N this paper, I will generalize the modified Newton method previously applied in Chow (1968) to the computation of full-information maximum likelihood estimates of parameters of a system of linear structural equations to the case of a system of nonlinear structural equations. The success of that method for linear systems 1 has stimulated my present attempt to generalize it for nonlinear systems. The subject of maximum likelihood estimation of nonlinear simultaneous equation systems has been studied by Eisenpress and Greenstadt (1966). There are three main differences between their approach and ours. First, their basic formulation is more general, assuming that all parameters in the system may appear in every equation,2 whereas we assume as the basic setup that there is a distinct set of parameters belonging to each equation. Second, partly because of the first, we are able to obtain simpler and more explicit expressions for the derivatives of likelihood function required in the calculations. Third, and also partly because of the first, we can conveniently deal with the important problem of linear restrictions on the parameters in the same equation or in different equations. A fourth feature of this paper, and a feature which has partly motivated it, is the contrast of the linear with the nonlinear case. As it will be shown, there are many similarities in the computations of both. This demonstration can enhance our understanding of the nature of the estimation equations. Two additional features of this paper are the treatments of identities in the system and of residuals which may follow an autoregressive scheme. We will derive in section II the estimation equations for nonlinear systems, under the assumptions that each structural equation contains a distinct set of parameters, that the parameters are not subject to any linear restrictions, and that the (additive) residuals are serially uncorrelated. Section III treats the special case when some equations are linear, and contrasts this case with the nonlinear case. Section IV deals with identities and linear restrictions on the parameters. Section V is concerned with the problem of autoregressive residuals.

They indicate that purchases of durable goods (DUR) are the third most volatile component behind inventory investment (INV I) and federal government expenditures (F GOV). The correlation coefficients between GNP and its components listed in the second row of table 1 show that durable goods purchases have the third hiighest covariance with GNP after inventory investment and nonresidential investment (NRI). Theoretically, these purchases represent either changes in the size of the household portfolio through changes in the flow of savings, or a reallocation of accumulated wealth among assets in response to changes in rates of return. Empirically, some effort has been given to examining the separate influences of rates of return and income. Hamburger (1967) found that interest rates, the price of durable goods relative to other prices faced by the consumer, and disposable personal income all have a significant impact on purchases of durable goods. portance of these variables as sources of fluctuation in purchases. Motley (1970) included a user cost of real assets variable in addition to the rate on savings deposits and expected income, and found in sharp contrast to Hamburger that none of them had a significant influence on the demand for the sum of durables and housing. The analysis of fluctuations in durable good purchases presented here differs from its prede-

Robert C. Blattberg, Evaluation of the Power of the Durbin-Watson Statistic for Non-First Order Serial Correlation Alternatives, The Review of Economics and Statistics, Vol. 55, No. 4 (Nov., 1973), pp. 508-515

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.

CONTROVERSIES involving inflation, particularly inflation in developing countries, have usually focused on Latin America.1 One major point which emerges from these controversies is the distinction between a fully anticipated, fully adjusted inflation and an inflation which proceeds with such irregularity that economic agents are able neither to anticipate nor to adjust completely. To the extent that individuals can anticipate and adjust to inflation, a higher rate of inflation will cause the income velocity of money to rise, as attempts are made to exchange money for hedges against inflation.' The influence of inflation on monetary velocity is less clear, however, under the conditions of imperfect anticipation and adjustment which prevail in Latin America. Not only are the rates of inflation in most Latin American countries high, but they also tend to be highly variable. In addition, most Latin American countries have less than perfect markets for hedging against inflation, and these are further restricted by the regulations often imposed on interest rates, prices and international trade in the wake of inflationary conditions.' The present paper examines the impact of inflation on the income velocity of money for sixteen Latin American countries over the period 1950-1969. Such an examination not only indicates the sensitivity of demand for real cash balances to changes in the price level, but also reflects the extent to which economic agents under conditions prevailing in Latin America can anticipate inflation and adjust by hedging. Aside from Cagan's well-known work on hyperinflation (Friedman, 1956, pp. 25117), most empirical studies of the demand for money in individual countries conclude that inflation does not have a significant impact on velocity.4 These studies generally argue that the small changes in the price level usually observed cannot be adequately anticipated or are not large enough to cover the costs of adjustment. A recent article by Melitz and Correa (1970) on international differences in income velocity, like most studies of individual countries (but contrary to theoretical expectations), also concludes that inflation does not influence velocity. This article, like the present study, uses international comparisons, but the findings differ substantially. Melitz and Correa find that the coefficient for the impact of inflation on velocity does not have the expected sign and therefore omit the inflation variable from further consideration. They argue that price changes are important only in cases of hyperinflation and that adjusting to mild inflation is too costly and difficult to be worthwhile. Having excluded inflation as an explanatory variable, Melitz and Received for publication May 24, 1972. Revision accepted for publication January 30, 1973. * The authors wish to thank Michael C. Lovell for many helpful comments and suggestions, Francisco Chaves for assistance with data collection and computational work, and the Wesleyan Computer Center for generous use of its facilities. ' Best known is the monetarist-structuralist controversy over the causes of inflation and the impact of inflation on economic development. See, for example, Baer and Kerstenetzky (1964), Johnson (1967, pp. 281-291) and Baer (1967). 2-Johnson (1967, pp. 104-142) identifies the tax on real cash balances as the essence of the quantity theory approach to inflation, and it is the efforts to escape this tax which cause monetary velocity to rise with inflation. 'In discussions of inflation and economic growth, more costs and benefits of inflation are attributed to structural imperfections rather than to the tax on real cash balances. Structuralists emphasize the benefits of inflation in circumventing market imperfections, while monetarists focus on the costs of inflation in conjunction with inappropriate government regulations. See Johnson (1967, pp. 281-291) and Baer (1967). 4 However, some studies in a collection edited by Meiselman (1970) provide limited support for a positive influence of inflation on velocity in several less-developed countries. Deaver (Meiselman, 1970, pp. 7-67), finds the rate of inflation to be a significant variable in explaining velocity changes in Chile during the period 1932-1955, while Campbell (Meiselman, 1970, pp. 339-386) finds a positive correlation between velocity and changes in the rate of inflation in a comparative study of South Korea and Brazil. In a cross-country study Perlman (Meiselman, 1970, pp. 297337) finds nominal interest rates or inflation rates (as proxies for the opportunity cost of holding money) to be significant in explaining international differences in liquid asset portfolios.

T HIS research provides an estimate of the demand for educational attainment across states within a framework of optimal community choice. The communi,ty is envisioned as having the ability to choose a level of educational attainment for its students which will maximize its net benefits subject to the prevailing technical relationship. The technical relationship specified in this paper considers separately the impact of school inputs, pupil inputs, and social characteristics on educational attainment. Most previous studies of the educational industry have failed to specify a structural model of educational attainment which simultaneously accounts for supply and demand factors. Those which have attempted to measure the effects of inputs on educational attainment have failed to standardize for demand conditions. Expenditure studies have either ignored supply conditions or have resulted in reduced form equations in which the structural parameters cannot be identified. McMahon (1970) estimated the relationship between expenditures and inputs using state data. His conceptual framework did not, however, permit the estimation of the price and income elasticities of demand.' Estimates of income elasticities have been made by Hirsch (1960), Brazer (1959), and Pryor (1968). Their estimates utilized a single equation expenditure function approach which failed to take account of variations in price and the simultaneous aspects of the determination of price and quantity. With the exception of a recent paper by Barlow (1970) estimates of price elasticities have not been published in studies of educational expenditures.2 While Barlow's demand function is similar in form to our own, his single equation estimation procedure ignored the effects of supply changes and thereby introduced the possibility of simultaneous equation bias. The approach utilized in this research is to take account explicitly of the simultaneous nature of demand and supply. Our conceptual framework permits the identification of both price and income elasticities as well as output elasticities for the inputs. Our price and income elasticities are statistically significant and consistent with theoretical expectations. We find that school inputs, pupil inputs, and community characteristics all have important impacts on educational attainment.