Knowledge that Transforms

Given n observations on a system of linear stochastic difference equations with appropriate initial conditions, and given a prior density (possibly diffuse) of its parameters, this paper obtains the predictor of the time series k periods into the future with minimum mean squared error. Completely analytical solution is given for predictions from the first-order univariate system, and, in the general higher-order multivariate case, for k up to 5. econometric equations to produce forecasts were not designed for the purpose of forecasting. In this paper, it is argued that these estimation methods may be inadequate if the resulting estimates are to be used to make ex ante predictions for more than one period ahead, and if the accuracy of the predictions is measured, as it usually is, by the mean squared errors. A formulation of the multiperiod prediction problem is presented. It will then become clear that the same set of parameter estimates cannot be optimal in making predictions for different time periods into the future, when optimality is defined by minimum mean squared errors in small samples. Recently there has been much interest in comparing different econometric models, or different versions of the same econometric model obtained by applying different estimation techniques, in terms of how well they would have forecasted the dependent variables during the sample period, given the true values of the exogenous variables and given the values of the dependent variables lagged one or more periods. It has now become clear that, as judged by ex post forecasting for the sample period, models or techniques that perform better for one-period predictions may do worse for multiperiod predictions. For example, purely auto- regressive models could do better than models based on structural equations in ex post forecasting for one quarter ahead, but were worse in forecasting three or four quarters ahead, as documented in Hickman (4). Klein (9) and Fair (3) have compared multiperiod predictions of the sample data by different estimation techniques applied to the same econometric model. A related, though different, question naturally arises as to whether different parameter estimators should be used to produce ex ante forecasts for different periods into the future. The former topic is one of fitting equations to a set of data. The latter topic is one of statistical decision, and is the subject of this paper.'

MUCH APPLIED ECONOMETRIC work is based on the correlation and regression of ratio variables which have the same denominator. The denominator generally deflates the various sets of measurements in order to make them comparable. In certain cases deflation is a way of obtaining efficient estimators. Briggs [1] has studied the effect of errors in the deflating variable on the correlation between ratios. The effect of errors in the deflating variable on a scatter diagram of measurements on ratio variables is to displace each representative point along a ray from the origin passing through the representative point of the error-free measurement. This suggests that with error in the deflating variable the ordinary least squares (OLS) estimators of a bivariate regression are biased toward indicating a relationship of proportionality between the ratio variables. It can be demonstrated that this conjecture is generally valid for multiple linear regression with error in the deflator. In most cases in which deflation is used, it is reasonable to suppose that the deflator is a random variable distributed independently of the ratio variables; in this case the conditional expectation of the undeflated independent variables is proportional to the value of the deflator. Under this assumption it can be established that when there is error in the deflator the estimators of the slopes of the regression are inconsistent unless (i) the ratio regression has zero intercept, or (ii) the mean of each of the independent variables is zero, or (iii) the error in the deflator is systematic (i.e., the error term has zero variance). The estimator of the intercept may be inconsistent even when the estimators of the slopes are consistent; for the intercept estimator to be consistent it is necessary that either (i) the intercept

[Two principal questions are treated: (i) Which equilibrium conditions (or, which types of factor demand equations) based on the neoclassical single-capital-good model are unchanged when there are many different capital goods, only one of which is used at any one time? (ii) Are there any "pseudo" productions functions (of "capital" and labor) which correctly describe behavior by profit-rate maximizing entrepreneurs in the many-capital-goods model? A new surrogate production model is developed to resolve these questions. It is shown, inter alia, that if one wishes to predict changes in labor and value capital requirements in response to changes in factor prices, the neoclassical marginal rate of substitution relationship can be justified as the basis for the econometric specification.]

IN TESTING HYPOTHESES on the coefficients of a linear regression model with stochastic regressors it is well known that the usual t test and F test are applicable if the stochastic regressors are statistically independent of the disturbances [3, p. 268; 5, pp. 27-28]. Also, there are cases in which economic hypotheses can be stated in terms of the independence of stochastic regressors and disturbances, the best known examples being the current versus the permanent income hypotheses and the recursiveness hypothesis in a simultaneous equations model. Therefore, it is desirable to develop a procedure that can be used to test the hypothesis that the stochastic regressors and disturbances are independent. In this paper, we examine four alternative tests of independence between the stochastic regressors and disturbances. In the rest of this section we specify the stochastic model and state the hypotheses to be tested. In Section 2 we present two finite sample tests. In Section 3 two alternative asymptotic tests are given and asymptotic power functions of all four tests are examined. In Section 4 we give examples of applications of the test in econometrics. We consider the following linear model:

This paper examines the possible explanations for the changes in output, capital, and labor input of a sample of manufacturing plants over a number of years. Apart from the scale of operation, these changes could be attributed to three causes: technology diffusion, substitution, and improvements in X-efficiency. The empirical findings indicate that a diffusion model modified to incorporate X-efficiency improvements provides the best explanation. This suggests the need for a new approach to the specification of production

EXPERIMENT DESIGN THEORY for regression analysis is becoming important in econometric work, most notably in the context of negative income tax [1] and other subsidy experiments (such as housing and health subsidy experiments being planned). The usual regression design model requires the experimenter to specify the functional form of the behavioral equation under investigation. An apparent difficulty is that the experimenter does not know the true functional form. This paper suggests new procedures for handling the difficulty. The procedures were stimulated by work in planning the New Jersey negative tax experiment. To the planners of the New Jersey experiment, the relevant statistical design literature was not applicable, cookbook fashion, because the practical design guidelines were constructed for simpler situations. However, by combining ingredients from the statistical literature, the New Jersey experimenters were able to define a well-behaved mathematical programming model whose solution would tailor a design to their situation [3]. (The references [2 and 4] provide a useful entry to the relevant statistical literature.) The new procedures suggested here build on the New Jersey design model, which is reviewed in Section 2. Though the exposition runs in terms of subsidy experiments, the material is more generally applicable.