1. MBA 532 Business Statistics by Rushan Abeygunawardana Department of Statistics, University of Colombo

21. Example: Moving Average Method 23 40 25 27 32 48 33 37 37 50 40 1 2 3 4 5 6 7 8 9 10 11 Sales Year

34. Example: Fitting an ARIMA Model The series show an upward trend . The first several autocorrelations are persistently large and trailed off to zero rather slowly  a trend exists and this time series is nonstationary (it does not vary about a fixed level) Idea: to difference the data to see if we could eliminate the trend and create a stationary series.

35. Example: Fitting an ARIMA Model… A plot of the differenced data appears to vary about a fixed level. Comparing the autocorrelations with their error limits, the only significant autocorrelation is at lag 1. Similarly, only the lag 1 partial autocorrelation is significant. The PACF appears to cut off after lag 1, indicating AR(1) behavior. The ACF appears to cut off after lag 1, indicating MA(1) behavior  we will try: ARIMA(1,1,0) and ARIMA(0,1,1) A constant term in each model will be included to allow for the fact that the series of differences appears to vary about a level greater than zero.

36. Example: Fitting an ARIMA Model… The LBQ statistics are not significant as indicated by the large p-values for either model. ARIMA(1,1,0) ARIMA(0,1,1)

37. Example: Fitting an ARIMA Model… Finally, there is no significant residual autocorrelation for the ARIMA(1,1,0) model. The results for the ARIMA(0,1,1) are similar. Therefore, either model is adequate and provide nearly the same one-step-ahead forecasts.

38. The first sample ACF coefficient is significantly different form zero. The autocorrelation at lag 2 is close to significant and opposite in sign from the lag 1 autocorrelation. The remaining autocorrelations are small. This suggests either an AR(1) model or an MA(2) model. The first PACF coefficient is significantly different from zero, but none of the other partial autocorrelations approaches significance, This suggests an AR(1) or ARIMA(1,0,0) ARIMA The time series of readings appears to vary about a fixed level of around 80, and the autocorrelations die out rapidly toward zero  the time series seems to be stationary .

39. Both models appear to fit the data well. The estimated coefficients are significantly different from zero and the mean square ( MS ) errors are similar. ARIMA AR(1) = ARIMA(1,0,0) MA(2) = ARIMA(0,0,2) A constant term is included in both models to allow for the fact that the readings vary about a level other than zero. Let’s take a look at the residuals ACF …

40. ARIMA Finally, there is no significant residual autocorrelation for the ARIMA(1,0,0) model. The results for the ARIMA(0,0,2) are similar. Therefore, either model is adequate and provide nearly the same three-step-ahead forecasts. Since the AR(1) model has two parameters (including the constant term) and the MA(2) model has three parameters, applying the principle of parsimony we would use the simpler AR(1) model to forecast future readings.

43. Introduction to ARIMA models