Step 4: Add level 2 explanatory variables

Do large firms pay higher wages than small firms? Do they also pay higher returns on education than small firms? Answer these two questions by estimating models with firm size.

Tip

It is always useful to try various versions of some variables. For firm size, the natural log transformation and recoding size into small and large firms are both worth trying. The latter is recommended as a starting point since it makes the interpretation of the cross-level interaction easier. The -2LLs for the models with the size alternatives could indicate the empirically best version.

Question

Why might lnsize (the natural log of size) work better in the analysis than size?

Answer

Size is very right-skewed and the natural log shrinks the right tail of the distribution.

SPSS Mixed

First, we have to create the alternative versions of size and the cross-level interaction terms.

Compute lnsize = ln(size).

SPSS Frequencies can be used to find the quartiles:

FREQUENCIES VARIABLES=size
/FORMAT=NOTABLE
/NTILES=4
/ORDER=ANALYSIS.

Table 4.15. Statistics

Let us use 70 as the cut-off point between small and large firms:

Recode size (low thru 70 =0)(71 thru high=1)into large.

The cross-level interactions:

Compute edsize = edyears*size.
Compute edlnsize = edyears*lnsize.
Compute edlarge = edyears*large.

Firstly, trying out the three versions of the main effect of firm size indicates that lnsize is the best version, followed by large. Next, estimate models with the cross-level interactions.

With edlarge:

Table 4.16. Estimates of Fixed Effectsa

Table 4.17. Estimates of Covariance Parametersa

With lnlarge:

Table 4.18. Estimates of Fixed Effectsa

Table 4.19. Estimates of Covariance Parametersa

The two main conclusions are as follows:

Firm size has a direct or main effect on the wage level, as large firms pay better. Firm size does not seem to affect the return on education, however, although the latter seems to vary among firms. Interpretation is easiest for the model with large. On average, large firms pay about NOK 5 more per hour. The marginal effect of education is almost identical in small (4.428) and in large (4.429) firms. The estimate of the variance in the slope residuals can be used to calculate a confidence interval for the effects of education among firms. The standard error is 2.13, which means a confidence interval of 4.43 +- 4.18 = 0.25 - 8.61.

Stata xtmixed

First, we have to create the alternative versions of size and the cross-level interaction terms.

generate lnsize = ln(size)
generate large =.
replace large = 0 if (size < 71)
replace large = 1 if (size > 70)

The cross-level interactions:

generate edsize = edyears*size
generate edlnsize = edyears*lnsize
generate edlarge = edyears*large

First, trying out the three versions of the main effect of firm size indicates that lnsize is the best version, followed by large. Next, estimate models with the cross-level interactions.

With edlarge:

. xtmixed wage edyears age agesqr female large edlarge || firmno: edyears, mle variance

Table 4.20.

With edlnsize:

Table 4.21.

The two main conclusions are as follows:

Firm size has a direct or main effect on the wage level, as large firms pay better. Firm size does not seem to affect the return on education, however, although the latter seems to vary among firms. Interpretation is easiest for the model with large. On average, large firms pay about NOK 5 more per hour. The marginal effect of education is almost identical in small (4.428) and in large (4.429) firms. The estimate of the variance in the slope residuals can be used to calculate a confidence interval for the effects of education among the firms. The standard error is 2.13, which means a confidence interval of 4.43 +- 4.18 = 0.25 - 8.61.

Final note

The file also contains a private versus public sector indicator. I leave it is an open exercise to try adding this firm level variable to the model. Another possibility is to redo the analysis with the natural logarithm of wage as the dependent variable.

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