We study the optimal monetary policy in a two-country open-economy model under two monetary arrangements: (a) multiple currencies controlled by independent policy makers; (b) common currencies with a centralized policy maker. Our findings suggest that: (i) monetary policy competition leads to higher long-term inflation and interest rates with large welfare losses; (ii) the inflation bias and the consequent losses are larger when countries are unable to commit to future policies; (iii) the welfare losses from higher long-term inflation dominates the welfare costs of losing the ability to react optimally to shocks.

Recent studies have shown that the dynamics of firms (growth, job reallocation, and exit) are negatively correlated with the initial size of the firm and its age. In this paper we analyze whether financial factors, in addition to technological differences, are important in generating these dynamics. We introduce financial-market frictions in a basic model of industry dynamics with persistent shocks and show that the combination of persistent shocks and financial frictions can account for the simultaneous dependence of firm dynamics on size (once we control for age) and on age (once we control for size).

This paper develops asymptotic prediction functions that approximate the shape of the density of future observations and correct for parameter uncertainty. The functions are based on extensions to a definition of predictive likelihood originally suggested by Lauritzen and Hinkley. The prediction function is shown to possess efficiency properties based on the Kullback-Leibler measure of information loss. Examples of the application of the prediction function and the derivation of relative efficiency are shown for linearnormal models, nonnormal models, and ARCH models.

[In this paper we consider the estimation of a model with time varying structure. The parameters of the model are assumed to be subject to permanent and transitory changes over time. Estimation methods are developed, and the asymptotic properties of the estimates are derived.]

A NY econometric equation representing a complex behavioral or technical relationship is, of necessity, an approximation of reality. As such, it is subject to errors in specification and structural change over time. This problem is well recognized by econometricians. Duesenberry and Klein (1965) point out that *'. . . as technology, institutional arrangements, tastes and managerial techniques change over time, the relationships represented by our equations inevitably change. Furthermore, when statistical tests are applied to econometric relationships, the hypothesis of structural stability is frequently rejected.' Some methods for dealing with structural change have evolved. Quandt (1957) has developed a maximum likelihood technique for estimating a point of structural change within a sample.2 Klein and Evans (1967) adjust the intercepts of the Wharton Model to account for structural change.3 The purpose of this paper is to test the robustness of Adaptive Regression (1973) to specification errors causing structural change over time, relative to ordinary least squares analysis with and without the autoregressive correction.4 Since econometricians are inevitably faced with structural change and errors in specification, they should use a technique which is robust rela-tive to such problems. The device most commonly used is to assume that the disturbances are subject to an autoregressive process. The autoregressive correction may frequently ameliorate the effects of misspecification and structural change, but it is doubtful whether such processes, except in rare instances, describe the true distribution of the disturbances. The economics literature seldom gives any justification for this scheme except that omitted variables may be subject to an autoregressive process or the structure of the model may be changing.5 We suspect the reasons for the widespread use of the autoregressive correction are that it is a simple hypothesis, explains serial correlation in the disturbances, and can be dealt with efficiently. The adaptive regression model considered in this paper is equally simple but more general, explains serial correlation, and can also be dealt with efficiently.6 In the next section the adaptive regression model is presented and the Bayesian estimators are developed. In section II the results of a Monte Carlo Study are presented. Two models are considered for which data are generated by eleven different schemes. The estimation and forecasting efficiency of adaptive regression, and ordinary least squares with and without the autoregressive correction are compared. Section III contains an analysis of the role of time trends in econometric relationships. In section IV the relative forecasting ability of the three estimation techniques is tested on real data. The three models suggested by Received for publication February 10, 1972. Revision accepted for publication November 30, 1972. *The authors acknowledge helpful comments of Professors F. G. Adams, R. Roll and R. Summers and the participants of the NBER conference on Bayesian Statistical Inference in Economics. Computations were executed on the University of Pennsylvania computer. 'Examples of such tests include Brown (1966), Goldfield (1969) and Howrey (19-70). One of the most extensive studies was done by Duffy (1969). 'The Quandt technique is limited by the fact that it is mainly useful for finding stable subsamples. If structural change occurs often, it is not very useful. Rosenberg (1968) has used stepwise composition to develop the computationally efficient Aitken estimates of a model subject to structural change over time. His procedure, however, requires that the true covariance matrix of the disturbances be known up to a constant scale factor. 'Adjusting the intercepts is an ad hoc method for keeping the model on track for ex ante forecasting. The intercepts are not assumed to change over the sample period which is always much longer than the forecasting period. 'The autoregressive correction assumes the error is subject to a first or second order autoregressive scheme. See Dhrymes (1969) for the maximum likelihood approach and Zellner and Tiao (1965) for the Bayesian development. The latter approach is used in this paper. 'In fact, if omitted variables are subject to an autoregressive process, the disturbances will, in general, be subject to a more complicated process. 6 A test with sufficient power to differentiate betweer these two models (or others which result in serial correlation) using sample sizes generally available to econometricians does not appear to exist. Further, if one did, its usefulness would be limited as neither structure is likely to be an exact representation of reality. That one structure is more likely on the basis of the data does not imply thai it will forecast better if, in fact, a third structure is generating the data.

Money is incorporated into a real business cycle model using a cash-in-advance constraint. The model economy is used to analyze whether the business cycle is different in high inflation and low inflation economies and to analyze the impact of variability in the growth rate of money. In addition, the welfare cost of the inflation tax is measured and the steady-state properties of high and low inflation economies are compared.

In most natural sciences (physics, chemistry, biology) theories are validated by controlled experiment. However, in other natural sciences (astronomy, meteorology), and in most social sciences, including economics, the data are characteristically generated not by experiment but by measurement of uncontrolled systems. In economics, theories take the form of restrictions on the models assumed to generate the data, and statistical methods replace experimental controls in testing these restrictions. And here is the difficulty: in economics, particularly macroeconomics, the theory used to derive tests ordinarily does not generate a complete specification of which variables are to be held constant when statistical tests are performed on the relation between the dependent variable and the independent variables of primary interest. Accordingly, in such cases there will be a set of often very different candidate regression-based tests, each of which has equal status with the others since each is based on a different projection of the same underlying multivariate model. Except in the unlikely event that the explanatory variables are mutually orthogonal, the conditional regression coefficients, which generally form the basis for the test statistic, will depend on the conditioning set. We conclude from this that, if a theory which does not generate a complete specification of the regression test is nonetheless to have testable implications, these implications must be robust over the permissible alternative specifications. If the restrictions indicated by the theory are satisfied in some projections, but not in others that have an equal claim to represent implications of the theory, one cannot conclude that the theory has been confirmed. The fact that the observable implications of valid theories must obtain over a broad (but usually incompletely specified) set of regressions rather than for a single regression introduces a large and unavoidable element of imprecision into hypothesis testing in macroeconomics. Generally it appears to be appropriate to weaken the statistical criterion for rejecting theories. Consider, for example, the theory of money demand, which will engage our attention in this paper. The Tobin-Baumol square root formula implies that the elasticity of money demand with respect to the interest rate is exactly one-half. But which interest rate? Should wealth be held constant? Inflation? In view of such uncertainties it would be inappropriate to insist in a literal-minded fashion on rejecting the Tobin-Baumol model if in some regression the measured interest elasticity differed from one-half by more than two standard deviations, and only then. Obviously a more flexible approach is called for. The practice has been to conclude that the statistical evidence is consistent with the Tobin-Baumol model as long as the interest rate coefficient is negative. If it is negative and significant, or negative and insignificantly different from minus one-half, that would provide somewhat stronger confirmation. But a positive coefficient, particularly a significantly positive coefficient, would be viewed as raising questions about the validity of the theory. In macroeconomics generally, as in the money demand application, the typical response to specification uncertainty has been to regard a theory as supported if the signs of the estimated coefficients agree with those expected from theory, and as disconfirmed otherwise. There is no theoretical justification for this procedure, but it seems to be a reasonable course to follow. The point that economic theory ordinarily generates incompletely specified statistical *University of California, Santa Barbara. We have received helpful comments from Andrew Abel, Robert Clower, Michael Darby, Robert Engle, Stephen Goldfeld, David Laidler, Edward Leamer, Robert Lucas, Frederic Mishkin, and Edward Prescott. Thomas Hall provided able research assistance.