THE UNIVERSITY OF SYDNEY
Mathematical Statistics: STAT3925/STAT4025 - Semester 1 - 2022
Time Series Analysis : Problem Set - Week 13 (Tutorial and Revision Problems)
Reminder:
There will be a Computer Quiz today (Monday 23 May) at 16.00 in your class time.
Submit your answers through turnitin by due time.
There will be a Non-computer Quiz on Friday 27 May (Week 13) at 11.00 in your lecture time.
Submit your answers through turnitin by due time.
Attempt these questions before your class and discuss any issues with your tutor.
1. A stationary process {Xt} is said to be an ARCH(1) if it satisfies
Xt = t?t,
2
t = ?0 + ?1X
2
t1,
where {?t} is a sequence of iid random variables with mean zero and variance 1, ?0, ?1 > 0.
(i) Assuming ?t = X2t 2t is a sequence of uncorrelated random variable and E(?t) = 0 for all t, show that
X2t follows an AR(1) process.
(ii) Given that Xt is weakly stationary, find = E(X2t ) and explain why 1 does not satisfy ?1 1.
(iii) Show that E[(X2t )(X2tk )] = ?k1E[(X2t )2]; k 1.
(iv) Given that {Xt} is stationary up to order 4, explain why E[(X2t )Xtk] = ?k1E(X3t ).
(v) Find the sdf of X2t .
2. Suppose that r1, r2, · · · , rn are observations of a return series that follows the following
AR(1)-GARCH(1,1) model given by
rt = μ+ rt1 +Xt, Xt = t?t, 2t = ?0 + ?X
2
t1 +
2
t1,
where || < 1,?0, ?, > 0, {?t} is an iid sequence satisfying E(?t) = 0 and V ar(?t) = 1.
Assuming both {rt} and {Xt} are stationary, find
(i) E(Xt) and E(X2t ),
(ii) E(rt) and E(r2t ).
3. Suppose that {X1,t} and {X2,t} are formed from
X1,t = 0.6X1,t1 0.2X2,t1 + Z1,t,
X2,t = 0.4X1,t1 0.4X2,t1 + Z2,t,
where {Zi,t} ?WN(0, 1) with ? = cor(Z1t, Z2t) for i = 1, 2 and for all t .
(i) Show that Xt = (X1,t , X2,t)0 may be expressed as Xt Xt1 = Zt for suitably chosen 2? 2 matrix
where Zt = (Z1,t , Z2,t)0 (0 stands for transpose operation).
(ii) Determine whether this bivariate VAR(1) is stationary.
(iii) Given n observations on Xt, find the h step ahead forcast function, X?t+h.
(iv) Determine whether X?t+h exist as h!1.
4. Suppose that {X1,t} and {X2,t} are two time series satisfying
X1,t = 0.5X1,t1 + 0.3X2,t1 + Z1,t 0.4Z1,t1
X2,t = 0.6X1,t1 + 0.4X2,t1 + Z2,t 0.5Z2,t1
where {Zi,t} ?WN(0,2i ) and Cov(Zi,t, Zj,t) = ij for i = 1, 2 and for all t.
(i) Find 2? 2 matrices and ? such that Xt is expressed as a bivariate ARMA(1,1) process satisfying
Xt = Xt1 + Zt +?Zt1,
where Xt = (X1,t, X2,t)0 and Zt = (Z1,t, Z2,t)0.
(ii) Determine whether the vector process {Xt} in (i) is stationary.
(iii) Find the `step-ahead forecast function, X?t+` for all ` 1 from the time origin t.
PTO for Revision Problems. These problems will be discusses today (if time permits) and tomorrow
(24 May) during the lecture.
1
.Revision Problems - From 2021 Exam
1. An insurance company in Sydney have a collection daily insurance claims (in thousands of dollars) for the last
30 years. The manager wants to analyse and find a suitable model using the data x1, x2, · · · , x180 from the last
six months. The time series plot indicates that the series has a clear upward quadratic trend. As a result, a
time series consultant suggests to use the following model for further analysis:
Xt = f(t) + Zt, Zt ?WN(0,2),
where f(t) = a+ bt+ ct2 is a deterministic function of t with constants a, b, c.
(i) What do you expect from the shape of the ACF plot of this series?
(ii) A senior statistician plans to use a filter in the form of (1B)r to remove this quadratic trend from the
series. What is the value of r to be used?
(iii) Using a mathematical argument, justify your choice of r in (ii).
(iv) Find the mean and variance of the resulting series Wt = (1B)rXt in (iii).
(v) Write down a sequence of R commands to obtain the resulting series in {Wt} from the original {Xt}.
2. Suppose that MA(2) process given by
Xt = 10 + Zt + 0.80Zt1 0.60Zt2,
where {Zt} ?WN(0,2).
(i) Find the mean and variance of the series {Xt}.
(ii) Find its acf ?k for all k 0.
(iii) Sketch the correlogram in (ii).
(iv) Determine whether this MA(2) process is invertible.
3. A company uses the following stationary ARMA(2, 2) model to forecast its daily net profit (in thousands of
dollars):
Xt = 10 + 0.6Xt1 0.5Xt2 + Zt + 0.7Zt2, where{Zt} ? NID(0, 1.52).
Let {xt; 1 ? t ? n} be the time series of n daily readings from 1 January 2020.
(i) Find the ` step-ahead forecast function from the time origin n, X?n+`, ` 1.
(ii) What is the long term forecast value (ie. `!1) for the daily profit through this model?
(iii) Given that n = 300, x300 = 15.7, x299 = 13.2 and the last two estimated residuals are z?300 = 0.8, z?299 =
1.2, find the first two forecast values from the time origin t = 300.
(iv) Find the two-step-ahead forecast error and its variance.
4. Consider an ARMA(1, 2) process given by
Xt 0.5Xt1 = Zt 0.6Zt1 + 0.4Zt2,
where {Zt} ?WN(0,2).
(i) Show that the spectral density function (sdf) of {Zt}, fZ(!) = 22? , ? < ! < ?.
(ii) Find the sdf of {Xt}, fX(!), ? < ! < ?.
(iii) Show that the sdf fX(!) is a continuous function of !.
5. Suppose that {Xt} follows ARFIMA(1, , 0) process generated by
(1 ?B)(1B)Xt = Zt, where |?| < 1, 2 (0, 0.5) and {Zt} ?WN(0,2) with sdf fZ(!).
(i) Show that the sdf fX(!) does not exist as ! ! 0.
(ii) Find constants j , j 0 in terms of the gamma function, () such that Xt =
P1
j=0 jZtj .
(iii) When ? = 0, show that V ar(Xt) <1.
2
6. (a) A stationary process {Xt} is said to be a GARCH(2, 1) process if it satisfies
Xt = t?t,
2
t = ?0 + ?1X
2
t1 + ?2X
2
t2 + 1
2
t1, (?)
where ?0 > 0, ?1,?2 0, 1 > 0 and {?t} is a sequence of iid random variables with mean zero and
variance 1.
(i) Given that ?t = X2t 2t is a martingale di?erence series, find the values of r, s such that X2t follows
an ARMA(r, s) process.
(ii) Let i, i = 1, 2, . . . , r be the corresponding AR coecients in (i). If Xt is weakly stationary, find
E(X2t ) and show that the corresponding AR coecients i satisfy
Pr
i=1 i < 1.
(b) Suppose that {X1,t} and {X2,t} are formed from
X1,t = 0.5X1,t1 + 0.2X2,t1 + Z1,t + 0.5Z1,t1
X2,t = 0.3X1,t1 + 0.4X2,t1 + Z2,t + 0.4Z2,t1,
where {Zi,t} ?WN(0,2i ) and Cov(Zi,t, Zj,t) = ij for i = 1, 2 and for all t.
(i) Find 2? 2 matrices and ? such that Xt is expressed as a bivariate ARMA(1,1) process such that
Xt = Xt1 + Zt +?Zt1,
where Xt = (X1,t, X2,t)T and Zt = (Z1,t, Z2,t)T , where T stands for the transpose of a vector.
(ii) Determine whether the vector process {Xt} in (i) is stationary.
(iii) Find the `step-ahead forecast function, X?t+` for all ` 1 from the time origin t.