ECMT6006 2019S1 Mid-Semester Exam
18 April 2019
Instructions: This is a closed book exam. Please answer all questions. The
total mark is 100 and the breakdown is shown in square brackets. The duration
of the exam is 1 hour and 10 minutes, including 10 minutes reading time and 1
hour writing time. Please do NOT write your name but only your SID on the
answer booklet(s).
Problem 1. [30 pionts] Let Pt be the price of a stock at time t, and assume the
stock pays no dividend. Let Rt+1 be the single-period gross return from time t
to t+ 1.
(i) Given the information set Ft at time t, show that a (point) forecast of the
price Pt+1 can be derived from a (point) forecast of Rt+1. [5]
(ii) Given the information set Ft at time t, show that the conditional variance
of the price Pt+1 can be derived from the conditional variance of Rt+1. [5]
(iii) What is the log return from time t to t + 1? [2] Why is it also called the
continuously compounded return? [4]
(iv) What is the relationship between the log return and the simple net return?
[4] Why is it often convenient to use log returns? [4]
(v) Explain at least three limitations of using the normal distribution to model
the gross returns Rt. [6]
Problem 2. [35 points] Answer the following questions on the test for return
predictability using historical data.
(i) First, you simply run an OLS regression of your returns on a constant
rt = β + εt,
for t = 1, 2, . . . , T . What is the OLS estimate for β? [3] Write down the
test statistic that you would use to test the null hypothesis H0 : β = 0. [4]
How would you make the decision for this test? [3] What is the implication
if your null hypothesis is rejected? [3]
(ii) Next, you add the lagged value of return as another regressor,
rt = β0 + β1rt−1 + εt,
for t = 2, 3, . . . , T . What is the relationship between the true parameter
β1 in this regression and the first-order autocorrelation of {rt}? [5] What
is the relationship between the return autocorrelation and the return pre-
dictability? [2]
(iii) The table below presents the results of joint tests for autocorrelation in
the daily Nasdaq returns and 3-month T-bill interest rate returns. “LB”
denotes Ljung-Box test and “Robust” denotes the robust test.
Table 1: Joint tests for serial correlation of log returns
L = 5 L = 10 L = 20
95% CV 11.07 18.31 31.41
Test Stat LB Robust LB Robust LB Robust
S&P 500 11.80 6.31 20.54 16.98 31.26 30.52
T-Bill 207.42 32.58 222.23 38.31 376.69 87.69
Answer the following.
(a) Describe how the robust test is conducted. [5]
(b) What is the difference between a LB test and a robust test? [5]
(c) Interpret the results in the table. What is your conclusion on the
predictability in these two return series? [5]
Problem 3. [35 points] Consider the following AR(1)-ARCH(1) model for
index stock returns,
rt = φ0 + φ1rt−1 + εt, εt = σtνt
σ2t = ω + αε
t−1, νt|Ft−1 ∼ F (0, 1)
where ω > 0, α ≥ 0, σt > 0, Ft−1 denotes the information set up to time t− 1,
and F (0, 1) denotes some distribution with mean 0 and variance 1.
(i) Show that {εt} is a white noise process, and {ε2t} is an AR(1) process. [10]
(ii) What does “ARCH” stand for? [2] What is the key difference between this
model and a simple AR(1) model without ARCH specification? [3]
(iii) What empirical evidence shown in the index stock returns constrains the
use of a simple ARMA model and motivates the ARCH specification?
Please explain. [5]
(iv) ε2t is sometimes used as a proxy for σ
t . What is the relationship between
the processes {ε2t} and {σ2t }? [5]
(v) What is the optimal one-step ahead two-standard-deviation interval fore-
cast for the return using this model? [6] Explain how you obtain the
feasible version of this forecast. [4]