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4. Given below is the estimation results of an ARIMA model with its correlogram of residuals. Y denotes the monthly series of the number of air passengers in Beijing (in thousands of people). Dependent Variable: DIY,1,12) Method: ARMA Conditional Least Squares (Marquardt - EViews legacy) Sample (adjusted): 2011M02 2017M12 Included observations: 83 after adjustments MA Backcast 2010M01 2011M01 Variable Coefficient Std. Error 1-Statistic Prob -0.334757 -0.887161 0.101864 0.033848 -3.286325 -20.36586 0.0015 0.0000 MA(1) SMA(12) Root MSE Mean dependent var S.D. dependent var Akaiko info criterion Schwarz criterion Hannan-Quinn criter 17.61812 R-squared -0.432169 Adjusted R-squared 26.66651 SE of regression 8.623926 Sum squared resid 8.682211 Log likelihood 8.647341 Durbin-Watson stat 0.558174 0.552720 17.83430 25763,04 -355.8929 1.901020 Sample: 2010101 2017M12 Included observations: 83 O-statistic probabilities adjusted for 2 ARMA terms Autocorrelation Partial Correlation PAC Q-Stat Prob 1 -0.001 -0.001 0.0001 2 -0.022 -0.022 0.0428 3 0.035 0.035 0.1533 0.695 4 0.073 0.072 0.6251 0.732 5 -0.091 -0.090 1.3809 0.710 6 -0.057 -0.056 16778 0.795 7 -0.068 -0.078 2.1110 0.834 8 -0.030 -0.032 2. 1958 0.901 9 -0.188 -0.178 5.5603 0.592 10 -0.043 -0.045 5.7382 0.677 11 0.068 0.062 6.1857 0.721 12 -0.117 -0.125 7.5365 0.674 a) Is the estimated process an adequate model? Explain b) Write the model in terms of the backshift operator and without backshift operator c) Table A below gives the last 13 observations of the series (December 2016 to December 2017). Table B on the other hand gives the last 13 residuals generated by the model. Compute the forecasts for January and February of 2018 and 95% forecast intervals (given the forecast standard error: January 2018=17.9145, February 2018=21.5241). 2016 December 644.8 January 694.7 July 739.8 Table A: Observed values of y 2017 February March April May June 670.5 703.8 6971 699.3 658.9 August September October November December 773.6 7246 760.5 725.4 714.8 2016 December 18.0859 January 14.0280 July -0.5367 Table B: Residuals 2017 February March April -20.6552 5.0320 -2.8319 August September October 9.9016 13.8680 17.2985 May June 4.4813 -23.7153 November December 23.1671 0.7232 4. Given below is the estimation results of an ARIMA model with its correlogram of residuals. Y denotes the monthly series of the number of air passengers in Beijing (in thousands of people). Dependent Variable: DIY,1,12) Method: ARMA Conditional Least Squares (Marquardt - EViews legacy) Sample (adjusted): 2011M02 2017M12 Included observations: 83 after adjustments MA Backcast 2010M01 2011M01 Variable Coefficient Std. Error 1-Statistic Prob -0.334757 -0.887161 0.101864 0.033848 -3.286325 -20.36586 0.0015 0.0000 MA(1) SMA(12) Root MSE Mean dependent var S.D. dependent var Akaiko info criterion Schwarz criterion Hannan-Quinn criter 17.61812 R-squared -0.432169 Adjusted R-squared 26.66651 SE of regression 8.623926 Sum squared resid 8.682211 Log likelihood 8.647341 Durbin-Watson stat 0.558174 0.552720 17.83430 25763,04 -355.8929 1.901020 Sample: 2010101 2017M12 Included observations: 83 O-statistic probabilities adjusted for 2 ARMA terms Autocorrelation Partial Correlation PAC Q-Stat Prob 1 -0.001 -0.001 0.0001 2 -0.022 -0.022 0.0428 3 0.035 0.035 0.1533 0.695 4 0.073 0.072 0.6251 0.732 5 -0.091 -0.090 1.3809 0.710 6 -0.057 -0.056 16778 0.795 7 -0.068 -0.078 2.1110 0.834 8 -0.030 -0.032 2. 1958 0.901 9 -0.188 -0.178 5.5603 0.592 10 -0.043 -0.045 5.7382 0.677 11 0.068 0.062 6.1857 0.721 12 -0.117 -0.125 7.5365 0.674 a) Is the estimated process an adequate model? Explain b) Write the model in terms of the backshift operator and without backshift operator c) Table A below gives the last 13 observations of the series (December 2016 to December 2017). Table B on the other hand gives the last 13 residuals generated by the model. Compute the forecasts for January and February of 2018 and 95% forecast intervals (given the forecast standard error: January 2018=17.9145, February 2018=21.5241). 2016 December 644.8 January 694.7 July 739.8 Table A: Observed values of y 2017 February March April May June 670.5 703.8 6971 699.3 658.9 August September October November December 773.6 7246 760.5 725.4 714.8 2016 December 18.0859 January 14.0280 July -0.5367 Table B: Residuals 2017 February March April -20.6552 5.0320 -2.8319 August September October 9.9016 13.8680 17.2985 May June 4.4813 -23.7153 November December 23.1671 0.7232