I am grateful to Meghnad Desai, Harry G. Johnson, Marcus H. Miller, Marc Nerlove, R. D. Terrell, J. J. Thomas, and the editors of this journalfor comment and discussion during the preparation of this paper. Although not explicitly referred to in the text, the more theoretical survey article on distributed lags by Griliches [28] also clarified my thoughts in a number of places. Errors of omission and comnmission are, as usual, my own responsibility.

The implications for applied econometrics of the assumption that unobservable expectations formed rationally in Muth's sense examined. The statistical properties of the resulting models and their distributed lag and time series representations described. Purely extrapolative forecasts of endogenous variables can be constructed, as alternatives to rational expectations, but less efficient. Identification and estimation considered: an order condition is that no more expectations variables than exogenous variables enter the model. Estimation is based on algorithms for nonlinear-in-parameters systems; other approaches surveyed. Implications for economic policy and econometric policy evaluation described. EXPECTATIONS VARIABLES ARE WIDELY USED in applied econometrics, since the optimizing behavior of economic agents, which empirical research endeavors to capture, depends in part on their views of the future. Directly observed expectations or anticipations relatively rare, hence implicit forecasting schemes used. Most commonly expectations taken to be extrapolations, that is, weighted averages of past values of the variable under consideration. However, these are almost surely inaccurate gauges of expectations. Consumers, workers, and businessmen ... do read newspapers and they do know better than to base price expectations on simple extrapolation of price series alone (Tobin [31, p. 14]). An alternative approach is offered by the rational expectations hypothesis of Muth [15], which assumes that in forming their expectations of endogenous variables, economic agents take account of the interrelationships among variables described by the appropriate economic theory. Price movements observed and experienced do not necessarily convey information on the basis of which a rational man should alter his view of the future. When a blight destroys half the midwestern corn crop and corn prices subsequently rise, the information conveyed is that blights raise prices. No trader or farmer under these circumstances would change his view of the future of corn prices, much less of their rate of change, unless he is led to reconsider his estimate of the likelihood of blights, again quoting Tobin. This paper examines the implications of the rational expectations hypothesis for applied econometrics, and argues that its full force has yet to be appreciated in empirical work. The discussion is quite general, proceeding in terms of the standard linear simultaneous equation system, and pays little attention to specific applications of the hypothesis, such as the efficient markets literature and

Univariate autoregressive moving average models for the endogenous variables of a dynamic simultaneous equations system can be interpreted as a form of solution of that system. This paper considers the interrelationships between the various representations of the system, and develops joint estimation and model selection procedures for the multiple time series model which arises as a multivariate representation of the individual autoregressive moving average models. A test of the restriction of common autoregressive parameters is incorporated. Two empirical examples are presented, the first concerned with a model of the hog cycle and the second with a model of the United States economy previously considered by Zellner and Palm.

A test for fourth order autocorrelation in the error term of a regression equation estimated from quarterly data is described. The development draws on the finite sample results of Durbin and Watson and illustrates how their procedure for the first order case can be generalized. In the model y = X,B + u where X is a matrix of fixed regressors and u, = put-4 + Bt, an appropriate test statistic for Ho: p = 0 is the statistic d4 = {z- )2/z2 computed from the least squares regression residuals z = y - Xb. Bounds to the significance points of d4 are tabulated. Maximum likelihood estimation methods are described; these are equally appropriate when lagged values of the dependent variable appear among the regressors, and they provide asymptotic tests for general-autoregressive error structures, as well as for the special case ut = oe1u_-1 + 04ut4 - aLa4ut-5 + et. Examples from the empirical literature are presented. THE POSSIBILITY THAT the errors in a regression equation estimated from quarterly data possess fourth order autocorrelation was considered, among other things, in a recent paper [28], and a non-parametric test was proposed. Appropriate generalized least squares estimation methods were also described. In this paper we first present a more rigorous solution to the problem of testing for fourth order autocorrelation, which utilizes the approach introduced by Durbin and Watson [6 and 7]. We then describe non-linear estimation methods which simultaneously estimate the regression coefficients and the parameters of the simple fourth order or more general autoregressive error structures. The usual interpretation of the error or disturbance term in econometric models is that it represents the effect of omitted or unobservable variables on the dependent variable. The error term might thus be expected to display certain features of observed economic variables, in particular, when quarterly data are employed, seasonal variation. Equally, when seasonally unadjusted data are being employed in order that one may attempt to explain seasonal variation in the dependent variable, along with other types of variation, by means of explanatory economic or seasonal dummy variables, then the presence of non-systematic seasonal variation, or an incomplete accounting for seasonality by the regressors, will produce seasonal effects in the error term, with the possible consequence of fourth order autocorrelation. Thus we require a test for correlation not between the errors 1 Some of the results contained in this paper were reported in my paper Estimation and Tests for Quarterly Regression Equations with Autocorrelated Errors presented at the Second World Congress of the Econometric Society in Cambridge, September, 1970. I am grateful to David Hendry for comment and discussion and, in Section 3, for the use of his computer program, to Zvi Griliches and an anonymous referee for comments, to Andrew Tremayne for research assistance, and to M. I. Nadiri and Michael Parkin for supplying their data. Added in proof: After this was written an unpublished paper by H. D. Vinod entitled Generalization of the Durbin-Watson Statistic for Higher Order Autoregressive Processes was brought to my attention; this considers statistics similar to d4 for tests of higher order autocorrelation in the non

Kenneth F. Wallis, Lagged Dependent Variables and Serially Correlated Errors: A Reappraisal of Three-Pass Least Squares, The Review of Economics and Statistics, Vol. 49, No. 4 (Nov., 1967), pp. 555-567

IN RECENT years the Durbin-Watson statistic has been used uncritically to test for serial correlation the of relationships containing lagged endogenous which are estimated by single or simultaneous equations methods. When lagged endogenous are included an equation estimated by ordinary least squares, however, the Durbin-Watson statistic is asymptotically biased towards 2 (the value which it should have if no serial correlation is fact present). is doubtful, therefore, that the statistic should be used either to test for serial correlation the or to provide any indication of the extent of such correlation when the estimated equation contains lagged values of any endogenous variable. The widespread use of the Durbin-Watson statistic inappropriate situations may stem from misinterpretation of a remark one of Durbin's later papers. the original papers setting forth their test, Durbin and Watson stated: It should be emphasized that the tests described this paper apply only to regression models which the independent can be regarded as 'fixed variables'. They do not, therefore, apply to autoregressive schemes and similar models which the lagged values of the dependent variable occur as independent variables [2, p. 159]. a subsequent paper, however, showing that the statistic could be used with some slight modification systems of simultaneous equations, Durbin wrote: In some formulations certain of the x's coincide with lagged values of the y's. The theory becomes much more complicated such cases, and we shall not consider them except to point out that the results obtained later the paper, which are exact for the model specified above, may be expected to hold approximately for models containing lagged dependent variables [1, p. 370]. This later paper discussed the distribution of the Durbin-Watson statistic under the null hypothesis of no serial correlation and did not cover the test's power against alternatives. While the original Durbin and Watson papers showed that the test has high power against Markov alternatives, the asymptotic result, derived from a result obtained by Malinvaud and presented below, indicates that this conclusion does not hold when lagged endogenous are included. a paper which expressions for the asymptotic bias of least squares estimates of regression coefficients various models containing lagged dependent and serially correlated were derived, Griliches commented that in most cases the addition of the lagged dependent variable to the regression will reduce the serial correlation of the residuals and hence increase the Durbin-Watson statistic [3, p. 70]. The asymptotic results were extended to cover the bias the estimated 235