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st: selection in continuous time survival framework without instruments
I have a dataset that is perfectly set up for continuous time survival
analysis. In real time data we observe whenever someone is arrested for a
drug offence and whenever someone attends drug treatment. We wish to test
the hypothesis that after one arrest, being sent to drug treatment reduces
the time to the next drug arrest. The problem here is that it is likely
that there are person-level characteristics that make folks more likely to
be sent to treatment, correlated with the probability of arrest AND THESE
CHARACTERISTICS ARE UNOBSERVED TO THE RESEARCHER (caps intentional). E.g.
demeanor in court. The best fit of the data is a log-normal distribution,
which is usual in studying recidivism.
To control for this potential unobserved heterogeneity bias one may be
tempted to use a two-stage least squares or instrumental variables approach.
However, no good candidates for identifying instruments are apparent.
Here are my questions:
1. Does anyone know of a way of using an individual fixed-effects (in
economics lingo) that would help handle this bias in a continuous time
framework? I can picture doing this straightfowardly in a discrete
framework, where the probability of next arrest in a given time frame is a
function of whether the person attended drug treatment. In that case, one
could do a simple clogit, and assume that the unobserved heterogeneity does
not change systematically over time. But how could this be done in a
continuous time setting? I really don't want to have to arbitrarily pick a
period and create an artificial panel - why should I have to do that?!
2. Any other ideas on controlling for this kind of unobserved heterogeneity
bias? Again, remember there are no obvious identifying instruments. And
note that I think this is subtly different from modeling 'frailty' (could be
wrong on that though).
I'd be grateful for your thoughts
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