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Duke Shu, a Willamette MBA interested in business forecasting, data mining, and strategic marketing, uploaded a series of his works to "cast a brick to attract a jade"--hoping to hear more constructive, brilliant feedback from industrial experts while networking with them.
ARIMA Exercise Author: Duke Shuhttp://www.linkedin.com/in/dukeshu
Double Exponential Smoothing ForGNP Smoothing model is not a perfect fit as seen from the graph. Violation of Independence is indicated by two low-order spikes in autocorrelation for HotRes. Residual vs Fit chart reinforces the conclusion of Independence violation because of six isolated groups of parallel downward trends in residuals.
Difference to Attain StationarityBased on Time Series plot of GNP, intuitively ARIMA model will include a Constant. ACF does not die down rapidly for GNP (non-stationary), but does for diff1 (stationary).
Model identification Exploration on Difference1 Tentative model identification: AR(1) on first differences or ARIMA (1,1,0) (p=1, d=1, q=0) Only the two-order autocorrelations are significantly different from zero in ACF for diff1. Therefore, the first differences appear stationary. Further, it appears that the autocorrelation function DAMPS and the partial autocorrelation function appears to CUT after one lag (first-order autoregressive model, AR (1)).
Exploration on Difference2Although independence is not violated in ACF diff2, it fails to show an obvious pattern (damp or cut) compared to ACF diff1. Like ACF diff2, PACF diff2 also fails to show anobvious pattern compared to PACF diff1. If we have to pick up a pattern, we choose
Model ARIMA(1,2,1) Mildly explosive case because MA coef is greater than 1. Parsimony principle indicates we should drop this model and consider ARIMA(1,1,1).
Model ARIMA (1,1,1)Drop MA, because it is not a significant term to confirm the tentative identification[ARIMA (1,1,0)] and refit the model.
Assessment of ARIMA(1,1,0)All model parameters, including the constant term, are significant. This is demonstratedby high p-values in Ljung-Box as well. MS=22.97 slightly improved compared to theprevious over-fit Model ARIMA (1,1,1), where MS=23.27. Although it is slightly largerthan MS=21.77 in ARIMA (1,2,1), parsimony principle indicates this model is the best.The model has a strong adequacy:principles of independence, meanzero, normality, and constant variance are held. Independence is reinforced inautocorrelation within boundaries.
ARIMA (1,1,0) no constantBased on the Time Series plot intuitively we had concluded that ARIMA modelshould include the constant term. Indeed our previous model ARIMA (1,1,0) withthe constant showed, that it was significant with p-value almost equal to zero. Forcomparison purpose we computed ARIMA (1,1,0) without the constant. It has alarger MS of 25.96 therefore is not as good as our final model ARIMA (1,1,0) withconstant.
Forecasting for ARIMA (1,1,0) withconstant Final Model yt=2.7012+1.6155yt-1-0.6155yt-2+Ɛt The model is good for Forecasting purposes it has narrow Predication intervals.