STAT 443 slides07

# STAT 443 slides07 - STAT 443: Forecasting ARMA Forecasting...

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Unformatted text preview: STAT 443: Forecasting ARMA Forecasting h-step forecasting Forecasting ARIMA ( p , d . q ) models ARIMA and Holt-Winters Forecasting STAT 443: Forecasting ARMA Forecasting h-step forecasting Forecasting ARIMA ( p , d . q ) models ARIMA and Holt-Winters Forcasting • More on one step forecasting for ARMA models • h-step ahead forecasting for ARMA models • Forecasting with ARIMA models • Relationship between Holt-Winters and ARIMA modelling STAT 443: Forecasting ARMA Forecasting h-step forecasting Forecasting ARIMA ( p , d . q ) models ARIMA and Holt-Winters Forecasting • Have algorithms for computing P ( X n + 1 | X n ,..., X 1 ) based on ACVF γ ( h ) • For AR ( p ) use Yule-Walker or Durbin-Levinson • For MA ( q ) and ARMA ( p , q ) use innovation algorthim • With data replace ACVF γ ( h ) in algorithm with sample ACVF b γ ( h ) STAT 443: Forecasting ARMA Forecasting h-step forecasting Forecasting ARIMA ( p , d . q ) models ARIMA and Holt-Winters Forcasting ARMA ( p , q ) • For an ARMA ( p , q ) process φ ( B ) X t = θ ( B ) Z t where Z t ∼ WN ( ,σ 2 ) and let m = max ( p , q ) • Then P ( X n + 1 | X n ,..., X 1 ) is given by ∑ n j = 1 b b nj ( X n + 1- j- b X n + 1- j ) 1 ≤ n < m b φ 1 X n + ··· + b φ n X n + 1- p + ∑ q j = 1 b b nj ( X n + 1- j- b X n + 1- j ) , n ≥ m • where b X n + 1 = P ( X n + 1 | X n ,..., X 1 ) b φ and b b ij are determined by the innovation algorithm using the sample ACVF function STAT 443: Forecasting ARMA Forecasting h-step forecasting Forecasting ARIMA ( p , d . q ) models ARIMA and Holt-Winters Forcasting AR ( p ) • Using this in the case of AR ( p ) model φ ( B ) X t = Z t get • P ( X n + 1 | X n ,..., X 1 ) is given by b φ 1 X n + ··· + b φ n X n + 1- p when n > p ....
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## This note was uploaded on 03/21/2012 for the course STAT 443 taught by Professor Yuliagel during the Winter '09 term at Waterloo.

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STAT 443 slides07 - STAT 443: Forecasting ARMA Forecasting...

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