Unless provided with information to the contrary, assume for each question below that the Classical Linear Model assumptions hold.

Size: px

Start display at page:

Download "Unless provided with information to the contrary, assume for each question below that the Classical Linear Model assumptions hold."

Luke Boone
5 years ago
Views:

1 Economics 345: Applied Econometrics Section A01 University of Victoria Midterm Examination #2 Version 1 SOLUTIONS Spring 2015 Instructor: Martin Farnham 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 bubble form. 1) 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. ˆβ 0 is an estimate of A) how many cigarettes a day it takes to lower birth weight by 1 ounce, on average B) how many ounces an extra cigarette a day lowers birth weight, on average. C) how many ounces the average baby weighs, when cigs=0 and faminc=0. D) the standard error of cigs. E) none of the above. 2) Suppose that you estimate the model y = β 0 x + u. You calculate residuals and find that the explained sum of squares is 400 and the total sum of squares is The R-squared is A) 1/4 B) 1/3 C) 1/2 D) 2/3 E) 3/4 1

2 3) In linear regression, the assumption of homoscedasticity is needed for I. unbiasedness II. simple calculation of variance and standard errors of coefficient estimates. III. the claim that the OLS estimator is BLUE. D) II and III only. 4) Which of the following is an advantage of multivariate regression over univariate regression? A) One can control for other determinants of y that may be correlated with the primary x variable of interest. B) The zero conditional mean assumption is more likely to hold with multivariate regression. C) One can often reduce bias by moving to multivariate regression. D) One can often obtain a model that better fits the data with more right-hand-side variables in the model. E) All of the above are advantages of multivariate regression over univariate regression. 5) Suppose the true population model of y is given by y = β 0 x 1 x 2 + β 3 x 3 + u. Which of the following will lead to a higher variance of the OLS estimator, ˆβ 3? A) A smaller sample size. B) Less variation in x 3. C) Greater variation in u. D) Higher correlation between x 1 and x 3. E) All of the above will lead to a higher variance of the OLS estimator, ˆβ 3. 2

3 6) 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. D) I and II only. Questions 7-8 share the following setup: Suppose the true population model of y is given by (1) y = β 0 x 1 x 2 + u, but you estimate the following model instead: (2) y = γ 0 +γ 2 x 2 + v. [The gammas (γ ) are just constant parameters like the betas ( β )] 7) Suppose x 1 and x 2 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( ˆβ 2 ) III. The error variance of model (1) will be smaller than that of model (2). D) I and III only. 3

4 8) Suppose x 1 and x 2 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( ˆβ 2 ) III. E[ ˆγ 2 ]>E[ ˆβ 2 ]. D) I and III only. END SECTION 1. Section 2: Short Answers Answer each question as clearly and concisely as possible on the exam paper. 1) 26 points total Note: This question has multiple parts, but each can be answered on its own, so if you get stuck on one part, move onto another. Suppose you are interested in the effect of classroom size (number of students) on student achievement an important question for education policymakers. You have classroom-level data for 4 th grades across Canada that includes average test scores for each classroom (from a standardized test), and the number of students in each classroom. For example, a 4 th grade class at one school may have 80 students divided into three classrooms: one with 24 students, one with 26 students, and another with 30 students. In order to determine the effect of classroom size, you estimate the following regression by OLS: Score i = β 0 Size i + u i where i denotes classroom, Score is the standardized test score for classroom i, and Size is the number of students in classroom i. 4

5 a) (4 points) In this real world setting, is the OLS estimate, ˆ β 1, likely to be unbiased? Explain. Explanation required for credit. No, it s likely to be biased. This is because there are likely to be many things that affect Score, and some of them are likely to be correlated with Size. If they are omitted from the model, they are present in the error term, and those that are correlated with Size will cause problems. In particular, their presence in the error term will cause the error term to be correlated with Size. This represents a violation of the zero conditional mean assumption. Suppose you learn that the true model of test scores is Score i = β 0 Size i hhinc i + v i (note: this is almost certainly not the true model, but just assume that it is). hhinci is the average household income of a child in classroom i. b) (4 points) Assume that all schools are financed out of local property taxes, and that some school districts are rich and some are poor. Are Size and hhinc likely to be correlated? How? Explain. Explanation required for credit. Yes, Size and hhinc are likely to be negatively correlated. This is because parents in high income areas can afford to pay high taxes to support good schools. Well-funded schools are likely to have lower class sizes. Thus areas with high income will likely have low values of Size. c) (4 points) Now just assume that Size and hhinc are negatively correlated (one can tell stories for the correlation to go either way, so just assume this) and that hhinc affects Score. What effect does including hhinc in the regression have on bias of ˆβ 1? Explain clearly what happens to the estimator when you include hhinc. Explanation required for credit. This question threw a lot of people. Remember the basic omitted variables bias story. If the true model should include hhinc, but we leave it out, that will cause the estimated coefficient on Size to be biased. The bias will be negative, because hhinc and Size are negatively correlated and because beta2 is positive (kids in areas with higher income tend to perform better in school on average). 5

6 If omitting hhinc causes negative omitted variables bias, then including hhinc should cause the negative omitted variables bias to go away. Another way of putting this is that ˆβ 1 should become larger (in expected value) once we include hhinc. Many people said that adding hhinc would cause bias. But adding variables doesn t cause bias. If anything, it reduces bias. d) (4 points) Can you see any potential disadvantage of adding hhinc to the model? Explain. Yes. Consider the formula for variance of the OLS slope estimators: Var( β ˆ 1 ) = n i =1 σ 2 (x 1i x 1 ) 2 (1 R 1 2 ) If you include hhinc, the error variance in the model (the numerator of the above expression) will fall. Holding the denomimator constant, this would cause variance to fall, which is not a problem. However, the denominator isn t likely to stay constant. This is because the R-squared term will likely go up. This is the R-squared obtained by regressing Size on hhinc, and that s likely to be greater than zero. Without hhinc, the R-squared is zero (because Size regressed on a constant term has an R-squared of zero). But with hhinc, the R- squared goes up. A higher R-squared means a small denominator which makes the Variance go up (holding the numerator constant). If the effect of the higher R-squared term dominates the effect of the lower sigmasquared term, then variance will go up overall. Higher variance would be a disadvantage of adding hhinc to the model. e) (4 points) Suppose that Size is number of students in a classroom and Score is a number between 1 and 100. If you obtain ˆβ 1 = 1.2, what does that mean? Interpret this estimate clearly (as if trying to explain the effect of class size on test scores to a friend who doesn t know econometrics). Use proper units in explaining what that coefficient estimate means. A short (1 sentence) answer will suffice. This means that if you increase the number of students in a class by 1, the test score will fall by 1.2 points, on average. 6

7 Suppose you estimate Score i = β 0 Size i hhinc i + v i. You obtain the following standard errors and parameter estimates: ˆβ 0 = 95; se( ˆβ 0 ) = 30 ˆβ 1 = 1.2; se( ˆβ 1 ) = 0.4 ˆβ 2 = 0.003; se( ˆβ 2 ) = n = 25 f) 6 points Conduct the following hypothesis test at the 1% significance level: H 0 : β 1 = 0 H 1 : β 1 < 0 In your answer draw the distribution of the test statistic that you use, and clearly define the rejection region(s), indicating the critical value(s). t = ˆ β 1 0 se( β ˆ 1 ) = = 3 degrees of freedom are n-k-1=25-2-1=22 We need to find the critical value for a one-sided alternative at the 1% level, from a t- distribution with 22 degrees of freedom. The t-critical value is Since our t-statistic is more negative than t-critical, we reject the null hypothesis in favour of the alternative. (picture to follow) END SECTION 2. END OF EXAM. 7

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

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):