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32_TimeSeriesB_handout

# 32_TimeSeriesB_handout - Time Series II Wooldridge 11,18...

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Time Series - II Wooldridge 11,18 73-261 Y. Kryukov CMU, Tepper School of Business November 17, 2010 Y. Kryukov (CMU) 73-261: Time Series Nov-17 1 / 14

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Overview 3.2 Time series: residuals Stationary processes °MA, AR Non-stationary process °random walk: Testing: dickey fuller Fixing: di/erences Serially correlated residuals: Testing: Durbin-Watson Fixing: di/erences & FGSL Y. Kryukov (CMU) 73-261: Time Series Nov-17 2 / 14
Stationary processes Stochastic process: x t , t = 1 , 2 , ... Process is stationary if "distribution of x t " does not depend on t . Take several points in time: t 1 , t 2 , ..., t m and a gap h ° 1. Stationarity = joint distribution of ( x t 1 , x t 2 , ..., x t m ) is same that of ( x t 1 + h , x t 2 + h , ..., x t m + h ) . Covariance-stationary process: E ( x t ) = μ V ( x t ) = σ 2 Cov ( x t , x t + h ) = c h Weak dependence: c h ! 0 as h ! Y. Kryukov (CMU) 73-261: Time Series Nov-17 3 / 14

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Moving Average (MA) process MA(1): x t is Moving Average of degree 1 if x t = e t + α e t ± 1 where e t °i.i.d. with E ( e t ) = 0 , V ( e t ) = σ 2 e MA(1) is stationary: E ( x t ) = ( 1 + α ) E ( e t ) = .... V ( x t ) = ° 1 + α 2 ± σ 2 e cov ( x t , x t + 1 ) = .... cov ( x t , x t + h ) = .... if h ° 2 MA( M ) °Moving average of degree M : x t = e t + α 1 e t ± 1 + ... + α M e t ± M Stationary for any M Y. Kryukov (CMU) 73-261: Time Series Nov-17 4 / 14
Autoregressive (AR) process AR(1): y t is AutoRegressive of degree 1 if y t = ρ y t ± 1 + e t Stationary if j ρ j < 1: E ( y t ) = .... , V ( y t ) = ² 1 1 ± ρ 2 ³ σ 2 e cov ( y

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32_TimeSeriesB_handout - Time Series II Wooldridge 11,18...

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