Introduction to Econometric Methods - Problem Set
Q1. Let kids denote the number of children ever born to a women, and let educ denote years of education for the women. A simple model relating fertility to years of education is
Kids = β0 + β1educ + u,
where u is the unobserved error.
(i) What kind of factors are contained in u? Are these likely to be correlated with level of education?
(ii) Will a simple regression analysis uncover the ceteris paribus effect of education on fertility? Explain.
Q2. In the simple linear regression model y = β0 + β1x + u, suppose that E(u) ≠ 0. Letting α0 = E(u), show that the model can always be rewritten with the same slope, but a new intercept and error, where the new error has a zero expected value.
Q3. The data set BWGHT.RAW contains data on birth to women in the United States. Two variables of interest are the dependent variable, infant birth weight in ounces (bwght), and an explanatory variable, average number of cigarettes the mother smoked per day during pregnancy (cigs). The following simple regression was estimated using data on n = 1,988 births:
(bwght)^ = 119.77 - 0.514 cigs
(i) What is the predicted birth weight when cigs = 0? What about when cigs = 20 (one pack per day)? Comment on the difference.
(ii) Does this simple regression necessarily capture a casual relationship between the child's birth weight and the mother's smoking habits? Explain.
(iii) To predict a birth weight of 125 onuces, what would cigs have to be? Comment.
Q4. The data in 401K.RAW are a subset of data analyzed by Papke (1995) to study the relationship between participation in a 401(k) pension plan and the generosity of the plan. The variable prate is the percentage of eligible workers with an active account; this is the variable we would like to explain. The measure of generosity is the plan match rate, mrate. This variable gives the average amount the firm contributes to each worker's plan for each $1 contribution by the worker. For example, if mrate = 0.50, then a $1 contribution by the worker is matched by a 50¢ contribution by the firm.
(i) Find the average participation rate and the average match rate in the sample of plans.
(ii) Now, estimate the simple regression equation
(prate)^ = β^0 + β^1mrate,
and report the results along with the sample size and R-squared.
(iii) Interpret the intercept in your equation. Interpret the coefficient on mrate.
(iv) Find the predicted prate when mrate = 3.5. Is this a reasonable prediction? Explain what is happening here.
(v) How much of the variation in prate is explained by wrote? Is this a lot in your opinion?
(vi) Test the null hypothesis that β1 ≤ 0 against the alternative that β1 > 0 using a t-test with significance level of 5%. What is the p-value of your test? How does your choice of critical value depend on whether you can assume the population is normally distributed? Compute a 90% non-rejection region (confidence interval) for your test. See the lecture notes on D2L for details on how to compute a non-rejection region for a one-sided test.
Q5. Use the data in WAGE2.RAW to estimate a simple regression explaining monthly salary (wage) in terms of IQ score (IQ).
(i) Find the average salary and average IQ in the sample. What is the sample standard deviation of IQ? (IQ scores are standardized so that the average in the population is 100 with a standard deviation equal to 15.)
(ii) Estimate a simple regression model where a one-point increase in IQ changes wage by a constant dollar amount. Use this model to find the predicted increase in wage for an increase in IQ of 15 points. Does IQ explain most of the variation in wage?
(iii) Now, estimate a model where each one-point increase in IQ has the same percentage effect on wage. If IQ increases by 15 points, what is the approximate percentage increase in predicted wage?
Q6. Use the data in ATTEND.RAW for this exercise. We want to study the relationship between attendance rate (atndrte), which is measured as a percent, and score on an achievement test (ACT), which has a maximum possible value of 32.
(i) Find the smallest and largest values of atndrte and ACT in the sample.
(ii) In the population model
atndrte = β0 + β1ACT + u,
interpret the coefficient β1. Is the sign of β1 obvious? Explain.
(iii) Use the data in ATTEND.RAW to estimate the model from part (ii). Report the estimated equation in the usual way, including the sample size and R-squared. Does ACT explain a lot of the variation in the attendance rate?