Lec10 - STAT 626 Outline of Lecture 10(Partial Correlogram ACF and PACF of ARMA Models(3.4 1 Review One-Sided MA or Causal Process Is a time series

Lec10 - STAT 626 Outline of Lecture 10(Partial Correlogram...

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STAT 626: Outline of Lecture 10 (Partial) Correlogram: ACF and PACF of ARMA Models ( § 3.4) 1. Review: One-Sided MA( ) or Causal Process: Is a time series involving only the past and present values of a white noise (shocks, inputs): x t = X j =0 ψ j w t - j with absolutely summable coefficients. Its autocovariance function is given by γ ( h ) = σ 2 w X j =0 ψ j + h ψ j . 2. Autoregressive and Moving Average (ARMA ( p, q )) Models: x t = φ 1 x t - 1 + . . . + φ p x t - p + w t + θ 1 w t - 1 + . . . + θ q w t - q . OR φ ( B ) x t = θ ( B ) w t , where φ ( z ) , θ ( z ) are the AR and MA polynomials, respectively. Focus on ARMA(1,1) Models: x t = φx t - 1 + w t + θw t - 1 . 3. Causal Solution: When x t can be written as a one-sided MA or linear process, i.e. in terms of the past and present values of the WN: w t , w t - 1 , . . . . This is important for computing the ACF of various ARMA models. 4. Invertible ARMA: w t can be written in terms of the past and present values , i.e. x t , x t - 1 , . . . , OR w t = x t + π 1 x t - 1 + π 2 x t - 2 + . . . = X j =0 π j x t - j . This is important for parameter estimation and computing predictors. 1
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5. What is the partial correlation between X and Y adjusted for the effect of Z ? 6. What is the partial autocorrelation function (PACF) of a stationary time series? The correlation coefficient between x t and x t + h after removing the linear effects of the intervening variables { x t +1 , . . . , x t + h - 1 } is called the lag- h partial autocorrelation of a stationary time series and denoted by φ hh , h = 1 , 2 , . . . , . The sequence or the function φ hh , h = 1 , 2 , . . . , is called the partial autocorrelation function (PACF) of the time series. The plot of φ hh vs h = 1 , 2 , . . . is call the partial correlogram of the time series. What is the shape of the partial correlogram of an AR(1)? AR(p)? What is the shape of the partial correlogram of and MA(q)? What is the shape of the partial correlogram of and ARMA(p,q)? 2
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102 3 ARIMA Models with initial conditions ψ j - j X k =1 φ k ψ j - k = θ j , 0 j < max( p, q + 1) . (3.41) The general solution depends on the roots of the AR polynomial φ ( z ) = 1 - φ 1 z - · · · - φ p z p , as seen from (3.40). The specific solution will, of course, depend on the initial conditions. Consider the ARMA process given in (3.27), x t = . 9 x t - 1 + . 5 w t - 1 + w t . Because max( p, q + 1) = 2, using (3.41), we have ψ 0 = 1 and ψ 1 = . 9 + . 5 = 1 . 4. By (3.40), for j = 2 , 3 , . . . , the ψ -weights satisfy ψ j - . 9 ψ j - 1 = 0. The general solution is ψ j = c . 9 j . To find the specific solution, use the initial condition ψ 1 = 1 . 4, so 1 . 4 = . 9 c or c = 1 . 4 /. 9. Finally, ψ j = 1 . 4( . 9) j - 1 , for j 1, as we saw in Example 3.7. To view, for example, the first 50 ψ -weights in R, use: 1 ARMAtoMA(ar=.9, ma=.5, 50) # for a list 2 plot(ARMAtoMA(ar=.9, ma=.5, 50)) # for a graph 3.4 Autocorrelation and Partial Autocorrelation We begin by exhibiting the ACF of an MA( q ) process, x t = θ ( B ) w t , where θ ( B ) = 1+ θ 1 B + · · · + θ q B q . Because x t is a finite linear combination of white noise terms, the process is stationary with mean E ( x t ) = q X j =0 θ j E ( w t - j ) = 0 , where we have written θ 0 = 1, and with autocovariance function γ ( h ) = cov ( x t + h , x t ) = cov q X j =0 θ j w t + h - j , q X k =0 θ k w t - k = ( σ 2 w q - h j =0 θ j θ j + h , 0 h q 0 h > q.
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  • Fall '08
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  • Autocorrelation, Stationary process, ACF, Autoregressive moving average model, Time series analysis, Partial autocorrelation function

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