End-of-year Examinations, 2019
No electronic/communication devices are permitted. Students may take exam question paper away after the exam.
Mathematics and Statistics EXAMINATION
End-of-year Examinations, 2019
STAT317 / ECON 323 – 19S2 (C) Time Series Methods
STAT456 / ECON614 – 19S2 (C) Time Series and Stochastic Processes
Examination Duration: 120 minutes Exam Conditions:
Restricted Book exam: Approved materials only. Calculators with a ‘UC’ sticker approved. Materials Permitted in the Exam Venue: Restricted Book exam materials.
One A4, double sided, handwritten page of notes.
Materials to be Supplied to Students:
1 x Standard 16-page UC answer book
Instructions to Students:
Use black or blue ink only (not pencil).
Students in STAT456 and ECON614 have to work on ALL SIX questions.
Students in STAT317 and ECON323 have to CHOOSE FIVE out of SIX questions.
Show ALL working.
If you use additional paper this must be tied within the exam booklet and remember to write your name and student number on it.
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End-of-year Examinations, 2019
Questions Start on Page 3
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STAT317/456 ECON323/614 Students in STAT456 and ECON614 have to work on ALL 6 questions.
Students in STAT317 and ECON323 have to CHOOSE 5 out of 6 questions. Only 5
questions will be marked.
Each question is worth 16 marks.
Q.1 Basic Concepts [16 marks]
(a) Give the definition of weak white noise. [3 marks]
(b) Assume we apply the Ljung-Box test to a time series Xt, t = 1,2,…,n and
get the p-value of p = 0.043. How would you interpret this result? [3 marks]
(c) In a time series decomposition a time series is usually separated into three
components. Name these three components. [3 marks]
(d) What is a trend in a time series? [2 marks]
(e) Give the definition of the autocovariance function of an arbitrary time series Xt, when Xt is not necessarily stationary. [1 mark]
(f) Defineanestimatoroftheautocorrelationfunctiongivenasamplex1,x2,…,xn when Xt is stationary. If your definition involves other estimators then also de- fine these. It must be clear in the end how the estimator of the autocorrelation function is computed from the sample. [4 marks]
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Q.2 Random Walk and Stationarity [16 marks]
(a) Random Walk
i. Give the definition for a random walk Xt, t = 1,2,3,…,n with drift δ,
volatility σ, and initial value X0 = 0.
ii. Give an example for an application of the random walk model.
iii. Show that the estimator for the drift
δ= 1 (Xn−X1)
i. What are the conditions for a time series Xt, t = 1,2,3,…,n to be weakly stationary? [3 marks]
ii. Is a random walk stationary? Give a reason for your answer. [2 marks]
iii. Is the AR(1) process Xt = 2 + 0.5Xt−1 stationary? Give a reason for your
answer. [2 marks]
iv. Would you assume that the GDP of shown in the plot below is a stationary time series? Give a reason for your answer. [2 marks]
iv. Give the formula for an unbiased estimator of the squared volatility σ2.
[4 marks] [1 mark]
[1 mark] [1 mark]
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Q.3 Time Series Regression [16 marks]
We consider the multiple linear regression model
Xt =β0 +β1Zt,1 +β2Zt,2,…,βqZt,q +εt.
(a) Give the four Gauss-Markov assumptions for multiple linear regression.
Remember: These are the conditions that ensure that OLS is the best linear unbiased estimators. [4 marks]
(b) Which of the Gauss-Markov assumptions are typically violated for time series data? [2 marks]
(c) What are the two major drawbacks for the OLS estimator in time series re- gression with violated Gauss-Markov assumptions? [2 marks]
(d) Assume we observe a time series Xt, t = 1,2,3,…,n that appears to have an exponential trend. Explain how we can use linear regression to estimate this nonlinear trend? Write down the corresponding regression model. [4 marks]
(e) Write down a regression model that accounts for a trend and for seasonality that repeats after four observations. Avoid multicollinearity in the model. Show how all four seasonal components can be estimated with your model.
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Q.4 Periodogram [16 marks]
(a) A monthly time series for t = 1,…,n is to be simulated from the following
2πt π Xt=10sin 12+4 +Wt
where Wt ∼ W N (0, σW2 ). What are the following quantities:
i. amplitude; ii. frequency;
iii. period of the cycle; iv. phase in radians; and
v. phase in time units.
[1 mark] [1 mark] [1 mark] [1 mark]
(b) A general regression model for fitting a sine wave at a fixed frequency f is given by:
Xt =Acos(2πft+φ)+Wt which has a non-linear parameter φ. Use the relation:
cos(A ± B) = cos(A) cos(B) ∓ sin(A) sin(B)
to reformulate the above regression model as linear in the unknown coefficients.
(c) Consider the following periodogram, identify the period of possible cycles in
the underlying time series.
Series: x Raw Periodogram
0.0 0.1 0.2
frequency bandwidth = 0.00241
(d) The Nyquist frequency is the highest frequency considered in the periodogram. How does a sine wave behave at the Nyquist frequency? [2 marks]
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0 20 40 60 80
STAT317/456 ECON323/614 (e) In the following diagram the time sefreiqe=s1i/1s6taondb17e/1o6 bserved at times t = 1, . . . , 16.
● ● ●
−1.0 −0.5 0.0 0.5 1.0
0 5 10 15
Use the diagram to explain the concepts of aliasing. [3 marks]
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Q.5 AR(1) and MA(1) Models [16 marks]
(a) An AR(1) model can be written as
Xt = φ0 + φ1Xt−1 + Wt.
i. How is this model related to the random walk?
ii. Under the assumption of stationarity, derive the mean of Xt.
iii. Under the assumption of stationarity, derive the variance of Xt. [2 marks]
iv. Under the assumption of stationarity, show that the autocovariance func- tion of Xt at lags |h| = 0,1,… is given by:
σW2 |h| γX(h)=Cov(Xt,Xt−h)=1−φ2φ1 .
[3 marks] v. State a condition on the values of φ1 that will make the AR(1) a causal
(b) An MA(1) model can be written as
Xt = μ + Wt + θWt−1.
i. Derive the mean and variance of Xt.
ii. Derive the autocovariance of Xt.
iii. State a condition that ensures the MA(1) model is invertible and explain why this property is useful. [2 marks]
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Q.6 Model Selection [16 marks]
(a) Explain the key features of the dependence observed in the following plot of a
0 100 200
300 400 500
0 5 10 15 20
5 10 15 20
(b) Explain how the features mentioned in part (a) are shown in the sample auto- correlation and partial autocorrelation functions below. [2 marks]
(c) Use these plots to identify whether a suitable model could be an AR(p), MA(q) or mixed ARMA(p, q). Explain your choice and suggest the order of the model. [2 marks]
(d) Write down the backshift (or characteristic) polynomials for the ARMA(p,q) model:
Xt −φ1Xt−1 −···−φpXt−p =Wt +θ1Wt−1 +···+θqWt−q
(e) What are the conditions for invertibility and causal stationarity for an ARMA process? [2 marks]
(f) What condition is needed to avoid parameter redundancy for an ARMA pro- cess.
[1 mark] 9 of 10 pages TURN OVER
−0.5 0.0 0.5 1.0 Xt
−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4
−4 −2 0 2 4
(g) Identify the order of the following ARMA(p,q) model and check it satisfies these three conditions.
Xt = 0.7Xt−1 − 0.1Xt−2 + Wt + 0.2Wt−1
If the model has parameter redundancy then rewrite the model without the
You may find it useful to know that the roots of the quadratic az2 + bz + c are
given by z = −b±b2 −4ac. [5 marks] 2a
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