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# a425_arma - Time Series - ARIMA Models APS 425 - Advanced...

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Time Series - ARIMA Models APS 425 - Advanced Managerial Data Analysis (c) Prof. G. William Schwert, 2002-2010 1 APS 425 – Winter 2010 Time Series Analysis: ARIMA Models Instructor: G William Schwer Instructor: G. William Schwert 585-275-2470 [email protected] Topics •Typical time series plot • Pattern recognition in auto and partial autocorrelations • Stationarity & invertibility • Stochastic Seasonality (seasonal ARIMA)

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Time Series - ARIMA Models APS 425 - Advanced Managerial Data Analysis (c) Prof. G. William Schwert, 2002-2010 2 Partial autocorrelations • Used to identify pure AR models • Estimate a sequence of AR(k) models, and report the last coefficient estimate, kk , for each lag k: AR(k): Z t = + 1k Z t-1 + . . . + kk Z t-k + a t Graph the pacf coefficients, kk , and see where they become zero, which implies that the right model is a (k-1) th order AR process 1 AR(1): Z t = + 1 Z t-1 + a t 1 = .9 Note: exponential decay -0.4 -0.2 0 0.2 0.4 0.6 0.8 Autocorrelation -1 -0.8 -0.6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 A Lag k Auto Partial
Time Series - ARIMA Models APS 425 - Advanced Managerial Data Analysis (c) Prof. G. William Schwert, 2002-2010 3 1 AR(1): Z t = + 1 Z t-1 + a t 1 = .5 -0.4 -0.2 0 0.2 0.4 0.6 0.8 utocorrelation -1 -0.8 -0.6 A Lag k Auto Partial 1 AR(1): Z t = + 1 Z t-1 + a t 1 = -.5 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Autocorrelation -1 -0.8 -0.6 Lag k Auto Partial

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Time Series - ARIMA Models APS 425 - Advanced Managerial Data Analysis (c) Prof. G. William Schwert, 2002-2010 4 1 AR(1): Z t = + 1 Z t-1 + a t 1 = -.9 -0.4 -0.2 0 0.2 0.4 0.6 0.8 utocorrelation -1 -0.8 -0.6 A Lag k Auto Partial Note: oscillating autocorrelations 4 5 AR(1): Z t = + 1 Z t-1 + a t 1 = .9 -2 -1 0 1 2 3 -5 -4 -3 0 1 02 03 04 05 06 07 08 09 0 1 0 0 Note: smooth long swings away from mean
Time Series - ARIMA Models APS 425 - Advanced Managerial Data Analysis (c) Prof. G. William Schwert, 2002-2010 5 4 5 AR(1): Z t = + 1 Z t-1 + a t 1 = -.9 -1 0 1 2 3 -3 -2 0 1 02 03 04 05 06 07 08 09 0 1 0 0 Note: jagged, frequent swings around mean 1 AR(2): Z t = + 1 Z t-1 + 2 Z t-2 + a t 1 = 1.4 2 = .45 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Autocorrelation -1 -0.8 -0.6 A Lag k Auto Partial

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Time Series - ARIMA Models APS 425 - Advanced Managerial Data Analysis (c) Prof. G. William Schwert, 2002-2010 6 1 AR(2): Z t = + 1 Z t-1 + 2 Z t-2 + a t 1 = .4 2 = .45 -0.4 -0.2 0 0.2 0.4 0.6 0.8 utocorrelation -1 -0.8 -0.6 A Lag k Auto Partial 1 AR(2): Z t = + 1 Z t-1 + 2 Z t-2 + a t 1 = .7 2 = -.2 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Autocorrelation -1 -0.8 -0.6 Lag k Auto Partial
Time Series - ARIMA Models APS 425 - Advanced Managerial Data Analysis (c) Prof. G. William Schwert, 2002-2010 7 1 AR(2): Z t = + 1 Z t-1 + 2 Z t-2 + a t 1 = -.7 2 = .2 -0.4 -0.2 0 0.2 0.4 0.6 0.8 utocorrelation -1 -0.8 -0.6 A Lag k Auto Partial 4 6 AR(2): Z t = + 1 Z t-1 + 2 Z t-2 + a t 1 = 1.4 2 = .45 -6 -4 -2 0 2 -10 -8 0 1 02 03 04 05 06 07 08 09 0 1 0 0 Note: smooth long swings away from mean

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## This note was uploaded on 03/03/2010 for the course APS 425 taught by Professor Schwert during the Spring '10 term at Rochester.

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a425_arma - Time Series - ARIMA Models APS 425 - Advanced...

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