ECMT6006 2019S1 Mid-Semester Exam 2019
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ECMT6006 2019s1 Mid-semester Exam
18 April 2019
Instructions: This is a closed book exam. please answer all questions. The total mark is 1oo and the breakdown is shown in square brackets. The duration of the exam is 1 hour and 1o minutes, including 1o 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. [3opionts] 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 gTOSS return from time t to t + 1.
(i) Given the information set ：t 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 ：t 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 gTOSS 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 term
Tt = β + Et ,
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 = o. [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,
Tt = β0 + β1 Tt- 1 + Et ,
for t = 2, 3, . . . , T. what is the relationship between the true parameter β1 in this regression and the irst-order autocoTTelation of {Tt }？ [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 = 1O |
L = 2O |
|||
|
95% CV |
11.O7 |
18.31 |
31.41 |
|||
|
Test Stat |
LB |
Robust |
LB |
Robust |
LB |
Robust |
|
S&P 5OO |
11.8O |
6.31 |
2O.54 |
16.98 |
31.26 |
3O.52 |
|
T-Bill |
2O7.42 |
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 diference 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,
Tt = φ0 + φ1 Tt- 1 + Et , Et = σt Ⅴt
σt(2) = w + aEt(2)- 1 , Ⅴt |Ft- 1 “ F (O, 1)
where w 持 O, a > O, σt > O, Ft- 1 denotes the information set up to time t - 1, and F (O, 1) denotes some distribution with mean O and variance 1.
(i) show that {Et } is a white noise process, and {Et(2)} is an AR(1) process. [1O]
(ii) what does “ARCH” stand for？ [2] what is the key diference between this model and a simple AR(1) model without ARCH speciication？ [3]
(iii) what empirical evidence shown in the index stock returns constrains the use of a simple ARMA model and motivates the ARCH speciication？ please explain. [5]
(iv) Et(2) is sometimes used as a proxy for σt(2). what is the relationship between the processes {Et(2)} and {σt(2)}？ [5]
(v) what is the optimal one-step ahead two-standard-deviation inteTval fore- cast for the return using this model？ [6] Explain how you obtain the feasible version of this forecast. [4]
2023-08-12