ECOM037 Quantitative Techniques 2020
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Main Examination Period 2020 – January - Semester A
ECOM037 Quantitative Techniques
Question 1
Let the two variable regression model Yt 
1 
2Xt 
ut be the model of interest.
a) Derive the formula for the ordinary least squares estimators for 
1 and 
2 .
[10 marks]
b) What is the interpretation of the parameters 
1 and 
2 ?
[10 marks]
c) Define the concept of the coefficient of determination R2 and interpret it. Hint: To interpret the R2 , use its two extreme values and accompany your answer with the corresponding figures.
[10 marks]
d) Let Z be a test statistic and ζ be the calculated test statistic from the sample. What is the p-value of the test and how would you compute it? Explain your answer diagrammatically.
[10 marks]
e) Use the concepts of “critical value” and “p-value” to describe how you would test the null hypothesis H0 : 
2 
0 versus the alternative hypothesis H1 : 
2 
0 by means of a t-test.
[10 marks]
Question 2
a) Define the concept of Fama-French (1993) alpha.
[10 marks]
b) How would you test whether a mutual fund delivers a positive Fama-French
[10 marks]
c) Assume that the returns Rt of stock ABC are described by the following regression model:
Rt 
0 
1X1t 
2X2t 
ut .
How would you test whether an announcement about unemployment affects the returns of the stock? Hint: Reformulate the regression model by using dummy variables.
[10 marks]
d) Let the regression model Rt 
0 
1X1t 
2X2t 
ut , be the model of interest, where ut 
ut
1 
t and 
t is an independent and identically distributed (i.i.d.) error term. Describe how you would correct for autocorrelation in the disturbance term by transforming the regression model.
[10 marks]
e) Comment on the following sentence: “The presence of autocorrelation in the disturbance term of the model shown in Question d) above may give rise to Type II errors in the case where hypothesis testing is performed by means of a standard t-test where the standard errors of the ordinary least squares are used.”
Question 3
a) Let a sample consist of T observations. Consider the k-variables multiple regression model Y 
X
u , where Y and u are (T 
1) vectors, X is a (T 
k) matrix and 
is a (k 
1) vector. Prove that the vector of the ordinary least squares estimators 
ˆ is given by 
ˆ 
(X'X)
1X'Y .
[10 marks]
b) Consider the Cobb-Douglas function Yi 
1X
X
eui where Yi is a firm’s i output, and X2i, X3i are its capital and labour inputs, respectively, 
1 , 
2 , 
3 are constant parameters, e is the Euler’s number (e=2.71828), and ui is the disturbance term. How can you estimate 
1 , 
2 , 
3 by ordinary least squares and describe what the interpretation of these three parameters will be?
[10 marks]
c) In question a) above, use the notions of restricted and unrestricted models and describe in detail how you would test the null hypothesis H0 : 
2 
3 
1 versus the alternative Ha : 
2 
3 
1 .
[10 marks]
d) A trader trades ABC stock. She wishes to hedge its market risk by using a futures contract written on the ABC stock. Derive the optimal hedge ratio of the hedging strategy.
[10 marks]
e) In question d) above, discuss how the trader can estimate the optimal hedge ratio by ordinary least squares.
[10 marks]
2022-01-13