Rational Expectations
Estimate problems

There are now a variety of methods of estimating linear econometric models which incorporate rational expectations. Researchers have sought estimation techniques which are statistically efficient or precise, easy to compute and applicable to a wide range of models. Despite many claims to the contrary, it appears that it is not possible to satisfy all three criteria. Although standard estimation procedures can be used to obtain efficient estimates of econometric models with rational expectations, these procedures lack generality. They are applicable only to models in which the only variables about which rational expectations are formed are dated in the current period: difficulties arise if rational expectations are formed concerning the value of a variable during some future period.

In Discussion Paper No. 111, Research Fellow Michael Wickens explores estimation techniques for a number of models which have future-dated rational expectations. Wickens shows that before an efficient method of estimation can be devised it is necessary to know what type of solution the model has. Three types of solution are possible in rational expectations models: a globally stable solution, which in general is not unique, a saddlepoint solution, and a globally unstable solution. The last two may or may not be unique, and may not even exist. Each of these solutions (where they exist) can be given both a 'backward' and a 'forward' representation. These solutions can be expressed in a fairly general form, which may give the impression that a general way of obtaining an efficient estimator is also available. But each type of solution imposes a different set of restrictions on coefficients in the general representation, and efficient estimation requires that these restrictions be taken into account. The exception is the case of globally stable solutions, which impose no coefficient restrictions and are therefore not unique. Thus an efficient method of estimation is not generally available.

Efficient estimation, Wickens argues, is not possible unless there is some reason to impose a particular type of solution on the model. One possibility is to assume at the outset that the solution is unique. Efficient estimation will then be possible, but will require knowledge of the restrictions that need to be imposed. One can then obtain fully efficient estimates of the model's coefficients. The coefficient restrictions imposed in order to obtain the unique solution can then be tested by re- estimating the model without the restrictions and carrying out either a Likelihood Ratio or a Lagrange Multiplier test. In the absence of such an assumption of uniqueness, Wickens notes, it will be necessary to discover what type of solution the model possesses before an efficient estimator can be obtained. This will require preliminary estimation and hypothesis testing.
Wickens analyses a number of commonly used models. He discusses different ways of representing the solutions to these models and relates these representations to previous solutions that have appeared in the literature. Each solution is given one or more backward and forward representations, and the coefficient restrictions associated with each representation and each type of solution are given. Wickens proposes both fully efficient and less efficient estimators for each model and for each type of solution.


The Estimation of Linear Models with Future Rational Expectations by Efficient and Instrumental Variable Methods
Michael Wickens

Discussion Paper No. 111, June 1986 (IM/ATE)