ARCHIVED – Social Capital and Employment Entry of Recent Immigrants to Canada

10. Appendix B

Log likelihood test of panel-level variance component in the random effects model
Assuming that the unobserved individual effects zi in the general model are unrelated to the observed explanatory variables xit: Cov (xit, zi) =0, t=1, 2, …, T, so that the conditional distribution f (zi| xit) is independent on xit, I get the random effects model:

Mathematical equation  i=1,…, n, t=1,…, Ti.

E (vit | xit) =0,

where Mathematical equation  

Mathematical equation

and Mathematical equation are iid logistic distributed with mean zero and variance Mathematical equation, independently of zi.

The proportion of the total variance contributed by the panel-level (i.e. subject level) variance component is Mathematical equation. When Mathematical equation is zero, the panel-level variance component is unimportant, and the panel estimator is not different from the pooled estimator. A likelihood-ratio test of the null hypothesis that Mathematical equation equals zero compares the pooled estimator with the random effects estimator. In our analysis, a likelihood ratio test of this is included at the bottom of the Stata output of the random effects estimation (e.g. see following output for the random estimation of the employment probability of male immigrants):

> /*Random-effects logit regression*/
> xtlogit em swpa swsd refugee other
>              age agesq married nkid nkid4_14 nykid
>              Atlantic Quebec Prairies BC noncma
>              bregion1 bregion3-bregion5
>              min1-min8
>              ed1-ed3 ed5 insch
>              Eng Fre
>              prework  lengthca lengthsq
>              jobarranged visitbf workbf studybf
>              spwkcur spwage relative nr rlnear fsdensity
>              friend frnear newfri nfoutwk frdiv frdensity
>              pgo ngo godiv godensity govo
>              if male==1 & lf==1, i(id) re;

 

Random-effects logistic regression              Number of obs      =      7632
Group variable (i): id                                    Number of groups   =      4239

Random effects u_i ~ Gaussian              Obs per group: min =         1
                                                               avg =       1.8
                                                               max =         2

                                                                 Wald chi2(58)      =    831.86
Log likelihood  = -3839.4768                    Prob > chi2        =    0.0000

em Coef. Std. Err. z P>z [95% Conf. Interval]
swpa -.1475564 .1960424 -0.75 0.452 -.5317926 .2366797
swsd -.3746873 .2039548 -1.84 0.066 -.7744313 .0250567
refugee -.8966477 .199611 -4.49 0.000 -1.287878 -.5054172
other .0392413 .2225304 0.18 0.860 -.3969102 .4753928
age .133618 .0283932 4.71 0.000 .0779684 .1892677
agesq -.2100867 .0344855 -6.09 0.000 -.2776769 -.1424964
married -.0579413 .1197569 -0.48 0.629 -.2926605 .1767778
nkid .0307523 .0682012 0.45 0.652 -.1029196 .1644242
nkid4_14 -.1334155 .0772561 -1.73 0.084 -.2848346 .0180037
nykid -.0653268 .1039146 -0.63 0.530 -.2689957 .1383421
Atlantic -.2547173 .4284288 -0.59 0.552 -1.094422 .5849877
Quebec -1.00029 .1457717 -6.86 0.000 -1.285998 -.7145831
Prairies .2874948 .1151471 2.50 0.013 .0618107 .5131789
BC -.3100151 .1035028 -3.00 0.003 -.5128768 -.1071534
cma7 .4066035 .1910132 2.13 0.033 .0322246 .7809824
bregion1 -.2648533 .2925672 -0.91 0.365 -.8382744 .3085678
bregion3 .0995336 .305917 0.33 0.745 -.5000527 .69912
bregion4 -.3280924 .228842 -1.43 0.152 -.7766145 .1204298
bregion5 -.702198 .244845 -2.87 0.004 -1.182085 -.2223106
min1 -.9100354 .2500677 -3.64 0.000 -1.400159 -.4199117
min2 .0407751 .2390026 0.17 0.865 -.4276613 .5092115
min3 -.1819008 .2183392 -0.83 0.405 -.6098377 .2460361
min4 .6222302 .2898216 2.15 0.032 .0541902 1.19027
min5 -.5215466 .3444505 -1.51 0.130 -1.196657 .1535639
min6 -.2768887 .1967725 -1.41 0.159 -.6625557 .1087783
min7 -.8182016 .2831731 -2.89 0.004 -1.373211 -.2631926
min8 -.065734 .3900471 -0.17 0.866 -.8302123 .6987443
ed1 .1318664 .1350965 0.98 0.329 -.1329179 .3966507
ed2 .1330504 .1757951 0.76 0.449 -.2115017 .4776025
ed3 -.0327392 .1278272 -0.26 0.798 -.2832758 .2177974
ed5 -.0111976 .0995632 -0.11 0.910 -.2063379 .1839427
insch -1.140078 .0864846 -13.18 0.000 -1.309585 -.9705717
Eng .1298138 .115577 1.12 0.261 -.0967131 .3563406
Fre .0156922 .1419002 0.11 0.912 -.2624271 .2938114
prework .4016835 .1404277 2.86 0.004 .1264502 .6769167
lengthca .1853822 .0685596 2.70 0.007 .0510079 .3197565
lengthsq -.5139186 .2567438 -2.00 0.045 -1.017127 -.01071
jobarranged 1.743389 .2073841 8.41 0.000 1.336924 2.149855
visitbf -.0391938 .1179006 -0.33 0.740 -.2702747 .191887
workbf .2173325 .2750612 0.79 0.429 -.3217776 .7564426
studybf .649554 .2066795 3.14 0.002 .2444696 1.054638
spwkcur .7706661 .1185297 6.50 0.000 .5383523 1.00298
spwage -.0003937 .0002214 -1.78 0.075 -.0008277 .0000402
relative -.1288294 .2097851 -0.61 0.539 -.5400005 .2823418
nr -.0072518 .0648803 -0.11 0.911 -.1344149 .1199113
rlnear .275572 .198389 1.39 0.165 -.1132632 .6644072
fsdensity .3784671 .1665552 2.27 0.023 .0520248 .7049094
friend -.1587796 .1272786 -1.25 0.212 -.408241 .0906818
frnear .2339187 .1186878 1.97 0.049 .001295 .4665425
newfri .1444655 .1722342 0.84 0.402 -.1931073 .4820382
nfoutwk -.1062178 .0265518 -4.00 0.000 -.1582584 -.0541771
frdiv .3795836 .1718096 2.21 0.027 .0428428 .7163243
frdensity .1736812 .1589156 1.09 0.274 -.1377877 .4851501
pgo .2223838 .2611897 0.85 0.395 -.2895385 .7343062
ngo .0097016 .1699385 0.06 0.954 -.3233718 .3427749
godiv 1.358319 2.41174 0.56 0.573 -3.368604 6.085243
godensity -.5040376 .3178618 -1.59 0.113 -1.127035 .1189601
govo -.0037153 .1373188 -0.03 0.978 -.2728551 .2654246
_cons -1.991571 .6896534 -2.89 0.004 -3.343267 -.6398755
             
/lnsig2u .2068822 .1191101     -.0265693 .4403338
             
sigma_u 1.10898 .0660454     .9868032 1.246285
rho .2721057 .0235915     .2283914 .3207094
             

 

Likelihood-ratio test of rho=0: chibar2(01) =  87.23 Prob >= chibar2 = 0.000

We can think of rho ( Mathematical equation) as being the (analogous) equivalent of the intra-cluster correlation (icc) in a multilevel model. Therefore when Mathematical equation is zero the panel model is not a significant improvement on the pooled one. Here, the p value of the likelihood-ratio test of Mathematical equation= 0 tells us that the null hypothesis is rejected and there exists unobserved heterogeneity so that panel data model is favoured over the pooled estimator.

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