lOMoARcPSD|63158501
4 STA3704
JAN/FEBR 2023
Question 1 [31 Marks]
A time series of the Civil cases recorded and summonses issued for debt (Business enterprises and private per-
sons cases recorded) in South Africa, seasonally adjusted, from January 2014 to October 2022 (Source: StatsSA,
www.statssa.gov.za) was analyzed using JMP SAS. The output is at the end of this question paper on pages 6 to
10.
(a) What conclusions, as to the stationarity of the series, can be made from the time series, the AC F and the
P AC F plots? (3 marks)
(b) Consider the output from the analysis of the differenced series, i.e., as from Difference:.1 B/^1 in the
output.
(i) What conclusions can be made from the time series, the AC F and P AC F plots as to the stationarity
of the series and model identification? (3+5 marks)
(ii) What can you deduce from the first 15 lags of the ACF and the PACF plots as to the information
contained in the model? (3 marks)
(c) (i) Justify the adequacy or otherwise of the fitted models. (2 marks each = 6 marks)
(ii) Based on your knowledge of time series ARIMA modelling, choose the appropriate model for the time
series of the civil cases recorded and summonses issued for debt in South Africa, with appropriate
justification. (8 marks)
(d) Write down the mathematical equation for the appropriate significant model chosen in (c). (3 marks)
Question 2 [21 Marks]
(a) Explain the general purpose of Time Series Analysis. (4 marks)
(b) (i) List the steps involved in model identification. (4 marks)
(ii) Describe two ways by which a time series can be nonstationary. (2 marks)
(iii) For each way in c(ii), how can the series be stationarised? (2+2 marks)
(c) In which situations are the following processes applicable?
(i) an AR, and
(ii) an MA. (2 + 2 marks)
(d) Why is it necessary to do diagnostic checking on the residuals after a time series model has been fitted?
(3 marks)
[TURN OVER]
Downloaded by stephen musyoka ()
, lOMoARcPSD|63158501
5 STA3704
JAN/FEBR 2023
Question 3 [26 Marks]
2
(a) For each of the ARIMA models below, with et N 0; e and t > 0 , give the values for E.rYt / and
V ar .rYt /:
(i) Yt D Yt 1 C et 0:5et 1 : (2 + 2 marks)
(ii) Yt D 3 C Yt 1 C et 0:57et 1 C 0:8et 2 (2 + 2 marks)
(b) Suppose that an A R M A.2; 2/ model is given as: Yt D 32 C 0:6Yt 1 0:4Yt 2 C et 0:5et 1 C 0:3et 2 .
(i) Test the stationarity and invertibility of the model. (3 + 3 marks)
(ii) Substantiate, using (i), whether or not the model is from a general ARMA model. (2 marks)
(c) Consider the model Yt D et et2 1 : Assume that the white noise series is normallly distributed with mean
zero, variance 2e ; E.et3 1 / D 0 and E.et4 1 / D 3 4e : Note that V ar .et2 / D E.et4 1 / [E.et2 1 /]2 :
(i) Find the autocorrelation function for fYt g: (8 marks)
(ii) Is fYt g stationary? Justify your answer. (1+1 marks)
Question 4 [12 Marks]
(a) Verify that the autocorrelation function for an MA(1) process does not change when the parameter is replaced
with its inverse. (4 marks)
(b) Consider the MA process: Z t D at 0:6at 1 0:5at 2 ; where a1 ; a2 ; ::: are independent and identically
distributed random variables with mean 0 and variance a2 :
1 k D0
0:1863 k D1
(i) Show that the autocorrelation function (ACF) of this process is k D (4 marks)
0:3106 k D2
0 k >2
(ii) Using (i) or otherwise, find the value of the second order partial autocorrelation 22 : (2 marks)
(c) For an AR(1) model with Yt D 25; D 0:55; and D 16:
(i) Find the forecast values YOt .l/ for l = 1, 2, and 8. (2+2+2 marks)
(ii) Assuming the AR(1) model has independently and identically distributed random variables et with
mean 0 and variance 2e D 0:6: Calculate the 95% confidence limits for the forecasts YOt .1/ and YOt .2/
calculated in (i) above. (3+3 marks)
TOTAL [100 Marks]
[TURN OVER]
Downloaded by stephen musyoka ()
, lOMoARcPSD|63158501
6 STA3704
JAN/FEBR 2023
Output for Question 1.
[TURN OVER]
Downloaded by stephen musyoka ()
, lOMoARcPSD|63158501
7 STA3704
JAN/FEBR 2023
[TURN OVER]
Downloaded by stephen musyoka ()
4 STA3704
JAN/FEBR 2023
Question 1 [31 Marks]
A time series of the Civil cases recorded and summonses issued for debt (Business enterprises and private per-
sons cases recorded) in South Africa, seasonally adjusted, from January 2014 to October 2022 (Source: StatsSA,
www.statssa.gov.za) was analyzed using JMP SAS. The output is at the end of this question paper on pages 6 to
10.
(a) What conclusions, as to the stationarity of the series, can be made from the time series, the AC F and the
P AC F plots? (3 marks)
(b) Consider the output from the analysis of the differenced series, i.e., as from Difference:.1 B/^1 in the
output.
(i) What conclusions can be made from the time series, the AC F and P AC F plots as to the stationarity
of the series and model identification? (3+5 marks)
(ii) What can you deduce from the first 15 lags of the ACF and the PACF plots as to the information
contained in the model? (3 marks)
(c) (i) Justify the adequacy or otherwise of the fitted models. (2 marks each = 6 marks)
(ii) Based on your knowledge of time series ARIMA modelling, choose the appropriate model for the time
series of the civil cases recorded and summonses issued for debt in South Africa, with appropriate
justification. (8 marks)
(d) Write down the mathematical equation for the appropriate significant model chosen in (c). (3 marks)
Question 2 [21 Marks]
(a) Explain the general purpose of Time Series Analysis. (4 marks)
(b) (i) List the steps involved in model identification. (4 marks)
(ii) Describe two ways by which a time series can be nonstationary. (2 marks)
(iii) For each way in c(ii), how can the series be stationarised? (2+2 marks)
(c) In which situations are the following processes applicable?
(i) an AR, and
(ii) an MA. (2 + 2 marks)
(d) Why is it necessary to do diagnostic checking on the residuals after a time series model has been fitted?
(3 marks)
[TURN OVER]
Downloaded by stephen musyoka ()
, lOMoARcPSD|63158501
5 STA3704
JAN/FEBR 2023
Question 3 [26 Marks]
2
(a) For each of the ARIMA models below, with et N 0; e and t > 0 , give the values for E.rYt / and
V ar .rYt /:
(i) Yt D Yt 1 C et 0:5et 1 : (2 + 2 marks)
(ii) Yt D 3 C Yt 1 C et 0:57et 1 C 0:8et 2 (2 + 2 marks)
(b) Suppose that an A R M A.2; 2/ model is given as: Yt D 32 C 0:6Yt 1 0:4Yt 2 C et 0:5et 1 C 0:3et 2 .
(i) Test the stationarity and invertibility of the model. (3 + 3 marks)
(ii) Substantiate, using (i), whether or not the model is from a general ARMA model. (2 marks)
(c) Consider the model Yt D et et2 1 : Assume that the white noise series is normallly distributed with mean
zero, variance 2e ; E.et3 1 / D 0 and E.et4 1 / D 3 4e : Note that V ar .et2 / D E.et4 1 / [E.et2 1 /]2 :
(i) Find the autocorrelation function for fYt g: (8 marks)
(ii) Is fYt g stationary? Justify your answer. (1+1 marks)
Question 4 [12 Marks]
(a) Verify that the autocorrelation function for an MA(1) process does not change when the parameter is replaced
with its inverse. (4 marks)
(b) Consider the MA process: Z t D at 0:6at 1 0:5at 2 ; where a1 ; a2 ; ::: are independent and identically
distributed random variables with mean 0 and variance a2 :
1 k D0
0:1863 k D1
(i) Show that the autocorrelation function (ACF) of this process is k D (4 marks)
0:3106 k D2
0 k >2
(ii) Using (i) or otherwise, find the value of the second order partial autocorrelation 22 : (2 marks)
(c) For an AR(1) model with Yt D 25; D 0:55; and D 16:
(i) Find the forecast values YOt .l/ for l = 1, 2, and 8. (2+2+2 marks)
(ii) Assuming the AR(1) model has independently and identically distributed random variables et with
mean 0 and variance 2e D 0:6: Calculate the 95% confidence limits for the forecasts YOt .1/ and YOt .2/
calculated in (i) above. (3+3 marks)
TOTAL [100 Marks]
[TURN OVER]
Downloaded by stephen musyoka ()
, lOMoARcPSD|63158501
6 STA3704
JAN/FEBR 2023
Output for Question 1.
[TURN OVER]
Downloaded by stephen musyoka ()
, lOMoARcPSD|63158501
7 STA3704
JAN/FEBR 2023
[TURN OVER]
Downloaded by stephen musyoka ()
