neighborhood choice, including public housing in the neighborhood equation only.
4.3 Two probit estimates of neighborhood effects and public housing effects
A simultaneous model of two probits of unemployment and neighborhood choice is estimated by
a classical likelihood maximization method, with the same exogenous variables as in the three
probit model and gives very similar estimated coefficients. Given that the correlation term
between the residuals of the two equations is significant, neighborhood effect on unemployment
must be calculated as the difference in conditional probabilities, that themselves are calculated
on the basis of joint probabilities. For instance, the effect on unemployment probability of living
in a deprived quarter is:
P(yii = 1∣y2 = 1) - P(yu = ι∣ya = 0) = P y - P yP y1 y.. 0 (4.1)
This formula accounts both for the direct effect of neighborhood type and for the effect due to
the correlation of unobservables between the two equations.
The public housing variable, as each exogenous variable affecting the probability to live
in a deprived neighborhood, has an indirect effect on unemployment probability that may be
calculated as:
P (yi1 = 1∣x = 1) - P (yi1 = 1∣x = 0) = (P (yi1 = 1, yi2 = 1∣x = 1) + P (yi1 = 1, yi2 = 0∣x = 1))
(4.2)
- (P (yi1 = 1, yi2 = 1∣x = 0) + P (yi1 = 1, yi2 = 0∣x = 0))
Table 6 shows the effects of neighborhood and public housing accommodation on un-
employment following several specifications that differ by the type of model (simple probit,
seemingly unrelated probits and simultaneous bivariate probits) and by the presence of the pub-
lic housing variable in the neighborhood equation. As suggested by Wooldridge (2001, p. 467),
predicted effects are calculated for each individual and averaged over the sample. The standard
errors of these effects are calculated by the delta method.20
As far as neighborhood type is concerned, column 1 displays the “naive” effect of +2.13
probability points that is calculated on the basis of the simple probit. In column 2, we take
the correlation between unobservables into account by estimating a seemingly unrelated probit
model; that is, we do not include neighborhood into the unemployment equation, but neigh-
borhood type may still influence unemployment probability through the correlation between
20The delta-method allows to approximate the variance of a vector-valued function of a random vector X . It
is based on the following general result: Var(G(X))=(∂G∕∂X)'Var(X)(∂G∕∂X) where X is the mean of X,
Var(X) is the variance-covariance matrix of X, G() is a vector function and G'() its matrix of first derivatives.
17