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Unformatted text preview: elation function given by
ρ(s) =
ˆ ¯
¯
(Xt − X )(Xt +s − X )
¯
(Xt − X )2 ¯
(Xt +s − X )2 If the β  < 1 then this quantity approximates the true
autocorrelation function.
26 / 52 Introduction Stationary Processes Nonstationary Processes Spurious Regressions Testing for Nonstationarity Cointegration
Autocorrelation function ADF Autocorrelation function 27 / 52 Introduction Stationary Processes Nonstationary Processes Spurious Regressions Testing for Nonstationarity Cointegration
Autocorrelation function ADF Autocorrelation function 28 / 52 Introduction Stationary Processes Nonstationary Processes Spurious Regressions Testing for Nonstationarity Cointegration
Autocorrelation function ADF Autocorrelation function The problems are
It is hard to differentiate between processes where β2 = 1
which is nonstationary and β2 = 0.95 which is stationary.
It is hard to interpret strange graphs which may result from
random walk 29 / 52 Introduction Stationary Processes Nonstationary Processes Spurious Regressions Testing for Nonstationarity Cointegration
Autocorrelation function ADF ADF
Suppose we have the model:
Xt = β2 Xt −1 + εt
We would like to test the hypothesis:
H0 : β2 = 1 vs H1 : β2 < 1
Note that we are not interested in negative unit roots or β > 1.
To test this hypothesis, we use a trick  we can transform the
equation:
∆Xt = (β2 − 1)Xt −1 + εt 30 / 52 Introduction Stationary Processes Nonstationary Processes Spurious Regressions Testing for Nonstationarity Cointegration
Autocorrelation function ADF ADF
Suppose we have the model:
Xt = β2 Xt −1 + εt
We would like to test the hypothesis:
H0 : β2 = 1 vs H1 : β2 < 1
Note that we are not interested in negative unit roots or β > 1.
To test this hypothesis, we use a trick  we can transform the
equation:
∆Xt = (β2 − 1)Xt −1 + εt 31 / 52 Introduction Stationary Processes Nonstationary Processes Spurious Regressions Testing for Nonstationarity Cointegration
Autocorrelation function ADF ADF
Suppose we have the model:
Xt = β2 Xt −1 + εt
We would like to test the hypothesis:
H0 : β2 = 1 vs H1 : β2 < 1
Note that we are not interested in negative unit roots or β > 1.
To test this hypothesis, we use a trick  we can transform the
equation:
∆Xt = (β2 − 1)Xt −1 + εt 32 / 52 Introduction Stationary Processes Nonstationary Processes Spurious Regressions Testing for Nonstationarity Cointegration
Autocorrelation function ADF ADF ∆Xt = (β2 − 1)Xt −1 + εt
We run this regression with OLS and we look at the coefﬁcient
of (β2 − 1). The hypothesis now is:
H0 : β2 − 1 = 0 vs H1 : β2 − 1 < 0
The test statistic is the Augmented Dickey Fuller statistic. The
problem with this statistic is that is has low power. It is hard to
differentiate between processes where β2 = 1 which is
nonstationary and β2 = 0.95 which is stationary.
One can allow for a time trend and drift term as well. 33 / 52 Introduction Stationary Processes Nonstationary Processes Spurious Regressions Testing for Nonstationarity Cointegration
Autocorrelation function ADF ADF ∆Xt = (β2...
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This document was uploaded on 03/12/2014 for the course ECON 202 at University of London University of London International Programmes (Distance Learning).
 Spring '13
 ChristopherDougherty
 Econometrics

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