Dummy Variables and Truncated Variables

8.1 Introduction

8.2 Dummy Variables for Changes in the Intercept Term

8.3 Dummy Variables for Changes in Slope Coefficients

8.4 Dummy Variables for Cross-Equation Constraints

8.5 Dummy Variables for Testing Stability of Regression Coefficients

8.6 Dummy Variables Under Heteroskedasticity and Autocorrelation

8.7 Dummy Dependent Variables

8.8 The Linear Probability Model and the Linear Discriminant Function

8.9 The Probit and Logit Models

8.10 Illustrative Example

8.11 Truncated Variables: The Tobit Model Summary

Exercises

8 Introduction

In the preceding chapters we discussed the estimation of multiple regression equations and several associated problems such as tests of significance, R, heteroskedasticity, autocorrelation, and multicollinearity. In this chapter we discuss some special kinds of variables occurring in multiple regression equations and the problems caused by them. The variables we will be considering are;

1. Dummy variables.

2. Truncated variables.

We start with dummy explanatory variables and then discuss dummy dependent variables and truncated variables. Proxy variables referred to in Machlups quotation are discussed in Chapter 11.

Dummy explanatory variables can be used for several purposes. They can be used to

1. Allow for differences in intercept terms.

2. Allow for differences in slopes.

3. Estimate equations with cross-equation restrictions.

4. Test for stability of regression coefficients.

We discuss each of these uses in turn.

8.2 Dummy Variables for Changes in the Intercept Term

Sometimes there will be some explanatory variables in our regression equation that are only qualitative (e.g., presence or absence of college education and racial, sex, or age differences). In such cases one often takes account of these effects by a dummy variable. The implicit assumption is that the regression lines for the different groups differ only in the intercept term but have the same slope coefficients. For example, suppose that the relationship between income and years of schooUng x for two groups is as shown in Figure 8.1. The dots are for group 1 and circles for group 2.

Let us remember the unfortunate econometrician who, in one of the major functions of his system, had to use a proxy for risk and a dummy for sex.

Fritz Machlup

Journal of Political Economy July/August 1974

Figure 8 . Regression lines with a common slope and different intercepts.

Note that the slopes of the regression lines for both groups are roughly the same but the intercepts are different. Hence the regression equations we fit will be

+ + a, + Px + M

for the first group for the second group

These equations can be combined into a single equation, = a, + (qj - a,)D + x +

where

(8.1)

(8.2)

for group 2 for group 1

The variable D is the dummy variable. The coefficient of the dummy variable measures the differences in the two intercept terms.

If there are more groups, we have to introduce more dummies. For three groups we have

a, + px + t/ a2 + x + + Px + M

for group 1 for group 2 for group 3

These can be written as

= a, + («2 - «,)£>, + («3 - ai)£>2 + Px + w

(8.3)