1U08p2c

housing from only one project. The likelihood of a household being treated by either project is:
P(T i = 1) = P(T i1 = 1) + P(T i2 = 1) − P(T i1 = 1)P(T i2 = 1)
where P(T ia = 1) is calculated by (7). Notice the adjustment for the fact that a household
cannot be treated more than once. In this framework, the expression P(T ia = 1), given by (7)
has to be interpreted as the project specific contribution to being treated, not the probability of
being treated by that project. For many projects, the probability is most simply expressed as
complement of the probability of being selected by none of the projects:
P(T i = 1) = 1 −
= 1 −
A
∏
a
A
∏
a
(P(T ia = 0)) (8)
Λ(−x i β − dis ia ρ) (9)
This expression, when estimated, gives a single solution to the coefficient ρ, a common effect
of distance for all housing projects, no matter how many different housing projects are used in
the estimation. I use this model to predict the probability of being treatment for a single period,
based on the housing projects that were built in that period.
The problem is complicated further by the use panel data: we’d like efficient estimates for
the probability of receiving housing in each period. Households cannot receiving housing more
than once, so the predicted probability of treatment should decline in a period after a household
had a high predicted probability of treatment, all things equal. We want to derive an expression
for the probability that a household has received housing at point in time up until the specific
period. I development a functional form that conditions the probability of receiving housing in
a particular time period on the probability of having received housing in previous periods.In
the interests of space, this method relegated to the Appendix, Section A.2. There I also develop
a multinominal estimator that predicts, the period in which a household will most likely be
treated.
In addition, I present Monte Carlo simulations using simulations with calibrated parameters
for ρ which gives the effect of proximity on the probability of receiving housing. I find that
the estimator developed here does a good job of recovering the true parameter value for ρ,
even in the presence of considerable noise and individual fixed effects. The average predicted
probabilities of treatment from this model match the rates of treatment in the simulated data.
I am able to estimate the equation given by the 8 and the time (wave) specific probability of
being treated using maximum likelihood techniques. It is to the results of these estimates that I
now turn.
4.3 First Stage Results
I have outlined the key elements on the first stage of my instrumental variables strategy. Taken
together I will estimate a system of equations taking the following form:
ỹ it = ˜λ t + ˜X it β + ˜T it τ + ˜ɛ it (10)
˜T it = ˜δ t + ˜X it δ 1 + ˜Ĝ it π + ṽ it (11)
Ĝ it = G(X it , Z it ; ̂ρ) (12)
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