parameter point in k space is less than the average squared distance of the unrestricted, ordinary least squares estimator from the same parameter point. The second sets up a test of betterness of the restricted estimator over the unrestricted estimator of E(YjX), where betterness is again defined in average squared distance.

Testing for fourth order autocorrelation in quarterly regression equations

Abstract: A test for fourth order autocorrelation in the error term of a regression equation estimated from quarterly data is described. The development draws on the finite sample results of Durbin and Watson and illustrates how their procedure for the first order case can be generalized. In the model y = X,B + u where X is a matrix of fixed regressors and u, = put-4 + Bt, an appropriate test statistic for Ho: p = 0 is the statistic d4 = {z- )2/z2 computed from the least squares regression residuals z = y - Xb. Bounds to the significance points of d4 are tabulated. Maximum likelihood estimation methods are described; these are equally appropriate when lagged values of the dependent variable appear among the regressors, and they provide asymptotic tests for general-autoregressive error structures, as well as for the special case ut = oe1u_-1 + 04ut4 - aLa4ut-5 + et. Examples from the empirical literature are presented. THE POSSIBILITY THAT the errors in a regression equation estimated from quarterly data possess fourth order autocorrelation was considered, among other things, in a recent paper [28], and a non-parametric test was proposed. Appropriate generalized least squares estimation methods were also described. In this paper we first present a more rigorous solution to the problem of testing for fourth order autocorrelation, which utilizes the approach introduced by Durbin and Watson [6 and 7]. We then describe non-linear estimation methods which simultaneously estimate the regression coefficients and the parameters of the simple fourth order or more general autoregressive error structures. The usual interpretation of the error or disturbance term in econometric models is that it represents the effect of omitted or unobservable variables on the dependent variable. The error term might thus be expected to display certain features of observed economic variables, in particular, when quarterly data are employed, seasonal variation. Equally, when seasonally unadjusted data are being employed in order that one may attempt to explain seasonal variation in the dependent variable, along with other types of variation, by means of explanatory economic or seasonal dummy variables, then the presence of non-systematic seasonal variation, or an incomplete accounting for seasonality by the regressors, will produce seasonal effects in the error term, with the possible consequence of fourth order autocorrelation. Thus we require a test for correlation not between the errors 1 Some of the results contained in this paper were reported in my paper "Estimation and Tests for Quarterly Regression Equations with Autocorrelated Errors" presented at the Second World Congress of the Econometric Society in Cambridge, September, 1970. I am grateful to David Hendry for comment and discussion and, in Section 3, for the use of his computer program, to Zvi Griliches and an anonymous referee for comments, to Andrew Tremayne for research assistance, and to M. I. Nadiri and Michael Parkin for supplying their data. Added in proof: After this was written an unpublished paper by H. D. Vinod entitled "Generalization of the Durbin-Watson Statistic for Higher Order Autoregressive Processes" was brought to my attention; this considers statistics similar to d4 for tests of higher order autocorrelation in the non

A remark on the core of an atomless economy

Abstract: In an atomless economy any allocation that is not blocked by "small" coalitions is in the core. Hence, in such an economy, a competitive equilibrium is characterized by the blocking power of part of the coalitions which excludes all "big" coalitions. THE CORE of an economy consists of all the allocations that are not blocked by any coalition. In this note we prove that for any positive number 8, the core of an atomless economy coincides with the set of allocations that are not blocked by any coalition of measure less than s. This result implies that even if the "large" coalitions cannot be formed, any unblocked allocation is still in equilibrium with respect to some price system. In particular, the formation of the coalition of all traders or any "large" coalition is not needed to insure the Pareto-optimality of final allocation. We prove here directly that the core is equal to the set of allocations that are not blocked by small coalitions. The same result can also be obtained by proving that Aumann's [1], Vind's [5], or Hildenbrand's [2] equivalence theorems hold with the additional restriction on the measure of the blocking coalitions. In any case the proof is a simple application of Liapunov's convexity theorem [3 and 4] (for the statement of the theorem see also [2, Appendix, p. 451]).

Some statistical implications of the log transformation of multiplicative models

Abstract: An attempt is made to set out the implications of the log transformation on the stochastic properties of the model, which are postulated in the original multiplicative relationship. The estimation of the mean of the dependent variable, given some vector of explanatory variables, is accomplished by minimizing the mean square error within a certain class of estimators allowing for biased estimators, assuming known variance. The resulting estimator is modified in order to face the problem of unknown variance. This modified estimator turns out to dominate (in MSE) the least-squares, the ML, and the MVU Bradu and Mundlak estimator.

A second remark on the core of an atomless economy

Abstract: In this note we shall show that we can further restrict the coalitions that are allowed to form and still have the above identity. Let the commodity space be of finite dimension 1, and let a > 0. The result is that an allocation is a Walras allocation if and only if it cannot be blocked by a coalition which is the union of at most 1 + 1 coalitions, each of which has measure and diameter less than a. That a coalition has measure and diameter less than a intuitively means that the coalition consists of relatively "few" agents, and that the agents in the coalition resemble one another in chosen characteristics, e.g., initial allocation, production possibilities,