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Lecture_5_ARMA_Falll_2009

# Lecture_5_ARMA_Falll_2009 - Lecture 5 Univariate time...

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Unformatted text preview: Lecture 5 Univariate time series modeling and forecasting (Box-Jenkins Method) Read: (WK Ch 7, Enders Ch 1,2) EC 413/513 Economic Forecast and Analysis (Professor Lee) Univariate Time Series Models Where we attempt to predict returns using ONE variable based on the information contained in their past values. We omit a trend and seasonality for simplicity, but can include them in ARMA time series models. We are focused on other stochastic factors. Overview Sta tionary ARMA (p, q) Model What is the ARMA model? y t = c + α 1 y t-1 + α 2 y t-2 + .. + α p y t-p + θ 1 u t-1 + θ 2 u t-2 + ... + θ q u t-q + u t [ ] [--------------- AR ------------- ] [--------- MA ------------------ ] [error] .. ARMA( p, q ) model Add Independent variables β X t to have ARMA- X models. y t = c + β X t + α 1 y t-1 + θ 1 u t-1 + u t .. ARMA(1,1)-X What is AR model? • An autoregressive model of order p , an AR( p ) can be expressed as y t = c + α 1 y t-1 + α 2 y t-2 + .. + α p y t-p + u t .. AR( p ) model e.g.) When we omit c for simplicity (mean zero). y t = α 1 y t-1 + u t AR(1) y t = α 1 y t-1 + α 2 y t-2 + u t AR(2) 2 What is MA model? • The q th order MA model is given as a function of past errors : y t = c + u t + θ 1 u t-1 + θ 2 u t-2 + ... + θ q u t-q .. MA( q ) model e.g.) When we omit c for simplicity (mean zero), we have y t = θ 1 u t-1 + u t MA(1) y t = θ 1 u t-1 + θ 2 u t-2 + u t MA(2) What is a White Noise Process, u t ? • A white noise process is one with (virtually) no discernible structure. It is an independently and identically distributed (iid) random variables… purely random ! • A white noise process, u t ( t = 1,2,3,...), is a sequence of independently and identically distributed (iid) random variables with E( u t ) = 0 Var( u t ) = σ 2 . • Thus, the ACFs and PACFs (see below) are all zero (not significantly different from 0). The plots of ACFs and PACFs will lie inside of the Confidence Interval (see below). Key Point of Estimating ARMA Models: → Allow for enough AR(p) & MA(q) terms so that the error term looks like a white noise process . ARIMA (p, d , q) Model If y t is non-stationary , we take a first-difference of y t so that ∆ y t becomes stationary. 3 ∆ y t = y t- y t-1 (d = 1 implies one time differencing. d = 1 is enough in most cases.) ∆ y t = c + α 1 ∆ y t-1 + .. + α p ∆ y t-p + θ 1 u t-1 + ... + θ q u t-q + u t .. ARIMA( p, 1 , q ) model Note: When d = 0, no difference is needed. We simply denote ARMA(p,q), instead of ARIMA(p,0,q). It is called an I(0) process. Important Remark: If the data is non-stationary (d = 1), the plot of ACFs is relatively flat and decline slowly. Unit root = non-stationary = needs first differencing = integrated process = I(1) process = needs AR I MA models Note: In unusual cases, we use a second difference (d = 2), implying a first difference of the first differenced series ∆ 2 y t = ∆ y t- ∆ y t- 1 = ( y t- y t- 1 29- ( y t- 1- y t- 2 ) = y t- 2 y t- 1 + y t-...
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Lecture_5_ARMA_Falll_2009 - Lecture 5 Univariate time...

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