dx,j

where

fij for the linear probability model

fijP.il - P,) for the logit model fijiZ,) for the probit model

and () is the density function of the standard normal.

In the case of the linear probability model these derivatives are constant. In the case of the logit and probit models, we need to calculate them at different levels of the explanatory variables to get an idea of the range of variation of the resulting changes in probabilities. If one is interested in the prediction of the effect on the log-odds ratio, then for the logit model, this effect is constant since

Measuring Goodness of Fit

There is a problem with the use of conventional R-type measures when the explained variable takes on only two values."* The predicted values are probabilities and the actual values are either 0 or 1. For the Unear probabiUty model and the logit model we have 2 >" = 2 » with the Unear regression model, if a constant term is also estimated. For the probit model there is no such exact relationship although it is approximately valid. We will see this in the illustrative example presented later.

There are several 7?-type measures that have been suggested for models with qualitative dependent variables. The following are some of them. In the case of the linear regression model, they are all equivalent. However, they are not equivalent in the case of models with qualitative dependent variables.

These are summarized in Maddala, Limited-Dependent, pp 37-41.

ables are significant, we need not make any changes in the estimated coefficients for the logit model. On the other hand, if the estimated model is going to be used for prediction purposes, an adjustment in the constant term, as suggested eariier, is necessary.

Prediction of Effects of Changes in the Explanatory Variables

After estimating the parameters p„ we would like to know the effects of changes in any of the explanatory variables on the probabilities of any observation belonging to either of the two groups. These effects are given by

1. = squared correlation between and y.

2. Measures based on residual sum of squares. For the linear regression model we have

R = I -

1 (y, - y.r It ( ,-

We can use this same measure if we can use 2."=i ( . ~ 1 as the measure of residual sum of squares. Effron- argued that we can use it. Note that in the case of a binary dependent variable

1.( .- = 1 1- nf = - n{

«,«2

Hence Effrons measure of is

, = 1

Amemiya argues that it makes more sense to define the residual sum of squares as

h y,(i - y,)

that is, to weight the squared error (y, - y,) by a weight that is inversely proportional to its variance.

3. Measures based on likelihood ratios. For the standard linear regression model,

= Po + E + « ~ IN(0, a) 1=1

let LuR be the maximum of the likelihood function when maximized with respect to all the parameters and Lr be the maximum when maximized with the restriction 3, = 0 for / = 1,2, ,k- Then

One can use an analogous measure for the logit and probit model as well. However, for the qualitative dependent variable model, the likeHhood function (8.15) attains an absolute maximum of 1. This means that

Lr < Lur < 1

B. Effron, "Regression and ANOVA with Zero-One Data: Measures of Residual Variation, Journal of itie American Staiistical Association, May 1978, pp 113-121 Amemiya, "Qualitative Response Models," p. 1504.

L" < 1 - < 1

Hence Cragg and Uhler suggest a pseudo i?: (It lies in [0, 1].)

Another measure of R is that of McFadden,* who defines it as

McFaddens R= I - --

log Fr

However, this measure does not correspond to any R measure in the hnear regression model.

4. Finally, we can also think of R in terms of the proportion of correct predictions. Since the dependent variable is a zero or 1 variable, after we compute the y, we classify the ith observation as belonging to group 1 if y, > 0.5 and classify it as belonging to group 2 if y, < 0.5. We can then count the number of correct predictions. We can define a predicted value y,, which is also a zero-one variable such that

1 ify, >0.5 0 ify, < 0.5

(Provided that we calculate y, to enough decimals, ties will be very unlikely.) Now define

„, number of correct predictions

count R =----7---

total number of observations

Although this is a useful measure worth reporting in all problems, it might not have enough discriminatory power. In the illustrative example in the next section, we found that the logit model predicted 41 of the total 44 cases correctly, whereas the probit and linear probability model each predicted 40 of the 44 correctly. However, looking at y„ the linear probability model had five observations with y, substantially greater than 1, thus outside the range of (0, 1).

"I. G. Cragg and R. Uhler, "The Demand for Automobiles," Canadian Journal of Economics, 1970, pp. 386-406. See also Maddala, Limited-Dependent, pp. 39-40 for a discussion of this Pseudo- .

*D. McFadden, "The Measurement of Urban Travel Demand," Journal of Public Economics, 1974, pp. 303-328.