Candidate Name: Student ID: Signature: The University of New South Wales School of Economics Business Forecasting ECON2209/5248 Final Examination, Session 1, 2006 1. Total time allowed: 2 hours 30 minutes. 2. Total number of questions: 4 3. Total marks for this exam: 60 4. Attempt all questions. The value for each part of the questions is given. 5. Calculators may be used. 6. Answers must be written in black or blue ink. Pencils may only be used for drawing, sketching or graphical work. 7. All answers must be written on the exam answer-booklet. Do NOT write any answers on this exam paper. 8. Some notation, useful formulae and statistical tables are given in the Appendix at the end of this paper. 9. This paper may NOT be retained by candidate. 1 Question 1 (a) State the classical decomposition (additive and standardized) for a time series, say {yt }. Characterize each component briefly. [4 marks] (b) Outline a procedure that seasonally adjusts a time series, say {yt }. [5 marks] (c) The following table contains the quarterly retail sales of a software company from 1998q1 to 1999q4. The MA(4)-smoothed series is also included. Actual Retail Sales 1998q1-1999q4 98q1 98q2 98q3 98q4 99q1 99q2 99q3 99q4 25.9 26.6 18.0 MA(4) 11.0 26.3 21.2 13.3 13.2 20.3 19.1 18.0 18.5 18.6 Here MA(4) is defined as (yt2 + yt1 + yt + yt+1 )/4. Use the MA(2*4) smoother to estimate the trend series. Report the trend series and the de-trended series with corresponding timing (ie, with an appropriate year and quarter). [3 marks] (d) The following table contains the monthly Yen/$AU exchange rate series from Apr/1999 to Aug/1999. Exchange Rate Apr/1999-Aug/1999 Apr May Jun Jul Aug 78.6 78.9 79.7 75.1 70.7 Use the exponential smoother with = .8 to smooth the above series up to Jul/1999 (inclusive). Based on the smoothed series, make a point forecast for the exchange rate at Aug/1999. Comment on the adequacy of the exponential smoother for long-horizon forecasts. [3 marks] 2 Question 2 (a) By saying that a time series is covariance stationary, what exactly do we mean? [3 marks] (b) Jan has collected a monthly time series consisting of 167 observations on the differenced log Yen/$AU exchange rate and plotted the sample ACF for the series (see below). Use the QLB (2) statistic to test whether or not the series is a white-noise process at the 5% level of significance. Write down your hypotheses, decision rule and conclusion. [3 marks] Lag -1.00 -0.60 -0.20 0.20 0.60 1.00 ACF |-+---------+---------+----0----+---------+---------+-| 1 | R 2 | | 3 | R | | -0.084039 4 | R| | -0.052353 5 | R| | -0.054732 6 | R | | -0.095489 7 | 8 | R |R | 0.001073 | 0.121080 | R| 0.044049 | -0.057184 9 | | R | 0.062974 10 | R | 0.009568 |-+---------+---------+----0----+---------+---------+-| Here 'R' represents the magnitudes of ACF values. (c) Suppose that we have a time series {yt }100 t=1 . How would you compute the sample partial autocorrelation function p( ) for = 1, 2, 3? [3 marks] (d) Consider the AR(1) process {yt }, where yt = yt1 + c + t with || < 1 and c being constant parameters. Here t iid WN(0, 2 ). (i) Find the mean and the autocovariance ( ) of the above process. [3 marks] (ii) Suppose that you know the values of T = {y1 , . . ., yT }, and c. Find the optimal point forecast (2-step ahead) under the MSFE. [3 marks] 3 Question 3 (a) Inspect the ACF and PACF plots of the observed time series {yt } below. Lag -1.00 -0.60 -0.20 0.20 0.60 1.00 ACF |-+---------+---------+----0----+---------+---------+-| 1 | 2 | + + | | + + 3 | + | + 4 | + | R 5 | + | 6 | + 7 | + 8 | 9 10 R R | 0.69096 R | 0.43271 R | 0.30402 | 0.19814 + | 0.13667 | R + | 0.06957 |R + | 0.04214 + | R + | 0.09846 | + | R + | 0.11726 | + | R + | 0.13686 |-+---------+---------+----0----+---------+---------+-| Lag -1.00 -0.60 -0.20 0.20 0.60 1.00 PACF |-+---------+---------+----0----+---------+---------+-| 1 | + | + 2 | +R | + R | 0.69096 3 | + 4 | + R| 5 | + 6 | + R| 7 | + |R + | 0.03507 8 | + | R | 0.12675 9 | + R + 10 | + | R+ | -0.08558 | R+ | + 0.07437 | -0.04275 |R + | + 0.02784 | -0.05811 | -0.00659 | 0.06543 |-+---------+---------+----0----+---------+---------+-| Here 'R' represents the magnitudes of ACF and PACF values and '+' represents the two-standard-deviation bands. Based on the above information, choose an ARMA(p,q) model for the series {yt }, ie, specify the values of p and q. Explain your choice briefly. [5 marks] (b) Finding the correct p and q for an ARMA(p,q) model based solely on the sample ACF and PACF may be difficult. Recommend an alternative approach to determine the values of p and q. List the steps of your recommendation. [5 marks] (c) When choosing a stationary ARMA model, we require that the model be a \"well-defined\" one. What do we mean by a \"well-defined\" ARMA model? [2 marks] (d) Suppose that a stationary time series with 50 observations can be fitted by an MA(1) model. How would you estimate the MA(1) model using the technique of least squares? [3 marks] 4 Question 4 An investment consultant firm believes that its operation profit is strongly influenced by the volume of shares traded on the New York Stock Exchange (NYSE). Two analysts are hired to model the volume series. The monthly log volume of the NYSE and its change (difference) are plotted in the diagrams below. Log Volume Traded 10 log(Volume) 8 6 4 2 0 0 100 200 300 Time 400 500 Differenced Log Volume Traded 0.8 0.6 log(Volume) 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0 100 200 300 Time 400 500 (a) One analyst has estimated three models from 537 monthly observations on 5 the differenced log volume of the NYSE, {yt }. The estimation results are summarized in the table below, where yt is the dependent (or left hand side) variable. Estimation Results Model yt1 yt2 AR(2) -.413 AR(3) ARMA(1,1) yt3 t1 c SSR -.218 .016 16.94 [.042] [.042] [.008] -.414 -.220 -.005 .016 [.043] [.046] [.043] [.008] .346 -.794 .007 [.066] [.043] [.002] 16.94 16.35 The standard errors for estimated coefficients are given in brackets. (i) Based on the above table, which model would you prefer? Explain briefly. [4 marks] (ii) Regardless of your answer to (i), use the estimated AR(2) model to make 1-step ahead and 2-step ahead point forecasts for the differenced log volume. It is known that the most recent observations yT = .23 and yT 1 = .15. [4 marks] (b) The other analyst has estimated two models from 537 monthly observations on the log volume of the NYSE, {yt }. The estimation results are summarized below, where the difference yt = yt yt1 is the dependent variable. Estimation Results Model yt1 yt1 yt2 t c SSR Model A -.141 -.325 -.165 .00147 .391 16.20 [.029] [.045] [.043] [.00030] [.078] -.413 -.218 .00001 .013 [.042] [.042] [.00005] [.016] Model B 16.94 The standard errors for estimated coefficients are given in brackets. 6 Based on the above table, should yt1 be included in the model? Explain briefly. [4 marks] (c) Given the models in (a) and (b), Comment on whether or not a combination of the forecasts from different models may improve forecast performance in this context. [3 marks] Appendix Some Notation MSFE: mean squared forecast error AR: autoregressive MA: moving average ARMA: autoregressive moving average SSR: sum of squared residuals ACF: autocorrelation function PACF: partial autocorrelation function c : constant term of a model t : time index T : sample size Useful Formulae MA(2*4): yt = (yt2 + 2yt1 + 2yt + 2yt+1 + yt+2 )/8 QLB (m) = T (T + 2) Pm =1 2 ( )/(T ) Exponential smoother: yt = yt + (1 ) yt1 , ARMA(p, q): xt = Pp i=1 i xti + t + Pq j=1 j tj BIC or SIC: BIC = log(SSR/T ) + k log(T )/T Useful Statistical Tables Dickey-Fuller Critical Values Level of Significance Critical Value 1% 2.5% -3.98 -3.68 7 5% 10% -3.42 -3.13 y1 = y1 +c Table 6. Upper-tail Chi-square Critical Values: 2(,) Chi2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 70 80 90 100 0.995 0.99 0.975 0.95 0.9 0.1 0.05 0.025 0.01 0.005 3.93E-05 0.0002 0.0010 0.0039 0.0158 2.7055 3.8415 5.0239 6.6349 7.8794 0.0100 0.0201 0.0506 0.1026 0.2107 4.6052 5.9915 7.3778 9.2103 10.5966 0.0717 0.1148 0.2158 0.3518 0.5844 6.2514 7.8147 9.3484 11.3449 12.8382 0.2070 0.2971 0.4844 0.7107 1.0636 7.7794 9.4877 11.1433 13.2767 14.8603 0.4117 0.5543 0.8312 1.1455 1.6103 9.2364 11.0705 12.8325 15.0863 16.7496 0.6757 0.8721 1.2373 1.6354 2.2041 10.6446 12.5916 14.4494 16.8119 18.5476 0.9893 1.2390 1.6899 2.1673 2.8331 12.0170 14.0671 16.0128 18.4753 20.2777 1.3444 1.6465 2.1797 2.7326 3.4895 13.3616 15.5073 17.5345 20.0902 21.9550 1.7349 2.0879 2.7004 3.3251 4.1682 14.6837 16.9190 19.0228 21.6660 23.5894 2.1559 2.5582 3.2470 3.9403 4.8652 15.9872 18.3070 20.4832 23.2093 25.1882 2.6032 3.0535 3.8157 4.5748 5.5778 17.2750 19.6751 21.9200 24.7250 26.7568 3.0738 3.5706 4.4038 5.2260 6.3038 18.5493 21.0261 23.3367 26.2170 28.2995 3.5650 4.1069 5.0088 5.8919 7.0415 19.8119 22.3620 24.7356 27.6882 29.8195 4.0747 4.6604 5.6287 6.5706 7.7895 21.0641 23.6848 26.1189 29.1412 31.3193 4.6009 5.2293 6.2621 7.2609 8.5468 22.3071 24.9958 27.4884 30.5779 32.8013 5.1422 5.8122 6.9077 7.9616 9.3122 23.5418 26.2962 28.8454 31.9999 34.2672 5.6972 6.4078 7.5642 8.6718 10.0852 24.7690 27.5871 30.1910 33.4087 35.7185 6.2648 7.0149 8.2307 9.3905 10.8649 25.9894 28.8693 31.5264 34.8053 37.1565 6.8440 7.6327 8.9065 10.1170 11.6509 27.2036 30.1435 32.8523 36.1909 38.5823 7.4338 8.2604 9.5908 10.8508 12.4426 28.4120 31.4104 34.1696 37.5662 39.9968 8.0337 8.8972 10.2829 11.5913 13.2396 29.6151 32.6706 35.4789 38.9322 41.4011 8.6427 9.5425 10.9823 12.3380 14.0415 30.8133 33.9244 36.7807 40.2894 42.7957 9.2604 10.1957 11.6886 13.0905 14.8480 32.0069 35.1725 38.0756 41.6384 44.1813 9.8862 10.8564 12.4012 13.8484 15.6587 33.1962 36.4150 39.3641 42.9798 45.5585 10.5197 11.5240 13.1197 14.6114 16.4734 34.3816 37.6525 40.6465 44.3141 46.9279 11.1602 12.1981 13.8439 15.3792 17.2919 35.5632 38.8851 41.9232 45.6417 48.2899 11.8076 12.8785 14.5734 16.1514 18.1139 36.7412 40.1133 43.1945 46.9629 49.6449 12.4613 13.5647 15.3079 16.9279 18.9392 37.9159 41.3371 44.4608 48.2782 50.9934 13.1211 14.2565 16.0471 17.7084 19.7677 39.0875 42.5570 45.7223 49.5879 52.3356 13.7867 14.9535 16.7908 18.4927 20.5992 40.2560 43.7730 46.9792 50.8922 53.6720 20.7065 22.1643 24.4330 26.5093 29.0505 51.8051 55.7585 59.3417 63.6907 66.7660 27.9907 29.7067 32.3574 34.7643 37.6886 63.1671 67.5048 71.4202 76.1539 79.4900 35.5345 37.4849 40.4817 43.1880 46.4589 74.3970 79.0819 83.2977 88.3794 91.9517 43.2752 45.4417 48.7576 51.7393 55.3289 85.5270 90.5312 95.0232 100.4252 104.2149 51.1719 53.5401 57.1532 60.3915 64.2778 96.5782 101.8795 106.6286 112.3288 116.3211 59.1963 61.7541 65.6466 69.1260 73.2911 107.5650 113.1453 118.1359 124.1163 128.2989 67.3276 70.0649 74.2219 77.9295 82.3581 118.4980 124.3421 129.5612 135.8067 140.1695 8