ISyE 6402 Georgia Institute of Technology Homework 2 Solutions Questions With Correct Solutions, Already Passed!!

ISyE 6402 Georgia Institute of Technology Homework 2 Solutions Question 1: 1a. From both Time Series and ACF plots, it is clear there is significant trend present and thus the non-constant mean and zero autocorrelation assumptions are both violated. Although there is one spike in the late 90s, variance appears constant for the most part. Looking at the time series plot, it appears mean is now constant. There may be some slight heteroscedasticity present, most notably witnessed with the graph narrowing in the late 70s and bulging in the 2000s. It is not very pronounced though, so further testing would be needed to confirm. ACF plot has a consistent pattern of alternating rise and falls, indication the presence of some minor seasonality. While not major in effect, this may indicate the series fails the zero-autocorrelation assumption. Differenced data is likely not stationary, but it comes closer to meeting the requirements than undifferenced data and as such would be more appropriate for use in analysis. Grading: Out of 4 1 mark for correct plots on original data 1 mark for identifying non-constant mean and concluding data is not stationary 1 mark for correct plots of differenced data 1 mark for identifying mean is now constant, that differenced data is better for use in analysis regardless of stationarity assumptions. 1b. ACF of the original data does not cutoff or trail off, so an ARMA(p,q) on undifferenced data is not the best fit for the series. In the ACF plot of the differenced data we notice that there are no clear significant lags and an alternating pattern. PACF plot trails off which suggests the process is either ARIMA(0,d,q) or ARIMA(p,d,q) process, but graphical analysis alone cannot determine order. By iterative selection using AIC to compare models, order (3,1,3) has the lowest AIC of 3613. Next lowest model has an AIC 3637 which is significantly larger (by more than 2), this indicates it is the best model to select by AIC metric. Autocorrelation plots of ARIMA(3,1,3) residuals: We see in both plots that while there is a little noise in the middle, there are no significant lags or trails in both autocorrelation plots. This indicates residuals are likely independent with respect to time. Box-Ljung and Box-Pierce tests both produce non-significant values for p. This means we fail to reject the assumption of uncorrelated residuals for the model. Grading: Out of 4 1 mark for considering lags in ACF and PACF plots of series student indicated was best in problem 1a; must comment on lags cutting off and/or trailing off and relate that to ARIMA order. 1 mark for comparing ARIMA models of orders in the range specified and selecting the one with the smallest AIC value (should be order (3,1,3)) . If not that order, you may give credit evidence provided this method was appropriately used but orders ended up different. 1 mark for ACF and PACF plots of residuals. Student must comment on the lags and whether they indicate significant autocorrelation. 1 mark for Box-Ljung test or Box-Pierce test (only one necessary). Student must properly interpret the p value such that a p .05 indicates a lack of significant autocorrelation of residuals 1c. Model: Note that in each of the coefficients are significance with a confidence level of .05. You can determine this by knowing DF = N (649) – K (6) – 1 = 642. At that point it is essentially Z-Score which would be significant at 1.96. Divide value by S.E. and you will see none are lower in absolute value than that. Model Equation: General Form: ∆Yt =