PPT14 ACF-PACF Fall 2010

PPT14 ACF-PACF Fall 2010 - McGill University Advanced...

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ACF/PACF Autocorrelation Partial Autocorrelation
ARIMA Models ARIMA = Autoregressive Integrated Moving Average AR = Autoregressive I = Integrated MA = Moving Average

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Stationary Time Series From an intuitive point of view, “a time series is said to be stationary if there is no systematic change in mean (no trend), if there is no systematic change in variance and if strictly periodic variations have been removed.” (Chatfield, p.13) A longitudinal measure in which the process generating returns is identical over time. http://financial-dictionary.thefreedictionary.com/Stationary+time+series
Non-stationary Time Series File: Sales 35, Table 10.5 Page 315 Variable: Sales = Y t Index Sales 33 30 27 24 21 18 15 12 9 6 3 160 140 120 100 80 60 40 20 0 Time Series Plot of Sales

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ACF for a non-stationary time series SALES 35 Lag Autocorrelation 9 8 7 6 5 4 3 2 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Autocorrelation Function for Sales (with 5% significance limits for the autocorrelations)
Differencing Nonstationary series have an ACF that declines slowly (exponentially) rather than quickly declining to zero. You must “difference” such a series until it is stationary before you can identify the process.

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Differencing First Difference D t = Y t – Y t-1 If Y t is non-stationary, D t will often be stationary. Occasionally, second differencing will be necessary to achieve stationarity.
Values of I in ARIMA The I in ARIMA refers to the degree of differencing. If I = 0 no differencing is necessary If I = 1 take first differences D t = Y t – Y t-1 If I = 2 take second differences Z t = D t – D t-1 Note: Z t = D t – D t-1 = (Y t – Y t-1 ) – ( Y t-1 – Y t-2 ) = Y t – 2Y t-1 + Y t-2 Note: It is rarely necessary to take higher order differences, i.e. I = 0, 1, or 2 is sufficient in most cases. c.f. Page 323

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Stationary Time Series File: Sales 35, Table 10.5 Page 315 Variable D t = Y t – Y t-1 Index Dt 33 30 27 24 21 18 15 12 9 6 3 15 10 5 0 -5 -10 Time Series Plot of Dt
ACF for D t Lag Autocorrelation 9 8 7 6 5 4 3 2 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Autocorrelation Function for Dt (with 5% significance limits for the autocorrelations)

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Correlation plots After a time series has been made stationary by differencing, the next step in fitting an ARIMA model is to determine whether AR or MA terms, or a combination of the two, are needed to correct any autocorrelation that remains in the differenced series. With software like Minitab, we can use a trial and error approach to see what works best. But there is a
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This note was uploaded on 02/28/2012 for the course MANAGEMENT MGSC 272 taught by Professor Smith during the Spring '12 term at McGill.

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PPT14 ACF-PACF Fall 2010 - McGill University Advanced...

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