2024 – Part I each question is worth 4 points Consider the following regression model yi 1 2xi ei

Part I (each question is worth 4 points): Consider the following regression model:
yi = 1 + 2xi + ei;
for i = 1;:::;N. Let eijxi 0;2 i. That is, conditional on xi, ei has a distribution with a meanof 0 and a variance of 2 i. Let
s2 x =
1 N
N X i=1
(xi x)2 ; where x = 1 N
N X i=1
xi; and y =
1 N
N X i=1
yi: (1)
Let also b1and b2 be the ordinary least-squares (OLS) estimates for 1 and 2, respectively. 1. Let e = yb1 b2x, then, (a) e = 0 always (b) e need not be equal to zero (c) if y > 0, then it must be that e > 0 (d) if y > 0 and x > 0, then it must be that e > 0
2. Assume that assumptions SR1-SR5 made in Chapter 2 hold, then:
(a) E (b2) = 2 only if in addition the distribution of ei is normal. (b) E (b2) = 2 always. (c) E (b2) = 2 only if in addition 2 i = 2 for all i, i.e., 2 i is a constant. (d) E (b2) 6= 2 even if all the assumptions hold. 3. Suppose one tested and rejected the null hypothesis H0: 2 = 0 against H1: 2 = 5 at the signicant level = 0:05, then one of the following must be true:
(a) He will reject H0: 2 = 0 against H1: 2 6= 0. (b) He will reject H0: 2 = 0 against H1: 2 < 1. (c) He will reject H0: 2 = 0 against H1: 2 < 4. (d) He will reject H0: 2 = 0 against H1: 2 > 0.
4. Suppose b2 = :75, se(b2) = :05, and N = 42. The 90% condence interval for b2 would be:
(a) [:721;:761]. (b) [:602;:901]. (c) [:666;:834]. (d) [:424;1:194].
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5. Suppose it is given that x = 10:5, y = 3:30, and b1 = 2:2. Then
(a) b2 = 19:8 (b) b2 = 19:8 (c) b2 = 10:5 (d) b2 = 0:105
6. Suppose that CI2 is that 100(1)% condence interval for 2. Assume that you want to test H0: 2 = 0 against H1: 2 6= 0 with being the type I error. Under what condition will one reject H0?
(a) There is not enough information provided to answer the question. (b) If and only if 0 (zero) is in CI2. (c) The condence interval CI2 provides no information that is useful for testing the above hypothesis. (d) If and only if 0 (zero) is not in CI2.
7. The larger is the variation in xi, i = 1;:::n, in the sample:
(a) The larger is the variance of b2. (b) The smaller is the variance of b2. (c) The variation in xi has no e⁄ect on the variance of b2. (d) The variation in xi only a⁄ect the point estimate b2.
8. Suppose that we ran a regression and we obtained b2 = 1. If we were to dene a new variable x = cx, then a regression of y on x will yield an estimate for 2, say b 2, that is equal to (a) b 2 = 1=c. (b) b 2 = c. (c) b 2 = 1 + c. (d) b 2 = 1c. 9. Consider the hypotheses H0: 2 = 0 against H1: 2 > 0. Using the statistic t = b2=b se(b2) for testing H0 against H1: (a) The smaller the sample size, the more likely it is that we reject H0. (b) The larger the sample size, the more likely it is that we reject H0. (c) The sample size has no e⁄ect on the likelihood of rejecting H0. (d) Sometimes answer (a) will be true and sometimes answer (b) will be true.
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10. Consider the SSE from a given regression. Which of the following statements is correct?
(a) There is no direct link between SSE and R2. (b) To determine how high R2 is we need to know both SSR and SSE. (c) The larger is the SSE from a regression the larger is R2. (d) The smaller is the SSE from a regression the larger is R2.
11. Consider a 100(1)% condence interval for 1 given by CI1. Then, (a) The larger is the larger is the length of the interval. (b) The smaller is the larger is the length of the interval. (c) does not have any e⁄ect on the length of the condence interval. (d) Whether or not have an e⁄ect is determine by its magnitude.
12. Suppose that b2 = 1:5, se(b2) = 10:5, and N = 20. Then the 90% condence interval for 2 is:
(a) [20:56;23:56]. (b) [1:25;1:75]. (c) [:5;2]. (d) [16:71;19:71]. 13. It is given that the 99% condence interval for 2 is given by [6:3717;7:4817], where N = 29. Then b2, the point estimate for 2, is
(a) b2 = 1:25 (b) b2 = 0:555 (c) b2 = 1:50 (d) b2 = 5:123
14. Consider the null hypothesis H0: 2 = 0 against the alternative hypothesis H1: 2 < 0. (a) If one were to reject H0 then one will accept H0: 2 = 0 against H1: 2 6= 0. (b) If one were to reject H0 then we cannot determine whether he/she will also reject H0: 2 = 0 against H1: 2 6= 0. (c) If one were to reject H0 then one will also reject H0: 2 = 0 against H1: 2 6= 0. (d) If one were to accept H0 then one will also accept H0: 2 = 0 against H1: 2 6= 0.
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15. Suppose one tested and rejected the null hypothesis H0: 2 = 0 against H1: 2 = 2 at the signicant level = 0:05, then one of the following must be true:
(a) He will reject H0: 2 = 0 against H1: 2 6= 0. (b) He will reject H0: 2 = 0 against H1: 2 < 0. (c) He will reject H0: 2 = 0 against H1: 2 > 0. (d) He will reject H0: 2 = 0 against H1: 2 < 2.
16. The rejection region consists the values of test statistic that have:
(a) Low probability of occurring when the null hypothesis is true. (b) High probability of occurring when the null hypothesis is true. (c) Low probability of occurring regardless of whether the null hypothesis is true or not. (d) High probability of occurring regardless of whether the null hypothesis is true or not.
17. The Gauss-Markov Theorem establishes that the OLS estimator is:
(a) The best among all possible estimators for 2. (b) The best among all linear unbiased estimators for 2. (c) The best among all linear unbiased estimators for 2 for which ei has a normal distribution. (d) The best among all possible estimators that minimize the SSE.
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Part II (each question is worth 3 points): Consider the following STATA output in which the summary statistics for four variables are provided:
yi = pizza = annual expenditure on pizza in dollars by an individual.
x2i = income = annual income in thousands of dollars of the individual.
x3i = age = age in years.
x4i = female = an indicator variable that take the value 1 if the person is a female, and take the vale 0 (zero) otherwise.
There is also the output for three alternative regressions (see the attached STATA output for this exam). These regressions are:
1: yi = 1 + 2x4i + ei 2: yi = 1 + 2x2i + ei 3: yi = 1 + 2x2i + 3x3i + vi 4: yi = 1 + 2x2i + 3x3i + 4x4i + ui
For each of the following questions determine whether it is true, false, or it is not possible to determine (Cannot be determined):
1. On average, a males annual expenditure on pizza is about $184 more than that of a female.
(a) True (b) False (c) Cannot be determined
2. A 95% condence interval for = 2 + 3 is given by [9:23;3:43]. (a) True (b) False (c) Cannot be determined
3. In Model 4, one can reject the hypothesis that 3 = 0 at any reasonable .
(a) True (b) False (c) Cannot be determined
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4. The variable x2i is economically more important than the variable x3i.
(a) True (b) False (c) Cannot be determined
5. The estimated covariance between the estimates for 2 and 3 is negative.
(a) True (b) False (c) Cannot be determined
6. In Model 2 the SSE is greater than the SSR.
(a) True (b) False (c) Cannot be determined
7. Model 1 is meaningless.
(a) True (b) False (c) Cannot be determined
8. One cannot have both age and income in the same regression.
(a) True (b) False (c) Cannot be determined
9. The results indicate that, holding age, income, and gender constants, a person spends, on average, $200 per year on pizza.
(a) True (b) False (c) Cannot be determined
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10. The R2 in Model 1 is an invalid measure of the goodness-of-t.
(a) True (b) False (c) Cannot be determined
11. From the output, one cannot determine the 90% condence interval for 3 in Model 3
(a) True (b) False (c) Cannot be determined
12. Comparing the results of Model 1 and Model 2 indicates that income is not an important determinant of expenditure on pizza for women.
(a) True (b) False (c) Cannot be determined