Step 3: Develop the random model

Should one or more of the regression coefficients be set free to vary among the firms? In other words, do one or more of the regression coefficients show significant variation among the firms? This step should be guided by theory, especially in situations with many x-variables. What about our model? Due to market imperfections, differences can be expected between firms in terms of the marginal wage premium for an extra year of education.

Question:

Use SPSS or Stata to check whether the effects of years of education on wages show statistically significant variation among the firms? Estimate the best model.

Tip:

Use the Wald (Z) test as an initial evaluation of the variance components. Next, use likelihood ratio (LR) tests to compare models with and without random coefficients.

χ2H = -2LLK-H - -2LLK

Answer SPSS Mixed

SPSS does not let us automatically perform the LR test. We have to estimate each model and compute the chi square as the difference between the -2LL for the models found in the information criteria table. The table of covariance parameters shows that this is the smaller model without any random coefficient.

Table 4.7. Information Criteriaa

Table 4.8. Estimates of Covariance Parametersa

The output for the models with the coefficient of education defined as random:

Table 4.9. Information Criteriaa

Table 4.10. Estimates of Covariance Parametersa

With the exception of the covariance term, the estimates divided by their standard errors are all greater than 2. The LR test statistic: 34,407.44 - 34,344.278 = 63.16. The test statistic is chi-square distributed, with the difference in the number of parameters as degrees of freedom. In this example, the random coefficient model has two more random terms. The outcome is statistically significant at any conventional level and the coefficient of education seems to vary among the firms.

Can we drop the covariance term UN(2,1)?

The LR test statistic: 34,346.014 - 34,344.278 = 1.74. This is chi-square distributed with one degree of freedom. The outcome is not statistically significant and the covariance can be dropped from the model.

The main output from the best model so far:

Table 4.11. Information Criteriaa

Table 4.12. Estimates of Fixed Effectsa

Table 4.13. Estimates of Covariance Parametersa

Answer Stata xtmixed

LR tests can be performed manually using this formula:

χ2H = 2(LLK - LLK-H)

In Stata, however, estimates from models can be stored and the results used to perform an LR test for edyears as follows:

/*Fitting random intercepts and storing results*/
quietly xtmixed wage edyears age agesqr female || firmno: , mle nolog
estimates store ri
/*Fitting random coefficients and storing results*/
quietly xtmixed wage edyears age agesqr female || firmno: edyears, mle nolog cov(un)
estimates store rc
/*Running the likelihood-ratio test to compare the two models*
/Lrtest ri rc

Output

Likelihood-ratio test: LR chi2(2) = 63.16
(Assumption: ri nested in rc): Prob > chi2 = 0.0000

The test statistic is chi-square distributed, with the difference in the number of parameters as degrees of freedom. In this example, the random coefficient model has two more random terms. The outcome is statistically significant at any conventional level and the coefficient of education seems to vary among the firms.

Do we need the covariance between the level 2 residuals? This should also be decided using an LR test. The outcome below shows that the term can be deleted:

Likelihood-ratio test: LR chi2(1) = 1.74
(Assumption: ri nested in rc): Prob > chi2 = 0.1877

The first model is based on the default settings for the covariance structure and, in the second model, the covariance structure is specified as unstructured.

The best model so far:

Table 4.14.

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