*K. F. Wallis. "Econometric Implications of the Rational Expectations Hypotheses." Econometrica. Vol. 48, 1980. pp. 49-74; N. S. Revankar. "Testing of the Rational Expectations Hypothesis." Economeirua. Vol. 48. 1980. pp. 1347-1663; D. L. Hoffman and P. Schmidt. "Testing the Restrictions Implied by the Rational Expectations Hypothesis." Journal of Econometrics. February 1981, pp. 265-288.

w,, and W2, in the supply function and the constraints on the parameters. (Note that the coefficients of w,, and iv,, involve no new parameters.) This is an omitted variable interpretation of the rational expectations models.

If the nature of the autoregressions for Z, and Zi, is specified, one can derive an alternative expression for p]. For purpose of simplicity of exposition, let us assume that Z, and Zb are first-order autoregressive as in equations (10.35). We then have

zl, = aiZ., i

Zb = a2Z2.»-l

Substituting these in (10.37), we get

P\ = R (7 2.>-1 - 7iaiZi.,-i) Pi - P2

Now we substitute this expression in (10.34) and get

Pi - P: Pi - P2

We now estimate this equation along with equation (10.33) and equations (10.35). Again, note that the coefficients of z,and Z2, i in equation (10.40) do not involve any new parameters. If higher-order autoregressions are used for Z, and Z2,. then p] will involve more of the lagged values of these variables.

Note that even if estimates a, and a, are obtained from (10.35) and substituted in (10.40). we still have to deal with cross-equation constraints. Tests of these cross-equation constraints have been often referred to as "tests for the rational expectations hypothesis."-" The restrictions, however, arise because the exogenous variables ;„ and Zi, are not known at time (/ - 1) and not from the rational expectations hypothesis as such. Moreover, the number of restrictions depends on the specification of the order of autoregression of the exogenous variables z„ and Z;,. In view of this, it might be inappropriate to name the tests for the restrictions as tests of the rational expectations hypothesis. Note also that if equation (10.39) is used, the restrictions do not depend on the specification of the order of autoregression of the exogenous variables.

We have used a simple demand and supply model to outline the problems that are likely to arise in the Muthian rational expectations models if the exogenous variables are not known at time (/ - 1).

Illustrative Example

We will illustrate the methods we have described, with the estimation of a demand and supply model. In Table 10.5 we present data for the 1964-1984 growing seasons on fresh strawberries in Florida. The variables in the table are

P, = price (cents per flat)

Q, = number of flats (thousands)

d, = food price deflator (1972 =1.00)

C, = production cost index (1977 = 100)

N, = U.S. population (millions) •

X, = real per capita food expenditures (dollars per year)

The model estimated was the following

a = ao + a,(F; - C) + a,e, , -f- l/„ supply

P, ~ d, = fio + - N,) + + p,/ + M,, demand

All variables are in logs except the time trend /. (Note that the data in Table 10.5 are not in logs.) P, is the expected price (as expected at time / - I). The different estimation methods depend on different specifications of P,.

Five different models have been estimated (except for the first model, the other models assume rational expectations):

would like to thank my colleague, J Scott Shonkwiler. for the data, model, and computations.

Summary •

There are three procedures we have outlined.

1. Just substitute p, for /7* and use lagged exogenous variables as instruments. This can be done for all expectational variables.

2,,Estimate autoregressions for Z, and z- Get the estimated residuals -,, and ,,. Use these as additional regressors as in equation (10.39) after substituting p, for p]. Estimate the equations using the parameter constraints (computer programs like TSP do this).

3. Get an expression for p] based on the structure of the model and the structure of the exogenous variables as in equation (10.40). Estimate the equations using the parameter constraints.

Note that method I uses equation (10.30), which says that p, = p + e,. Methods 2 and 3 use equation (10.31), which says that

p, = E(P,\I,- 1)

The latter methods use the structure of the model in deriving an expression for p,. Method 3 uses the structure of the exogenous variables as well.

1. The Cobweb model. In this model we substitute -i for P] and estimate the demand and supply functions by OLS.

2. The 2SLS (Two-Stage Least Squares) method. This is the procedure for the rational expectations model if the exogenous variables are assumed to be known at time / - 1 and the errors are not serially correlated. Since the rational expectations hypothesis implies

where v, is an error uncorrelated with the variables in the information set /, ,, we just substitute P, - v, for P, and combine the error v, with the error in the supply function. Since v, has the same properties as „, we just estimate the model by two-stage least squares (2SLS).

3. The IV (Instrumental Variable) method. This is the method we use to get consistent estimators of the parameters under rational expectations when the exogenous variables at time t are not known at time (/ - 1). In this case, the error v, can be correlated with the exogenous variables. We used the lagged exogenous variables as instruments.

4. The OV (Omitted-Variable) method. In this method we first estimated first-order autoregressions for the exogenous variables and used the residuals as additional explanatory variables in the supply function (esti-

Table 10.5 Data on Florida Fresh Strawberries. 1964-1984