ECMT3150: In-Semester Exam (2022s1)
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ECMT3150: In-Semester Exam (2022s1)
Time allowed: 2 hours
6 April 2022
The total score of this exam is 70 marks. Attempt all questions.
1. (Total: 12 marks) Discuss the advantage and disadvantage for each of the following model selection methods.
(a) (4 marks) Autocorrelogram
(b) (4 marks) Information criteria
(c) (4 marks) Regression R2
2. (Total: 2 marks) Carol has been analysing the stock return series. Carol identiÖed a time series model that can predict future returns using past squared returns pretty well. Which one of the following statements must be true?
(a) The stock return series is a martingale di§erence sequence
(b) The stock return series is not a martingale di§erence sequence
(c) Not sure
3. (Total: 2 marks) Carol has been analysing the stock return series. Carol identiÖed a time series model that can predict future returns using past squared returns pretty well. Which one of the following statements must be true?
(a) The stock return series is an iid sequence
(b) The stock return series is not an iid sequence
(c) Not sure
4. (Total: 2 marks) Robert decides to estimate an AR(9) model on monthly ináation rate series spanning 10 years. How many degrees of freedom does the Ötted model have?
5. (Total: 2 marks) Compute y^0 (3) for the following model, where et { wn(0; 0:01), i.e., a
white noise process with mean zero and variance 0.01. It is given that y0 = 1. yt = 1 + 0:5yt — 1 + et:
6. (Total: 20 marks) Let H denote the dummy variable for hospitalisation due to COVID-19. Let V denote the number of vaccinations an individual received.
|
|
H = 0 |
H = 1 |
|
V = 0 |
x |
0:1 |
|
V = 1 |
0:15 |
0:03 |
|
V = 2 |
0:25 |
0:02 |
|
V = 3 |
0:34 |
0:01 |
(a) (2 marks) Find x.
(b) (2 marks) Find the marginal distribution of V by specifying its probability mass func-
tion.
(c) (2 marks) What is the probability that an individual is hospitalised due to COVID-19?
(d) (3 marks) Given that John is hospitalised due to COVID-19, what is the probability that he had three vaccinations?
(e) (4 marks) Compute E(V IH = 1) and E(V IH = 0).
(f) (3 marks) Using (c) and the law of iterated expectations, compute E(V). (g) (4 marks) Compute Var(V IH = 1) and Var(V IH = 0).
7. (Total: 30 marks) Mimi, an ECMT3150 student, studies the following MA(1) process
yt = "t + 0:9"t — 1 ,
where "t { iid N(0; 0:09) (normal distribution with mean 0 and variance 0:09).
(a) (3 marks) Is (yt} a martingale di§erence sequence? Justify your answer with a proof. (b) (3 marks) Is (yt} stationary? Why or why not?
(c) (3 marks) Is (yt} invertible? Why or why not?
(d) (3 marks) Compute the unconditional mean and variance of (yt}. (e) (4 marks) Derive the autocorrelation function (ACF) of (yt}.
(f) (4 marks) Plot the ACF and partial autocorrelation function (PACF) of (yt}. (g) (4 marks) Derive the AR representation of (yt}. Show your steps.
(h) Little Bob studies the following AR(1) model instead:
zt = 0:9zt — 1 + "t ,
where "t { iid N(0; 0:09).
i. (2 marks) Plot the ACF and PACF of (zt}.
ii. (4 marks) Compare and discuss how a negative shock today will have an impact on the future values of yt and zt .
2023-04-06