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Unformatted text preview: ACTL2003 Week 10 Time Series  An Introduction I Residual analysis I Nonstationarity I ARIMA I SARIMA I Unit root test I Conintegrated processes I Markov property I Simulation Readings: Chan, Chapters 4 and 8 Reference: ACTED Chapter 12 and 13 CT6 1/40 Model building For a time series, model building can be classified into three stages: 1. Model selection (choosing ARIMA) 2. Model identification (estimation of parameters) 3. Model checking (diagnostic) (This Week) 2/40 Model checkingResidual Analysis I Principle : The residuals Z t = X t X t will form a good approximation to a white noise process, if the proposed model X t is a good approximation to the underlying time series process. Here, X t are the fitted values, which are computed using the estimated parameters. I Frequently used checks to see whether a set of data is a likely realisation of a white noise process. I Plot the residuals Z t against t : any evident pattern, whether in the average level of the residuals (trend) or in the magnitude of the fluctuations about 0, means that the model is inadequate. 3/40 Frequently used checks to see whether a set of data is a likely realisation of a white noise process (continued): I Plot the sample ACF (SACF) of Z t I Rationale: Under the null hypothesis that Z t WN (0 , 1), it is can be shown that the ACF of { Z t } Z ( h ) N (0 , 1 n ) . So the confidence intervals at significance 95% of the sample autocorrelations are 0 1 . 96 n . (Recall that for white noise process, the ACF is 1 at lag 0, and 0 at other lags.) I As a rule of thumb, one can use 2 n to determine if the SACF is significantly different from zero . (Note: only short lags of SACF are meaningful.) These bounds are usually provided automatically by the estimation software (such as Minitab, R, SPlus). I : If too many values of SACF are outside the range ( 2 n , + 2 n ), we can conclude that the fitted model does not fit well (more parameters will be needed). 4/40 Frequently used checks to see whether a set of data is a likely realisation of a white noise process (continued): Portmanteau Test : I Portmanteau statistic: Q = n ( n + 2) h X j =1 2 Z ( j ) n j . I Theorem: Under the null hypothesis that Z t WN (0 , 1), the Portmanteau statistic Q = n ( n +2) h X j =1 2 Z ( j ) n j 2 ( n p q ) approximately for large n , Note: In practice, h is chosen between 15 and 30, and n should be large, say n 100. I Reject the null if Q > 2 1 ( n p q ). 5/40 Nonstationarity In practice, different kinds of nonstationarity are often encountered. Roughly speaking, a nonstationary time series may exhibit nonstationarity in the level of mean, variance, or both....
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This note was uploaded on 06/12/2011 for the course ASB 2003 taught by Professor Kim during the Three '11 term at University of New South Wales.
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