Anon

140 The Basics of financial economeTrics

normal cumulative probability distribution, it is standard cumulative prob-
ability distribution of a distribution called the logistic distribution.
The general formula for the logit regression model is

==... ++++

=+−

YXXXFabX bX bX

e

P( 1,,,)( ... )

1/[1 ]

12 NN 11 22 N

W^

where W = a + b 1 X 1 + b 2 X 2 +... + bNXN.
As with the probit regression model, the logit regression model is esti-
mated with ML methods.
Using our previous illustration, W = –0.65. Therefore

1/[1 + e–W] = 1/[1 + e–(–0.65)] = 34.3%

The probability of default for the company with these characteristics is
34.3%.

Key Points

■ (^) Categorical variables are variables that represent group membership
and are used to cluster input data into groups.
■ (^) An explanatory variable that distinguishes only two categories is called
a dichotomous variable. The key is to represent a dichotomous categori-
cal variable as a numerical variable called a dummy variable that can
assume the value of either 0 or 1.
■ (^) An explanatory variable that distinguishes between more than two cat-
egories is called a polytomous variable.
■ (^) In a regression where there are dummy variables, the t-statistic applied
to the regression coefficients of dummy variables offer a set of impor-
tant tests to judge which explanatory variables are significant. The
p-value associated with each coefficient estimate is the probability of
the hypothesis that the corresponding coefficient is zero, that is, that the
corresponding dummy variable is irrelevant.
■ (^) The Chow test is an F-test that is used to gauge if all the dummy vari-
ables are collectively irrelevant. The Chow test is the F-test of the unre-
stricted regressions with and without dummy variables.
■ (^) A regression model can be interpreted as a conditional probability
distribution. A regression model where the dependent variable is a cat-
egorical variable is therefore a probability model.
■ (^) Three probability models most commonly used are the linear probabil-
ity model, the probit regression model, and the logit regression model.