End-of-year Examinations, 2018
STAT317/456-18S2 (C) / ECON323/614-18S2 (C)
Family Name First Name Student Number Venue
Seat Number
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No electronic/communication devices are permitted. Students may take exam question paper away after the exam.
Mathematics and Statistics EXAMINATION
End-of-year Examinations, 2018
STAT317-18S2 (C) Time Series Methods
STAT456-18S2 (C) Time Series and Stochastic Processes ECON323-18S2 (C) Time Series Methods
ECON614-18S2 (C) Time Series and Stochastic Processes
Examination Duration: 120 minutes
Exam Conditions:
Restricted Book exam: Approved materials only. Any scientific/graphics/basic calculator is permitted. Materials Permitted in the Exam Venue:
One A4 double sided, handwritten page of notes and formulas
Materials to be Supplied to Students:
1 x Write-on question paper/answer book
Instructions to Students:
This is a closed book examination
Use black or blue ink only (not pencil)
Attempt ALL 6 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.
For Examiner Use Only
Question Mark Q1
Q2 Q3 Q4 Q5 Q6
T otal
Page 1 of 24
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Q.1 Basic Concepts [16 marks]
(a) Give the definition of strong white noise. [2 marks]
(b) Name a statistical test that can give evidence for strong white noise. [1 mark]
(c) Explain what a cycle of length s in a time series Xt, t = 1,2,3,…,n is. [4 marks]
3 STAT317/456 ECON323/614
PLEASE TURNOVER
STAT317/456 ECON323/614 4
(d) Explain the growth rate of a time series Xt, t = 1,2,3,…,n and give a formula for how it is computed. [4 marks]
(e) Give the definition of the autocorrelation function of an arbitrary time series Xt, when Xt is not necessarily stationary. [1 marks]
5 STAT317/456 ECON323/614 (f) Defineanestimatoroftheautocorrelationfunctiongivenasamplex1,x2,…,xn
when Xt is stationary.
If your definition involves other estimators then also explain these. It must be clear in the end how the estimator of the autocorrelation function is computed from the sample. [4 marks]
PLEASE TURNOVER
STAT317/456 ECON323/614 6
Q.2 Random Walk [16 marks]
(a) Give the definition for a random walk Xt, t = 1, 2, 3, . . . , n with drift δ, volatil- ity σ, and initial value X0 = 0. [4 marks]
(b) Derive from the definition of the random walk the expectation E(Xt) and the variance V ar(Xt). [4 marks]
7 STAT317/456 ECON323/614 (c) Is a random walk stationary if δ = 0? Give reasons for your answer. [4 marks]
(d) Give at least one arguments for and one argument against the suggestion to model the log-exchange rate of NZD and USD by a random walk. [4 marks]
PLEASE TURNOVER
STAT317/456 ECON323/614 8
Q.3 Autocovariance [16 marks]
(a) Give the formula for the autocovariance function of a white noise process with variance σ. [4 marks]
(b) Explain in one or two sentences why the function takes these values. [4 marks]
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(c) The following plot shows an estimated autocovariance function for a white noise process using the estimator for stationary time series.
Explain why the estimation errors become smaller at the right hand side of the plot. Use the formula of the estimator in your argument. [4 marks]
PLEASE TURNOVER
STAT317/456 ECON323/614 10
(d) What can be learned from the autocovariance function of a time series and how is that different from the autocorrelation function? [4 marks]
Q.4 ARMA Models [16 marks]
(a) Write down the backshift (or characteristic) polynomials for the ARMA(p,q) model:
Xt −φ1Xt−1 −···−φpXt−p =εt +θ1εt−1 +···+θqεt−q
[2 marks]
(b) What are the conditions for invertibility and causal stationarity for an ARMA process? [2 marks]
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(c) What condition is needed to avoid parameter redundancy for an ARMA process.[1 mark]
PLEASE TURNOVER
STAT317/456 ECON323/614 12
(d) Identify the order of the following ARMA(p, q) models and check they satisfy these three conditions.
You may find it useful to know that the roots of the quadratic az2 + bz + c are √
given by z = −b± b2 −4ac. 2a
i. Xt = 1.1Xt−1 − 0.3Xt−2 + εt + 0.4εt−1 [4 marks]
ii. Xt = 0.9Xt−1 − 0.2Xt−2 + εt − εt−1 [4 marks]
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(e) The following ARMA(2,1) is overparameterised, write down a simplified version of this model:
Xt − 1.3Xt−1 + 0.4Xt−2 = εt − 0.5εt−1.
[3 marks]
PLEASE TURNOVER
STAT317/456 ECON323/614 14
Q.5 Model Selection [16 marks]
(a) Explain the key features of the dependence observed in the following plot of a
times series.
[2 marks]
0 200 400
600 800 1000
Time, t
Xt
−6 −4 −2 0 2 4 6
15 STAT317/456 ECON323/614
(b) Explain how the features mentioned in part (a) are shown in the sample auto- correlation and partial autocorrelation functions below. [2 marks]
Autocorrelation
0 5 10 15 20
Lag
Partial Autocorrelation
5 10 15 20
Lag
(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. [3 marks]
PLEASE TURNOVER
ACF
0.0 0.2 0.4 0.6 0.8 1.0
Partial ACF
−0.4 −0.2 0.0 0.2 0.4 0.6 0.8
STAT317/456 ECON323/614 16
(d) Consider the following pairs of autocorrelation and partial autocorrelation plots
of from pure AR and MA models only.
Explain the key features of the dependence shown in these plots, identify whether it is an AR or MA and it’s order. Also identify whether the coef- ficients will be positive or negative at each lag.
i.
[3 marks]
Partial Autocorrelation
5 10 15 20
Lag
Autocorrelation
0 5 10 15 20
Lag
ACF
0.0 0.2 0.4 0.6 0.8 1.0
Partial ACF
−0.4 −0.2 0.0 0.2 0.4 0.6 0.8
ii.
Autocorrelation
17
STAT317/456 ECON323/614 [3 marks]
Partial Autocorrelation
5 10 15 20
Lag
0 5 10 15 20
Lag
iii. Explain how a model fit statistic like the Akaike Information Criterion (AIC) could be used to determine the model order. [3 marks]
PLEASE TURNOVER
ACF
0.0 0.2 0.4 0.6 0.8 1.0
Partial ACF
−0.4 −0.2 0.0 0.2 0.4 0.6 0.8
STAT317/456 ECON323/614 18
Q.6 AR(1) and MA(1) Models [16 marks]
(a) A zero mean AR(1) model can be written as Xt = φXt−1 + εt.
i. Under the assumption of stationarity, derive the variance of Xt. [2 marks]
ii. Under the assumption of stationarity, derive the autocovariance function of Xt. [3 marks]
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iii. Calculate the autocovariance for an AR(1) with φ = 0.5 for lags 0, 1 and 2. [3 marks]
iv. State a condition on the values of φ that will make the AR(1) a causal stationary model. [2 marks]
PLEASE TURNOVER
STAT317/456 ECON323/614 20
(b) An MA(1) model can be written as
Xt = μ + θεt−1 + εt.
i. Derive the mean and variance of Xt. [2 marks]
ii. Derive the autocovariance of Xt. [2 marks]
21 STAT317/456 ECON323/614 iii. What happens to an MA(1) model if |θ| > 1? Explain why this is a
problem.
[2 marks]
END OF PAPER
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