Microsoft Word - Quiz3.docx
y James D. Wilson (University of San Francisco)
1. You fit a regression model to two different data sets at once (you’re a great multi-
tasker), and look at the quantile-quantile (qq) plots of the standardized residuals for
each fit. These are shown above.
(a) Comment (
iefly) on the model assumptions of the linear regression model for Fit
1 and Fit 2.
(b) What transformation (if any) would you apply to the data sets for each fit? Give
justification for your choices.
(c) Let’s now focus on the data associated with Fit 2. One way to formally test whether
or not the residuals are normal is to use the Kolmogorov-Smirnov (K-S) hypothesis
test. The null hypothesis of the test is that the e
ors are normally distributed.
You try (for hours) many transformations on the response variable associated with
Fit 2 and no matter how much you try, you keep rejecting the null hypothesis of the
K-S test. You decide to stop trying and live with the fact that you cannot get a
satisfactory result from the test. Describe specifically the effects that this result has
on the properties of the least squares estimators of your fitted regression model.
2. Now you fit three regression models on three different data sets and look at the
esiduals against the fitted values. These are shown above.
(a) Discuss each plot and what they suggest about the model assumptions for each fit.
(b) Describe how you would go about addressing the diagnoses you made in (a).
3. Let β1 be the LASSO coefficient estimates when λ = 1 and let β100 be the LASSO
coefficient estimates from the same data when λ = 100. What can we say about how the
L1-norm of β1 compares with that of β100?
4. Suppose Hank created an Etsy website to sell craft bowties. He collects data on
1000 people visiting his website and records 1200 different pieces of information for
each person (age, clicks, how did they end up on his site, how many bowties they looked
at, etc). Let X denote the 1000 × 1200 data matrix representing this data. He is interested
in understanding the relationship of the 1200 characteristics he recorded with the time
it takes him to sell a craft bowtie (Y). Answer the following questions:
(a) What can you say about (XTX)−1?
(b) Name 3 methods you could use for developing a predictive model that predicts the
time to sell a craft bowtie as a function of X.
(c) Say you fit a LASSO regression of Y on X and find that the LASSO estimator for age
is 0.50. What does this say about the effect of age on the time it takes to sell a craft
(d) Say you fit a Ridge regression of Y on X and find that the Ridge estimator for age is
0.25. What does this say about the effect of age on the time it takes to sell a craft