Table 10.4 Unrestricted Least Squares Estimates of Distributed Lags"

10.10 RATJONAL EXPECTATIONS 431

Consider equation (10.11). One way of generalizing it is to add more lags on the left-hand side and right-hand side of the equation, and have the parameters different. If p is the number of lags on the right hand side and q the number of lags on the left-hand side, we have a rational distributed lag model which we denote by RDL {p, q). For instance, the RDL (3, 2) model corresponding to (10.11) is )

A similar generalization can be given for the partial adjustment model (10.16). We have

y, + Piy/~i + 2 ,-2 = « ? + a,yt, + 1{+ .

As before, we have to somehow eliminate the unobserved variables x* and yf. Since the algebraic details are similar to those in Section 10.7, we will not pursue them here." Lucas and Rapping" use the RDL model for price expectations, instead of the adaptive expectations model, but they estimate it in the autoregressive form ignoring serial correlation.

The rational distributed lag model and its use in expectations should not be confused with the rational expectations models we discuss in the next section.

10.10 Rational Expectations

In Sections 10.2 and 10.3 we discussed some naive models of expectations and the adaptive expectations model. We also saw how the latter can be generalized using some polynomial lags. During the last decade a new theory, that of "rational expectations," has taken a strong hold in almost all econometric work on expectations.

The basic idea of "rational expectations" comes from a pathbreaking paper by John Muth,- who observed that the various expectational formulas that were used in the analysis of dynamic economic models had little resemblance to the way the economy works. If the economic system changes, the way expectations are formed should change, but the traditional models of expectations do not permit any such changes. The adaptive expectations formula, for instance, says that economic agents revise their expectations upward or downward based on the most recent error. The formula says

"The estimation of the RDL model in the distributed lag form is discussed in G. S. Maddala, Econometrics (New York: McGraw-Hill, 1977), pp. 366-367.

™R. E. Lucas, Jr., and L. A. Rapping, "Price Expectations and the Phillips Curve," The American Economic Review, Vol. 59, 1969, pp. 342-350. They warn that this should not be confused with the "rational expectations" model as defined by Muth (which we discuss in the next section).

"John F. Muth, "Rational Expectations and the Theory of Price Movements," Econometrica. Vol. 29, 1961, pp. 315-335.

/, ~ :~ = Hy.i - yUx) 0<X<1

where y* is the expectation for y, as formed at time / - 1. In this formula X is a constant. Of course, one can modify this formula so that X depends on whatever variables produce changes in the economic system.

There are some reasonable requirements that y* or the predicted value of y, should satisfy. Consider the prediction error

e, = y, - y,*

It is reasonable to require that expectations be unbiased, that is,

E{e,) = 0

Otherwise, there is a systematic component in the forecast error which forecasters should be able to correct.

Muth also required that the prediction error be uncorrelated with the entire information set that is available to the forecaster at the time the prediction is made. If we denote by /, , the information available at time (r-1) and write

y, = y: + E, (10.30)

they yl depends on /, , and y, is uncorrelated with e,. One implication of this equation is that varCv,) = varCv*) + var(e,) and hence var(y,) > var(y*). If the prediction error is correlated with any variables in /, ,, it implies that the fore-Caster has not used all the available information. Taking (mathematical) expectations of all variables in (10.30), we get

yJ = £CvJ/,-,) (10.31)

The left-hand side of (10.31) should be interpreted as the subjective expectation and the right-hand side of (10.31) as the objective expectation conditional on data available when the expectation was formed. Thus there is a connection between the subjective beliefs of economic agents and the actual behavior of the economic system.

Formula (10.31) forms the basis of almost all the econometric work on rational expectations. There are three assumptions involved in the use of formula (10.31).

1. There exists a unique mathematical expectation of the random variable y, based on the given set of information

2. Economic agents behave as if they know this conditional expectation and equate their own subjective expectation of y, to this conditional expectation. Note that this implies that the economic agents behave as if they have full knowledge about the model that the econometrician is estimating, that is, they behave as if they know not only the structure of the model but the parameters as well.

3. The econometrician, however, does not know the parameters of the model but has to make inferences about them based on assumption 2 about the behavior of economic agents and the resulting stochastic behavior of the system.