Neighborhood Effects, Public Housing and Unemployment in France

Individual contributions to the likelihood can be written as follows:

P (yi1, yi2, yi3) = Φ3[qi1 (α1Xi1+βyi2+γyi3), qi2(α2X2i+δyi3), qi3(α3X3i), qi1qi2ρ12, qi1qi3ρ13, qi2qi3ρ23]

(2.6)
where qij = 2yij - 1 is equal to 1 whenever yij is 1 and to -1 whenever yij is 0, subscript i
denotes individual i and Φ3(.) is the trivariate normal cumulative distribution function. The
log-likelihood function is then:

N

lnL =      lnP (yi1, yi2, yi3)                                 (2.7)

i

The calculation of individual contributions requires to integrate over the distribution of the
vector of three error terms, which means the calculation of a triple integral. Simulated maximum
likelihood methods have been developed to circumvent this problem. One of the simulators
commonly used is the GHK (for Geweke-Hajivassiliou-Keane) simulator.⁷ The accuracy of the
GHK simulator is good as soon as the number of random draws is equal to or higher than
the square root of the sample size (Cappellari and Jenkins, 2003). With a sample of 10,473
individuals, we use 600 replications for each estimation, which is far above this threshold.

⁷The principle of this simulator is to use the lower triangular Cholesky decomposition of the covariance matrix
of error terms to replace correlated random variables by uncorrelated ones, which are drawn from truncated normal
density functions. Individual contributions to the likelihood are calculated as averages over several repeats of the
random draw. See for example Bolduc, 1999 for a presentation of the GHK simulator and its use in a multinomial
probit model.

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