Econometric Analysis of Cross Section and Panel Data - Free

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Chapter 15
15.4 Maximum Likelihood Estimation of Binary Response Index Models
Assume we have N independent, identically distributed observations following the
model (15.8). Since we essentially covered the case of probit in Chapter 13, the discussion
here will be brief. To estimate the model by (conditional) maximum likelihood,
we need the log-likelihood function for each i. The density of y i given x i can be
written as
f ðy j x i ; bÞ ¼½Gðx i bÞŠ y ½1 Gðx i bÞŠ 1 y ; y ¼ 0; 1 ð15:16Þ
The log-likelihood for observation i is a function of the K 1 vector of parameters
and the data ðx i ; y i Þ:
l i ðbÞ ¼y i log½Gðx i bÞŠ þ ð1 y i Þ log½1 Gðx i bÞŠ ð15:17Þ
(Recall from Chapter 13 that, technically speaking, we should distinguish the ‘‘true’’
value of beta, b o , from a generic value. For conciseness we do not do so here.)
Restricting GðÞ to be strictly between zero and one ensures that l i ðbÞ is well defined
for all values of b.
As usual, the log likelihood for a sample size of N is LðbÞ ¼ P N
i¼1 l iðbÞ, and the
MLE of b, denoted ^b, maximizes this log likelihood. If GðÞ is the standard normal
cdf, then ^b is the probit estimator; ifGðÞ is the logistic cdf, then ^b is the logit estimator.
From the general maximum likelihood results we know that ^b is consistent
and asymptotically normal. We can also easily estimate the asymptotic variance ^b.
We assume that GðÞ is twice continuously di¤erentiable, an assumption that is
usually satisfied in applications (and, in particular, for probit and logit). As before,
the function gðzÞ is the derivative of GðzÞ. For the probit model, gðzÞ ¼fðzÞ, and for
the logit model, gðzÞ ¼expðzÞ=½1 þ expðzÞŠ 2 .
Using the same calculations for the probit example as in Chapter 13, the score of
the conditional log likelihood for observation i can be shown to be
s i ðbÞ 1 gðx ibÞxi 0½y i
Gðx i bÞ½1
Gðx i bÞŠ
Gðx i bÞŠ
ð15:18Þ
Similarly, the expected value of the Hessian conditional on x i is
E½H i ðbÞjx i Š¼
½gðx i bÞŠ 2 x 0 i x i
fGðx i bÞ½1 Gðx i bÞŠg 1 Aðx i; bÞ ð15:19Þ
which is a K K positive semidefinite matrix for each i. From the general conditional
MLE results in Chapter 13, Avarð ^bÞ is estimated as