mating an equation like (10.39)]. imposing no parameter constraints. After obtaining initial estimates of the parameters in the demand and supply functions the predicted value of P] was obtained as in equation (10.38). This was introduced in the supply function which was then reestimated by OLS. This is just a two-stage method of imposing the parameter constraints in (10.39).

5. The joint estimation method. In this model the explicit expression for the rational expectation P, is derived and then equation (10.40) is estimated jointly with the demand function imposing parameter constraints.

The results from these five methods of estimation are presented in Tables 10.6 and 10.7. Regarding the estimates of the supply function, the price coefficient is significant only when the estimation is done by the 2SLS method or the joint estimation method. The IV and OV methods which are less efficient methods for the estimation of rational expectations models gave rather worse results (the results, however, for the two methods are very close to each other).

Table 10.6 Estimates of the Parameters of the Supply Function

"Figures in parentheses are standard errors. »DW is the Durbin-Watson test statistic.

The Cobweb model gave the worst results. Of some concern is the fact that the coefficient a, of Q, , is not significantly different from I, and the DW statistic is low for a model with a lagged dependent variable. Both these results indicate that the model is misspecified.

Regarding the estimates of the demand function (presented in Table 10.7), the Cobweb model and the 2SLS, OV, and joint estimation methods for the rational expectations model, all gave very similar results. It is only the IV method that gives slightly different results. The results are not very encouraging, although p, and p, both have expected signs. The negative coefficient of implies a gradual downward shift of the demand function. .

Overall, we cannot say that the data are very informative about the models. All we can say, perhaps, is that the rational expectations model appears to be more appropriate than the Cobweb model. Many coefficients are not significant (although they have signs that one would expect a priori). The computations, however, illustrate the methods described. In estimating the rational expectations models, we used progressively more information about the models in the four methods described. Although the results are not much different in this case, one would be able to find differences in other cases. The estimation of the hyperinflation model described in Section 10.5 is left as an exercise.

10.13 The Serial Correlation Problenn in Rational Expectations Models

Until now we have assumed that the errors are serially uncorrelated. If they are serially correlated, we cannot use variables in the information set /, , as instruments. The following example illustrates the problem. Consider the model

y, - cur; + , + , (10.41)

z, is an exogenous variable, but x, can be an exogenous or endogenous variable. We will assume that z, is not a part of the information set ,. This means that Z, should also be treated as endogenous if we substitute x, for ,*. Finally, u, follow a first-order autoregressive process

«, = pWi-i + V. IpI < 1 and V, ~ 1N(0, a)

Let us write, as usual, equation (10.41) as

y, = py,-i + oa, - 1 + Pz, - Ppz,-, + V,

Using the rational expectations assumption and substituting x = x, - e„ we get

>/ = py,-i + ax,- apx,., + Pz, - ppz,-, + w, (10.42)

where w, = v, - a(e, - pe, ,). We would like to estimate this equation by instrumental variables. Note that

Summary

1. In this chapter we have discussed three types of models of expectations:

(a) Naive models of expectations.

(b) The adaptive expectations model.

(c) The rational expectations model.

Naive models merely serve as benchmarks against which other models or survey data on expectations are judged.

2. Two methods of estimation of the adaptive expectations model have been discussed:

(a) Estimation in the autoregressive form.

(b) Estimation in the distributed lag form.

The first method is easy to use because one can use an OLS regression program. However, this method is not advisable. The second method is the better one. It can be implemented with an OLS regression program by using a search method (see Section 10.4).

3. Many economic variables when disturbed from their equilibrium poshion do not adjust instantly to their new equilibrium position. There are some lags in adjustment. The partial equilibrium model has been suggested to handle this problem. Recently, a generalization of this model, the error correction model, has been found to be more useful (see Section 10.6). One can combine these partial adjustment models with the adaptive (or any other) model of expectations (see Section 10.7).

4. Besides the partial adjustment and error correction models, two other models are commonly used in the literature on adjustment lags. These are:

(a) The polynomial lag (also known as the Almon lag) (Section 10.8).

(b) The rational lags (Section 10.9).

"A concise discussion of methods of obtaining consistent estimates of the parameters in single-equation models with rational expectations can be found in B. T. McCallum. "Topics Concerning the Formulation. Estimation and Use of Macroeconomic Models with Rational Expectations." Proceedings of the Business and Economic Statistics Section. American Statistical Association. 1979. pp. 65-72.

x, ( correlated with e, , and hence w,. Thus although ,- :, i, and z, , are definitely in the information set ,, we cannot use them as instrumental variables.

The solution to the problem is to use the set of variables in /, , rather than /,..,. Thus we obtain the predicted values x„ x,.,, f„ z,-i, y,-i by regressing each of these variables on the variables in the information set /, , and use these as instruments for estimating equation (10.42). Note that if the degree of autoregression in u, is of a higher order, we have to use the variables in the information set of a higher order of lag. For instance, if u, is a second-order autoregression, we use and so on.

Thus with serial correlation in errors we should not use the variables in the information set /, „, to construct the instrumental variables in models with rational expectations."