regress .y - .y* on Z only and test whether the coefficient of z, is significantly different from zero. Recall that this is what we do in tests for weak rationality where we regress y, - y, on y, ,.

Suppose that (y - y*) is uncorrelated with z, but not with z;. so that in equation (10.32), 2 Pi 0. This implies that the expectations y* are not rational. Now suppose that Z\ and Zz are uncorrelated. We can still get the result that the coefficient of Z in a regression of - * on z, is zero. Thus if we do not reject the hypothesis that this coefficient is zero, it does not necessarily mean that we can accept the hypothesis of rationality.-* This implies that tests of weak rationality are indeed weak. However, in practice, it is highly unlikely that the variables in the information set /, , are uncorrelated. Hence one can have sufficient confidence in the weak tests and one need not worry that all the variables in the information set , are not included in the tests.

10.12 Estimation of a Demand and Supply Model Under Rational Expectations

We will illustrate the estimation of econometric models with rational expectations using equations (10.30) and (10.31) by considering a simple demand and supply model. Consider the following model (all variables considered as deviations from their means):

/ = PiPf + + «1, demand function (10.33)

Qi = 2 / + yjZb + «2/ supply function (10.34)

where q, = quantity demanded or supplied (both are same by the assumption of equilibrium) p, = market price

p* = market price at time / as expected at time (f - 1);

we assume that there is one period lag between production decisions and market supply Z„ Zb = exogenous variables

u„, = serially uncorrelated disturbances with the usual properties , (zero mean, constant variances, and constant covariance)

We consider two cases: Zi, and Z;, known, and unknown, at time (/ - 1).

Case 1

Z, and Z2, are known at time (/ - I), that is, they are in the information set ,. In this case the estimation of the model is very simple. The rational expectations hypothesis implies that

*A. Abel and F. S. Mishkin. "An Integrated View of Tests of Rationality. Market Efficiency and Short-Run of Monetary Pobcy," Journal of Monetary Economics, Vol. II, 1983, pp. 3-24

p, = p, + e,

where ( ,) = 0 and cov(e,, z„) = cov(f„ Zj,) = 0. Thus e, has the same stochastic properties as „ and . Now substituting p] = p, - t, in (10.34) we

= + yySv + (« - P:e») (10.34)

The composite error in this equation has the same properties as the error term in (10.34) (zero mean, and zero correlation with z„ and Z;,).

Thus the implication of the "rational" expectations hypothesis is that all we have to do is to substitute p, for p* and proceed with the estimation as usual. Thus the rational expectations hypothesis greatly simplifies the estimation procedures. This conclusion holds good in any simultaneous equations model involving expectations of current endogenous variables if the current exogenous variables are assumed to be known at time (/ - 1) (or at the time expectations are formed), and the errors are serially independent.

There is, however, one problem that the "rational" expectations hypothesis creates. Suppose that the variable Z:, is missing in equation (10.34). Then with any exogenous specification of the expectational variable p, (as in the adaptive expectations formulation), the equation system would be identified. On the other hand, under the "rational" expectations hypothesis the demand function is not identified. Thus, in the general simultaneous-equations model with expectations of current endogenous variable, the identification properties will depend on the identification properties of the simultaneous equations model resulting from the substitution of y, for y,, where y] is the expectation of the endogenous variable y,. This result makes intuitive sense since what the rational expectations hypothesis does is to make the expectations endogenous.

Case 2

Zu and Z:, are not known at time (/ - 1). That is, the exogenous variables at time / have to be forecast at time U - 1). What will happen if we just substitute pjejpi-.- e, as before? Since e, is not uncorrelated with z,, and Z2, [they are not in the information set at time (r - 1)], the composite error term in (10.34) will be correlated with z„ and z:,.

How do we estimate this model, then? What we have to do is to add equations for z„ and Z:,. A common procedure is to specify them as autoregressions. For simplicity we will specify them as first-order autoregressions. Then we have

Zu = aiM-i + v„ (10.35)

We now estimate equations (10.33), (10.34). and (10.35) together."

. R. Wiekens, "The Efficient Estimation of Econometric Models with Rational Expectations." Review of Economic Studies, Vol. 49, 1982, pp. 55-67. who calls this method the "errors in variables method" (EVM).

where

"2, = "2, + ~(Wi, - «2,)

The error term ul, has the same properties as the error term Mj, except in the special case where ,, and ,, are independent. Thus the consequence of the Muthian rational expectations hypothesis is the inclusion of the extra variables

An alternative way of looking at this is to say that we estimate the demand and supply model given by (10.33) and (10.34) by replacing pj by p, and using lagged values of z„ and Zi, as instrumental variables.

There is an alternative method of estimating the model with rational expectations which is somewhat more complicated. This is called the suhstitution method (SM). in this method we derive an expression for p, using the relationship

p, = £(p,/, ,)

and substitute it in the model and then estimate the model. For this we proceed as follows.

Equating demand and supply we get the equilibrium condition.

PiP, + 7iZ„ + M„ = ; + , + "1, (10.36)

Now take expectations throughout, conditional on the information set /, ,. Note that F(w„/, ,) = 0 and Eiu.JI,,) = 0. Hence we get

PiP; + y,Zu = p2P; + 72 (10.37)

where

Zu = £(z,J/,-,) and zl, = E(Z2,\I,-,)

These are, respectively, the expectations of Zu and Zi, as of time (r - 1). Let us define

Zu = Zu + Wu

Z2, = zi, + 2,

Subtracting (10.37) from (10.36), we get or

P, = P, + -hiu - 72 2, + Uu - «2,) (10.38)

We now substitute this expression in (10.34) and get

, = P2P, + 722, + ~yu - ~2, + "2, (10.39)

Pi pi