120 The Basics of financial economeTrics
their respective squared errors. The p-values associated with each coefficient
estimate is the probability of the hypothesis that the corresponding coeffi-
cient is zero, that is, that the corresponding variable is irrelevant.
We can also use the F-test to test the significance of each specific dummy
variable. To do so we can run the regression with and without that variable
and form the corresponding F-test. The Chow test^2 is the F-test to gauge
if all the dummy variables are collectively irrelevant. The Chow test is an
F-test of mutual exclusion, written as follows:
(^) F
nk
k
=
[]−+ −
()+
SSR SSR SSR
SSR SSR
() 12 ()
122
(6.9)
where SSR 1 =the squared sum of residuals of the regression run with data
in the first category without dummy variables
SSR 2 =the squared sum of residuals of the regression run with data
in the second category without dummy variables
SSR=the squared sum of residuals of the regression run with
fully pooled data without dummy variables
The test statistic F follows an F distribution with k and n – 2k degrees of
freedom. Observe that SSR 1 + SSR 2 is equal to the squared sum of residuals
of the regression run on fully pooled data but with dummy variables. Thus
the Chow test is the F-test of the unrestricted regressions with and without
dummy variables.
Illustration: predicting corporate bond Yield spreads To illustrate the use of
dummy variables, we will estimate a model to predict corporate bond
spreads.^3 The regression is relative to a cross section of bonds. The regres-
sion equation is the following:
Spreadi = β 0 + β 1 Couponi + β 2 CoverageRatioi + β 3 LoggedEBITi + εi
(^2) Gregory C. Chow, “Tests of Equality between Sets of Coefficients in Two Linear
Regressions,” Econometrica 28 (1960): 591–605.
(^3) The model presented in this illustration was developed by FridsonVision and is
described in “Focus Issues Methodology,” Leverage World (May 30, 2003). The data
for this illustration were provided by Greg Braylovskiy of FridsonVision. The firm
uses about 650 companies in its analysis. Only 100 observations were used in this
illustration.