辅导 ORBS7260/MFFM7040 Applied Time Series / Times Series Analysis Assignment 2 Semester One, 2024/25辅

Department of Mathematics

Assignment 2

Answer ALL FOUR questions in the provided answer booklet.

Question 1 (20 marks)

Assume  {at } is white noise process, which is a sequence of identical independent distributed

random variables with zero mean and constant variance σa(2) .

(a) Evaluate the mean, variance and covariance function and hence determine the for the stationarity for each of the following processes.

(i) Xt = μ + 2tat

(ii) Wt =▽Yt = Yt - Yt-1    where Yt = t + at

(b) Consider a stationary moving average process, Xt = 3 + at + θat -1    where   -∞ < θ< ∞ .

(i)    Evaluate the mean, variance and the covariance function.

(ii)   Hence, find the first and second order autocorrelation coefficient for the process.

Question 2 (20 marks)

Consider a stationary time series Zt which follows an ARMA(1, 1) process, Zt = φ0 + φ1Zt -1 + at + 0.1at -1 ,

where at is a white noise process with a variance of σa(2) . Find the mean, variance and the

autocorrelation function (ACF) of an ARMA(1,1) process in terms of σa(2) ,φ0  and φ1   .

Question 3 (40 marks)

Consider the ARMA(2,3) process,

(1-0.1B)(1-0.8B)Zt = 9+(1-0.1B)(1+0.6B)(1+0.3B)at ,

where at is a white noise process with unit variance. It is known that the above process is over- estimated.

(a)    Suggest a parismony model ARMA(1,2) for the above process.

(b)   Hence, determine the stationarity and invertibility of the process.

(c)   Find the mean, the variance and the first two lags of the autocovariance function of the process.

(d)   Find the first three lags of the autocorrelation function (ACF) for the process.

(e)   Find the first two lags of the partial autocorrelation coefficients.

Question 4 (20 marks)

Consider a time series with sample mean and sample variance are 1.4 and 4, respectively. The values of the first two lags for sample autocorrelation function (ACF) coefficients are 0.4 and 0.1 respectively. It is known that the time series Zt follows an ARMA(1,1) process,

Zt = φ0  + φ1Zt -1  + θ1at -1  + at ,

where at is a white noise process with an unknown constant variance of σa(2) .

Assume the ARMA(1,1) process is a stationary and invertible process.

By using the values of the  sample autocorrelation coefficients, estimate the corresponding

unknown parametersφ0 ,φ1 ,θ1 andσa(2) for the ARMA(1,1).