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University of Warwick

January 2021

Quantitative Methods for Finance


Instructions
This is an OPEN BOOK examination.
Time allowed: 2 hours
You should answer ALL questions in Section A and ANY TWO questions from Section B. Section A is worth a
maximum of 40 marks. Section B is worth a maximum of 60 marks.




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Section A – Answer ALL questions

Question 1

Consider the following results, of a linear regression estimated by ordinary least squares (OLS),
investigating the determinants of the excess returns of an individual company’s stock:

̂ =
0.531
(0.174)
+
0.732
(0.049)

−
0.249
(0.051)
+
0.601
(0.042)
(1)

2 = 0.451, . = 124

where is the excess return on a stock,
is the excess return on an aggregate stock market
index, is the small minus big and the high minus low risk factors of Fama and French.
The first coefficient is the estimate of the intercept and the coefficients next to the variable names
are the estimated respective slope parameters. The numbers in brackets beneath the estimated
coefficients are their standard errors.


a) Provide an interpretation of the estimated coefficient on the
variable. (2 marks)

b) What is the key assumption when calculating the variance of OLS estimators? Explain
carefully and show how the expression for the variance of the OLS estimator is derived.
(3 marks)

c) If decreases by 2% points how does the excess return on the individual stock change
on average, ceteris paribus? (1 mark)

d) Describe, providing mathematical expressions, the difference between the adjusted and
unadjusted R-squared. Does a low R-squared invalidate a regression? Explain carefully.
(4 marks)


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Question 2

Consider again the estimation results in equation (1) (from Question 1).

a) If you were to test the null hypothesis that the coefficient on
= 1 against the one-sided
alternative that it is less than 1, would you reject or fail to reject the null at any of the three
1%, 5% and 10% significance levels? Explain clearly how you reached the conclusion.
(Hint: the critical values of the t-student distribution −−1 at the 1%, 5% and 10%
significance level are equal to 2.327, 1.645, and 1.281, respectively) (2 marks)

b) Do the independent variables in equation (1) help or not help explain ? Construct a joint
test of this hypothesis and provide mathematical formulas where appropriate. (Hint: the
critical values of the F-student distribution ,−−1 with degrees of freedom q=3 and n-k-1
= 120 are 2.13, 2.68 and 3.78 at the 10%, 5% and 1% significance level respectively)
(4 marks)

c) Explain the general principle of joint hypothesis testing using an F-statistic. Provide
mathematical expressions where appropriate. (4 marks)


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Question 3

Consider again the estimation results from equation (1) (from Question 1)

a) Say we have omitted from the regression a risk factor measuring momentum. What are the
relevant facts we need to consider when determining if this omission causes the estimated
coefficient on the variable to be biased? What might be direction of the bias of the OLS
estimated coefficient for ? (4 marks)

b) What effect does the degree of correlation among the explanatory/independent variables in a
linear regression have on the standard errors of the estimated coefficients? Provide examples
and mathematical formulas where appropriate. (3 marks)

c) Construct the 90% confidence interval for coefficient on the term (three decimals are
enough). State in one sentence how you interpret this confidence interval. Provide mathematical
formula where appropriate. (Hint: the 90th critical value c of the t-student distribution −−1 is
equal to 1.64. (3 marks)




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Question 4

We want to estimate the following linear regression model by OLS:

= 0 + 11 + 22 + (2)

We are concerned about the potential problem of heteroscedasticity in the regression error.

a) What happens to OLS estimators and their variances if we introduce heteroscedasticity but
retain all other assumptions of the classical linear regression model? How would you test for
the presence of heteroscedasticity using White’s test? State carefully the null hypothesis of
the test and write down any regression equations you would need to estimate. (4 marks)


b) What are the potential sources of heteroscedasticity? Provide an example of at least one of
the sources. (2 marks)

c) Describe and define the concept of a weakly dependent time-series. Give an example and
explain why this property is important for time-series regression analysis. Explain carefully
and provide mathematical expressions where appropriate. (4 marks)







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Section B – Answer ANY TWO questions

Question 5

We seek to analyse the effect of loan to income ratios and age on the probability of defaulting on a
loan. Hence, we are interested in the following relationship:

̂ = 0 + 1_ + 2 + (3)

Where the P=1 if the individual defaulted and is zero otherwise, _ is the loan payments to
income ratio and is the age of an individual. We estimate equation (3) using a Logit model by
maximum likelihood estimation (MLE) on N=1629 observations, with the following results:

̂ = −1.96 + 4.832 × _ + 0.0139 × + (4)

Where the log-likelihood for the unrestricted model given by (4) is -980.64, for the restricted model
excluding is -996.21 and for the restricted model excluding both ly_ratio and age i.e. with just
a constant, is -1050.21.

a) Compute the partial effect at the average of the _ on the probability of defaulting
on the loan (Hint: the mean of _ and age are 0.291 and 41.529 respectively). Three
decimals are enough. Explain clearly how you reached the conclusion, comment on your
results and provide mathematical formulation where appropriate. (5 marks)

b) What are the main features of the Logit and Probit models which potentially make them
attractive alternatives to the Linear Probability Model (LPM) when estimating probabilities?
Explain providing an example. Use mathematical formulas where appropriate. (10 marks)

c) Suppose we want to know whether being male or female makes a difference to the
probability of defaulting on a loan. To answer this question, we introduce an additional
variable into the Logit regression, female, which takes the value of 1 if the individual is
female and zero otherwise. The estimated equation is:

̂ = −1.84 + 4.719 × _ + 0.0122 × + 1.23 × + (5)


What is the difference in the probability of defaulting on the loan between female and male
borrowers? Assume _ and age are 0.25 and 40 respectively. Two decimal places are
enough. Carefully explain you answer, provide an interpretation, and use mathematical
formulas where appropriate. (10 marks)
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d) Returning to the regression which does not include the female dummy, equation (4), test the
null hypothesis 0: ̂2 = 0 (i.e. the coefficient on age) against the two-sided
alternative 1: ̂2 ≠ 0. Explain clearly how you reached the conclusion and comment on
your results. Does the fit of the equation improve when you include the ly_ratio and age
variables? (Hint: the critical values of the =1
2 distribution are equal to 2.70, 3.84 and 6.63
at the 10%, 5% and 1% significance level, respectively.) (5 marks)





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Question 6

a) Is the following process stationary?

= −1 + ℎ . . . (0,
2) (6)

State why this is or is not the case, with specific reference to the definition of weak stationarity
regarding the mean, variance, and covariance. Use mathematical derivations where appropriate.
(6 marks)

b) We wish to test for the presence of a unit root in the natural logarithm of a time-series, . To
this end we estimate, using linear regression, the following equation (using 244 observations):

∆ =
0.0246
(3.52)
+
0.0003
(2.49)
−
0.0360
(−2.49)
−1 +
0.343
(5.66)
∆−1 (7)

Where we report t-statistics in the brackets, ∆ = − −1 and is a linear deterministic time
trend.

Outline the testing procedure, stage by stage, of how you would test for a unit root. Does contain
a unit root? Why would we include a time-trend in such a regression? Provide a detailed description
and use mathematical formulas where appropriate. (Hint: the 10% and 5% Dickey-Fuller critical
values, where a time trend is included, are -3.13 and -3.43 respectively) (10 marks)

c) Describe, using an empirical example or simulation experiment, what is meant by a spurious
regression. Carefully explain and provide mathematical formulation where appropriate. (7 marks)

d) How might the use and application of cointegration among two or more variables resolve the
spurious regression problem? Provide a detailed example and outline the basic concept using
mathematical formulation where appropriate. (7 marks)




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Question 7


a) What are the main issues to be considered when using pooled OLS to estimate a panel data
structure? Carefully state why FE and RE panel data methods might be preferable.
(8 marks)

b) Using an example, explain the idea and describe how you would apply the difference-in
difference approach to examine the effect of a policy change (e.g. a tax reform) or an event
(e.g. building a factory). Assume two cross sectional data sets are available, one at a point
in time before the policy change or event and one after. Each cross-sectional data set has
independent variables which either are unaffected or are affected by the policy change or
event. (8 marks)

c) Describe the fixed effects and the first-differenced estimators for panel data. What criteria
might you use to decide which of the two approaches, fixed effects and first differencing, to
use for estimation? Provide mathematical formulation where appropriate. (8 marks)


d) Say we estimate a panel model with just one independent variable, using both fixed effects
and random effects methods, so that we have two estimates of the same coefficient, ̂ =
0.533 and ̂ = 0.379 with associated standard errors of 0.159 and 0.130 respectively.
Construct a test to choose which of the two estimators you would use. Explain in detail the
implementation, provide an interpretation, and use mathematical formulation where
appropriate. (Hint: the critical values of the =1
2 distribution are equal to 2.70, 3.84 and
6.63 at the 10%, 5% and 1% significance level, respectively.) (6 marks)