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Other Limited Dependent Variable Models 467

Standard Errors for Predicted Probabilities

For simplicity, consider the case of a single regressor in the probit model. Then the pre-
dicted probability at a fixed value of that regressor, x, is pn(x) = Φ(bnM0 LE + bnM1 LEx), where
bn0MLE and bn1MLE are the MLEs of the two probit coefficients. Because this predicted prob-
ability depends on the estimators bn0MLE and bn1MLE, and because those estimators have a
sampling distribution, the predicted probability will also have a sampling distribution.

The variance of the sampling distribution of pn(x) is calculated by approximating the
function Φ(bnM0 LE + bn1MLEx), a nonlinear function of bnM0 LE and bn1MLE, by a linear function of
bnM0 LE and bnM1 LE. Specifically, let

pn(x) = Φ(bnM0 LE + bn1MLEx) ≅ c + a0(bnM0 LE - b0) + a1(bn1MLE - b1) (11.19)

where the constant c and factors a0 and a1 depend on x and are obtained from calculus.

[Equation (11.19) is a first-order Taylor series expansion; c = Φ(b0 + b1x); and

a0 and a1 are the partial derivatives, a0 = 0 Φ(b0 + b1x) > 0b0 ͉ bN M0 LE, bN MLE and a1 =
1

0 Φ(b0 + b1x) > 0b1 ͉ bN M0 LE, bN MLE .] The variance of pn(x) now can be calculated using the approx-
1

imation in Equation (11.19) and the expression for the variance of the sum of two random

variables in Equation (2.31):

var3pn(x)4 ≅ var3c + a0(bnM0 LE - b0) + a1(bn1MLE - b1)4
= a20var(bn0MLE) + a12var(bnM1 LE) + 2a0a1cov(bn0MLE, bn1MLE). (11.20)

Using Equation (11.20), the standard error of pn(x) can be calculated using estimates of the
variances and covariance of the MLEs.

A p p e n d i x

11.3 Other Limited Dependent Variable Models

This appendix surveys some models for limited dependent variables, other than binary vari-
ables, found in econometric applications. In most cases the OLS estimators of the parameters
of limited dependent variable models are inconsistent, and estimation is routinely done using
maximum likelihood. There are several advanced references available to the reader inter-
ested in further details; see, for example, Ruud (2000) and Wooldridge (2010).

Censored and Truncated Regression Models

Suppose that you have cross-sectional data on car purchases by individuals in a given year.
Car buyers have positive expenditures, which can reasonably be treated as continuous