• This is a take-home test over 8 hours.
You will answer a total of five questions, each containing one to eight sub-questions. Make sure to answer all sub-questions. You must show your work to get the full grade.
• This is open book and open internet.
You are prohibited from collaborating in any way with other people on the exam. This means no discussing the questions in person, by phone, using chat services, question boards, email, etc.
If you draw your answer from another source (e.g. the textbook, the internet), you must reference it. The normal rules of plagiarism apply to this exam.
Instructor help on the test will be limited to clarification questions only. I will be available via zoom from 10 am to 10.20 am, from 1 pm to 1.20 pm. Please use the corresponding zoom link. You can also contact me via email; I will answer within an hour.
I have attached the t-distribution and F-distribution tables (at the end of this document). You can use them to calculate critical values.
Submit your exam to Gradescope when complete. Note that you can submit as many times as you wish before the deadline. Late submissions will receive a zero grade.
You are required to hand-write your responses to each question on paper (i.e. do not use a word processor unless you have accommodations through ALC).
• I would suggest answering each sub-question on a separate piece of paper.
• Sign your name below the response to each question.
• Upload your hand-written responses to Gradescope in one of two ways:
– Take a legible photo of your response to each question separately, and then upload each image. Make sure to tag your questions.
– Scan your questions into a single PDF, then upload it and tag each question. Make sure to tag your questions.
For each statement below, evaluate whether it is true, false, or uncertain, and fully explain your answer. a)(5 points) In a multiple regression, when control variables are correlated with the unobserved error term, the OLS slope estimate on the variable of interest is always biased.
b)(5 points) A high (but not perfect) amount of collinearity between independent variables in a regression model will result in higher standard errors of the OLS slope estimates.
c)(5 points) If the variables x1 and x2 are uncorrelated with each other, the multiple regres- sion of y on x1 and x2 will always produce different slope estimates from separate simple regressions of y on x1 and y on x2.
d)(4 points) Suppose you estimate the parameters of the model y = β0 + β1x1 + β2x2 + u. Based on an F-test, you reject the null hypothesis that β1 = 0 and β2 = 0. The appropriate conclusion is that both β1 ƒ= 0 and β2 ƒ= 0.
e)(5 points) The intuition behind the homoskedasticity-only F-test is that if we impose restric- tions on the model, and as a result the fit improves, we accept the restrictions.
Consider the regression model relating crop yield to two types of fertilizer:
yield = β0 + β1fertilizer1 + β2fertilizer2 + u
Use “approach 2” to transform the regression so that you can use a t-statistic to test the hypothesis that fertilizer 1 has three times the effect fertilizer 2 has on yields. Define the null and alternative hypotheses, as well as ALL the steps you would follow to perform the test.
Suppose you are given the following sample regression function:
ln(h^wage) = 1.5 + 0.1exper − 0.2 female − 0.1 immigrant − 0.05 (female*immigrant) where ln(hwage) is the natural logarithm of hourly wages, exper is years of experience, female is a variable equal to 1 if the person is female and 0 if the person is male, and immigrant is a variable equal to 1 if the person is immigrant and 0 if it is non-immigrant. The sample size is 85.
a)(6 points) Precisely interpret the estimate on female, the estimate on immigrant, and the interaction term between female and immigrant.
b)(6 points) Suppose you wanted to test whether the regression function that relates hourly wages with experience and gender is the same for immigrants and non-immigrants. Explain the null and alternative hypotheses you would use for the test, as well as the steps you would follow to perform the test. Make sure to find the corresponding critical value when possible.
Suppose you are given the following sample regression function:
price = β0 + βscoreScore + βF ranceFrance + βAustraliaAustralia + u (1) where price is the wine price measured in dollars, score is the score given by a wine expert (scale 1 to 100 points), Europe and Australia are dummy variables indicating the wine’s country of origin.
The reference category is United States (USA). The sample size is 50.
a)(5 points) Suppose that all of the wines in the sample came from France and USA. What problem, if any, would this cause for estimating regression (1) using OLS?
You estimate regression (1) and obtain the following estimates. The standard errors are shown in parenthesis below the estimate.
p^rice = 5 + 0.1 Score + 1.2 France − 0.5 Australia
(0.10) (0.045) (0.70) (0.30)
b)(5 points) Precisely interpret the estimate on the intercept, and the slope estimates on the dummy variables France and Australia.
c)(4 points) Using an F-test, test the null hypothesis that the effect of score on wine prices is zero against the alternative that it is not at the 5% significance level.
d)(4 points) Construct a 95% confidence interval for the effect of a 10-point increase in scores on wine prices.
e)(5 points) Jointly test the null hypothesis that βAustralia = 0 and βF rance = 0, at a 5% significance level. Assume the t-statistics for testing each restriction individually have a correlation of 0.5.
f)(5 points) Suppose you instead wanted to use France as the reference category. Write down the new regression model and find the estimates for the intercept and all slope coefficients. g)(4 points) Suppose you decide that oak ageing matters for determining the price of a wine, so you add it to the regression model to get
price = β0 + βscoreScore + βF ranceFrance + βAustraliaAustralia + βoakoak + u (2) where oak represents the number of years of oak ageing. Would the R2 and the adjusted R2 in regression (2) be higher or lower than the one without oak (i.e., regression (1))? Explain.
h)(4 points) Suppose you wanted to extend regression (1) to let the effect of scores on wine price to depend on the level of score. Write down the regression model that accounts for this effect.
a)(5 points) Interpret the coefficient on ln(income) in regression (1), and explain why it is larger in absolute terms than in regressions (2) and (3).
b)(5 points) Precisely explain what is the effect of age on birthweight. Use the results from regression (2).
c)(5 points) Test, at a 5% significance level, the null hypothesis that the effect of age on birthweight is linear.
d)(5 points) In regression (3), for the OLS estimate of βln(income) to be unbiased, you can make one of two assumptions: 1) E(u ln(income), age, age2, smoker) = 0 (mean independence of the error), or 2) E(u ln(income), age, age2, smoker) = E(u age, age2, smoker)(conditional mean independence of the income.
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