Since this involves an infinite series and we do not observe the infinite past values of x„ we have to tackle this problem somehow. What we do is to break up the series into the observed and unobserved past. We will write the infinite series as

The first part is observed and we will denote it by ZuOO- We use X since it depends on K. The second part can be written as

eo

\ 2 (1 - X)X"x,, = X S (1 - Wxj (writing J = i - t)

Notice that the second part of this expression is nothing but (10.9) with t = 0, that is x*i (the expected price for the first period). If we treat this as an unknown parameter and define Zi, = we can write

X,+ i = Zu + CZz,

Thus equation (10.1) that we wish to estimate can be written as

y, = a + bx;+i + u,

= a + bizM + cZb) + u,

= a + bzu + cz2, + u, (10.15)

with c = be. Note that z„ and Zo, depend on X. Actually, we are not interested in the parameter c. The estimation proceeds as follows. For each value of X in the range (0, 1) we construct the variables

Zu = S (1 - % ,

1 = 0

and Z2, = X. Thus

Z„ - (1 - X)x,

Zn = (1 - X)(X2 + Xx,)

Zn = (1 - X)(JC3 + XX2 + Xx,)

and so on. We estimate equation (10.15) by ordinary least squares and obtain the residual sum of squares. Call this RSS(X). We choose the value of X for which RSS(X) is minimum and obtain the corresponding estimates of a and b as the desired least squares estimates.

Note that since z,, and Z2, are nonlinear functions of X, the estimation of (10.15) involves nonlinear least squares. What we have done is to note that/or given X we have a linear least squares model. Thus we are using a search procedure over X. In practice one can choose X in intervals of 0.1 in the first stage and then intervals of 0.01 in the second stage.

10.5 TWO ILLUSTRATIVE EXAMPLES 413

10.5 Two Illustrative Examples

As mentioned earlier, the adaptive expectations model was used by (among others) Nerlove for the analysis of agricultural supply functions, and by Cagan for the analysis of hyperinflations. The hyperinflation model involves more problems than have been noted in the literature, and hence we will discuss it in greater detail. In the analysis of agricultural supply functions, we have to deal with the expectation of a current endogenous variable p„ whereas in the hyperinflation model, we have to deal with the expectation of a future endogenous variable Consider the supply function

a = a + ,* + IZ, + u,

where p* is the price expected to prevail at time t (as expected in the preceding period), Q, the quantity supplied, z, an exogenous variable, and u, a disturbance term. The adaptive expectations model implies that

p] - xp;„, = (1 - x)p,„,

Thus

Qi - -t = a - aX + ( , - XpJ-i) + yiz, - Xz,„,) + u, - Ku,, Eliminating p, we get

Q, = a(l - X) + Xft-i + (1 - X)p,, + 7z, - x7z, , + «, - , ,

This equation can be estimated by the methods described in Section 10.4. Computation of the estimates for this model using the data in Table 9.1 is left as an exercise. We will consider Cagans hyperinflation model in greater detail because it involves more problems than the agricultural supply model.

The hyperinflation model says that the demand for real cash balances is inversely related to the expected rate of inflation. That is, the higher the expected rate of inflation, the lower the real cash balances that individuals would want to hold. Of course, the demand for real cash balances depends on other variables like income, but during a hyperinflation the expected inflation rate is the more dominant variable. We specify the demand for money function as

m, - p, = a + b(p,y - p,) + u, b < 0

where m, is the log of the money supply, p, the log of the price level, p*+, the expectation of p,, as expected at time t, and u, the error term. Since the variables are in logs, pj+, - p, is the expected inflation rate. We will denote it by it+i. The actual rate of inflation is ,+ , = p,+, - p,. Let us define y, = m, -Pf. Then the demand for money model can be written as

where

1 + 6(1 - X) 1 + b(l - X) 1 + (1 - X)

e, = . . . -- V, =

1 + fc(l - X) I + (1 - X)

Thus the appropriate equation to estimate for the estimation of the hyperinflation model under adaptive expectations is equation (10.15") and not equation

This is similar to equation (10.1) and the estimable equation we derived in Section 10.4 is equation (10.14). This equation applies here with , in place of x,. The equation we get is

y, = a{\ - X) + Xy, , + b{\ - \) , + V, (10.14)

where v, = u, - Xw, ,. Cagan estimated this equation by OLS but the problem with the OLS estimation of this equation is that it gives inconsistent estimators because the equation involves the lagged dependent variable y, , and the errors are serially correlated. One can use Kleins method or estimation in the distributed lag form to avoid this problem. The equation we estimate is similar to equation (10.15) and is

y, = a + bzu + czb + u, (10.15)

where

Zu = 2 (1 - X)X4-,

1 = 0

and Z2, - X. Thus for each value of X we generate z„ and Zz, and estimate (10.15) by OLS. We choose the value of X for which the residual sum of squares is minimum, and the corresponding estimates of a, b, and c.

However, even though this is the correct method of estimation for the agricultural supply functions, it is not the correct method for the hyperinflation model. The reason is that the model is one where money supply m, is the exogenous variable and price level p, is the endogenous variable. By defining the variable y, = m, - p, and writing the equation in a form similar to (10.1) we have missed this point. Equation (10.15) cannot be estimated by OLS because TT, = p, - p, , in Zu is correlated with the error term u,.

We can solve this problem by moving the variable p, to the left-hand side. Let us define W, = Zu - (1 - X)p,. Now W, involves p, i and higher-order lagged values of p, and does not involve current p,. We can write equation (10.15) as

m, - p, = a + b(l - k)p, + bW, + cz-i, + u, Collecting the coefficients of p, and simplifying, we can write this equation as p, = Go + e,m, + el, + %yZ2, + V, (10.15")