1. Nonstationary Time Series Data and Cointegration
modified JJ
Prepared by Vera Tabakova, East Carolina University

2. Chapter 12: Nonstationary Time Series Data and Cointegration
12.1 Stationary and Nonstationary Variables
12.2 Spurious Regressions
12.3 Unit Root Tests for Stationarity
12.4 Cointegration
12.5 Regression When There is No Cointegration
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3. 12.1 Stationary and Nonstationary Variables
Figure 12.1(a) US economic time series
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4. 12.1 Stationary and Nonstationary Variables
Figure 12.1(b) US economic time series
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5. 12.1 Stationary and Nonstationary Variables
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6. 12.1.1 The First-Order Autoregressive Model
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7. 12.1.1 The First-Order Autoregressive Model
(12.2b)
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8. 12.1.1 The First-Order Autoregressive Model
(12.2c)
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9. 12.1.1 The First-Order Autoregressive Model
Figure 12.2 (a) Time Series Models
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10. 12.1.1 The First-Order Autoregressive Model
Figure 12.2 (b) Time Series Models
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11. 12.1.1 The First-Order Autoregressive Model
Figure 12.2 (c) Time Series Models
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12. Random Walk Models
(12.3a)
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13. Random Walk Models
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14. (12.3b)
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15. Random Walk Models
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16. Random Walk Models
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17. (12.3c)
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18. Random Walk Models

19. UNIT ROOT Process-Estimation
If the true data generating process is a random walk and you estimate an AR(1) by regressing y(t) on y(t-1), the OLS estimator is no longer asymptotically normally distributed and does not converge at the standard rate sqrt(T).
Instead it converges to a limiting NON-NORMAL distribution that is tabulated by Dickey-Fuller
Also , the OLS estimator converges at rate T and is SUPERCONSISTENT
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20. UNIT ROOT Process-Estimation
AS a consequence, the t-ratio statistics are not asymtptically normally distributed Instead, they have a NON-NORMAL limiting distribution, that was tabulated by Dickey-Fuller If you regress one unit root process on another the outcomes are meaningless (spurious) unless the processes are cointegrated
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21. 12.2 Spurious Regressions
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22. Figure 12.3 (a) Time Series of Two Random Walk Variables
12.2 Spurious Regressions
Figure 12.3 (a) Time Series of Two Random Walk Variables
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23. Figure 12.3 (b) Scatter Plot of Two Random Walk Variables
12.2 Spurious Regressions
Figure 12.3 (b) Scatter Plot of Two Random Walk Variables
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24. 12.3 Unit Root Test for Stationarity
Dickey-Fuller Test 1 (no constant and no trend)
(12.4)
(12.5a)
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25. 12.3 Unit Root Test for Stationarity
Dickey-Fuller Test 1 (no constant and no trend)
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26. 12.3 Unit Root Test for Stationarity
Dickey-Fuller Test 2 (with constant but no trend)
(12.5b)
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27. 12.3 Unit Root Test for Stationarity
Dickey-Fuller Test 3 (with constant and with trend)
(12.5c)
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28. 12.3.4 The Dickey-Fuller Testing Procedure
First step: plot the time series of the original observations on the variable.
If the series appears to be wandering or fluctuating around a sample average of zero, use test equation (12.5a).
If the series appears to be wandering or fluctuating around a sample average which is non-zero, use test equation (12.5b).
If the series appears to be wandering or fluctuating around a linear trend, use test equation (12.5c).
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29. D-F test
In each case you consider the t-statistic on the coefficient of y(t-1), called gamma, which is equal to zero if the true data generating process process is a unit root process
That t-ratio is called “tau” and the critical values for tau are given in the table:
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30. 12.3.4 The Dickey-Fuller Testing Procedure
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31. 12.3.4 The Dickey-Fuller Testing Procedure
An important extension of the Dickey-Fuller test allows for testing for unit root in AR(p) processes: AUGMENTED Dickey-Fuller.
The unit root tests with the intercept excluded or trend included in AR(p) have the same critical values of tau as the AR(1)
(12.6)
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32. 12.3.5 The Dickey-Fuller Tests: An Example
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33. Order of Integration
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34. COINTEGRATION (Engle-Granger)
A regression of a unit root process on another(s) makes sense ONLY if they
are cointegrated i.e. satisfy a long run equilibrium and any departure from that equilibrium is eliminated by an Error Correction Model .
If the processes are cointegrated, the residuals of the regression are stationary
That regression represents the long run equilibrium
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35. 12.4 Cointegration (12.7) (12.8a) (12.8b) (12.8c)
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36. 12.4 Cointegration
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37. 12.4.1 An Example of a Cointegration Test
(12.9)
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38. 12.4.1 An Example of a Cointegration Test
The null and alternative hypotheses in the test for cointegration are:
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39. 12.5 Regression When There Is No Cointegration
First Difference Stationary
The variable yt is said to be a first difference stationary series.
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40. 12.5.1 First Difference Stationary
(12.10b)
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41. 12.5.2 Trend Stationary where and (12.11)
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42. Trend Stationary
To summarize:
If variables are stationary, or I(1) and cointegrated, we can estimate a regression relationship between the levels of those variables without fear of encountering a spurious regression.
If the variables are I(1) and not cointegrated, we need to estimate a relationship in first differences, with or without the constant term.
If they are trend stationary, we can either de-trend the series first and then perform regression analysis with the stationary (de-trended) variables or, alternatively, estimate a regression relationship that includes a trend variable. The latter alternative is typically applied.
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43. Keywords Augmented Dickey-Fuller test Autoregressive process
Cointegration
Dickey-Fuller tests
Mean reversion
Order of integration
Random walk process
Random walk with drift
Spurious regressions
Stationary and nonstationary
Stochastic process
Stochastic trend
Tau statistic
Trend and difference stationary
Unit root tests
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