Appendix C
Regressions Performed
Using natural logs and cross section data for 100 Census Area Units, we performed the following three regressions:
Regression
Number |
Dependent Variable |
Independent Variables |
| 1 |
Average income (Y) |
Proportion of workforce with higher educational attainment (EDU), proportion of workforce employed in advanced business services (ABS), employment density (DENS) |
| 2 |
Average income (Y) |
Proportion of workforce employed in advanced business services (ABS), employment density (DENS) |
| 3 |
Average income (Y) |
Proportion of workforce with higher educational attainment (EDU), employment density (DENS) |
Results of the Regression Analysis
Regression 1
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
| Intercept |
10.545 |
0.061 |
174.044 |
7.766E-122 |
10.425 |
10.666 |
| ABS |
0.075 |
0.024 |
3.078 |
0.0027 |
0.026 |
0.123 |
| EDU |
0.117 |
0.041 |
2.870 |
0.0051 |
0.036 |
0.198 |
| DENS |
0.088 |
0.011 |
7.709 |
1.167E-11 |
0.066 |
0.111 |
| Regression Statistics |
| Multiple R |
0.738 |
| R Square |
0.544 |
| Adjusted R Square |
0.530 |
| Standard Error |
0.118 |
| Observations |
100 |
ANOVA
|
df |
SS |
MS |
F |
Critical F |
Significance F |
| Regression |
3 |
1.584 |
0.528 |
38.201 |
2.699 |
2.458E-16 |
| Residual |
96 |
1.327 |
0.014 |
|
|
|
| Total |
99 |
2.911 |
|
|
|
|
Since the F-statistic exceeds the critical value at the 5% level of significance, we must reject the hypothesis that the coefficients are jointly equal to zero. However, given the finding of heteroscedasticity outlined below, this conclusion is not robust.
White Test
| nR^2 |
Critical Chi-Square (0.05, 9df) |
| 17.01 |
16.92 |
Since the nR^2 value exceeds the critical Chi-Square value, we must reject the hypothesis of homoskedasticity.
Due to the finding of heteroscedasticity, the error terms are biased, making the tests and inferences surrounding the estimated parameters unreliable.
Regression 2
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
| Intercept |
10.456 |
0.054 |
194.222 |
1.767E-127 |
10.349 |
10.563 |
| ABS |
0.099 |
0.023 |
4.221 |
5.480E-05 |
0.053 |
0.146 |
| DENS |
0.086 |
0.012 |
7.288 |
8.439E-11 |
0.063 |
0.110 |
| Regression Statistics |
| Multiple R |
0.7107 |
| R Square |
0.5051 |
| Adjusted R Square |
0.4949 |
| Standard Error |
0.1219 |
| Observations |
100 |
ANOVA
|
df |
SS |
MS |
F |
Critical F |
Significance F |
| Regression |
2 |
1.470 |
0.735 |
49.492 |
3.090 |
1.534E-15 |
| Residual |
97 |
1.441 |
0.015 |
|
|
|
| Total |
99 |
2.911 |
|
|
|
|
Since the F-statistic exceeds the critical value at the 5% level of significance, we must reject the hypothesis that the coefficients are jointly equal to zero.
White Test
| nR^2 |
Critical Chi-Square (0.05, 5df) |
| 2.416 |
11.070 |
Since the nR^2 value is lower than the critical Chi-Square value, we cannot reject the hypothesis of homoskedasticity.
Auxiliary Regressions to Detect Multicollinearity: Regression of ABS on DENS
| Ri |
Critical F (0.05, 1df, 98df) |
| 10.093 |
3.938 |
Since the Ri exceeds the critical F at the 5% level of significance, the two independent variables are collinear.
Regression 3
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
| Intercept |
10.449 |
0.054 |
192.899 |
3.424E-127 |
10.342 |
10.557 |
| EDU |
0.161 |
0.040 |
4.058 |
0.0001 |
0.082 |
0.240 |
| DENS |
0.099 |
0.011 |
8.713 |
8.024E-14 |
0.076 |
0.122 |
| Regression Statistics |
| Multiple R |
0.707 |
| R Square |
0.499 |
| Adjusted R Square |
0.489 |
| Standard Error |
0.123 |
| Observations |
100 |
ANOVA
|
df |
SS |
MS |
F |
Critical F |
Significance F |
| Regression |
2 |
1.453 |
0.727 |
48.339 |
3.090 |
2.723E-15 |
| Residual |
97 |
1.458 |
0.015 |
|
|
|
| Total |
99 |
2.911 |
|
|
|
|
Since the F-statistic exceeds the critical value at the 5% level of significance, we must reject the hypothesis that the coefficients are jointly equal to zero.
White Test
| nR^2 |
Critical Chi-Square (0.05, 5df) |
| 9.325 |
11.070 |
Since the nR^2 value is lower than the critical Chi-Square value, we cannot reject the hypothesis of homoskedasticity.
Auxiliary regressions to detect multicollinearity: Regression of ABS on DENS
| Ri |
Critical F (0.05, 1df, 98df) |
| 0.313 |
3.938 |
Since the Ri is lower than the critical F at the 5% level of significance, the two independent variables are not collinear.
Geographical Plot of Residuals
We also plotted the residuals from the third regression against census area units. These are shown in the figure below. A negative number means that the regression under-predicts average earnings whereas a positive number means that the regression over-predicts average earnings.
Figure C.1: Residuals plotted against census area unit
→ Full size version of Figure C.1 [128 kB JPG]
The above picture shows some "clustering" of colours throughout the isthmus. As mentioned above, this suggests some degree of spatial autocorrelation may be present (meaning that locations close to each other exhibit more similar values than those further apart).
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