```CITY UNIVERSITY OF HONG KONG
________________________________________________________________________
Course code & title
:
Session
:
Semester A, 2004/2005
Time allowed
:
Three hours
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:
COALi   o  1 FIS i   2 FEU i   3 PCOALi   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 ei ' s, i  1,....95, are the least squares residuals:
94
 ei  1.7458 ;
i 1
94
 ei2  621.188 ; e1  13.33682;
i 1
95
e e
i 2
i 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.
95
 95

Note: DW  
(8 marks)
(ei  ei 1 ) 2 /
ei2 


i 1
 i 2



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:
COˆ AL*i  16.245  75.29FISi*  100.26FEUi*  38.98PCOAL*i  105.99PGASi*
where COAL*i  COALi  0.45COALi 1 ;
FIS i* , FEU i* , PCOAL*i and PGASi* are defined analogously.
Obtain estimates of the  ' s in (1.1).
d)
(4 marks)
Predict COALT for the following values of the explanatory variables and residuals:
FIST = 96.4,
PGAST = 121.5
and
FEUT = 102.3,
eT-2 = -1.7458
PCOALT = 117.8,
(3 marks)
Question 2 (30 marks)
a)
Consider the ARIMA(p, d, q) model (1  B) yt  0.892875 
(1  0.58935B) t
, where
(1  0.74755B)
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 yt without using the backward shift operator. (4 marks)
b)
iii) Assume that yo  0. Find E ( yt )
(7 marks)
iv) Show that  t is stationary.
(5 marks)
Consider the ARIMA(1,0,1) process
yt  yt 1   t   t 1
;
 t ~ i.i.d.(0,  2 )
i) Calculate E ( yt ) and Var ( yt ) for   0.3 and   0.9 .
(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 and
updating equations at   yt  (1   )(at 1  bt 1 ) and bt   (at  at 1 )  (1   )bt 1 to fit
a time series and obtained:
y11  yˆ10  5.4;
yˆ9  108.49;
b9 =2.003 ;
a10  109.90;
( ŷ10 is the forecast value of y11 obtained in period 10)
i) Calculate yˆ11 , the prediction for y12 based on information up to period 11.
(8 marks)
ii) What are the forecast values of y13 , y14 and y15 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 yt , t = 1, 10, is fluctuating randomly around a constant level. The
sum of these ten observations is 204. Predict yt 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 ( yt , in units
of tubes) over n = 22 weeks in 2001. He then applied the Box-Jenkins’ method to yt and
obtained the following results:
y  709.7928 , s.e.( yt )  17.9318 , cov( yt , yt 1 )  222.446 , cov( yt , yt  2 )  199.707
cov( yt , yt 3 )  225.660 , cov( yt , yt  4 )  163.887 , cov( yt , yt 5 )  158.989 , cov( yt , yt 6 )  166.938 ,
cov( yt , yt 7 )  135.698 , cov( yt , yt 8 )  127.925 , cov( yt , yt 9 )  124.962
a)
Construct the sample autocorrelation function of yt 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 yt be considered as stationary? Carefully explain your answers. (2 marks)
c)
Construct the partial autocorrelation function of yt using the following
information:
r22  0.27328, r33  0.43557, r44  0.22777, r55  0.09727
r66  0.02307, r77  0.05215, r88  0.04612, r99  0.00341
( rkk is the sample partial autocorrelation at lag k; s.e.( rkk )  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 (loge(L)) of the estimated equation be -286.17
Calculate the AIC and SBC values of the estimated model.
(Note: AIC = -2 loge(L) + 2g, SBC = -2 loge(L) + g loge(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
(5 marks)
h)
In practice, what other diagnostic tests would you perform to examine the validity
of the model?
(2 marks)
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