1. Naive models of expectation.

2. Adaptive models of expectation.

3. Rational expectations models.

In each case we concentrate on the econometric problems involved.

Since expectations play an important role in economic activity, there are many surveys conducted by different organizations to find out what consumers expectations are regarding different economic variables. For instance, the Survey Research Center at the University of Michigan conducts surveys regarding consumers attitudes toward purchases of different durable goods and their forecasts about future inflation rates. There are now survey data available on forecasts of a number of economic variables: wages, interest rates, exchange rates, and so on. An important question that arises regards how we can make use of these survey data. Many econometricians have also investigated the question of how well these survey data forecast the relevant economic variables. After discussing the three models of expectations mentioned above, we discuss the usefulness of survey data on expectations.

10.2 Naive Models of Expectations

The earliest models of expectations involved using past values of the relevant variables or simple extrapolations of the past values, as measures of the expected variables. Consider, for instance, an investment equation

y, = a + bx;i + u, (10.1)

where y, = investment in period /

x,+, = expected profits during period / + 1

u, = error term

Unless otherwise noted, expectations are formed in the previous time period. Thus x, denotes expectations of profits for period t as formed at period / - 1. Let X, be the actual profits for period Then a naive model for is

x.i = X, (10.2)

That is, the firm believes that the profits next period will be the same as profits this period. A simple extrapolative model would be to say that profits next period will increase by the same amount as the latest increase. This gives

and so on. It is, therefore, important to study models of expectations and how these models are estimated.

In the following sections we study three different models of expectation formation:

x;,, = (10.4)

In all these cases we estimate equation (10.1) after substituting the relevant formula for x from (10.2), (10.3), or (10.4). Since the formula for * is derived from outside and does not consider the equation (10.1) to be estimated, these expectations are considered exogenous (derived from outside the economic model under consideration). We will, in the following sections, discuss cases where expectations are endogenous (i.e, derived by taking account of the economic model we are considering).

The previous formulas for need to be changed suitably if we have quarterly data or monthly data. In these cases there are quarterly or monthly fluctuations, called seasonal fluctuations. For instance, December sales this year would be comparable to December sales last year because of the Christmas season. Hence formula (10.4) would be written as

Xj + j X,

for quarterly data

(10.5)

for monthly data

where, again, jc* denotes expected profits and x, actual profits. Note that we compare the corresponding quarters or months and take the most recent percentage gain as the benchmark.

Formulas like (10.5) were used by Ferber to check the predictive accuracy of railroad shippers forecasts with actual values. He compared the shippers forecasts with actual values and also forecasts given by formula (10.5) with the actual values and found that the railroad shippers forecasts were worse than those given by the naive formula. For the comparison he used average absolute error AAE given by

AAE = - 5] [actual - predicted

Robert Ferber, The Railroad Shippers Forecasts (Urbana, 111.: Bureau of Economic Research, University of Illinois, 1953).

or "

xU, = 2x, - x,, (10.3)

Another extrapolative model would be to say that profits will increase by the same percentage as the latest rate of increase. This gives

Xf+i X,

Hirsch and LovelF did a similar study based on Manufacturers Inventory and Sales Expectations Survey of the Office of Business Economics, U.S. Department of Commerce. However, they found that the anticipations data were more accurate than predictions from naive models.

Yet another naive formula that is often used, and used by Ferber, is that of regressive expectations. In this formula there are two components:

1. A growth component based on recent growth rates as in (10.5).

2. A return-to-normal component, also called the regressivity component.

Formula (10.5) would now be written as

Xi + i - , 1

- al - 1

(10.6)

a is the "reversal coefficient." Ferber estimated an equation like (10.6) using the railroad shippers forecasts for x, and the actual values for x,. He found an estimate of = 0.986 and a = 0.556 and concluded that expectations were regressive. Hirsch and Lovell also found that the data they considered also showed regressivity in expectations, but they argue that this is because the actual data are also regressive.

The naive models that we have considered are by no means recommended. However, they are often used as benchmarks by which we judge any survey data on expectations.

10.3 The Adaptive Expectations Model

The models considered in Section 10.2 use only a few of the past values in forming expectations. Some other models use the entire past history, with the past values receiving declining weights as we go farther into the distant past. These models are called distributed lag models of expectations. Consider

= + -, + • • + -, (10.7)

This is called a finite distributed lag since the number of lagged (past) values is finite. Po> Pi • • , are the weights that we give to these past values. The naive model (10.2) corresponds to Po = 1 and p, = P2 = • • = = 0. Distributed lags like (10.7) have a long history. Irving Fisher was perhaps the first one to use them. He suggested arithmetically declining weights

fOt + 1 - OP for 0 < / < 0 for / >

-A. A. Hirsch and M. C. LovelL Salei Anticipations and Inventory Befiavior (New York: Wiley. I%9).

I. Fisher, "Our Unstable Dollar and the So-Called Business Cycle." Journal of tite American Statistical Association. Vol. 20, 1925; also I. Fisher, "Note on a Short-Cut Method for Calculating Distributed Lags," Bulletin of the International Statistical Institute, Vol. 29, 1937, pp. 323-327.