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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 * * For searches and help try: * http://www.stata.com/help.cgi?search * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

**Follow-Ups**:**Re: st: System of equations, structural models and maximum likelihood in STATA***From:*Stas Kolenikov <skolenik@gmail.com>

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