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日期：2021-11-23 10:05

Empirical Methods – Problem Set 1

Professor Martin Lettau

Due November 21, 2021, 6:00pm PST, to be submitted via bCourses

Note: Use basic Python commands (e.g. matrix multiplication) for all questions in this

problem set. Do NOT use built-in packages (e.g. statsmodels or pandas regression commands)!

Download the data file (link). The spreadsheet includes monthly returns for the CRSP-VW

index (Rm), Proctor & Gamble (RP G), Unilever (RUL) and a Consumer Goods index (RHH). In

the lecture, we discussed regressions for Proctor & Gamble using the finite sample results under

normality. The problem set asks you to do the same analysis for Unilever returns.

1. Run the regression

RULt ? Rf t = α + β(Rmt ? Rf t) + et

(a) Report the coefficient estimates, the R2 and the adjusted Rˉ2

.

(b) Construct a scatterplot of UL returns (on the y-axis) and CRSP-VW returns (on the

x-axis) as well as the regression line.

(c) Compute the variance-covariance matrix of the OLS coefficients under the assumption

of homoskedasticity.

(d) Use t-tests to test the null hypothesis that each regression coefficient is individually

equal to 0.

(e) Assess whether there is significant evidence for heteroskedasticity.

(f) Compute standard errors and the 90%, 95% and 99% confidence intervals under the

assumption of homoskedasticity.

(g) Compute the variance-covariance matrix of the OLS coefficients under the assumption

of heteroskedasticity.

(h) Compute standard errors and the 90%, 95% and 99% confidence intervals using the

White variance-covariance matrix.

(i) Use t-tests to test the null hypothesis that each regression coefficient is individually

equal to 0 under the assumption of heteroskedasticity.

(j) Compute the AIC, BIC and Hannah-Quinn ICs.

(k) Compute the Durbin-Watson and Breusch-Godfrey test statistics. What do these tests

tell you?

(l) Use QQ plots and formal tests to check whether the errors normally distributed.

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(m) Run rolling regressions with 60-month windows and plot the β coefficients along with

their 95% confidence intervals. What do you learn from these regressions?

2. Run the regression

RULt ? Rf t = α + β(Rmt ? Rf t) + γ(RHHt ? Rf t) + et

and repeat the analysis in Q1.

3. Based on the results in Q1 and Q2, evaluate and compare the two regressions above. What

is you preferred model? Why?

4. You work for a hedge fund and your boss asks you what the current market βs are for

PG and UL. Using the evidence in the lecture notes and this problem set to give him/her

a comprehensive and well-reasoned answer keeping in mind the your boss is extremely

nitpicky and does not accept “opinions” without facts.

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