1
General Instructions
Please answer ALL questions in All Parts. Each part carries equal weight for the
final mark.
Question 1
[25 marks]
1. Define the following terms
• Stochastic process
• Weak stationarity
• Autocovariance
• Random walk
• Granger causality
2. Consider the following stochastic process
yt = 0.4 + 0.3 yt-1 + et-1
where et-1 is white noise with variance equal to 1. Find E[yt], E[yt|yt-1], Var(yt),
Corr(yt , yt+1), and Corr(yt , yt+2). Document all your calculations.
[25 marks]
3. Briefly discuss why vector autoregressions (VAR) have become popular for applications
in Finance and Economics.
[25 marks]
4. Give an example from Finance, where an error correction model specifies the data
generating process for a set of two (or more) variables. Explain, with reference to the error
correction term, why such a model is expected to exist.
[25 marks]
Question 2
Consider the simple regression model
(1) Δ�! = � + � Δ�! + �!
where �! is the natural logarithm spot price of the Apple stock, �! is the natural
logarithm of the forward price, and �! is a white noise error term.
2
1. Explain what is meant by Best Linear Unbiased Estimator (BLUE). Which assumptions
about the error term �! must hold, in order for ordinary least squares (OLS) to be
BLUE?
[25 marks]
2. Using formulas where appropriate, explain why OLS estimates of Eq.1 are of interest
for financial risk management. Why is Eq.1 specified in first differences rather than
the levels of the variables?
[30 marks]
3. Table 1 reports ordinary least squares estimates of Eq. 1. Describe the testing
procedure for a t-test for statistical significance of the OLS coefficient estimates.
Clearly describe the null and alternative hypotheses and the testing procedure. Fill in
the missing t-statistics and p-values in Table 1 and interpret the test results.
[30 marks]
Table 1 Reports OLS estimates of Eq 1
4. In light of your answers to question 2 and the results in Table 1, interpret the estimated
coefficient β+.
[15 marks]
Question 3
Assume that you have a sample of size N=1,000 generated from the AR(2) model
(2) �! = 0.2 + 0.9 �!”# + 0.01 �!”$ + �!
where �!~���(0, �$).
1. Given the sample data, explain how you can test if the AR(2) model of Eq. 2 has a
unit root or not. Clearly describe the procedure, the variables used, and the test
statistic and its hypotheses.
[25 marks]
Quick|Group statistics|Descriptive statistics|Common sample…
The command will prompt a dialogue window into which you can enter the
variables names for which you want to compute descriptive statistics.
Answer the following questions:
1. What is the average growth rate of spot and future price?
2. What are the standard deviations of the growth rates?
3. What is the correlation between spot and futures growth rates? [Hint:
Go to Quick|Group statistics|Correlations … ]
1.2 OLS estimates and hypothesis tests
To obtain the optimal hedge ratio, we need to estimate Eq. 1 using OLS. To
run an OLS regression in Eviews go to
Quick|Estimate equation…
and specify
spot_return c futures_return
and then press return. The regression output is shown in Table 1
Dependent Variable: SPOT_RETURN
Method: Least Squares
Sample (adjusted): 1999M7 2019M12
Included observations: 246 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.363302 0.444369
FUTURES_RETURN 0.123860 0.133790
R-squared 0.013422 Mean dependent var 0.004168
Adjusted R-squared -0.002238 S.D. dependent var 0.043333
S.E. of regression 0.460232 Akaike info criterion 7.916378
Sum squared resid 51.6860 Schwarz criterion 7.887879
Log likelihood -175.7145 Hannan-Quinn criter. 7.904903
F-statistic 0.857070 Durbin-Watson stat 2.969363
Answer the following questions:
3
3
2. What test result would you expect? Does your expectation change, if you only had a
sample of size T=30.
[15 marks]
3. Assume that you have fitted an AR(1) model to the data generated by Eq. 1. The
estimated model is
(3) �0! = 0.19 + 0.92�!”#
Discuss why you might prefer the estimated model of Eq. 3 over estimates of an
AR(2) model if you wanted to use the estimated models for forecasting. How does
your answer change, if you knew that your data has been generated by Eq. 2?
[20 marks]
4. Assume that the observed value of the time-series in period t is given by �! = 1.65
Using equations were appropriate, explain can how you obtain a 1-period ahead
forecast �!$# given the estimated model of Eq. 3? What is the forecasted value?
[25 marks]
5. How does your answer change, if you wanted to make a h-period ahead forecast with
h=2 and, respectively, h=100?
[15 marks]