During the period for which a time series is observed, it is sometimes the case that a change occurs (e.g. tax laws, reporting methods, etc.) which affects the level of the series . Simple Intervention Model : Y t = wX t + W t , t=1, . . . , n, where X t is a deterministic function of time. Section 10.2 and Example 6.6.3 Application to Intervention Analysis 3.16 Application to Intervention Analysis 46 Examples of intervention models: Pulse at fixed time T: momentary effect on the level of the series. Step change at fixed time T: change in level occurs after a time T. ≠ = = T. t if 0, T, t if , 1 t X < ≥ = T. t if 0, T, t if , 1 t X T T
t-2 , . 48 • Open the file sbl.tsm and select Regression > Specify, click on the Include auxiliary variables imported from file box and click on the browse button. Open the file called sblin.tsm and indicate that you have only 1 regressor. Make sure you check the intercept and uncheck polynomial buttons before clicking OK on the dialog box.
49•Now click on theGLSicon and inspect the plots of the current data set and its ACF/PACF.•These plots clearly suggest a strong seasonal component with period 12. Difference the data and considerXt= Yt– Yt-12, = b g(t) + Nt, t =13, …, 120,whereg(t)=1 for t=99,…,110, and0 otherwise, Nt=Wt– Wt-12modeled as an ARMA process. •Open the file sbld.tsmand select Regression > Specify,click on theInclude auxiliary variables imported from file box and click on the browse button. Open the file calledsbldin.tsm and indicate that you have only1 regressor.•Press the blue GLS button•Sample ACF/PACF of residuals suggests MA(13) or AR(13). Use autofitwith AR and MA orders up to 13.•Update gls estimates (click on blue GLS button). Press the MLE button and repeat. Final model isXt= -328.45 g(t) + Nt,whereNtis an MA(13) process is 50 .
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- Regression Analysis, Maximum likelihood, Likelihood function, zt, ARMA