Lec18.pdf - ECO441K Introduction to Econometrics Chapter...

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ECO441K Introduction to Econometrics Chapter 10/11: Regression with Time Series Data Stephen Donald Stephen Donald () Chapter 10/11: Time Series 1 / 26
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Lecture 18 - Outline Stationarity for a Time Series Process Lagged dependent variables and autoregression models Random Walk model Using Non-Stationary Variables in Regression Serial correlation and e/ect on OLS Tests for serial correlation Stephen Donald () Chapter 10/11: Time Series 2 / 26
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Stationarity Many practical and theoretical results depend on whether or not a Time Series process is °Weakly Stationary± Examples: Large Sample Results Regression Analysis involving such Time Series Variables Permanence of shocks Stephen Donald () Chapter 10/11: Time Series 3 / 26
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Stationarity Let y t for t = 1 , .... be a time series process Typically we will have the ²rst T components of this process in our data eg: GDP, Dow Jones value etc Could be any sort of time scale ³annual, quarterly, monthly, daily, second, nano-second ..... Stephen Donald () Chapter 10/11: Time Series 4 / 26
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Stationarity For such a process we can de²ne the mean, variance and autocovariance The mean is simply μ t = E ( y t ) The variance is simply, σ 2 t = E (( y t ° μ t ) 2 ) and the auto-covariance is, γ t , s = E (( y t ° μ t )( y s ° μ s )) Stephen Donald () Chapter 10/11: Time Series 5 / 26
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Stationarity Basically a process is °Weakly Stationary± if 1 μ t = μ for all t 2 σ 2 t = σ 2 for all t 3 γ t , s = γ j t ° s j Basically the mean and variance are constant ³not time dependent The auto-covariance can depend on the di/erence between the time indices ³that is, γ t + p , t = γ s + p , s = γ p There can be dependence in the process but it only depends on the lagged di/erence Stephen Donald () Chapter 10/11: Time Series 6 / 26
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Weak Dependence Without Random Sampling we need data to be °Weakly Dependent± in order to apply LLN and CLT to allow inference A series is °Weakly Dependent± and Stationary if in addition to the conditions for stationarity we have that, Corr ( y t , y t ° p ) = γ p σ 2 !
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