IMMIGRATION POLICY AND THE AGRICULTURAL LABOR MARKET: THE EFFECT ON JOB DURATION

Suppose that the log of the spell is normally distributed with mean lnτ_ij and variance σ_j . Then,
the survival function is expressed as

^zln t - ln τ?
—-

< ^σ j √

The hazard rate, the rate at which the spell is completed after duration t, is

'ln t - ln τ_t '
' —-

< ^{σ j} √

^tσ jSj^{( t )}

where φ(∙) is the probability density for the standard normal distribution. Next, we assume that
the mean duration of a spell (lnτ_ij) depends on independent variables zi (gender, marital status,
age, age squared, education, US farm experience, English speaking ability, race, ethnicity,
availability of free housing, task, region (California, Florida, and other), the year of the interview
(after 2001 or not), dummy variable for seasonal workers) so that

lnτij = zi^'βj .

Then, the duration can be expressed as lnt_ij = z_i^'β_j + u_ij where u_ij ~ N(0,σ_j ). However, duration
t_ij is observed only if person i has legal status j. This is a typical case for selection bias. Assuming
e_i and u_ij are bivariately normally distributed with correlation coefficient ρ, the mean of the log
of the duration conditioned on the legal status of person i is corrected as

E[ln t_ij | ln t_ij is observed] = z_i^'β_j + ρσ_jλ_ij

where λ_ij is the correction term for the selection bias which is given as⁵

⁵ Correction term is set to zero for native-born citizen.

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