This paper studies the random walk, in a general time series setting that allows for weakly dependent and heterogeneously distributed innovations. It is shown that simple least squares regression consistently estimates a unit root under very general conditions in spite of the presence of autocorrelated errors. The limiting distribution of the standardized estimator and the associated regression t statistic are found using functional central limit theory. New tests of the random walk hypothesis are developed which permit a wide class of dependent and heterogeneous innovation sequences. A new limiting distribution theory is constructed based on the concept of continuous data recording. This theory, together with an asymptotic expansion that is developed in the paper for the unit root case, explain many of the interesting experimental results recently reported in Evans and Savin (1981, 1984).

This paper studies pure exchange economies with infinite dimensional commodity spacces in the setting of Riesz dual systems. An Edgeworth equilibrium is an allocation that belongs to the core of every replication of the ec onomy. Under some mild conditions, it is shown that (1) Edgeworth equ ilibria exist, (2) an allocation is an Edgeworth equilibrium if and o nly if it is an approximate quasiequilibrium, and (3) if preferences are uniformly proper, then every Edgeworth equilibrium is a quasiequi librium. The obtained results specialize to most exchange economies t hat have appeared in the literature of general equilib rium theory. Copyright 1987 by The Econometric Society.

THIS PAPER DISCUSSES the problem of aggregating probability judgements and shows that Arrow-type paradoxes arise in this context, just as in the context of aggregating preferences. The need to aggregate probability judgements may arise in several different sorts of situation. Two examples which concern decision making under risk are: (i) when an individual, prior to making a decision, consults a number of experts who differ in their assessments of the probabilities of alternative states of nature; (ii) when the individuals constituting a society have to make a joint decision on the basis of identical utility functions but, again, differing assessments of the probabilities of alternative states of nature.2 We illustrate (i) above by means of a simple example: An individual consults two legal experts, who give him their probability judgements concerning the success or failure (there are just two possible outcomes) of proposed litigation. Scenario 1: The individual knows that litigation will succeed if judge J presides and barrister B defends, but will not succeed otherwise. From the experts' probability judgements the individual is able to infer that one expert has received a message that judge J will preside, and the other a message that barrister B will defend. (Either message, or both, may be false.) Given these messages, the individual updates in Bayesian fashion the prior probabilities he assigns to success and failure. Scenario 2: Success or failure depends on interpretation of the law. Pooling the information on which the experts' probability judgements are based is, we suppose, impracticable. (Perhaps they meet to discuss the case and cannot agree.) Aggregation of probability judgements in situations such as Scenario 1 (and according to axioms which we discuss in the next section) is clearly a wrong procedure which may lead to totally erroneous results. In situations such as Scenario 2 such aggregation may however have a useful role. Formulae for aggregating probability judgements have been discussed by, among others, Wagner (1982), Bordley (1982), Genest, Weerahandi, and Zidek (1984), and Fishburn and Rubinstein (1984). In this paper we generalize some results contained in the latter two papers, before going on to consider paradoxes. An early, Arrow-type impossibility result, based on rather strong assumptions, is due to Dalkey (1972). In Section 2 notation and axioms are introduced, and some results given. In Section 3 three propositions are stated and proved.

The relationship between co-integration and error correction models, first suggested in Granger (1981), is here extended and used to develop estimation procedures, tests, and empirical examples. If each element of a vector of time series x first achieves stationarity after differencing, but a linear combination a'x is already stationary, the time series x are said to be co-integrated with co-integrating vector a. There may be several such co-integrating vectors so that a becomes a matrix. Interpreting a'x,= 0 as a long run equilibrium, co-integration implies that deviations from equilibrium are stationary, with finite variance, even though the series themselves are nonstationary and have infinite variance. The paper presents a representation theorem based on Granger (1983), which connects the moving average, autoregressive, and error correction representations for co-integrated systems. A vector autoregression in differenced variables is incompatible with these representations. Estimation of these models is discussed and a simple but asymptotically efficient two-step estimator is proposed. Testing for co-integration combines the problems of unit root tests and tests with parameters unidentified under the null. Seven statistics are formulated and analyzed. The critical values of these statistics are calculated based on a Monte Carlo simulation. Using these critical values, the power properties of the tests are examined and one test procedure is recommended for application. In a series of examples it is found that consumption and income are co-integrated, wages and prices are not, short and long interest rates are, and nominal GNP is co-integrated with M2, but not M1, M3, or aggregate liquid assets.