1. Nonstationary Time Series Data and Cointegration
Chapter 12
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
Principles of Econometrics, 3rd Edition

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
(12.1c)
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6. 12.1 Stationary and Nonstationary Variables
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7. 12.1.1 The First-Order Autoregressive Model
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8. 12.1.1 The First-Order Autoregressive Model
(12.2b)
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9. 12.1.1 The First-Order Autoregressive Model
(12.2c)
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10. 12.1.1 The First-Order Autoregressive Model
Figure 12.2 (a) Time Series Models
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11. 12.1.1 The First-Order Autoregressive Model
Figure 12.2 (b) Time Series Models
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12. 12.1.1 The First-Order Autoregressive Model
Figure 12.2 (c) Time Series Models
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13. Random Walk Models
(12.3a)
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14. Random Walk Models
(12.3b)
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15. Random Walk Models
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16. Random Walk Models
(12.3c)
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17. Random Walk Models
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18. 12.2 Spurious Regressions
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19. 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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20. 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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21. 12.3 Unit Root Test for Stationarity
Dickey-Fuller Test 1 (no constant and no trend)
(12.4)
(12.5a)
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22. 12.3 Unit Root Test for Stationarity
Dickey-Fuller Test 1 (no constant and no trend)
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23. 12.3 Unit Root Test for Stationarity
Dickey-Fuller Test 2 (with constant but no trend)
(12.5b)
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24. 12.3 Unit Root Test for Stationarity
Dickey-Fuller Test 3 (with constant and with trend)
(12.5c)
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25. 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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26. 12.3.4 The Dickey-Fuller Testing Procedure
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27. 12.3.4 The Dickey-Fuller Testing Procedure
An important extension of the Dickey-Fuller test allows for the possibility that the error term is autocorrelated.
The unit root tests based on (12.6) and its variants (intercept excluded or trend included) are referred to as augmented Dickey-Fuller tests.
(12.6)
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28. 12.3.5 The Dickey-Fuller Tests: An Example
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29. Order of Integration
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30. 12.4 Cointegration (12.7) (12.8a) (12.8b) (12.8c)
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31. 12.4 Cointegration
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32. 12.4.1 An Example of a Cointegration Test
(12.9)
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33. 12.4.1 An Example of a Cointegration Test
The null and alternative hypotheses in the test for cointegration are:
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34. 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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35. 12.5.1 First Difference Stationary
(12.10b)
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36. 12.5.2 Trend Stationary where and (12.11)
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37. 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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38. 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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