I. Introduction, 403. — II. Demand equations and evidence for acceleration, 404. — III. The nonstochastic case: no oscillations without acceleration, 406. — IV. Autocovariances of a linear stochastic system, 408. — V. Spectral densities of a linear stochastic system, 411. —VI. Conclusions, 417.

An approximate solution, based on the method of dynamic programming, is provided for the optimal control of a system of nonlinear structural equations in econometrics with unknown parameters using a quadratic loss function. It generalizes the methods previously proposed by the author for the control of a nonlinear econometric model with constant parameters and of a linear econometric model with uncertain parameters. It is an improvement over the method of certainty equivalence which replaces the unknown parameters by their mathematical expectations and utilizes the solution for the resulting model. Since the solution is given in the form of feedback control equations, many of the useful concepts and techniques developed in the theory of optimal feedback control for linear systems are now applicable to the control of nonlinear systems using the method proposed, including the calculation of the expected loss of the system under control by analytical rather than Monte Carlo techniques. IN THIS PAPER, I present an approximate solution to the optimal control of a system of nonlinear structural equations using a quadratic welfare loss function when the parameters of the system are unknown. This is a generalization of ths solution given in Chapter 12 of Chow [2] for the control of nonlinear econometric systems with known parameters. It is also a generalization of the solution given in Chow [1] for the control of linear econometric systems with unknown parameters. The method of dynamic programming is applied to solve an optimal control problem involving a nonlinear econometric system with unknown parameters. As it turns out, the solution amounts to linearizing the nonlinear model about some nearly optimal control solution path and then applying a method for controlling the resulting linear model with uncertain parameters. This paper advances the state of the art in the control of nonlinear econometric systems as it improves upon the certainty-equivalence solution which is obtained by replacing the random parameters in a system by their mathematical expectations. It provides for a set of numerical feedback control equations based on a system of nonlinear structural equations in econometrics. It will show that many useful analytical concepts and tools developed in the theory of control of linear systems are indeed applicable to the control of nonlinear systems. Furthermore, in the derivation of an approximate solution using the method of dynamic programming, it will indicate precisely where the approximation takes place and why an exact solution is difficult to achieve. In Section 2, we set up the control problem and provide an exact solution to the optimal control problem for the last period. In Section 3, we give an approximate solution to the multiperiod control problem using dynamic programming. In Section 4, the mathematical expectations required in the solution of Section 3

Given n observations on a system of linear stochastic difference equations with appropriate initial conditions, and given a prior density (possibly diffuse) of its parameters, this paper obtains the predictor of the time series k periods into the future with minimum mean squared error. Completely analytical solution is given for predictions from the first-order univariate system, and, in the general higher-order multivariate case, for k up to 5. econometric equations to produce forecasts were not designed for the purpose of forecasting. In this paper, it is argued that these estimation methods may be inadequate if the resulting estimates are to be used to make ex ante predictions for more than one period ahead, and if the accuracy of the predictions is measured, as it usually is, by the mean squared errors. A formulation of the multiperiod prediction problem is presented. It will then become clear that the same set of parameter estimates cannot be optimal in making predictions for different time periods into the future, when optimality is defined by minimum mean squared errors in small samples. Recently there has been much interest in comparing different econometric models, or different versions of the same econometric model obtained by applying different estimation techniques, in terms of how well they would have forecasted the dependent variables during the sample period, given the true values of the exogenous variables and given the values of the dependent variables lagged one or more periods. It has now become clear that, as judged by ex post forecasting for the sample period, models or techniques that perform better for one-period predictions may do worse for multiperiod predictions. For example, purely auto- regressive models could do better than models based on structural equations in ex post forecasting for one quarter ahead, but were worse in forecasting three or four quarters ahead, as documented in Hickman (4). Klein (9) and Fair (3) have compared multiperiod predictions of the sample data by different estimation techniques applied to the same econometric model. A related, though different, question naturally arises as to whether different parameter estimators should be used to produce ex ante forecasts for different periods into the future. The former topic is one of fitting equations to a set of data. The latter topic is one of statistical decision, and is the subject of this paper.'

[Given n observations on a system of linear stochastic difference equations with appropriate initial conditions, and given a prior density (possibly diffuse) of its parameters, this paper obtains the predictor of the time series k periods into the future with minimum mean squared error. Completely analytical solution is given for predictions from the first-order univariate system, and, in the general higher-order multivariate case, for k up to 5.]

A natural generalization of least squares is proposed to estimate parameters in simultaneous linear equations. Full-information maximum likelihood is shown to be identical with this generalization. The extent to which other estimators deviate from the generalization is discussed. A paradox of Strotz is resolved, and application of canonical correlation theory to structural equations is indicated. 1. SUMMARY THIS PAPER compares various estimators of the parameters of linear simultaneous

A two-step procedure for estimating linear simultaneous structural equations with unit roots is presented. It generalizes the procedure of differencing for univariate time series via G. Box and J. M. Jenkins (1970). First, one finds the number of unit roots and the canonical variables that are stationary. Second, one retains only the stationary canonical variables and estimates a stationary model by standard methods. The procedure is easy to use. It is robus t against the difficult testing problem of finding the correct number of unit roots. A multiplier-accelerator model is estimated with interesting conclusions.

Using data on stock price and dividends, and on long-term and short-term interest rates, the authors test an important implication of present value models--that current value is a linear function of the conditional expectations of the next-period value and the current determining variable . This implication, combined with rational expectations, is strongly rejected. Combined with adaptive expectations, it is accepted. The latter model can also explain the observed negative relation between the rate of return and stock price. Thus the rational expectations assumption should be used with caution; the adaptive expectations assumption may be useful in econometric practice. Copyright 1989 by MIT Press.(This abstract was borrowed from another version of this item.)

N the study of optimal economic policy using a linear econometric model and a quadratic welfare function the parameters of the model are often assumed to be known for certain. Under this assumption the solution in the form of an optimal feedback equation can be obtained easily. Although one recognizes that in a realistic situation the parameters of an econometric model are never known for certain, he might still apply the above solution, using a set of estimates of the parameters as if they were the true values, if he believes that it is a good approximation to optimal policy. Such a procedure is well known to be a certainty equivalence solution. In a recent paper (Chow, 1973b), I have presented a method of obtaining the optimal feedback equations and the associated welfare costs by allowing for uncertainty in the parameters as expressed in the posterior density function computed from data available at the time of the current decision but not for possible future revision of this posterior density in the derivation of the current policy. Because future learning about the model is not explicitly taken into account in the design of the current policy, the above method is not truly optimal. However if the sample period is long as compared with the planning period, this method will probably be close to being optimal. The purposes of this paper are to present an approximate solution to optimal when learning is taken into account and to contrast this solution with the first two solutions. In the literature, the term control is used for a problem having the dual purpose of improving the system performance and of learning more about the system for the sake of future control. Numerous approximate solutions to this problem have been suggested.' The solution of this paper appears to be the simplest in conception, and yet it incorporates all theoretical elements in the calculations. It contains a logical structure which brings out clearly the effect of learning on the optimization process and enables the effect to be measured numerically. It provides useful contrasts to the certainty equivalence solution and the solution for unknown parameters without learning, being a natural generalization of these two solutions. We will set up the problem and describe the method of solution in section II. This method will be compared in section III with the two other methods just mentioned, both in conceptual terms and in terms of computations. Two simpler, modified versions of the method will also be briefly described. They are simpler to compute but they still take learning partially into account. Some numerical results using a simple one-equation model will be presented in section IV to bring out the effects of learning on the optimal solution. This paper is confined mainly to presenting the method and providing some illustrative calculations. A comprehensive study of the effect of learning on optimal policies using the method of this paper remains to be undertaken. From the viewpoint of economics in general, other than the study of quantitative economic policy using econometric models, the content of this paper may also be relevant. Maximization is in the heart of economics. Most of economic theory assumes maximization to take place in Received for publication December 26, 1973. Revision accented for publication July 8, 1974. * I am much indebted to Andrew Abel for extremely able research assistance, to Edison Tse, Ray C. Fair and several members of the Econometric Research Program seminar at Princeton for valuable suggestions and discussions, to a referee for comments on an early draft, and to the National Science Foundation for financial support through Grant GS32003X. 1 The references in the literature are too numerous to cite. In the economics literature, Prescott (1972) deals with the problem of learning using a very simple model but provides no new method of solution; its results were computed by complete enumeration. MacRae (1972) and Tse (1974) provide interesting approximations to the optimal solution and are highly recommended to the reader.