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###### 日期：2022-06-09 07:33

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.

There will be a Non-computer Quiz on Friday 27 May (Week 13) at 11.00 in your lecture 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.

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