1. Analysis of Financial Data Spring 2012 Lecture 5: Time Series Models - 3 Priyantha Wijayatunga Department of Statistics, Umeå University Priyantha.wijayatunga@stat.umu.se Course homepage: http://www8.stat.umu.se/kursweb/vt012/staa2st017mom2/ http://www8.stat.umu.se/kursweb/vt012/staa2st017mom2/

2. Some Demonstrations

3. ACF and PACF ACF and PACF (that cuts-off at lag 1 ) look like if the AR(1) model can fit the data

4. Best Fitting Model Stationary R-squared should be big! Check the significance of the residual autocorrelation with the Ljung–Box test Model Statistics a ModelNumber of Predictors Model Fit statisticsLjung-Box Q(18) Number of Outliers Stationary R-squaredStatisticsDFSig. x-Model_10,61420,82217,2340 a. Best-Fitting Models according to Stationary R-squared (larger values indicate better fit). ARIMA Model Parameters a EstimateSEtSig. x-Model_1xNo TransformationConstant50,239,203247,646,000 ARLag 1,783,01456,311,000 a. Best-Fitting Models according to Stationary R-squared (larger values indicate better fit). Model Description Model Type Model IDxModel_1ARIMA(1,0,0)

5. Another Time Series

6. ACF and PACF ACF and PACF (that cuts-off at lag 2 ) show that if AR(2) model can fit the data

7. Best Fitting Model Stationary R-squared should be big! Check the significance of the residual autocorrelation with the Ljung–Box test Model Description Model Type Model IDxModel_1ARIMA(2,0,0) Model Statistics Model Number of Predictors Model Fit statisticsLjung-Box Q(18) Number of Outliers Stationary R- squaredRMSEStatisticsDFSig. x-Model_11,8181,94919,14716,2610 ARIMA Model Parameters EstimateSEtSig. x-Model_1xNo TransformationConstant48,829,99449,121,000 ARLag 1,810,02236,359,000 Lag 2,104,0224,685,000 DAY, not periodicNo TransformationNumeratorLag 0,001 1,319,187

8. Residual ACF and PACF

9. Another Time Series

10. ACF of Time Series Since ACF is positive until large lags, it is an indication of nonstationarity. Differencing is needed

11. ACF and PACF of 1-Differenced Time Series

12. Model Description Model Type Model IDxModel_1ARIMA(2,1,0) Model Statistics ModelNumber of Predictors Model Fit statisticsLjung-Box Q(18) Number of Outliers Stationary R-squaredRMSEStatisticsDFSig. x-Model_10,7791,98011,83516,7550 ARIMA Model Parameters EstimateSEtSig. x-Model_1xNo TransformationConstant,248,418,593,553 ARLag 1,781,02235,119,000 Lag 2,114,0225,110,000 Difference1