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(Y/X), where betterness is again defined in average squared distance.

Tables of critical points for the noncentral F are presented with noncentrality equal to 1/2 of numerator degrees of freedom for denominator degrees of freedom of 1-30, 40, 60, 120, 200, 400, and 1,000, and numerator degrees of freedom of 1-30, 40, 60, 120, and 200, and type one errors of 0.05, 0.10, 0.25, and 0.50. These critical points can be used to test the second weak MSE criterion discussed in the companion paper [2]. An approximation is suggested for noncentral F(θ), and accuracy checks are given. An appendix provides a Fortran function for the approximation. The approximation is intended for using the first weak MSE test discussed in the companion paper.

"Inl statistical inference proper, the model is never questioned.... The methods of mathe-matical statistics do not provide us with a means of specifying the model. " 1

A reduced-form equation relating length of "formal schooling" to market, endowment, and ability parameters was derived for a life-cycle human capital accumulation model with alternative assumptions: (a) equal borrowing and lending rates and (b) no loans for human capital investment. Length of "formal schooling" increases when loans are unavailable. For both cases, length of "formal schooling" varies directly with length of work life, a Hicks-neutral "ability" index, and the ratio of the human capital rental rate to the price of associated inputs, and varies inversely with the discount rate, deterioration rate, and initial human capital stock.

In this paper we derive and present optimal critical points for pre-tests in regression using a minimum average relative risk criterion. We use the same type risk functions as Sawa and Hiromatsu [8] who, in a recent paper in this journal, derived pre-test critical values using a minimax regret criterion. Since James-Stein type estimators can be shown to dominate any pre-test estimator for the risk functions used here and in [8], no normative claims are made for the critical values we give. However, the use of pre-testing procedures continues in practice and the results given here, contrasted with other results, add to information about the character of costs and returns to such practices.