lec4.pdf - ARMA models Financial time series \u2013 lecture four Financial time series \u2013 lecture four ARMA models Outline ACF and PACF Detailed accounts

lec4.pdf - ARMA models Financial time series u2013 lecture...

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ARMA models Financial time series – lecture four Financial time series – lecture four ARMA models
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Outline ACF and PACF. Detailed accounts for ACF and PACF of ARMA models. Need to understand basic concepts and definitions of ACF and PACF. Know how to compute ACF for ARMA models. MLE and LS estimations for ARMA models. Understand the standard MLE and LS approach. Able to distinguish exact and conditional estimates. Able to write out the optimization form of these two approach for ARMA models. ARMA model prediction. Understand the general form of prediction for linear time series models. Able to conduct multiple step ahead forecast for AR, MA and simple ARMA models. Able to compute the prediction error and it’s variance. (2.4.4., 2.5.4 and 2.6.4 of Tsay book) Financial time series – lecture four ARMA models
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The autoregressive model AR(p) First order AR X t = φ 0 + φ 1 X t - 1 + w t where w t WN (0 , σ 2 ). Properties of AR(1) process: E ( X t ) = φ 0 Var ( X t ) = σ 2 1 - φ 2 1 if | φ 1 | < 1 ρ ( h ) = φ h 1 ACF examples of AR(1) processes. ts.sim = arima.sim(list(order = c(1,0,0), ar = 0.8), n = 200); acf(ts.sim,lag.max=12); Financial time series – lecture four ARMA models
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AR(1) with positive φ 1 . 0 2 4 6 8 10 12 0.0 0.2 0.4 0.6 0.8 1.0 Lag ACF Series ts.sim Figure: ACF when φ 1 =0.8 Financial time series – lecture four ARMA models
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AR(1) with negative φ 1 . ts.sim = arima.sim(list(order = c(1,0,0), ar = -0.8), n = 200); acf(ts.sim,lag.max=12); 0 2 4 6 8 10 12 ï 0.5 0.0 0.5 1.0 Lag ACF Series ts.sim Financial time series – lecture four ARMA models
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AR(1) with small φ 1 . Time ts.sim 0 50 100 150 200 ï 3 ï 2 ï 1 0 1 2 3 Figure: ACF when φ 1 =0.2 Financial time series – lecture four ARMA models
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AR(1) with large φ 1 . What are the differences? Time ts.sim 0 50 100 150 200 ï 15 ï 10 ï 5 0 5 10 Figure: ACF when φ 1 =0.99 Financial time series – lecture four ARMA models
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Properties of AR(1) The stationary condition for AR(1) process is | φ 1 | < 1. X t = φ 0 + φ 1 X t - 1 + w t Var ( X t ) = φ 2 1 Var ( X t - 1 ) + σ 2 If X t is stationary, (1 - φ 2 1 ) Var ( X t ) = σ 2 , thus | φ 1 | < 1. Condition on the past, we have E ( X t | X t - 1 ) = φ 0 + φ 1 X t - 1 , Var ( X t | X t - 1 ) = Var ( w t ) = σ 2 . Financial time series – lecture four ARMA models
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MA representation of AR(1) For simplicity, consider the mean zero series X t = φ 1 X t - 1 + w t = φ 1 ( φ 1 X t - 2 + w t - 1 ) + w t = φ 2 1 X t - 2 + φ 1 w t - 1 + w t = · · · = w t + φ 1 w t - 1 + φ 2 1 w t - 2 + · · · The shock at t - k has an exponentially decaying influence on X t . The associated coefficients are called the ψ weights. Financial time series – lecture four ARMA models
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Higher order AR models The zero mean AR(p) model is X t = φ 1 X t - 1 + φ 2 X t - 2 + · · · + φ p X t - p + w t , where w t is uncorrelated with X t - j for any j . Note that mean is zero here. Include the mean, the model is X t = φ 0 + φ 1 X t - 1 + φ 2 X t - 2 + · · · + φ p X t - p + w t , where φ 0 = μ (1 - φ 1 - · · · - φ p ), and μ = E ( X t ). Stationarity condition: all roots of 1 - φ 1 B - · · · - φ p B p = 0 are outside the unit disc. B is the backshift operator. Financial time series – lecture four ARMA models
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AR(2) with complex roots Consider an AR(2) X t = 1 . 5 X t - 1 - 0 . 75 X t - 2 + w t , the roots of 1 - 1 . 5 B + 0 . 75 B 2 are outside of unit circle. This will ensure a stationary AR(2) process.
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