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IB9X60 Quantitative Methods for Finance
创建日期：2024-04-24 15:32:41
所属分类：金融Finance
标签： IB9X60
Quantitative Methods for Finance

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IB9X60

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.

(Continued…/)


IB9X60

Page 2 of 9

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)


(Continued…/)








IB9X60

Page 3 of 9


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)


(Continued…/)



















IB9X60

Page 4 of 9

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)

IB9X60

Page 5 of 9

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)


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)
(Question 5 continued on next page)
IB9X60


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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