Question 1: Theory (20 points)
(a) (7 points) Suppose that you estimate a linear regression model using OLS twice, using
two separates samples drawn from the same population. You find that the coefficient
estimates that you obtain using the two samples differ. Is this surprising? Please explain
your reasoning.
(b) (4 points) What is the difference between the population regression function and the
sample regression function?
(c) (4 points) What is meant by the term auto-correlation?
(d) (5 points) What is meant by the term heteroskedasticity?
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Question 2: Applications (60 points) Suppose that you have just started your first job
as a junior investment analyst at a hedge fund. You have the data on average annual returns
(that is, yield) and risk levels for the following five securities that your fund is holding (e.g.
corporate bonds):
Security # Return (in %) Risk
1 2.5753 8
2 4.3934 12
3 14.2423 43
4 10.2303 22
5 3.3583 1
Table 1: Risk-return characteristics of five portfolio securities.
The risk index is a number between 0 and 100 that is increasing with security’s market
risk (e.g. risk index of 0 represents a risk-free security such as a government bond).
Suppose that you wish to use these data to estimate the following regression model:
Yi = β1 + β2Xi2 + ui
, (1)
for i = 1, …, 5, where Yi
is return (in percent)and Xi2 is the risk index for security i.
(a) (10 points) Use vector OLS to obtain estimate of the coefficients vector βˆ. Show your
work. You may use the fact that here,
(X
0X)
−1 =
0.47836 −0.01618
−0.01618 0.00094
. (2)
(b) (5 points) Using your estimate βˆ, what is the expected annual return to a risk-free
security?
(c) (5 points) Using your estimate βˆ, what is the expected annual return to a bond with a
risk index of Xi2 = 50?
(d) (6 points) On average, all else equal, by how much would the security return change if
the risk index were to increase by 1 unit?
(e) (7 points) In class we saw that the estimate of the variance-covariance matrix of βˆ can
be found using ˆcov(βˆ) = ˆσ
2
(X0X)
−1
, where ˆσ
2 =
uˆ
0uˆ
n−k
. Use these formulas to find the
variance-covariance matrix of βˆ.
(f) (7 points) With 95% confidence, by at most how much would the return change if you
were to increase the risk index by 1 unit?
(g) (7 points) You’re considering purchasing a bond that has a yield of 12%. No risk index
is available for this bond. Based on your estimates, what risk index can you expect to
be associated with such bond?
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(h) (6 points) Suppose that you’re considering adding a new bond to your portfolio. This
bond has a return of 8% and has a risk index of 8. On the basis of this return and risk
index only, it this a good investment proposition?
(i) (7 points) Suppose that you wish to test a hypothesis that risk has no effect on security’s
return against an alternative that it has a positive effect. Formulate the appropriate
hypothesis and using your estimates test it at 5% level of significance. Show all steps,
and clearly indicate your final decision whether to reject this H0.
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Question 3: GRETL (20 points) This question requires the use of GRETL. Download
the dataset titled “Securities returns data” from the “Data” folder of the course web-site and
import the dataset into GRETL. The dataset contains real financial data for twelve S&P
500 securities collected on October 3, 2014. The data are as follows:
sName sTicker sReturn sRisk sADV
Hershey Co HSY 17.42% 74 0.10
Johnson & Johnson JNJ 11.03% 64 0.64
Pepsico PEP 8.39% 63 0.38
M&T Bank Corp MTB 14.08% 66 0.07
Apple Inc AAPL 26.49% 105 6.48
Google GOOG 16.97% 75 0.87
Citi C 1.86% 97 0.86
McDonald’s Corp MCD 10.17% 55 0.62
Microsoft Corp MFT 12.38% 93 1.63
Procter & Gamble PG 7.64% 59 0.57
Goldman Sachs GS -0.39% 82 0.48
JP Morgan Chase JPM 5.90% 86 0.77
Table 2: Returns, risk and liquidity measures for twelve S&P 500 securities.
Variable sReturn is the average annual security return (i.e. yield) during the past five
years (in percent). sRisk is the corresponding risk index (estimated by NASDAQ) and sADV
is the average daily volume (in billions of U.S. dollars), that is often used to measure market
liquidity, that is, the ease with which one can find a buyer or a seller of that security. Write
a GRETL script that accomplishes the following:
(a) (5 points) Regress (using OLS) security returns on a constant, risk index and average
daily volume. Briefly comment on your results: do risk and liquidity affect returns? How
good is the overall model fit? (Please attach comments to GRETL output).
(b) (7 points) On the basis of your estimates, which of the securities in this sample represented the best investment opportunity relative to their liquidity and risk?
(c) (5 points) Regress (using OLS) security returns on risk and average daily volume but
without a constant. How did your results change? Recall that removing the constant
from a regression model amounts to assuming that β1 = 0. Do you think that it is safe
to assume so here? (Please attach comments to GRETL output).
(d) (3 points) Generate a scatter plot of residuals from model above (that is, model without
the constant) against the risk index. Is there anything unusual that you notice with respect to the behavior of these residuals?
Sample Solution
