## 2nd Hour Exam

Name _________KEY_________________________________

Directions: This is a closed book exam. Your work must be your own; you may neither give nor receive help. You must complete all questions. Unless you show (all) your work it will be impossible to receive part credit. You have 50 minutes for the exam.

1. (10 points) State the Central limit theorem and explain its relevance to multiple regression.

The Central Limit Theorem states that as the sample size becomes large the distribution of sample means approaches the Normal Distribution.

The least squares estimators can be interpreted as the parameters being estimated plus a term that is the weighted average (means) of the model error. Therefore, as the sample size becomes large the distribution of the LS estimator becomes normal. This is the basis for coinstructing such test statistics as Student's t and the F distribution.

2. (10 points) State the Gauss-Markov Theorem and explain its importance to regression analysis.

The Gauss Markov Theorem states that in the class of linear unbiased estimators the least squares estimator has minimum variance, i.e., it is the most efficient estimator in the class. The importance is that if your statement of the model is correct then there is no other linear unbiased estimator that has less variation in it, i.e., the LS estimator is the one that is most precise.

3. (10 points) What happens to  in a multiple regression model of the sort   as
1. The observations on the explanatory variable xj become more disperse?

From the formula sheet you can see that SSTx enters the denomiinator of the coefficient variance. SSTx is a measure of the dispersion of the variable x. Therefore as the dispersion of x becames greater the variance of the coefficient estimator becomes smaller. This cause the t-statistic to get larger. Alternatively, our estimator becomes more precise as the dispersion of the explanatory variable increases.

2. As the variable xj becomes more similar to the other explanatory variables?

In the denominator of the coefficient variance you will find (1-Rj2) where Rj2 is the coefficient of determination of the regression of xj on the other explanatory variables. As xj becomes more similar to the other variables the term in parentheses will get smaller, causing the variance of the coeffient to get larger. The larger variance of the coefficient indicates that our estimator of the coefficient is becoming less precise.

4. (10 points) Suppose that the correct regression model is  and your research assistant mistakenly estimates the coefficients of  . Later, when you discover the mistake, you recall that economic theory tells you that .  You also recall that x1 and x2 are inversely related.  What is the direction (sign) of the bias in your research assistant’s estimate of β1? How do you know?

Recall that in the instance of an omitted relevant variable the directio of the bias of the estimator is the sign of the omitted variable's coefficient times the sign of the correlation between the included variables and the omitted variable. Therefore, in this case the sign of the bias is negative.

5. (10 points) Explain the relationship between construction of a confidence interval and the test of an hypothesis.

With reference to your formula sheet, the upper or lower bound on a confidence interval involves the estimator plus or minus a chosen t-statistic times the standard error of the estimator. This could be rearranged to get an observed t-statistic in terms of the observed estimate of the coefficient and its standard error. Thus we recognize that a test of hypothesis is the dual to a confidence interval and we can achieve the same thing by using either approach.

 Table 1 Dependent Variable: SALARY Numbers in parentheses ( ) are standard errors. Numbers in [ ] are p-values. Model A Model B Constant 475.4765 (83.2535) [0.00] 638.4401 (87.0375) [0.00] CEOTEN (years as CEO) 48.9271 (14.8645) [0.0012] 46.8203 (16.3575) [0.0047] CEOTENSQ (square of CEOTEN) -1.3106 (0.4994) [0.0095] -1.2707 (0.5503) [0.0221] MKTVAL (market value of the firm) 0.0197 (0.0155) [0.2049] SALES (firm sales) 0.0177 (0.0108) [0.1034] PROFITS (firm profits) 0.0818 (0.2691) [0.7615] R-squared 0.2323 0.0496 Sum of Squared Residuals 46647708 57754227 S.E. of Regression 522.2965 576.1257

Table 2
Coefficient Covariance Matrix

 C CEOTEN CEOTENSQ MKTVAL SALES PROFITS C 6931.141 -976.5751 26.35528 -0.038740 -0.145225 0.109657 CEOTEN -976.5751 220.9544 -6.888693 -0.013729 0.002779 0.215158 CEOTENSQ 26.35528 -6.888693 0.249441 0.000264 9.59E-05 -0.006312 MKTVAL -0.038740 -0.013729 0.000264 0.000241 -1.62E-05 -0.003339 SALES -0.145225 0.002779 9.59E-05 -1.62E-05 0.000117 -0.001170 PROFITS 0.109657 0.215158 -0.006312 -0.003339 -0.001170 0.072416
1. (50 points) Mac Krell has asked his assistant, Ann Chovie, to construct an empirical model of CEO compensation in the seafood industry.  Using data for the CEOs of 177 firms she has produced the results shown above in Tables 1 and 2.
1. At the 1% level of test, is the coefficient on CEOTEN greater than zero? How do you know? Include in your answer the critical value for your test statistic and the relevant degrees of freedom.

Yes, it is greater than zero. In the BIG model t = 48.9/14.8 = 3.29 and in the little model t = 46.8/16.3 = 2.86. With 171 and 174 degrees of freedom the 1% critical t's are ~2.35. Since the observed t's are larger we reject the null.

Alternatively, the reported p-value in both models is smaller than .01 so we reject the null.

2. At the 1% level of test, is the coefficient on CEOTENSQ less than zero? How do you know? Include in your answer the critical value for your test statistic and the relevant degrees of freedom.

Using the same reasoning as in the previous part we have t = -1.31/.499 = -2.6 and t = -1.27/.55 = -2.31. Therefore, in the big model we would reject the null at the 1% level, but in the small model we would not.

You would come to the same conclusion based on the p-values. Note that EVIEWS reports the p-values as though you were doing a two tail test, but the question is stated as a one tail test.

3. Are there increasing, constant, or decreasing marginal returns to one’s tenure as a CEO? How do you know?

There are decreasing marginal returns to one's tenure as a CEO. We know this because of the sign pattern on CEOTEN and CEOTENSQ (+ and -, respectively) and we now know that the coefficients are significant.

4. Mr. Krell asks Ms. Chovie to test the hypothesis at the 5% level that the coefficients on MKTVAL and SALES are equal. Include in your answer the critical value for your test statistic and the relevant degrees of freedom.

This is a two tail test. The degrees of freedom = 171. The critical t = +/- 1.9. The observed statistic is t = (0.0197-0.0177)/sqrt(.000241+.000117-2(-1)(.000016)) = .002/.0197 = 0.10 This is a very small observed t-statistic. We do not reject the null.

5. In looking over the results of Table 1 Mr. Krell is struck by the fact that none of the coefficients on MKTVAL, SALES and PROFITS are individually different from zero. He asks you to test the hypothesis at the 5% level that the set of three coefficients are jointly zero. Include in your answer the critical value for your test statistic and the relevant degrees of freedom.
6.

This is an F-test for the joint significance of the three coefficients. The critical F at the 5% level and 3, 171 degrees of freedom is 2.65. The observed F is given by [(57754227-46647708)/3]/(46647708/171) = 13.57. We should reject the null that all three coefficients are zero.