1. Time series analysis - lecture 4 Consumer Price Index - Clothing and Footwear

2. Time series analysis - lecture 4 Consumer Price Index - Clothing and Footwear

3. Time series analysis - lecture 4 Seasonally differenced Consumer Price Index - Clothing and Footwear

4. Time series analysis - lecture 4 Seasonally differenced Consumer Price Index - Clothing and Footwear

5. Time series analysis - lecture 4 CPI Clothing and Footwear SARIMA (1, 0, 0, 0, 1, 0) Final Estimates of Parameters Type Coef SE Coef T P AR 1 0.7457 0.0522 14.29 0.000 Constant 0.2222 0.1783 1.25 0.214 Differencing: 0 regular, 1 seasonal of order 12 Number of observations: Original series 178, after differencing 166 Residuals: SS = 865.115 (backforecasts excluded) MS = 5.275 DF = 164

6. Time series analysis - lecture 4 CPI Clothing and Footwear SARIMA (1, 0, 0, 2, 1, 0) Final Estimates of Parameters Type Coef SE Coef T P AR 1 0.8145 0.0460 17.72 0.000 SAR 12 -0.6092 0.0830 -7.34 0.000 SAR 24 -0.2429 0.0843 -2.88 0.005 Constant 0.3275 0.1557 2.10 0.037 Differencing: 0 regular, 1 seasonal of order 12 Number of observations: Original series 178, after differencing 166 Residuals: SS = 651.883 (backforecasts excluded) MS = 4.024 DF = 162

7. Time series analysis - lecture 4 CPI Clothing and Footwear SARIMA (1, 0, 0, 2, 1, 0) residuals

8. Time series analysis - lecture 4 Models for multiple time series of data  Dynamic regression models  General input-output models  Models for intervention analysis  Response surface methodologies  Smoothing of multiple time series  Change-point detection

9. Time series analysis - lecture 4 Percentage of carbon dioxide in the output from a gas furnace

10. Time series analysis - lecture 4 The dynamic regression model where Y t = the forecast variable (output series); X t = the explanatory variable (input series); N t = the combined effect of all other factors influencing Y t (the noise); (B) = ( 0 + 1 B + 2 B 2 + … + k B k ), where k is the order of the transfer function

11. Time series analysis - lecture 4 Using the SAS procedure AUTOREG - regression in which the noise is modelled as an autoregressive sequence Consider a dataset with one input variable (gasrate) and one output variable (CO 2 ) data newdata; set mining.gasfurnace; gasrate1= lag1(gasrate); gasrate2= lag2(gasrate); gasrate3=lag3(gasrate); gasrate4= lag4(gasrate); run; proc autoreg data=newdata; model CO2 = gasrate/nlag=1; model CO2 = gasrate gasrate1/nlag=1; model CO2 = gasrate gasrate1 gasrate2/nlag=1; model CO2 = gasrate gasrate1 gasrate 2 gasrate3/nlag=1; model CO2 = gasrate gasrate1 gasrate2 gasrate3 gasrate4/nlag=1; output out=model4 residual=res; run;

12. Time series analysis - lecture 4 Predicted and observed levels of carbon dioxide in the output from a gas furnace - dynamic regression model with inputs time-lagged up to 4 steps

13. Time series analysis - lecture 4 No. air passengers by week in Sweden -original series and seasonally differenced data

14. Time series analysis - lecture 4 Intervention analysis where Y t = the forecast variable (output series); X t = the explanatory variable (step or pulse function); N t = the combined effect of all other factors influencing Y t (the noise); (B) = ( 0 + 1 B + 2 B 2 + … + k B k ), where k is the order of the transfer function