CITY UNIVERSITY OF HONG KONG
________________________________________________________________________
Course code & title : MS6215 Forecasting Methods for Business
: Semester A, 2004/2005 Session
Time allowed : Three hours
This paper has 6 pages (including this page)
Instruction to students:
1.
This paper consists of 4 questions.
2.
Attempt ALL FOUR questions.
3.
Start each question on a new page.
Materials, aids and instruments permitted to be used during examination:
Approved calculator
________________________________________________________________________
Special materials other than standard materials (e.g., answer book or supplementary sheets) to be supplied to students.
Nil

Question 1 (20 marks)
This question is concerned with the prediction of coal demand using regression technique.
The regression equation to be estimated is:
COAL i
  o
 
1
FIS i
 
2
FEU i
 
3
PCOAL i
 
4
PGAS
  i
; i

1 ,...
95 , (1.1) where COAL
 coal demand,
FIS = iron and steel production index,
FEU = utility production index,
PCOAL = price index for coal, and
PGAS = price index for natural gas. a) Upon estimating equation (1.1) by ordinary least squares, the following results are obtained, where e i
' s , i

1 ,....
95 , are the least squares residuals: i
94 

1 e i

1 .
7458 ; i
94 

1 e i
2 
621 .
188 ; e
1
 
13 .
33682 ;
95  i

2 e i e i

1

276 .
7389
Calculate the Durbin-Watson test statistic, and test the hypothesis of no first-order serial correlation at the 5% level of significance. Carefully set up the null and alternative hypotheses.
Note: DW
 i
95 

2
( e i
 e i

1
)
2
/ i
95 

1 e i
2
(8 marks) b) In the light of the test outcome in a), discuss how you would re-estimate the model using the Hildreth-Lu grid search procedure. (5 marks) c) Suppose the Hildreth-Lu procedure yields an estimate of 0.45 for the autocorrelation coefficient. Upon performing the autoregressive transformation and running the regression again, the following results are obtained:
C O AL
* i

16 .
245

75 .
29 FIS i
* 
100 .
26 FEU i
* 
38 .
98 PCOAL
* i

105 .
99 PGAS i
* where COAL
* i

COAL i

0 .
45 COAL i

1
;
FIS i
*
, FEU i
*
, PCOAL
* i and PGAS i
*
are defined analogously.
Obtain estimates of the

' s in (1.1). (4 marks) d) Predict COAL
T
for the following values of the explanatory variables and residuals:
FIS
T
= 96.4,
PGAS
T
= 121.5
FEU
T
= 102.3, PCOAL
T
= 117.8, and e
T-2
= -1.7458 (3 marks)

Question 2 (30 marks) a) Consider the ARIMA(p, d, q) model (1

B y t

0.892875

 t , where
B is the backward shift operator and
 t
' s are distributed as white noise. i) Determine p, d and q for this model. (2 marks) ii) Give an expression of y t
without using the backward shift operator. (4 marks) iii) Assume that y o

0.
Find E y t
) (7 marks) iv) Show that
 t
is stationary. b) Consider the ARIMA(1,0,1) process y t
  y t

1
  t
  t

1
;
 t
~ i.i.d.(0,
 2
)
i) Calculate ( t
) and ( ) for
 
0.3
and
 
0 .
9 .
(5 marks)
(4 marks)
ii) Show that the autocorrelation function of the process is given by

1

( 1
 
)(
  
)
1
  2 
2



 
 
1
;
 
2 , 3 , 4 ,......
(8 marks)

Question 3 (18 marks) a) Suppose an investigator used the Holt-Winter’s method with
 
0.4,
 
0.5
updating equations a t
  y t
(1

)( a t

1
 b t

1
) and b t
 
( a t
 a t

1
)
  
) b t

1 and
to fit a time series and obtained: y
11
 y
10

5.4; y
9
 b a
10

109.90;
(
ˆ
10 is the forecast value of y
11
obtained in period 10) i) Calculate y
11
the prediction for y
12
based on information up to period 11. ii) What are the forecast values of y
13
, y
14
and y
15
(8 marks)
if no further sample information is available beyond period 11? (2 marks) b) i) A time series with no trend is slowing changing over time. The forecast made in period 10 for the time series in period 11 is 20, and this forecast in fact over-estimates the actual value by 2. Suppose two-third of the weight is given to the most recent observation. Obtain a forecast for period 12.
(2 marks) ii) Suppose y t
, t = 1, 10, is fluctuating randomly around a constant level. The sum of these ten observations is 204. Predict y t
for any future period by the method of least squares.
(6 marks)

Question 4 (32 marks)
Mr. Rich Diamond has collected data on the weekly sales of a new toothpaste ( y t
, in units of tubes) over n = 22 weeks in 2001. He then applied the Box-Jenkins’ method to y t
and obtained the following results: y

709.7928
, . .( )

17.9318
, cov( , t

3
)

225.660
y y t

7

,
, cov( , t

4
 y y t

8
 y y t

1
)

222.446
,
, cov( , cov( , t

5
 t

2

,
, y y t

9
 cov( , t

6

, a) Construct the sample autocorrelation function of y t
for lags k

1,2,....,9.
Indicate if the autocorrelation coefficients are significantly different from zero (Hint: Let n

1/ 2 be the approximate standard errors of the sample autocorrelations). (8 marks) b) Should y t be considered as stationary? Carefully explain your answers. (2 marks) c) Construct the partial autocorrelation function of y t using the following information:
( r
22 r
66 r kk

0.27328, r
33

0.02307, r
77

0.43557, r
44

0.05215, r
88
 
0.22777, r
55
 
0.04612, r
99

0.09727
 
0.00341
is the sample partial autocorrelation at lag k; s.e.( r kk
)
 n

1/ 2 ) (5 marks) d) Suggest an appropriate ARIMA model on the basis of the above results. (2 marks) e) Suppose that the model identified in part d) is estimated and the Ljung-Box-Pierce test for the significance of the residuals are:
Q
*
(6)

54.59
with p-value = 0.0
Q * (12)

53.78
with p-value = 0.0 f)
So what do you conclude? What are the null and alternative hypotheses of the test?
(4 marks)
Let the log-likelihood value (log e
(L)) of the estimated equation be -286.17
Calculate the AIC and SBC values of the estimated model.
(Note: AIC = -2 log e
(L) + 2g, SBC = -2 log e
(L) + g log e
(n), g = number of parameters estimated) (4 marks) g) Suppose he then estimated an ARMA(3, 3) model which gave a log-likelihood value of -284.36. Which model do you prefer? The chosen model in part d) or the ARMA(3, 3) model? Carefully explain your answers. (5 marks) h) In practice, what other diagnostic tests would you perform to examine the validity of the model? (2 marks)