CITY UNIVERSITY OF HONG KONG ________________________________________________________________________ Course code & title : MS6215 Forecasting Methods for Business Session : Semester A, 2004/2005 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: 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 ) n1/ 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 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)

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# Exam - City University of Hong Kong