Stationary, Nonstationary

Stationary, Nonstationary - ApEc 8212 Applied Econometrics...

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1 ApEc 8212 Applied Econometrics -- Lecture #28 Time Series Analysis VII: Trends in Time Series Data (Enders, Chap. 4, Sections 1-7) I. Deterministic and Stochastic Trends In general, stochastic difference equations have 3 parts: y t = trend + stationary component + “noise” So far we have focused on the stationary component (Enders, Chapter 2) and on the noise (Enders, Chap. 3), assuming no trend . Yet many (most?) times series have trends and so are not stationary. This lecture shows how to analyze data that have trends. We can define a trend as anything that causes a time series variable not to have a long-run mean . Perhaps the simplest example of a time series with a trend is: y t = a 0 The solution to this linear difference equation is: y t = y 0 + a 0 t where y 0 is the initial condition at time period zero.
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2 Next, add a moving average component to this model: y t = y 0 + a 0 t + B (L)ε t = y 0 + a 0 t + ε t + β 1 ε t-1 + … + β p ε t-p Question: Is the moving average part, B (L)ε t , stationary? In this equation y t differs from its trend value (= y 0 + a 0 t) by B (L)ε t . B (L)ε t is stationary, so this deviation from trend is temporary. This model is called trend stationary : after removing the trend part, what remains is stationary. Trends can be both “ deterministic ” (non-stochastic) and stochastic . A simple example of the latter is: y t = a 0 + ε t where ε t is white noise (Var t ) = σ 2 ). This may seem to be a small change, but it has a big effect on y t . You can verify that the general solution to this difference equation is: y t = y 0 + Σ = t 1 i ε i + a 0 t Think of Σ = t 1 i ε t as a stochastic (random) intercept. Note: the effects of past ε’ s do not “decay” over time; these
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3 effects are “permanent”. Such a process has a stochastic trend (in addition to a 0 t, the deterministic trend): each ε t permanently changes the (conditional) mean of the series. Question : Is the y 0 + Σ = t 1 i ε t part of y t stationary? Four Time Series with Different Trends -8 -6 -4 -2 0 2 4 6 8 0 10 20 30 40 50 60 70 80 90 100 Time Random walk 0 8 16 24 32 40 48 56 0 10 20 30 40 50 60 70 80 90 100 Time Random walk plus drift 0 8 16 24 32 40 48 56 0 10 20 30 40 50 60 70 80 90 100 Time Trend stationary -8 -6 -4 -2 0 2 4 6 8 0 10 20 30 40 50 60 70 80 90 100 Time Random walk plus noise 1. Random Walk Model A special case of the stochastic trend model given above is the case where a 0 = 0. This gives the random walk :
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4 y t = y t-1 + ε t , which implies y t = ε t This is an AR(1) model, y t = a 0 + a 1 y t-1 + ε t , with a 0 = 0 1 = 1. The general solution, conditional on y t , is: y t = y 0 + Σ = t i 1 ε i The unconditional expectation (mean) is E[y t ] = y 0 . The conditional expectations, at time t, of y t+1 and y t+s are: E t [y t+1 ] = E t [y t + ε t+1 ] = y t E t [y t+s ] = E t [y t + Σ = s i 1 ε t+i ] = y t Random walk processes are not stationary because the (unconditional) variance is not constant over all t: Var(y t ) = Var( ε t + ε
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This note was uploaded on 02/24/2010 for the course ECON 570 taught by Professor Staff during the Fall '08 term at UNC.

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Stationary, Nonstationary - ApEc 8212 Applied Econometrics...

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