Wednesday, March 08, 2006

Re: st: rho in xtreg

Stas Kolenikov wrote:

There are two intrinsic impediments. First, -rho- is treated as an ancillary parameter in -xtreg-: it does not estimate its standard errors or anything. You can get around this with -mle- option that will estimate the sigma's, and then you would have to form -rho- and its standard error from there. Second, I am not aware of any statistically justified methodology of comparing two coefficient estimates in two models the way you want. In any nested model testing, you are imposing some restrictions, like setting the coefficient of gender equal to zero (effectively omitting it). This is not a restriction on -rho-, however. I don't think you can compare unrestricted estimates in two estimated models -- at least if they are estimated on the same data set. If they were estimated on independent data sets, then you could test them assuming covariance is equal to zero, so the test would be effectively a two-sample t-test.

A better way for you to go is to use some sort of structural equation modeling with multiple group comparisons where you would be allowing -sigma_u- and -sigma_e- to vary between males and females. If you are brave enough, you can dig into -xtreg- code (saving it under a different name, of course) to allow the variance parameters to be dependent on some covariates, like gender, and then test if those heteroskedastic models are significantly different from the regular -xtreg-. If not, you may want to (ask your students to) locate appropriate structural equation modeling software (AMOS, LISREL, M-plus, EQS) on your campus to run a model like that. This might also be doable in -gllamm-.

Note that you are assuming quite restrictive things here like known linear part of the model, known distribution, and known form of heteroskedasticity. While you can interpret the panel regression models as fitting linear regression models with some variations for the panel structure, with a nice feature of being the MLEs if the data just happened to be normal, you need to put more thought into the MLEs.

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Would it be illegitimate to assume that the covariances between sexes for sigma_u and for sigma_e are zero despite coming from the same dataset?

It seems as if there would be a gain in efficiency if sigma_u and sigma_e for both sexes could be estimated in the same model rather than separately. When fitted by -xtreg- in separate models, sigma_u is estimated with only low precision for women (see below) compared to when estimated in the same model. To go the route of a t-test of rho's separately estimated with -xtreg if sex/if !sex, vce(bootstrap/jackknife)- would seem weaker, given the low precision for women if estimated separately.

If I have everything correctly specified (at least the variances add up), then the -xtmixed- section of the do file below tests equality of separate Level-2 (random effects) variances between sexes estimated jointly. This assumes that the residual variance is the same between sexes. (In contrast, the post yesterday tested whether the residual variances differed, assuming that the Level-2 variances are the same between sexes.)

The final segment of the do-file below jointly tests both components of rho between sex, if I've got everything right. It uses -gllamm-, because -xtmixed- doesn't allow for structured residual covariance matrixes yet. Including sex among the fixed effects causes convergence problems for the -xtmixed- examples, so I omitted it from -gllamm- for consistency.

Stas's admonition about restrictive assumptions is well taken, but would the assumptions be any more onerous in this context than what they would be in other, more familiar, applications of linear mixed models fitted by iterative maximum likelihood methods?

Joseph Coveney

clear set more off sysuse bplong * * Tests equality between sexes of sigma_u--complements yesterday's * xi i.when xtreg bp _Iwhen, i(patient) mle nolog display _b[/sigma_u]^2 // Individual estimates of sigma_u xtreg bp _Iwhen_2 if sex, i(patient) mle nolog xtreg bp _Iwhen_2 if !sex, i(patient) mle nolog // Combined estimates of sigma_u xtmixed bp _Iwhen_2 || sex: R.patient || sex:, mle nolrtest nolog estat recov // Test reduced model estimates store A xtmixed bp _Iwhen_2 || sex: R.patient, mle nolrtest nolog // (Same as -xtmixed bp _Iwhen_2 || patient:, mle-) estat recov lrtest A . estimates drop _all * * Jointly tests equality between sexes of both sigma_u and sigma_e * quietly tabulate sex, generate(sex_) eq het: sex_1 sex_2 gllamm bp _Iwhen_2, i(patient sex) s(het) adapt nolog estimates store A gllamm bp _Iwhen_2, i(patient) adapt nolog lrtest A . exit

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