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

 From Stas Kolenikov To statalist@hsphsun2.harvard.edu Subject Re: st: System of equations, structural models and maximum likelihood in STATA Date Tue, 11 Sep 2012 13:54:32 -0500

You might want to look into David Roodman's -cmp-, which is very
thoroughly documented in The Stata Journal article. Type -findit cmp-.

On Tue, Sep 11, 2012 at 1:46 PM, Daniel Almar de Sneijder
<dasneijder@gmail.com> wrote:
> 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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--
-- Stas Kolenikov, PhD, PStat (SSC)  ::  http://stas.kolenikov.name
-- Senior Survey Statistician, Abt SRBI  ::  work email kolenikovs at
srbi dot com
-- Opinions stated in this email are mine only, and do not reflect the
position of my employer
*
*   For searches and help try:
*   http://www.stata.com/help.cgi?search
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*   http://www.ats.ucla.edu/stat/stata/