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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)!
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 R¯2
.
(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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