Knowledge that Transforms

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

[The nonlinear Kaldor theory of macroeconomic business cycles is combined with a classical growth mechanism to derive a deterministic model of business cycle phenomena. Sufficient conditions for the existence, uniqueness, and orbital asymptotic stability of a limit cycle are given. It is shown that the model also exhibits stochastic stability when the deterministic variables are randomly disturbed.]

In an atomless economy any allocation that is not blocked by coalitions is in the core. Hence, in such an economy, a competitive equilibrium is characterized by the blocking power of part of the coalitions which excludes all big coalitions. THE CORE of an economy consists of all the allocations that are not blocked by any coalition. In this note we prove that for any positive number 8, the core of an atomless economy coincides with the set of allocations that are not blocked by any coalition of measure less than s. This result implies that even if the large coalitions cannot be formed, any unblocked allocation is still in equilibrium with respect to some price system. In particular, the formation of the coalition of all traders or any large coalition is not needed to insure the Pareto-optimality of final allocation. We prove here directly that the core is equal to the set of allocations that are not blocked by small coalitions. The same result can also be obtained by proving that Aumann's [1], Vind's [5], or Hildenbrand's [2] equivalence theorems hold with the additional restriction on the measure of the blocking coalitions. In any case the proof is a simple application of Liapunov's convexity theorem [3 and 4] (for the statement of the theorem see also [2, Appendix, p. 451]).

[An attempt is made to set out the implications of the log transformation on the stochastic properties of the model, which are postulated in the original multiplicative relationship. The estimation of the mean of the dependent variable, given some vector of explanatory variables, is accomplished by minimizing the mean square error within a certain class of estimators allowing for biased estimators, assuming known variance. The resulting estimator is modified in order to face the problem of unknown variance. This modified estimator turns out to dominate (in MSE) the least-squares, the ML, and the MVU Bradu and Mundlak estimator.]

The fact that long term interest rates have been higher on average than short rates in the twentieth century has often been interpreted in the term structure literature as evidence of the existence of positive or term premiums. The purpose of this paper is to point out that an average differential between long and short rates does not necessarily represent a differential between returns realized by holders of long versus short term bonds. In particular, it is shown that if short term rates are positively autocorrelated, interest rate differentials overstate differentials in realized rates of return. We conclude that liquidity or term premiums properly estimated from the Durand yield curve data are not consistent with the liquidity preference theory. LONG TERM INTEREST RATES have been higher on average than short term rates during the twentieth century. The average differential over the period 1900-1958 between long term interest rates and spot one year rates in the Durand yield curve data increases monotonically with term to about half a percentage point at forty years. The interpretation given to these differentials in the term structure literature is that they are the additional rate of return earned on average by capital invested in long term bonds. The purpose of this paper is to point out that an average differential between long and short term interest rates does not necessarily represent a differential between realized rates of return. This is because the realized increment to capital invested in a sequence of short term bonds depends on the ex post product of uncertain future spot rates. The expected value of this ex post product will in general differ from the product of expected future spot rates. We show that if short term rates are positively autocorrelated, the interest rate differentials overstate differentials in realized rates of return. We shall refer to differentials in realized rates of return as term premiums. The proper interpretation of interest rate differentials is important to the assessment of the evidence for various theories of the term structure. The liquidity preference theory of J. R. Hicks [4, pp. 144-147] predicts that a term premium will be earned by capital invested in long term bonds because their holders require compensation for risk of capital fluctuation. Forward interest rates will exceed one period spot rates on average by the amount of premium which rises monotonically with horizon. The positive and monotonic average differentials between forward and spot one year rates in the Durand data have been