Then minimizing the sum of squares

J- I

we get a, 0,, and from the minimization of "Li and

% = Yj - a - fiiXij - fiiXij foTJ = , + 1 and , + 2

What we have considered here for tests for stability is the use of dummy variables to get within sample predictions. We can use the dummy variable method to generate out of sample predictions and their standard errors as well (a problem we discussed in Section 4.7). Suppose we have n observations on Y, Xx and Xj. We are given the (n + l)th observation on jc, and jCj and asked to get the predicted value of Y and a standard error for the prediction. The way we proceed is as follows.

Set the value of for the {n + l)th observation at zero. Define a dummy variable D as

Z) = 0 for the first n observations

= -1 for the (n + l)th observation

Now run a regression of on jc,, and D using the (n + 1) observations. The coefficient of D is the prediction „+, needed and its standard error is the standard error of the prediction. To see this is so, note that the model says:

Y = a + PiX, + fizXz + for the n observations

0 = a + p,x, + 2X2 - + for the (n + l)th observation

Minimizing the residual sum of squares amounts to obtaining the least squares estimators a, 0,, 2 from the first n observations, and

= a + p,x, + 02*2 for the (n + l)th observation

Thus, = f„+, and its standard error gives us the required standard error of

This method has been extended by Pagan and Nicholls* to the case of nonUnear models and simultaneous equation models.

8.6 Dummy Variables Under

Heterosl<edasticity and Autocorrelation

In the preceding sections we discussed the use of dummy variables but we need to exercise some caution in the use of these variables when we have heteroskedasticity or autocorrelation.

*A. R. Pagan and D. F. Nicholls, "Estimating Predictions, Prediction Errors and Ttieir Standard Deviations Using Constructed Variables," Journal of Econometrics, Vol. 24, 1984, pp. 293-310.

1«2 +

P,jc + «, for the first group 2 + «2 for the second group

Let var(M,) = and var(«2) = a. When we pool the data and estimate an equation like (8.4), we are implicitly assuming that a] - aj. If and al are widely different, then, even if «2 is not significantly different from a, and P2 is not significantly different from p,, the coefficients of the dummy variables in (8.4) can turn out to be significant. One can easily demonstrate this by generating data for the two groups imposing a, = «2 and p, = 2 but a] = Ibal (a] being chosen suitably), and estimating equation (8.4). The reverse situation can also arise; that is, ignoring heteroskedasticity can make significant differences appear to be insignificant. Suppose that we take a, = 2a2 and p, = ZPj. Then by taking a] = Ibaj (or a multiple around that) we can make the dummy variables appear nonsignificant. The problem is just the same as that of applying tests for stability under heteroskedasticity.

Regarding autocorrelation, suppose that the errors in the equations for the two groups are first-order autoregressive so that we use a first-order autoregressive transformation, that is, write

y, = ,- py,-i fory,

x = x,-px,i forxt

The question is: What happens to the dummy variables in equation (8.4)? These variables should not be subject to the autoregressive transformation and care should be taken if the computer program we use does this automatically. We can easily derive the appropriate dummy variables in this case.

Consider the case with , observations in the first group and «2 observations in the second group. We will introduce the time subscript t for each observation later when needed.

Define

aJ e a,(l - p)

«2 = «2(1 - P) Then equation (8.4) can be rewritten as

y* + { - a\)D, + p,x* + (p2 - ,)£>2 + e

where A will be defined as before with x in place of Xj and the random errors e, are defined by

u, - ,-, = e,

This point was brought to my attention by Ben-Zion Zilberfarb through a manuscript "On the Use of Autoregressive Transformation in Equations Containing Dummy Variables."

Consider first the case of heteroskedasticity. Suppose that we have the two equations

This equation, however, is all right for the observations in the first group and the last («2 - I) observations in the second group. However, the problem is with the first observation in the second group. For this observation, the p-differenced equation turns out to be

y, - /-1 = «2 - p«i + ( 2< - -i) + e,

y, = +

(«2* - «I) + p,x: + 02 - + e,

This means that the dummy variables D, and A have to be defined as follows:

1 - p 1

for all observations in the first group

for the first observation in the second group

for all the other observations in the second group

for all observations in the first group

for the first observation in the second group

for all the other observations in the second group

8.7 Dummy Dependent Variables

Until now we have been considering models where the explanatory variables are dummy variables. We now discuss models where the explained variable is a dummy variable. This dummy variable can take on two or more values but we consider here the case where it takes on only two values, zero or 1. Considering the other cases is beyond the scope of this book." Since the dummy variable takes on two values, it is called a dichotomous variable. There are numerous examples of dichotomous explained variables. For instance.

if a person is in the labor force otherwise

1 if a person owns a home 0 otherwise

There are several methods to analyze regression models where the dependent variable is a zero or 1 variable. The simplest procedure is to just use the usual

"Ttiis is a complete area by itself, so we give only an elementary introduction. A more detailed discussion of this topic can be found in G. S. Maddala, Limited-Dependent and Qualitative Variables in Econometrics (Cambridge: Cambridge University Press, 1983), Chap. 2, "Discrete Regression Models;" T. Amemiya, "Qualitative Response Models: A Survey," Journal of Economic Literature, December 1981, pp. 483-536; and D. R. Cox, Analysis of Binary Data (London: Methuen, 1970).