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Unformatted text preview: Dougherty: Introduction to Econometrics 3e Study Guide Chapter 13 Introduction to nonstationary time series Overview This chapter begins by defining the concepts of stationarity and nonstationarity as applied to univariate time series and, in the case of nonstationary series, the concepts of difference-stationarity and trend-stationarity. It next describes the consequences of nonstationarity for models fitted using nonstationary time-series data and gives an account of the Granger–Newbold Monte Carlo experiment with random walks. Next the two main methods of detecting nonstationarity in time series are described, the graphical approach using correlograms and the more formal approach using Augmented Dickey–Fuller unit root tests. This leads to the topic of cointegration. The chapter concludes with a discussion of methods for fitting models using nonstationary time series: detrending, differencing, and error-correction models. Further material Generalization of the Augmented Dickey Fuller test for unit roots In Section 13.3 it was shown how a Dickey–Fuller test could be used to detect a unit root in the process t t t t X X ε γ β β + + + = − 1 2 1 and an Augmented Dickey–Fuller test could be used for the same purpose when the process included an additional lagged value of X : t t t t t X X X ε γ β β β + + + + = − − 2 3 1 2 1 In principle the process may have further lags, the general form being t p s s t s t t X X ε γ β β + + + = ∑ = − + 1 1 1 One condition for stationarity is that the sum of the coefficients of the lagged X variables should be less than one. Writing our test statistic θ as 1 1 1 − = ∑ = + p s s β θ the null hypothesis of nonstationarity is H : θ = 0 and the alternative hypothesis of stationarity is H 1 : θ < 0. Two issues now arise. One is how to reparameterize the specification so that we can obtain a direct estimate of θ . The other is how to determine the appropriate value of p . From the definition of θ , we have ∑ = + − + = p s s 2 1 2 1 β θ β Substituting this into the original specification, we have t p s s t s t p s s t t X X X ε γ β β θ β + + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + + = ∑ ∑ = − + − = + 2 1 1 2 1 1 1 © Christopher Dougherty, 2007 The material in this book has been adapted and developed from material originally produced for the degrees and diplomas by distance learning offered by the University of London External System ( www.londonexternal.ac.uk ) Dougherty: Introduction to Econometrics 3e Study Guide Hence ( ) ( ) ( ) ( ) t p t p t p p s t t s p s t t s t t p s s t s p s s t p s t t s t t p s s t s p s s t p s s t p s s t t t p s s t s p s s t t t t t X X X X X X X t X X X X X t X X X X X t X X X X X ε γ β β β θ β ε γ β β β θ β ε γ β β β β θ β ε γ β β θ β + + − − − − − − − + = + + + − − − + = + + + − + − + = + + + − + = − − + − = − − + = − − + − = − + = + − = − − + −...
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This note was uploaded on 05/26/2010 for the course ECON 301 taught by Professor Öcal during the Spring '10 term at Middle East Technical University.
- Spring '10