Version: Spring 2018
EC200 Econometrics and Applications

Problem Set 4\

1. Suppose that $(Y_i,X_i)$ satisfy the three key least squares assumptions and, in addition, $u_i$ is $N(0,{\sigma }_u^2)$ and is independent of $X_i$. A sample of size $n=30$ yields

   $$\begin{array}{rlrlrlr}\hat{Y}& =43.2+& 61.5X& (10.02)& (7.4)R^2=0.54& SER=1.52& \end{array}$$

   where the numbers in parentheses are the homoskedastic-only standard errors for the regression coefficients.

   1. Construct a 95% confidence interval for ${\beta }_0$.\

   2. Construct a 90% confidence interval for ${\beta }_1$.\

   3. Test $H_0:{\beta }_1=55$ against $H_1:{\beta }_1\ne 55$ at the 5% level.\

   4. Test $H_0:{\beta }_1=55$ against $H_1:{\beta }_1>55$ at the 5% level.\

   5. Explain briefly why the test of $H_0:{\beta }_1=55$ against $H_1:{\beta }_1<55$ is trivial. You can use a picture if is helps make things clearer.\

2. In the 1980s, Tennessee conducted an experiment in which kindergarten students were randomly assigned to “regular” and “small” classes and given standardized tests at the end of the year. (Regular classes contained approximately 24 students, and small classes contained approximately 15 students.)
   Suppose that, in the population, the standardized tests have a mean score of 925 points and a standard deviation of 75 points. Let $SmallClass$ be a binary variable equal to 1 if the student is assigned to a small class and equal to 0 otherwise. A regression of $TestScore$ on $SmallClass$ yields $$\begin{array}{rlrlrlr}TestScore& =918.0+13.9& SmallClass& (1.6)& (2.5)R^2=0.01,& SER=74.6& \end{array}$$

   where the numbers in parentheses are the standard errors for the regression coefficients.

   1. Do small classes improve test scores? By how much? Is the effect large? Explain.\

   2. Is the estimated effect of class size on test scores statistically significant? Carry out a test at the 5% level.\

   3. Do you think that the regression errors are plausibly homoskedastic? Explain.\

   4. $SE(\widehat{{\beta }_1})$ was computed using the initial formula for standard errors (based on equations 5.3 and 5.4 in Stock and Watson). Would having heteroskedastic errors and using this formula affect the validity of your hypothesis tests? What if the errors are actually homoskedastic? Explain.\

3. Visit the Stock and Watson webpage (here: http://wps.aw.com/aw_stock_ie_3/178/45691/11696959.cw/index.html) and click on the “Additional Empirical Exercises.” tab. Complete Additional Empirical Exercise 5.3 using the data set CollegeDistance. Note that you can download this data from the Additional Emprical Exercises page.\

4. Finish Lab 3 - include do-file, log-file, and answers to questions.