[This paper addresses the problem of estimating unknown regression coefficients when erroneous data and other violations of the standard assumptions are possible. An estimator which has a limited sensitivity to these departures from the assumptions is presented, and some of its properties are derived. This estimator is shown to have a certain efficiency property relative to other estimators with the same sensitivity to erroneous data.]

This paper develops a test of the rational expectations hypothesis advanced by Muth [18]. The framework considered here allows for multiperiod expectations of several endogenous variables, with or without lagged exogenous variables. In conventional (linear) models, the hypothesis implies that the expectations are linear in certain relevant variables, and restricts the coefficients of these variables to be certain functions of the parameters in the imbedding model. The test is developed as a test of the validity of these restrictions. The paper also treats the estimation problem in some details, under the alternative hypothesis which is taken as simply the negation of the rational expectations hypothesis.

Events by Grunberg and Modigliani [3]. Economic forecasts are made to be used, and decisions based on them may affect their ultimate realization. Grunberg and Modigliani explored this problem in a model in which future aggregate supply was influenced by current decisions based on the predicted future price. They applied the Brouwer fixed point theorem to show that if the future equilibrium price is a bounded continuous function of its currently predicted value, a exists. In [8] this result was placed in a temporary equilibrium context and the notation of a correct prediction was extended to include probabilistic predictions based on estimation procedures. Again, a fixed point theorem was applied to show that the causal influence of a forecast does not always invalidate it. Although this result is a reassuring and necessary first step, the analysis is confined to a single realization of the exogeneous variables. Thus a forecast is a single point or probability distribution rather than a function or conditional distribution whose domain is the space of observable variables. Of course, if the complete exogenous specification of the economy is observable, this is no restriction since the theorems could be applied separately to each realization. However, it is more likely that the space of observable variables contains a mixture of exogenous and endogenous variables without containing the complete set of either. Then if the exogenous variables are generated stochastically, the results of the above mentioned papers do not guarantee the existence of a statistically forecasting procedure. What is needed are general equilibrium versions of the results in [10], where, in particular, the statistically forecast of a future price is derived as a function of current and past prices. It would seem natural to approach this as a fixed point problem in the space of joint distributions of the observable variables and the

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.

In this paper an explicit and computationally convenient expansion of the exact finite sample distribution function of a quasi-maximum likelihood spectral estimator is given. In the majority of practical situations it will be necessary to estimate certain nuisance parameters of the distribution. Therefore, a method of evaluating these parameters is suggested and some Monte-Carlo evidence concerning the practical implementation of the results is given. 1 INTRODUCTfON AN EXTENSIVE SET of asymptotic results relating complex statistical analysis to the problems that arise in estimating spectra from the discrete Fourier transforms of time series data has been well established; see, for example, Brillinger [3] and Goodman [6]; and Hannan [11] has recently extended these results to cover the modified Fourier coefficients, proposed by Bingham, Godfrey, and Tukey [1] and defined in (2.3) below. Unfortunately it seems likely that the sample sizes required for one to approach the asymptotic position will not be available when considering the analysis of many economic time series. In a recent article, however, Hatanaka [12] has shown that the elimination of leakage produced by the modified Fourier coefficients is effective for small finite realizations and that it may be possible to recover the loss of degrees of freedom associated with the familiar estimator obtained by averaging over the modified periodogram.2 The purpose of the present paper is to extend these results by using complex statistical analysis to derive expressions for both the form and exact finite sample distribution of a spectral estimator obtained from the modified Fourier coefficients. Thus in the following section a brief exposition of some basic theory is given and a quasi-maximum likelihood estimation procedure is suggested. In Section 3 an exact expression for the finite sample distribution of the proposed estimator is given and shown to incorporate an established distributional result as a particular special case. In the majority of practical situations it will be necessary to estimate certain nuisance parameters of this distribution if it is to be employed and a method of evaluating these parameters is also suggested. It is well known, however, that while the replacement of nuisance parameters by consistent estimates will generally leave asymptotic theory intact the consequences of estimating nuisance parameters are unlikely to be negligible in finite sample theory. Since it is not possible to determine analytically the effect that the estimation of these nuisance parameters will have, the results of some simple Monte-Carlo

[Under classical assumptions, characterizations are given for two classes of instrumental variable estimators of an equation in a simultaneous system. IV estimators where all instruments are nonstochastic are expressed in terms of multinormal random vectors in exactly the same way as the 2SLS estimator of a just-identified equation. These estimators have no finite moments of positive integral order. The second class, consisting of IV estimators based on certain stochastic instruments, includes the OLS, 2SLS, and modified 2SLS estimators. The inadmissibility (under squared-error loss) of some estimators in this class is considered when the equation being estimated contains two endogenous variables.]

FOLLOWING ON SOME informal conjectures by Dummett and Farquharson [3] and Vickery [20] we now have independent proofs by Gibbard [7] and Satterthwaite [17 and 18] that no collective choice rule exists whose social choice functions are singlevalued, strategy-proof, nondictatorial and have a range containing at least three alternatives. Because strategy-proof ness seems desirable and because it is closely related to mainstream economic theory issues of evaluating resource allocation institutions with respect to incentive compatibility (cf. Hurwicz [9]), their theorem has excited considerable attention [5, 6, 10, 11, 12, 13, 14, 15, 16, 19, and 21]. In this paper, the requirement of singlevaluedness is dropped and explorations are made of the consequences this has on the Gibbard-Satterthwaite results. Let E be the set of all alternatives (which must, by assumption, be mutually incompatible) and N= {1, 2,.. ., n} be the set of individuals. A nonempty subset, v, of E (i.e., an element of 2E _-0}) is an agenda. RE is the set of all complete and transitive binary relations on E; RE is the n-fold Cartesian product of RE. An element, u, of RE is called a profile and if u = (R1, R2,... , Rn), we say that R, is the preference ordering for individual i in u. In the usual way, we use Ri to define strict preference, Pi, and indifference, Ii: xPiy if and only if xRiy and not yRix; xIiy if and only if xRiy and yRix. A social choice function (on V) is a function, C, on Vc2E _{0} into 2E _{0} satisfying C(v) c v. Here V is the set of admissible agenda. The set of all social choice functions on V is called ST. A collective choice rule (on V, U) is a function, F, on UcRE into c6. Here U is the set of admissible profiles. The first constraints on the social choice function in the Gibbard-Satterthwaite theorem are domain restrictions. They admit only one agenda, V= {E} and then require the collective choice rule to work for all societies, U = RE. The most important constraint they use is singlevaluedness: for each v in V, C(v) contains exactly one element. Of course, there is only one V, namely E, in the Gibbard-Satterthwaite theorem. The importance of this constraint stems from its use in all the rest of the problem; singlevaluedness is used in their method of formalizing both nondictatorship and strategy-proofness. Let us deal first with nondictatorship. Using singlevaluedness, let C(v) be the unique member of C(v). Then a collective choice rule, F, is nondictatorial if for no i, i = 1, ... , n, is it true that for all (R1,. . . , Rn) =uE U and for all x C(v) in the range of C = F(u), C(v)Pix. Finally, we turn to strategy-proofness. A collective choice rule is strategy-proof at (v, u) if it is not manipulable at (v, u). F is manipulable at (v, u) if, when u = (R1, R2, ... , Rn), there is a u'=

This paper analyzes a one-commodity model in which alternative investment projects are characterized by return functions indicating the output intensities over time resulting from an initial unit investment. Saving is generated partly by households as a constant fraction of net income, and partly by business firms in accordance with a depreciation (or replacement) policy. It is shown that when a declining value depreciation policy is adopted, the ordering of consumption streams in terms of their present values, at any fixed interest rate for which these converge, induces an ordering of investment projects in terms of their internal rates of return. The same ordering of projects is also induced by applying the overtaking criterion to the consumption streams.