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From |
Weihua Guan <wguan@stata.com> |

To |
statalist@hsphsun2.harvard.edu |

Subject |
Re: st: Random effects probit |

Date |
Tue, 17 Sep 2002 14:44:50 -0500 |

In the recent conversation about the results from -xtprobit-, Wiji Arulampalam <Wiji.Arulampalam@warwick.ac.uk> and others queried how -xtprobit- obtains its starting values. As James Hardin <jhardin@stat.tamu.edu> nicely explained, [...] > When Stata starts to fit the full model, it takes the > values for a regular probit model and then calculates > the log-likelihood of the random effects probit model > using those coefficients with rho=0.0, 0.1, 0.2, ... It > continues to fit these models until the log-likelihood > fails to improve. It then starts the optimization using > the (regular) probit estimates along with the "best" > value of rho it found in this simpleminded search. > > So , several have asked why Stata doesn't use the > regular probit answers as the starting value. The > answer is that Stata does use those values, but > immediately tries to find a better starting value for > rho before starting the optimization process. [...] In this model, the starting value for "rho" can determine whether or not the optimization concludes successfully. Unfortunately, the grid search method used in -xtprobit- does not seem adequate for this particular problem. Since layering a non-linear optimization method on top of an approximation often brings difficulties in the estimation process, the importance of starting values is not too surprising. Still, we have been looking into other methods of optimization and quadrature techniques. With difficult data, e.g., when the quadrature check indicates unstable results, it may be possible to obtain better estimates by re-estimating the model using a one-dimensional grid of starting values. The starting values are constructed by augmenting the converged cross-sectional -probit- results with a grid of values for "rho". The optimization is then performed in a loop. Afterwards, if the largest log-likelihood from the converged results has a full rank estimated VCE, then we have a candidate solution that should be checked by -quadchk-. Here we implement this approach via a -forvalues- loop: ----------------------------Begin do-file----------------------------------- qui probit sue lague mat b = e(b) forvalues rho = 0.1(0.1).9 { local lnsig2u = ln(`rho'/(1-`rho')) mat b1 = b, `lnsig2u' di di as res "rho = " `rho' _c qui xtprobit sue lague, i(ind1) quad(20) from(b1, copy) di as txt _skip(10) "Log likelihood = " as res %8.0g e(ll) di as txt "coef." as res _col(15) %8.0g _b[lague] /* */ _col(30) %8.0g [sue]_cons _col(45) %8.0g [lnsig2u]_cons /* */ _col(60) %8.0g e(rho) di as txt "std. err." as res _col(15) %8.0g _se[lague] /* */ _col(30) %8.0g [sue]_se[_cons] /* */ _col(45) %8.0g [lnsig2u]_se[_cons] } ----------------------------End do-file------------------------------------- Since this is only an example, we want to make the output shorter. We achieve this by only displaying the estimated coefficients, their standard errors and the log-likelihood. In other implementations, it might be better to display all the output, but this is a matter of preference. Note that an estimated standard error of 0 indicates results with than full rank estimated VCE. The results from the above method are: lague _cons lnsig2u rho rho = .1 Log likelihood = -321.537 coef. .978465 -2.62413 .094874 .523701 std. err. .257994 .269102 .46741 rho = .2 Log likelihood = -324.476 coef. 1.50498 -2.1291 -1.44237 .191179 std. err. .179505 .065135 0 rho = .3 Log likelihood = -323.981 coef. 1.44582 -2.16482 -1.24305 .223905 std. err. .182259 .067259 0 rho = .4 Log likelihood = -323.855 coef. 1.4304 -2.17465 -1.1936 .232617 std. err. .182962 .067849 0 rho = .5 Log likelihood = -321.537 coef. .978465 -2.62413 .094874 .523701 std. err. .257992 .269095 .467398 rho = .6 Log likelihood = -321.537 coef. .978465 -2.62413 .094874 .523701 std. err. .257994 .269102 .46741 rho = .7 Log likelihood = -326.897 coef. 1.76802 -1.99957 -2.78606 .058082 std. err. .165957 .057566 0 rho = .8 Log likelihood = -325.316 coef. 1.60022 -2.0773 -1.80932 .140721 std. err. .174846 .062075 0 rho = .9 Log likelihood = -338.513 coef. .894369 -6.26237 3.17317 .959812 std. err. .194 0 .042211 The output shows that when the starting values for rho are 0.1, 0.5 or 0.6, the optimization converges with 20 quadrature points and the estimated VCE is of full rank. Since the solutions with the starting values for rho of 0.1, 0.5 and 0.6 are the same, the results from -quadchk- will also be the same. Choosing the solution produced by the starting values with rho=0.6, we can check the stability of the quadrature. . qui probit sue lague . mat b = e(b) . local rho = .6 . local lnsig2u = ln(`rho'/(1-`rho')) . mat b1 = b, `lnsig2u' . xtprobit sue lague, i(ind) quad(20) from(b1, copy) nolog Random-effects probit Number of obs = 2411 Group variable (i) : ind1 Number of groups = 500 Random effects u_i ~ Gaussian Obs per group: min = 1 avg = 4.8 max = 7 Wald chi2(1) = 14.38 Log likelihood = -321.53746 Prob > chi2 = 0.0001 ------------------------------------------------------------------------------ sue | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- lague | .9784651 .2579944 3.79 0.000 .4728054 1.484125 _cons | -2.624126 .2691015 -9.75 0.000 -3.151556 -2.096697 -------------+---------------------------------------------------------------- /lnsig2u | .0948743 .4674101 -.8212327 1.010981 -------------+---------------------------------------------------------------- sigma_u | 1.04858 .2450585 .6632413 1.657799 rho | .5237008 .11659 .305502 .7332122 ------------------------------------------------------------------------------ . quadchk, noout Refitting model quad() = 16 Refitting model quad() = 24 Quadrature check Fitted Comparison Comparison quadrature quadrature quadrature 20 points 16 points 24 points ----------------------------------------------------- Log -321.53746 -321.54191 -321.535 likelihood -.00444993 .00245667 Difference .00001384 -7.640e-06 Relative difference ----------------------------------------------------- sue: .97846512 .9789156 .97752771 lague .00045048 -.00093741 Difference .00046039 -.00095804 Relative difference ----------------------------------------------------- sue: -2.6241262 -2.6226448 -2.6260355 _cons .00148137 -.00190931 Difference -.00056452 .0007276 Relative difference ----------------------------------------------------- lnsig2u: .0948743 .09345529 .09850014 _cons -.00141901 .00362584 Difference -.01495678 .03821729 Relative difference ----------------------------------------------------- Unfortunately, the relative differences for lnsig2u:_cons appear to be too large to safely interpret them. At this point, we could either repeat our method with a different number of quadrature points or conclude that we cannot obtain interpretable results from this combination of model and data. Weihua Guan <wguan@stata.com> David M. Drukker <ddrukker@stata.com> * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

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