Problem Set 1 - BCUSE Solutions Spring 2024

Financial Econometrics Problem Set 1 – Review and Repeated Cross-sections Due Friday February 9 th @ 11:59pm Instructions: Please complete the Canvas Quiz associated with this assignment by 11:59pm on Thursday February 18 th . You may work with others, but you should complete the quiz on your own. Theory/BCUse Questions – These questions either require only theory without any data, or allow you to use the bcuse datasets. 1. In three sentences, discuss the consequences for bias, precision, and hypothesis testing of the following: a. Omitting an important variable: Omitting an important variable biases our coefficients as long as it is correlated with an independent variable (if you just said it biases the coefficients without the qualifier that’s fine, since “important” often means both correlated with the dependent variable and an independent variable). Precision is irrelevant since the estimate is biased, but omitting a variable generally has an unclear effect on precision – it saves a degree of freedom but may cost you explanatory power. Again, hypothesis testing is largely irrelevant, since estimates are biased. b. A sample correlation of 0.95 between two independent variables included in the model: This does not bias coefficients. It will result in imprecise estimates, and hypothesis tests will be less likely to find a significant result. c. Heteroscedasticity: Heteroscedasticity does not bias coefficients, but makes the estimates of an estimate’s precision, our standard errors, incorrect. This fact means that hypothesis tests are no longer valid. 2. The following equation describes the median housing price in a community as a function of the amount of nitric oxide (nox, a type of pollution) and rooms in the house. You do not need to turn in STATA output. log ( price ) = β 0 + β 1 log ( nox ) + β 2 rooms + u a. You suspect that nitric oxide negatively affects house prices, and the number of rooms positively affects house prices. You also suspect smaller houses with fewer rooms are built in areas with more pollution. If your assumptions are true, in what direction would the bias on β 2 be if you omitted nox from the regression? (Note: many publicly available housing datasets will have data on rooms, but not pollution, so this bias could show up in real life). β 2 would be biased upwards, since rooms would now be capturing both larger house size and less pollution. b. Use the bcuse dataset hprice2 to run the regression with and without nox. In reality, does it appear your assumptions from part a played out? Yes, the coefficient drops from 0.37 to 0.31. c. From the regression that includes nox and rooms, describe in one sentence your interpretation of β 1 (i.e., when nox goes up by X, price goes down by Y). When

nox goes up by 1 percent, the value of the house drops by 0.72 percent. This interpretation as an elasticity follows from the logging of both variables. d. True/False Explain. The estimate of β 2 you get in the regression that includes nox is definitely closer to the “true” value than the one that does not include nox. This is tricky. Of course, you have no way of knowing the true value, and it’s always possible that the true value is closer to the estimate from the “wrong” regression. We have a sample, and if for some reason the relationship in the sampled observations was weaker than in the entire population, than a upwardly biased estimate could be closer, at least in this one sample. 3. Using the BCUSE dataset GPA1 estimate the following model of college GPA (ColGPA) on high school GPA (hsGPA), ACT score, and the number of classes skipped per week. colGPA = β 0 + β 1 HSGPA + β 2 ACT + β 3 skipped + u a. What is the upper bound of the 90 percent confidence interval around β 1 ? The confidence interval is .26 to .57. I used the STATA option level(90) following the regression. b. Can you reject the hypothesis that β 1 =0 against the one-tailed alternative that it is greater than 0 at the 1% level? Yes, the p-value is less than 0.01 on the two- tailed test, so a one-tailed test would definitely reject. c. Can you reject the hypothesis that β 1 =1 against the two-sided alternative at the 5% level? Yes you can. I used the lincom command to do this in STATA, but you could just as easily note that the t-stat is (0.4118 – 1)/0.093 = -6.28 d. Use the lincom command in STATA to test the hypothesis that 10 extra ACT points would offset 1 skipped class (i.e., that 10* β 2 + β 3 =0). Can you reject the hypothesis against the two-tailed alternative? What is the p-value? No you cannot. The p-value is 0.542 which is much higher than the loosest standard of significance, at p equals 0.10. e. Although we did not discuss in lecture, this should be review: find the F-stat for the F-test of the hypothesis that the coefficient on ACT and skipped are jointly equal to zero ( β 2 = β 3 =0). The F-stat is 5.52. 4. For this question, use the BCUSE dataset intdef. You are interested in how the three- month T-bill rate relates to inflation (inf) and the federal deficit (def, as a % of GDP)), as well as whether a policy change in 1979 affects the interest rate. a. Do a static regression of the 3 month T-bill rate on inflation and the federal deficit in that year. What is the predicted interest rate when inflation is 4 percent and the deficit is 1 percent of GDP? About 4.7 percent. b. The dataset also contains lagged values for inflation and the deficit. Now, run the regression with inflation, inflation lagged one period, the deficit, the deficit lagged one period, and the post-1979 variable. Describe the sign and significance of the inflation and deficit variables and their lags. Does anything interesting come out of these additions? Inflation seems to have both a current and a lagged effect, whereas the effect of the deficit appears primarily through the lag. c. In the model with the lags, if inflation goes up permanently by 1 percentage point, how much would the interest rate go up? By about 0.73 percentage points.