Economics 371

Introductory Econometrics

Fall 2005

Problem Set 5

Problem 1 –

Let (Y1,…,Y81) be a random (i.e., i.i.d.) sample drawn from a population with E(Y) = μY and Var(Y) = . Suppose that

and .

  1. Consider the test of H0: μY =3 vs. HA: μY ≠ 3.
  1. What test statistic should be applied and what is its distribution under the

null hypothesis? Compute the value of the test statistic. Show your work.

  1. What is the p-value for the test? Explain how you derived this p-value.
  1. Do you reject the null hypothesis at the 5% level? Explain.
  1. Redo part (c) for the alternative hypothesis HA: μY > 3.
  1. Construct a 95-percent confidence interval and a 90-percent confidence

interval for μY. Show your work. Which is larger? Why?

  1. Suppose that μY = 4 and = 40. What are the values of E() and

Var()? Explain.

Problem 2 –

Consider the California school district data set that we have been using this semester. A California school district is considering allocating additional funds for the purchase of additional classroom computers in the hope of raising the district’s math test scores. An economist with the district used the California data to estimate a regression of average math test scores on the computer-student ratio, obtaining the following EViews output:

Dependent Variable: MATH_SCR
Method: Least Squares
Sample: 1 420
Included observations: 420
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable / Coefficient / Std. Error / t-Statistic / Prob.
C / 643.5867 / 2.057844 / 312.7481 / 0.0000
COMP_STU / 71.77327 / 14.37238 / 4.993832 / 0.0000
R-squared / 0.061797 / Mean dependent var / 653.3426
Adjusted R-squared / 0.059552 / S.D. dependent var / 18.75420
S.E. of regression / 18.18721 / Akaike info criterion / 8.644064
Sum squared resid / 138263.7 / Schwarz criterion / 8.663304
Log likelihood / -1813.254 / F-statistic / 27.53237
Durbin-Watson stat / 0.264858 / Prob(F-statistic) / 0.000000

Here, MATH_SCR = average math score

and COMP_STU = computers/enrollment = computers per students.

  1. What can you conclude about the result from a test of H0: 1 = 0 vs. HA:1 ≠ 0 at the 5-percent significance level? Explain.
  1. What can you conclude about the result from a test of H0: 1 = 0 vs. HA: 1 > 0 at the 5-percent signficance level? Explain.
  1. Construct a 99-percent confidence interval for 1. Show your work.
  1. Suppose the economist had not used the heteroskedasticity correction. Explain why the confidence interval constructed in (iii) would be different. Can you determine whether the interval would get bigger or smaller?
  1. Suppose that as a result of a data entry error, the economist had accidentally multiplied all of the observation of COMP_STU by 100. How would this affect the entry under “Coefficient” for COMP_STU? Explain.

Problem 3 – (24 points; 8 points each for parts (i)-(iii))

The economist discussed in Problem 2 decided to reestimate the relationship between math test scores and computers per student with average household income (AVGINC) as an additional explanatory variable. He obtained the following EViews output:

Dependent Variable: MATH_SCR
Method: Least Squares
Sample: 1 420
Included observations: 420
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable / Coefficient / Std. Error / t-Statistic / Prob.
C / 621.8610 / 2.209212 / 281.4854 / 0.0000
COMP_STU / 33.71514 / 0.0010
AVGINC / 1.756189 / 0.113490 / 15.47439 / 0.0000
R-squared / 0.502276 / Mean dependent var / 653.3426
Adjusted R-squared / 0.499889 / S.D. dependent var / 18.75420
S.E. of regression / 13.26269 / Akaike info criterion / 8.014904
Sum squared resid / 73349.89 / Schwarz criterion / 8.043763
Log likelihood / -1680.130 / F-statistic / 210.4072
Durbin-Watson stat / 0.838534 / Prob(F-statistic) / 0.000000
Wald Test:
Equation: Untitled
Null Hypothesis: / C(2)=0
C(3)=0
F-statistic / 131.0469 / Probability / 0.000000
Chi-square / 262.0938 / Probability / 0.000000
  1. Explain why the economist would add average income to the regression if all he/she cares about is how changes in computers per student affect average math scores? Can you tell from the regression output whether the economist’s concerns are valid? Explain.
  1. Notice that the standard error and t-statistic for COMP_STU have been “accidentally” omitted from the regression output. However, the economist can still determine the result from a test of H0: 1 = 0 vs. HA:1 ≠ 0 at the 5-percent significance level. How?
  1. Would you reject the null hypothesis that neither COMP_STU nor AVGINC affect MATH_SCR at the 5-percent signficance level? Explain. (Be specific about exactly which numbers you are using to arrive at your conclusion and why you are using those numbers.)