138 The Basics of financial economeTrics
A major drawback of the linear probability model is that the predicted
value may be negative. In the probit regression and logit regression models
described below, the predicted probability is forced to be between 0 and 1.
probit regression Model
The probit regression model is a nonlinear regression model where the
dependent variable is a binary variable. Due to its nonlinearity, one cannot
estimate this model with least squares methods. Instead, it is necessary to
use the maximum likelihood (ML) method described in Chapter 13. Because
what is being predicted is the standard normal cumulative probability distri-
bution, the predicted values are between 0 and 1.
The general form for the probit regression model is
P(Y = 1⎮X 1 , X 2 ,... , XK) = N(a + b 1 X 1 + b 2 X 2 +... + bKXK)
where N is the cumulative standard normal distribution function.
Suppose that the following parameters are estimated as follows:
β = –2.1 β 1 = 1.9 β 2 = 0.3 β 3 = 0.8Then
N(a + b 1 X 1 + b 2 X 2 + b 3 X 3 ) = N(–2.1 + 1.9X 1 + 0.3X 2 + 0.8X 3 )
Now suppose that the probability of default of a company with the fol-
lowing values for the explanatory variables is sought:
X 1 = 0.2 X 2 = 0.9 X 3 = 1.0
Substituting these values, we get
N(–2.1 + 1.9(0.2) + 0.3(0.9) + 0.8(1.0)) = N(–0.65)
The standard normal cumulative probability for N(–0.65) is 25.8%. There-
fore, the probability of default for a company with this characteristic is 25.8%.
Illustration: hedge Fund survival An illustration of probit regression is pro-
vided by Malkiel and Saha who use it to calculate the probability of the
demise of a hedge fund.^6 The dependent variable in the regression is 1 if a
(^6) Burton G. Malkiel and Atanu Saha, “Hedge Funds: Risk and Return,” Financial
Analysts Journal 22 (November–December 2005): 80–88.