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st: System of equations, structural models and maximum likelihood in STATA


From   Daniel Almar de Sneijder <dasneijder@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   st: System of equations, structural models and maximum likelihood in STATA
Date   Tue, 11 Sep 2012 14:46:40 -0400

Dear statalist,

How are you doing today? I have a question regarding the structure and
best method of solving a system of equations. More specific, I am
programming an option value model of retirement (has someone
experience with it?) and therefore use maximum likelihood to obtain
the parameters. As I am new to STATA I am not really sure how to
start. In the following I describe the problem.


First I describe the model, see also Stock and Wise (1990) for further
details, and then I will point out the exact questions.

The objective is to find probabilities of retiring in a specific year,
and the parameters \rho, \gamma, k, \beta in:


[1.] Pr[retire in year t] = Pr[ gt( r^{*}_t ) / Kt( r^{*}_t ) < - v_t ], where

[2.] v_{s} = \rho v_{s-1} + \epsilon_t, and

[3.] gt(r_t) = \sum^{r-1}_{s=t} \beta^{s-t} \pi(s|t) Et(Ys^{\gamma}) +
\sum^{S}_{s=r} \beta^{s-t} \pi(s|t) Et( [k Bs(r)]^{\gamma} })  -
\sum^{S}_{s=r} \beta^{s-t} \pi(s|t) Et( [ k Bs(t)]^{\gamma} }), and

[4.] Kt( r^{*}_t ) = \sum^{r-1}_{s=t} \beta^{s-t} \pi(s|t) \rho^{s-t}.

[5.] Also v_t = (\omega_t - \xi_t )


Here Ys are future wages and Bs(t) are retirement incomes with
\pi(s|t) the probability that a person will live in year s given that
she or he lives in year t. r^{*}_t is the year in which the value of
future stream of income is maximized. The value of future stream of
income if retirement is at age 'r' is given by:

[6.] Vt(r) = \sum^{r-t}_{s=t} \beta^{s-t} Uw(Ys) + \sum^{S}_{s=r}
\beta^{s-t} Ur[Bs(r)]

[7.] Uw(Ys) = Ys^{\gamma} + \omega_s,

[8.] Ur(Bs) = [ k Bs(r)]^{\gamma} + \xi_s

[9.] \omega_s = \rho \omega_{s-1} + \epsilon_{\omega,s} where E[disturbance] = 0

[10.] \xi_s = \rho \xi_{s-1} + \epsilon_{\xi, s} where E[disturbance] = 0



Particular points where I am struggling at are:

- What is the structure of a nonlinear maximum likelihood estimation
with STATA, i.e. how to set up the code.

- How can specific parameters be used in the context, such as equation
2. where we have v_t that depends on it's lag and a disturbance term
\epsilon_t

- How to estimate the probability such as in equation 1

- How to initialize and call the model.



I understand that this may be a somewhat big question, but I would
definitely appreciate it if someone can answer for at least some parts
of it. Also, it would help the community a lot, I think.


Thank you in advance everyone.



Regards,
Daniel
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