ECMT3150: Mid-Semester Exam (2017s1)
Lecturer: Simon Kwok
Time allowed: 1.5 hours
27 April 2017
The total score of this exam is 60 marks. Attempt all questions.
1. (Total: 12 marks) Short answer questions:
(a) (3 marks) What is strict stationarity?
(b) (3 marks) What is the di¤erence between unconditional and conditional forecast?
(c) (3 marks) What is spurious regression?
(d) (3 marks) What is volatility clustering?
2. (Total: 15 marks) Consider the time series model yt = 0:1 0:9yt
where "t wn(0; 2" = 0:09).
1 +"t ,
(a) (3 marks) Is the time series invertible? Explain.
(b) (3 marks) Is the time series stationary? Explain.
(c) (2 marks) Compute the unconditional mean of yt .
(d) (2 marks) Compute the unconditional variance of yt .
(e) (3 marks) Compute the autocorrelation function (ACF) of yt .
(f) (2 marks) Write down the partial autocorrelation (PACF) of yt .
3. (Total: 15 marks) Sam considered the time series model
yt
=
1 yt 1
at
=
t "t ;
2
t
= !+
where "t are iid N (0; 1). Let
to be estimated.
0
=(
1
+ at ;
2
1 at 1
1 ; !;
+
1;
2
1 t 1;
0
1)
be the parameter vector
(a) (3 marks) What is/are the requirement(s) on the elements of
ensure covariance stationarity of yt ?
0
to
(b) Using the time series data fyt g of length T = 500 and the joint
normal density as the likelihood function, Sam obtained the maximum likelihood estimate ^ = (0:6; 0:005; 0:1; 0:8)0 . Suppose that the
forecast origin is at time t = 500, with y500 = 0:2, y499 = 0:5 and
2
500 = 0:01.
i. (4 marks) Conditional on the information at the forecast origin,
what is the 1-step ahead forecast of 2501 ?
ii. (4 marks) Conditional on the information at the forecast origin,
what is the 1-step ahead forecast of y501 ? What is its standard
error?
(c) (4 marks) Sam was aware of empirical evidence (e.g., the presence of
outliers in the data) which suggested that the standardized errors "t
may not be normally distributed. Discuss the consequences of using
^ in part (b) for statistical inference on 0 .
4. (Total: 18 marks) The autocorrelation functions (ACFs) of six di¤erent
time series fut g; fvt g; fwt g; fxt g; fyt g and fzt g are shown in Figure 1.
(a) (12 marks, 2 or 0 for each of the six matches) Below is the list
of DGPs in terms of a generic time series fat g. The process f"t g is
iid with mean zero and variance 2" , where 0 < 2" < 1.
i.
ii.
iii.
iv.
v.
vi.
vii.
viii.
ix.
x.
xi.
at
at
at
at
at
at
at
at
= 0:1 + 0:7at 1 + "t :
= 0:05 + 0:9at 1 + "t :
= 0:9at 1 + "t :
= 0:3 + 1:7at 1 0:9at 2 + "t :
= 0:2 0:5at 1 + 0:2at 2 + "t :
= 0:1 + "t 0:9"t 1 :
= 0:1 + "t 0:8"t 1 + 0:4"t 2 :
= 0:1 + "t 0:5"t 1 0:3"t 2 0:2"t
0:4
at := (at at 1 )0:4 = "t :
at := at at 1 = "t :
at = 1:1at + "t :
3:
Pick one DGP from the above list that will yield the same ACF as
each of the six time series fut g; fvt g; fwt g; fxt g; fyt g and fzt g. Five
DGPs will be left unmatched. You would score zero for a particular time series if you picked more than one DGP or if you pick an
incorrect DGP for that time series.
To answer this question, copy the table on p.4 to your answer booklet
and …ll in the second column.
2
Figure 1: The autocorrelation functions of six time series.
3
Time series
fut g
fvt g
fwt g
fxt g
fyt g
fzt g
DGP (pick one from (i) to (xi))
Stationary? (yes/no)
(b) (6 marks) Determine whether each of the six time series is covariance stationary or not. Fill in the third column of the above table.
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