Question

Suppose a researcher, using data on class size (CS) and average test scores from 100 third-grade classes, estimates the OLS regression:

       

       

                                                      (20.4)      (2.21)

a. Construct a 95% confidence interval for , the regression slope coefficient.

b. Do you reject the two-sided test of the null hypothesis at the 5% level? At the 1% level? Explain.

c. Consider the two-sided test of null hypothesis , determine whether -5.6 is contained in the 95% confidence interval for .

d. Construct a 99% confidence interval for

e. Do you think that the regression errors are plausibly homoscedastic? Explain.

Answer 1

a)Construct 95% and 90% confidence intervals for Bi. Construct a 95% confidence interval for A, the regression slope coefficient Calculate the test statistic for the two-sided test of the null hypothesis .

b)Calculate the p-value for the two-sided test of the null hypotheis H. : B1 = 0.

c)Calculate the p-value for the hypotheses H : B1 = -5.6 v.s. H : B1 + -5.6. Without doing any additional calculations, determine whether -5.6 is contained in the 95% confidence interval for B1.

d)Calculate the p-value for the hypotheses H : B1 = -5.6 v.s. H : B1 + -5.6. Without doing any additional calculations, determine whether -5.6 is contained in the 95% confidence interval for B1.

e)

We say that the population error term ui is homoskedastic if var is constant for all i.We say that the population error term ui is homoskedastic if is constant for all i. Otherwise, we say that ui is heteroskedastic. An example during which the errors are likely heteroskedastic may be a regression of wages on schooling, as follows: wagesi = β0 + β1schoolingi + ui .

In particular, homoskedasticity would imply that is the same for all schooling levels. Another way of claiming this is often that the variability of wages around its mean is that the same no matter educational attainment.

Homoskedasticity isn't realistic during this case because likely that folks with more education have wider job opportunities, which could lead on to more variability in wages. In contrast, people with low education levels have fewer opportunities and doubtless work wage jobs, so there's less dispersion of wages among the uneducated. In sum, we might expect that variability in wages is higher for the highly educated, and therefore the variability in wages is low for those with low levels of schooling. Therefore, during this example, the errors ui are likely heteroskedastic.
The question asks whether the variability in test scores in large classes is that thesame because the variability in small classes. It is hard to say.

On the one hand, teachers in small classes could be ready to spend longer bringing all of the scholars along, reducing the poor performance of particularly unprepared students. On the opposite hand, most of the variability in test scores could be beyond the control of the teacher.
SE(βˆ 1) was computed using heteroskedastic standard errors. Suppose that the regression errors were homoskedastic: Would this affect the validity of the confidence interval constructed in part. The CI would still be valid. The SEs were computed using heteroskedastic-robust SEs, which are valid whether truthpopulation regression errors are homoskedastic or heteroskedastic.