# stovhastic processes

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STAT3702/8702 Assignments– Sem 2, 2017
page 2
_______________________________________________________________
Note: In all problems, we assume that } { t ω
is a white noise with mean zero and a constant
variance 2ω
σ .
Question 1 [Total: 10 marks]
(a) From a series of length n=100, we calculate the sample autocorrelations as r1 = -0.49, r2 = 0.31,
r3 = -0.21, r4 = 0.11 and |rk | ≤ 0.09 for k > 4. On the basis of this information alone, what
ARIMA model would we tentatively specify for the series ? [5 marks]
(b) Consider an MA(1) process for which it is known that the mean is zero. Based on a series of
length 3, we observe
x1 = 0, x2 = -1, x3 = 0.5
(i) Show that the conditional least squares estimate of θ is 0.5 [3 marks]
(ii) Find an estimate of the noise variance σw
2 [2 marks]
Question 2 [Total: 10 marks]
Suppose that the process {xt } develops according to
xt = xt −4 +ωt , for t > 4,
with xt =ωt for t=1,2,3,4.
Find
(a) the variance function for {xt }, [4 marks]
(b) autocorrelation function for {xt }, [4 marks]
(c) identify the model as a certain seasonal ARIMA model. [2 marks]
Question 3 [Total: 15 marks]
Based on quarterly data, a seasonal model of the form
−4 1 −1 2 −2 = + − − t t t t t x x ω θ ω θ ω
has been fitted to a certain time series.
(a) Find the first four Ψ weights for this model analytically. [5 marks]
(b) Suppose that 0.5, 0.25, 1 1 2 = = − = ω θ θ σ . Find the forecasts for the next four quarters if data for
the last four quarters are
STAT3702/8702 Assignments– Sem 2, 2017
page 3
Quarter I II III IV
Series 25 20 25 40
Residual 2 1 2 3
[5 marks]
(c) Find the 95% prediction intervals for the forecasts in part (b). [5 marks]
Question 4 [Total: 15 marks]
The dataset airpass2017.sav (airpass.dat) shows total monthly air-passenger-miles within the
United States from January 1960 through December 1977.
(a) Specify and estimate the parameters for at least two tentative ARIMA models for the period of
January 1960 to October 1977. Comment on each statistical output to support your tentative
models. [4 marks]
(b) Perform diagnostic checking on the models estimated in part (a), repeat the process in (a) until
you obtain a final model. [4 marks]
(c) Write down your final model in its mathematical form. Interpret the model. [3 marks]
(d) Use the model in (c) to forecast the total monthly air-passenger-miles within the United States in
November and December 1977 respectively, manually and using SPSS or R, including its 95%
confidence interval. [4 marks]
Question 5 [STAT8702 ONLY] [Total: 10 marks]
(a) Consider the ARIMA model
xt =ωt + Θωt −2
Show that the series is invertible for Θ < 1, and find the coefficients π in the representation
= Σ

=

k 0
ωt π k xt k [5 marks]
(b) Consider the ARCH(1) model with a constant mean:
( )
2 1,t-1 ( 2 )
0 1 1
2
1
; | y ~ N 0,
~ N 0, 1
0
t t t t
t t t
a a
y a a
σ α α σ
μ α
α

= +
= +
Show that the model, 2
0 1 −1 = + t t t y y α α ε , where εt
~ N(0,1) and is an independent white noise
series, is equivalent to an ARCH(1) model with 0 mean. [5 marks]
End of Assignment 2

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