Economics 345: Applied Econometrics Section A01 University of Victoria Midterm Examination #2 Version 1 SOLUTIONS Fall 2016 Instructor: Martin Farnham

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1 Economics 345: Applied Econometrics Section A01 University of Victoria Midterm Examination #2 Version 1 SOLUTIONS Fall 2016 Instructor: Martin Farnham Last name (family name): First name (given name): Student ID Number: Midterm Exam #2 Version 1 Unless provided with information to the contrary, assume for each question below that the Classical Linear Model assumptions hold. Section 1: Multiple Choice (3 points each) Select the most appropriate answer, and carefully bubble in the letter of the answer on your NCS marking card. Questions 1-3 refer to the following setup: Suppose you obtain the following fitted model: bwght = ˆβ0 + ˆβ 1 cigs + ˆβ 2 faminc, where bwght is child birth weight in ounces, cigs is the average daily number of cigarettes smoked per day by the mother during pregnancy, and faminc is family income measured in dollars. 1) is an estimate of! ˆβ 2 A) how many dollars of income it takes to increase birth weight by 1 ounce, on average B) how many ounces an extra dollar of income raises birth weight, on average. C) how many ounces the average baby weighs, when cigs=0. D) how many ounces the average baby weighs, when cigs is at its average value. E) none of the above. 1

2 2) If you re-estimate this model with cigs measured in packs (there are 20 cigarettes in a pack), how will the coefficient estimates change? A) The new coefficient estimates will be the same as the old estimates. B) The new coefficient estimates will be the old estimates divided by 20. C) The new coefficient estimates will be the old estimates multiplied by 20. D) Only ˆβ 0 will change. E) ˆβ 1 will be 20 times as large as in the original specification. 3) Suppose the true population model of y is given by y = β 0 + β 2 + β 3 x 3 + u. Which of the following will lead to a higher variance of the OLS estimator, ˆβ 3? A) A larger sample size. B) Less variation in x3. C) Less variation in u. D) Lower correlation between x1 and x3. E) All of the above will lead to a higher variance of the OLS estimator, ˆβ 3. 4) Which of the following is/are consequences of overspecifying a model (including irrelevant variables on the right-hand-side)? I. The variance of the estimators may increase. II. The variance of the estimators may stay the same. III. Bias of the estimators may increase. A) I only. D) I and II only. E) I, II, and III. Questions 5-6 share the following setup: Suppose the true population model of y is given by (1) y = β 0 + β 2 + u, but you estimate the following model instead: (2) y = γ 0 +γ 2 + v. [The gammas ( γ ) are just constant parameters like the betas ( β )] 2

3 5) Suppose x1 and x2 are uncorrelated, and β 1 0. Which of the following statements is/are DEFINITELY true? I. ˆγ 2 will be biased. II. Var( ˆγ 2 )<Var( ˆβ2 ) III. The error variance of model (1) will be less than that of model (2). A) I only. D) I and III only. E) I, II, and III. 6) Suppose x1 and x2 are negatively correlated, and! β 1 > 0. Which of the following statements is/are DEFINITELY true? I. ˆγ 2 will be biased. II. Var( ˆγ 2 )<Var( ˆβ2 ) III. E[ ˆγ 2 ]>E[ ˆβ2 ]. A) I only. D) I and III only. E) I, II, and III. 7) Assume that the five Gauss Markov assumptions hold. Which of the following is an advantage of a very large sample size for OLS estimation and inference? I. Coefficient estimates will tend to be very precise. II. Coefficient estimates will approach having a normal distribution. III. Inconsistent estimators will approach the true parameter value as n goes to infiniti. A) I only. D) I and II only. E) I, II and III. 3

4 8) Suppose the true model of y is given by y = β 0 + β 2 + u. Then, if you estimate the following mis-specified model, y = β 0 + v, and if corr(x1,x2)<0 and β2<0, which of the following statements is TRUE? A) ˆβ 1 will be negatively biased. B) ˆβ 1 will have positive inconsistency. C) ˆβ 1 will be unbiased. D) ˆβ 1 will be consistent. E) ˆβ 1 will be efficient. 9) Suppose you estimate the following model of child birthweight as a function of cigarette use and family characteristics: bwght = β 0 cigs + β 2 parity + β 3 faminc + β 4 motheduc + β 5 fatheduc + u, using a sample size of n=2006. Suppose that the R-squared for this model is When motheduc and fatheduc are dropped from the right-hand-side and the model is re-estimated, the R-squared falls to What is the F-statistic for the null hypothesis that the coefficients on motheduc and fatheduc are jointly equal to zero? Answers below are rounded to the nearest whole number. A) 1 B) 2 C) 3 D) 17 E) 25 10) Suppose you estimate the following model in EViews: y = β 0 + β 2 + β 3 x 3 + u. You want to test the following null hypothesis: H0: β 3 = 0 against H1: β 3 < 0. Along with an estimate of ˆ β 3, where the estimate is greater than 0, EViews gives you a p value of Consider the following statements: I. You fail to reject the null hypothesis at the 10% level. II. You can reject the null hypothesis at the 6% level. III. You can reject the null hypothesis at the 3% level. Which combination of statements is true? 4

5 A) I only. D) I and II only. E) II and III only. END SECTION 1. Section 2: Short Answers Answer the question as clearly and concisely as possible on the exam paper. Note that while this question has multiple parts, most parts can be answered without knowing the answers to other parts. So if you can t do one part, move on to a later part you can do. 1) 20 points total Suppose you estimate the following model by OLS: wage = β 0 educ + β 2 exper +u wage: hourly wage in dollars educ: years of education exper: years of experience You obtain the following fitted model using EViews, where the standard errors of coefficient estimates are given in parentheses below each estimate:! wage = educ +1.5exper (2.0) (0.7) (0.5) Number of Observations: 523 R-squared:

6 a) (6 points) At the 1% level, conduct a test of the hypothesis that β 2 = 0 against the alternative that β 2 > 0. Formally write out the null hypothesis; show how you calculate the appropriate test statistic; and make careful note of the degrees of freedom, the critical value of your test statistic, and whether or not you reject the null. H 0 :β 2 = 0 H 1 :β 2 > 0 df=n-k-1=520 ˆt = null ( ˆβ 2 β 2 ) se( ˆβ 2 ) = ˆβ 2 se( ˆβ 2 ) = = 3 t crit = This is a one-sided test, because we have a one-sided (positive) alternative. So the rejection region lies to the right of Therefore the t-statistic lies in the rejection region (because 3>2.326). So we reject the null hypothesis in favour of the alternative. b) (3 points) Draw a picture of the distribution of the test statistic used in (a), assuming the null hypothesis is true. Your clearly labeled drawing should denote the rejection region(s) and the value of the test statistic you obtained in (a). (see separate posted drawings; to follow with a delay) Note that we always assume the null hypothesis is true when we do hypothesis testing. The entire approach is based on the initial assumption that the null is true with rejection of the null occurring only if we get a realization of the t-statistic that is an extreme value (i.e., that lies in the rejection region). When we draw the t-distribution, we re drawing the t-distribution under the null which is the same as assuming the null hypothesis is true. This wording seemed to throw some people, but it is always done with that wording implicit at least. 6

7 Note that failure to understand that numbers increase in value as we move from left to right along an x-axis (and from down to up along a y-axis) constitutes a serious misunderstanding that you should remedy before the final exam. No student should have got to this point at UVic without realizing this basic convention, and yet about 5% of students in the course made this mistake on the exam. Ask me if you have questions. c) (2 points) What does your finding in (a) prove (if anything)? We re testing at the 1% significance level. The fact that we reject the null holds out two possibilities. Either 1) We got an unlucky draw and accidentally rejected the null when it s really true. This should happen 1% of the time in repeated samples. or 2) The null is false, and our t-test has led us to correctly reject the null in favour of the alternative. Since case 1 should only occur 1% of the time, we will interpret our finding as strong evidence that experience is a positive determinant of the wage. Technically, this doesn t absolutely prove anything, but (assuming the Gauss-Markov assumptions hold) most people would interpret this as strong evidence that more experience causes wage increases. d) (6 points) Now, at the 5% level, conduct a test of the null hypothesis that experience and education jointly have no effect on the wage, against the alternative that this is not true. Formally state the null and alternative hypothesis. Show how you calculate the appropriate test statistic, and make careful note of the degrees of freedom, the critical value of your test statistic, and whether or not you reject the null. 7

8 Anytime you see a call for a test of joint significance (at least in this course) you should be thinking F-test. You can t test for joint significance with a t-test, because a t-test will only accommodate a single-equation null hypothesis. H 0 :β 1 = 0, β 2 = 0 H 1 : Null is false Here is the unrestricted equation: wage = β 0 educ + β 2 exper +u The R-squared for the unrestricted equation is Here is the restricted equation (the equation that holds under the null): wage = β 0 +u The R-squared for the unrestricted equation is 0, because it has no explanatory variables. To conduct the F-test first construct the F-statistic: R 2 2 ( ur R r )/2 F = 2 ( 1 R ur )/ n k 1 ( ) = ( )/2 0.55/ numerator degrees of freedom: 2 denominator degrees of freedom: n-k-1=520 The F-critical value: 3 Since F=212.7>Fcrit=3, we reject the null hypothesis. e) (3 points) Draw a picture of the distribution of the test statistic used in (d), assuming the null hypothesis is true. Your clearly labeled drawing should denote the rejection region(s) and the value of the test statistic you obtained in (d). (see separate posted drawings; to follow with a delay) 8

9 END SECTION 2. END OF EXAM. This Page Intentionally Left Blank (for scratch work) 9