Chapter 17
Discrete Choice
132 Greene • Econometric Analysis, Seventh Edition
◼ Exercises
1. The log likelihood is
lnL = 0,0lnProb[y = 0, d = 0] + 0,1 lnProb[y = 0, d = 1] + 1,0
lnProb[y = 1, d = 0] + 1,1 lnProb[y = 1, d = 1]
where i,j indicates the sum over observations for which y = i and d = j. Since there are no other
regressors, this reduces to lnL = 24ln(1 − F()) + 32ln(1 − F()) + 28lnF() + 16lnF(). Although
it is straightforward to maximize the log likelihood directly in terms of and , an alternative,
Simple manipulations produce the two solutions A = 28/(24 + 28) = 0.539 and D = 16/(32 + 16) = 0.333.
Then, these functions can be inverted to produce the MLEs of and . Thus,
ˆ
= F −1(A) and
ˆ
= F −1(D)
−
ˆ
. The two inverse functions are −1(A) for the probit model, which must be approximated, and
ln[F/(1-F)] for the logit model. The estimates are,
We will compute the asymptotic covariance matrix for
ˆ
and
ˆ
directly using (17-22) for the probit
1 0
y d + d / − /(1 − ) 01
0 0 0.098 0.539 0.397 0.737 − 0.861 −0.636
Chapter 17 Discrete Choice 133
The estimated asymptotic covariance matrix is the inverse of the estimate of −E[H].
1 0 1 0 1 1 1 1
24(0.636) 28(0.636) 32(0.597) 16(0.597) .
0 0 0 0 1 1 1 1
− = + + +
H
the square roots of the diagonal elements, which are 0.1739 and 0.2552, respectively. To test the
and let nij equal the number of observations in each cell. Then, the unrestricted log likelihood is
lnL = 24ln 0.461 + 28ln 0.539 + 32ln 0.667 + 16ln 0.333 = −66.442. The likelihood ratio statistic
Prob[y = 0, d = 0] = P00 = 1/(1 + e0.156) = 0.462
Prob[y = 1, d = 0] = P10 = 1 − P00 = 0.538
2. Using the usual regression statistics, we would have
= − = − − − 2
, ( )( )/ ( ) .
i i i i i
a y bx b x x y y x x
For data in which y is a binary variable, we can decompose the numerator somewhat further. First,
divide both numerator and denominator by the sample size. Second, since only one variable need be
134 Greene • Econometric Analysis, Seventh Edition
3. The model was estimated using Newton’s method as described in the text. The estimated coefficients
and their standard are shown below:
ˆ*y
= − 0.51274 + 0.15964X
4. The derivatives of the log likelihood are given in (17-18) and (17-21). If all coefficients except the
constant term are zero, then the first-order condition for maximizing the log likelihood would be
lnL/ = i(yi − )(1) = 0 since with no regressors, i will not vary with i. This leads to the
constrained maximum
ˆ
= i yi /n = P, which might be expected. Thus, we estimate the constant term
5. (The section on grouped data for binary choice modeling does not appear in the 7th edition of the
text.) To estimate the coefficients, we will use a two-step FGLS procedure. Ordinary least squares
estimates based on the moment equation Pi = ( + Ti) are consistent, but inefficient. The OLS
regression produces
The predicted values from this regression can then be used to compute weights
2
ˆ
ˆˆ
(1 ) /[ ]
i i i
n −
(see Section 21.4.6 from the 5th edition). The weighted least squares regression produces
In order to achieve a predicted proportion of 95%, we will require zi = 1.645. The T required to
achieve this is
6. This is similar to Exercise 1. It is simplest to prove it in that framework. Since the model has only a
dummy variable, we can use the same log likelihood as in Exercise 1. But, in this exercise, there are
no observations in the cell (y = 1, x = 0). The resulting log likelihood is, therefore,
7. We will do this more generally for any model F(). Since the “model” contains only a constant, the log
likelihood is logL = 0log[1 − F()] + 1logF() = n0log[1 − F()] + n1logF(). The likelihood equation
8. Look at the two cases. Neither case has an estimator, which is consistent in both cases. In both cases,
the unconditional fixed effects estimator is inconsistent, so the rest of the analysis falls apart. This is
136 Greene • Econometric Analysis, Seventh Edition
Application
? ============================================================================
? Application 17.1 Binary Choice for Extramarital Affairs Using Redbook Data
? ============================================================================
?
Create ; A = (Yrb > 0) $
Namelist ; X = one,v1,v2,v5,v6 $
+———————————————+
| Binomial Probit Model |
| Maximum Likelihood Estimates |
+———————————————+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
———+Index function for probability
Constant| 1.43453507 .15493583 9.259 .0000
+——————————————-+
| Partial derivatives of E[y] = F[*] with |
+——————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]|Elasticity|
+——–+————–+—————-+——–+——–+———-+
———+
Constant| .27876593 .01081795 25.769 .0000
+———————————————+
| Binary Logit Model for Binary Choice |
| Maximum Likelihood Estimates |
| Dependent variable A |
+———————————————+
+——–+————–+—————-+——–+——–+———-+
Chapter 17 Discrete Choice 137
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+——–+————–+—————-+——–+——–+———-+
———+Characteristics in numerator of Prob[Y = 1]
+——————————————-+
| Partial derivatives of probabilities with |
+——————————————-+
+——–+————–+—————-+——–+——–+———-+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]|Elasticity|
+——–+————–+—————-+——–+——–+———-+
———+Marginal effect for variable in probability