Exercises

I. Explain the meaning of each of the following terms.

(a) Adaptive expectations.

(b) Regressive expectations.

(c) Rational expectations.

The latter should not be confused with the rational expectations model. Rational here means the ratio of two polynomials.

5. In the polynomial lags one has to choose the length of the lag and the degree of the polynomial. Given the lag length, some procedures have been suggested for these two choices. All lag distributions imply some restrictions on the coefficients. Before imposing these restrictions, it is always best to estimate unconstrained lag models by OLS.

6. The rational expectations model suggested by Muth has been very popular. The idea behind the rational expectations hypothesis is that the specification of expectations should be consistent with the rest of the model rather than being ad hoc. A proper terminology for this is "model-consistent expectations" rather than "rational expectations." Two investigators starting with two different models can arrive at two different expressions of the "rational expectation" of the same variable. However, we continued to use the word "rational" rather than "model consistent" because the former is the commonly used term.

7. The essence of the rational expectations hypothesis is that the difference between the realized value and the expected value should be uncorrelated with all the variables in the information set at the time the expectation is formed. Based on this, several tests for "rationality" have been applied to survey data on expectations. These tests are described in Section 10.11. In a large number of cases these tests reject the "rationality" of survey data on expectations.

8. There are. broadly speaking, two methods of estimation for rational expectations models. One procedure involves substitution of the realized value for the expected value and using some appropriate instrumental variables. The other procedure involves obtaining an explicit expression for the expected value from the model, substituting this in the model and then estimating the model using any parameter constraints that are implied. These procedures are illustrated with reference to a demand and supply model in Section 10.12.

9. With serially correlated errors one has to be careful in the choice of appropriate instruments when estimating rational expectations models. A careful examination of which variables are correlated or uncorrelated with errors will reveal what variables are valid instruments. This is illustrated with a first-order autoregression in Section 10.13.

10. Some empirical examples are provided for the Almon lag, but computation*-of the Koyck model and some rational expectations models are left as exercises."

(d) Koyck lag.

(e) Almon lag.

(f) Rational lags.

(g) Partial adjustment model.

(h) Error correction model.

(i) Testing rationality.

0) Estimation in autoregressive form, (k) Estimation in distributed lag form.

2. What are the problems one encounters in the OLS estimation under adaptive expectations in the following models?

(a) Models of agricuUural supply.

(b) Models of hyperinflation.

(c) Partial adjustment models.

(d) Error correction models.

3. Answer Exercise 2 if, instead of assuming adaptive expectations, we assume the following.

(a) Naive expectations y, = y, ,.

(b) Rational expectations.

4. In the demand and supply model for pork discussed in Section 9.3 (data are in Table 9.1). assume that supply depends on expected price, P,. Estimate the model assuming the following.

(a) Naive expectations P, = ,. This is the Cobweb model.

(b) Adaptive expectations.

(c) Rational expectations.

5. Estimate the hyperinflation model for Hungary and Germany using the data in Tables 10.1 and 10.2 and assuming the following.

(a) Naive expectations P,**, = P,.

(b) Adaptive expectations.

(c) Rational expectations.

6. Answer Exercise 4 with the Australian wine data discussed in Section 9.5 (data are in Table 9.2). Assume that the supply depends on expected price.

7. Explain tests for rationality based on

(a) y,-.,.

(b) var y and var y,.

(c) Tests for equality of coefficients.

8. What is meant by weak tests for rationality? Are these tests really weak?

9. Future prices have often been used as a proxy for expected prices in the cases of estimation of supply functions for agricultural commodities. How do you test whether the expectations implied in the future prices are rational?

Errors in Variables

11.1 Introduction

11.2 The Classical Solution for a Single-Equation Model with One Explanatory Variable

11.3 The Single-Equation Model with Two Explanatory Variables

11.4 Reverse Regression

11.5 Instrumental Variable Methods

11.6 Proxy Variables >

11.7 Some Other Problems -Summary

Exercises

I I.I Introduction

Since the early 1970s there has been a resurgence of interest in the topic of errors in variables models and models involving latent variables. This late interest IS perhaps surprising since there is no doubt that almost all economic variables are measured with error. « s -

Three early papers that sparked interest in this area are: A. Zellner. "Estimation of Regression Relationships Containing Unobservable Independent Variables," International Economic Renew. October 1970. pp 441-454. A S Goldberger. "Maximum Likelihood Estimation of Regression Models Containing Unobservable Variables. " International bionomu Review January 1972. pp 1-15 and Zvi Griliches. "Errors in Variables and Other Unobservables. Econometnca. November 1974, pp 971-998