There have been many criticisms of the rational expectations hypothesis as implied by equation (10.31). The basic argument is that "rational" economic agents need not behave this way. We will not go through these criticisms here. The whole controversy surrounding the rational expectations hypothesis could have been avoided if the term "rational" was not used to describe the mechanism of expectation formation given by (10.31). A more appropriate term to characterize (10.31) is model consistent, because the expression fory, derived from (10.31) depends on the particular model we start with. The coinage of the term "rational" is rather unfortunate. We will, however, continue to use the word "rational" to describe "model consistent" or Muthian expectations formed using (10.31).

The requirement that y* should completely summarize the information in /, , has led Lovell" to suggest the term "sufficient expectations" because it is related to the statistical concept of a "sufficient estimator" which may be loosely defined as an estimator that utilizes all the information available in the sample.

There are, basically, two methods of estimation for rational expectations models. The first method involves using the definition (10.31) and writing

y, = y, - V,

where v, is uncorrelated with all the variables in the information set /, ,. We then estimate this model using the instrumental variable methods using the appropriate instruments. This method is very easy to apply and gives consistent, though not efficient estimators. We will illustrate this using the hyperinflation model discussed in Section 10.5.

The second method uses information on the structure of the model to derive an explicit expression fory,. This method involves the following steps:

1. Derive the equation for y, from the model you start with.

2. Take expectations of y, conditional on /, , and substitute the resulting expression fory in the model.

3. Reestimate the model with this substitution.

We will illustrate this with a simple demand and supply model in Section 10.12. We will also present the results from the instrumental variable method for purposes of comparison.

The instrumental variable method, although not efficient, gives consistent estimators of the parameters in all rational expectations models and is worth using at least as an initial step. Consider, for instance, Cagans model of hyperinflation. The model is

m, - p, = a + b(pU, - p,) + u,

Now substitute ,. - v,, for p,\,. where v,, is uncorrelated with all variables in the information set We can write the model as

m,-p, = a + b{p,, - p,) + , - bv,.i

°M. C. Lovell, "Tests of the Rational Expectations Hypothesis," Tfte American Economic Review, Vol. 76. March 1986. pp. 110-124.

10.11 Tests for Rationality

There is a considerable amount of literature on what is known as "tests for rationality." In this literature we do not start with any economic model. Usually, we have observations on y* from survey data or other sources and we test whether the forecast error y, - y] is uncorrelated with variables in the information set It is customary to start with a test of unbiasedness by estimating the regression equation

y, = + 1 + /

and testing the hypothesis 3o = 0, 3, = 1.

Further, since y, , is definitely in the information set /, „ the following equation is estimated:

- = «0 + a,y, , + ˆ,

and the hypothesis Oo = 0, a, = 0 is tested. Rationality implies that a, = 0. An alternative equation that can be estimated is

.y, - - , = «0 + a,(y, , - y; ,) + e,

Again, rationality implies that a, = 0. If the forecast errors exhibit a significant nonzero mean and serial correlation (significant a,), then this implies that the information contained in past forecast errors was not fully utilized in forming future predictions.

Tests based on y, , and (y, , - y* ,) examine what is known as the weak version of the rational expectations hypotheses and are tests for weak rationality. The strong version says that the forecast error (y, - yJ) is uncorrelated with all the variables known to the forecaster.

Yet another test for rationality is that var(y,) > var(y*). As we saw earlier if

y, = y, + e,

and e, is uncorrelated with yl, then var(y,) = var(y) + var(E,). Hence var(y,) S: varCv,*). Lovell- considers these two tests:

"Lovell. "Tests."

This equation cannot be estimated by OLS because p,. i is correlated with v,+, and p, with u,. If m, and p, are known at time f. then v,, will be uncorrelated with these variables. One needs an instrument that is uncorrelated with ii, and v,.,. Valid instruments are w, ,, p, -and higher-order lags of w, and p,. One can regress p,, - P, on lagged values of m, and p, and use the 2SLS method.

The estimation of the hyperinflation model under rational expectations using the above-mentioned instrumental variable method using the data in Tables 10.1 and 10.2 is left as an exercise.

10.11 TESTS FOR RATIONALITY 435

1. Tests based on y, . "

2. Tests based on vaitv,) > vaitv,).

He examines the evidence from a number of surveys on sales and inventory expectations, price expectations, wage expectations, data revisions, and so on, and argues that in a majority of cases the tests for rationality reject the hypothesis of rationality.

Some studies that test for rationality of survey forecasts like those by Pe-sando, Carlson, and Mullineaux for price expectations-" and tests by Friedman for interest-rate expectations- use a different procedure. What they do is the following:

1. Regress on the variables in the information set /, ,.

2. Regress y* on the same variables in the information set /, ,.

3. Test the equality of coefficients in the two regressions using the Chow test described in Chapter 4.

However, note that the variance of the error terms in the two equations are likely to be unequal, since varty,) is expected to be greater than var(y,"). Hence some adjustment should be made for this before applying the Chow test. (Divide each data set by the estimated standard deviation of the error term.)

This test is formally identical to the test suggested earlier of regressing the forecast error y, - y] on the variables in /, , and checking that their coefficients are zero. Without any loss of generality, suppose that there are two variables Z and Z: in the information set /, , (z, and Zj can be sets of variables). We can write

= PiZ, -I- PjZj -I-

* = PIz, + + * Then ,

- ) = (Pi - p;)z, -I- (p2 - + in - «*) (10.32)

Thus a regression of (y - y*) on Z and Zj and testing whether the coefficients of z, and Z: are significantly different from zero is the same as testing the equality of the coefficients of z, and z; in the regressions of and y* on these variables. There is something to be said in the favor of the test based on (10.32) instead of the one based on separate regressions for and y*. With the test based on (10.32), the fact that var(w) and var(M*) are different does not matter.

What happens if in the tests for rationality we use only a subset of the variables in the information set /, ,? For instance, in equation (10.32) we just

"J. E. Pesando, "A Note on the Rationality of the Livingston Price Expectations," Journal of Political Economy. Vol. 83, August 1975. pp. 849-858; J. A. Carlson. "A Study of Price Forecasts," Annals of Economic and Social Measurement. Vol. 6, Winter 1977. pp. 27-56: D. J. Mullineaux. "On Testing for Rationality: Another Look at the Livingston Price Expectations Data," Journal of Political Economy. Vol. 86. April 1978. pp. 329-336.

"B. M. Friedman, "Survey Evidence on the Rationality of Interest Rate Expectations," Journal of Monetary Economics. Vol. 6. October 1980, pp. 453-466.