where y, such that 0 < y < 1, is known as the coefficient of expectation. Hypothesis (17.5.2) is known as the adaptive expectation, progressive expectation, or error learning hypothesis, popularized by Cagan14 and Friedman.15

What (17.5.2) implies is that "economic agents will adapt their expectations in the light of past experience and that in particular they will learn from their mistakes.''16 More specifically, (17.5.2) states that expectations are revised each period by a fraction y of the gap between the current value of the variable and its previous expected value. Thus, for our model this would mean that expectations about interest rates are revised each period by a fraction y of the discrepancy between the rate of interest observed in the current period and what its anticipated value had been in the previous period. Another way of stating this would be to write (17.5.2) as

13Sometimes the model is expressed as

14P. Cagan, "The Monetary Dynamics of Hyperinflations," in M. Friedman (ed.), Studies in the Quantity Theory of Money, University of Chicago Press, Chicago, 1956.

15Milton Friedman, A Theory of the Consumption Function, National Bureau of Economic Research, Princeton University Press, Princeton, N.J., 1957.

16G. K. Shaw, Rational Expectations: An Elementary Exposition, St. Martin's Press, New York, 1984, p. 25.

Models: Autoregressive and Distributed-Lag Models


which shows that the expected value of the rate of interest at time t is a weighted average of the actual value of the interest rate at time t and its value expected in the previous period, with weights of y and 1 _ y, respectively. If y = 1, X* = Xt, meaning that expectations are realized immediately and fully, that is, in the same time period. If, on the other hand, y = 0, X* = X*_1, meaning that expectations are static, that is, "conditions prevailing today will be maintained in all subsequent periods. Expected future values then become identified with current values.''17 Substituting (17.5.3) into (17.5.1), we obtain

Now lag (17.5.1) one period, multiply it by 1 _ y, and subtract the product from (17.5.4). After simple algebraic manipulations, we obtain where vt = ut _ (1 _ Y)ut-1.

Before proceeding any further, let us note the difference between (17.5.1) and (17.5.5). In the former, j1 measures the average response of Y to a unit change in X~, the equilibrium or long-run value of X. In (17.5.5), on the other hand, yP1 measures the average response of Y to a unit change in the actual or observed value of X. These responses will not be the same unless, of course, y = 1, that is, the current and long-run values of X are the same. In practice, we first estimate (17.5.5). Once an estimate of y is obtained from the coefficient of lagged Y, we can easily compute j 1 by simply dividing the coefficient of Xt ( = yPi) by y.

The similarity between the adaptive expectation model (17.5.5) and the Koyck model (17.4.7) should be readily apparent although the interpretations of the coefficients in the two models are different. Note that like the Koyck model, the adaptive expectations model is autoregressive and its error term is similar to the Koyck error term. We shall return to the estimation of the adaptive expectations model in Section 17.8 and to some examples in Section 17.12. Now that we have sketched the adaptive expectations (AE) model, how realistic is it? It is true that it is more appealing than the purely algebraic Koyck approach, but is the AE hypothesis reasonable? In favor of the AE hypothesis one can say the following:

It provides a fairly simple means of modelling expectations in economic theory whilst postulating a mode of behaviour upon the part of economic agents which

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