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**Nonstationary Time Series Data and Cointegration**

Chapter 12 Prepared by Vera Tabakova, East Carolina University

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**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

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**12.1 Stationary and Nonstationary Variables**

Figure 12.1(a) US economic time series Principles of Econometrics, 3rd Edition

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**12.1 Stationary and Nonstationary Variables**

Figure 12.1(b) US economic time series Principles of Econometrics, 3rd Edition

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**12.1 Stationary and Nonstationary Variables**

(12.1c) Principles of Econometrics, 3rd Edition

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**12.1 Stationary and Nonstationary Variables**

Principles of Econometrics, 3rd Edition

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**12.1.1 The First-Order Autoregressive Model**

Principles of Econometrics, 3rd Edition

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**12.1.1 The First-Order Autoregressive Model**

(12.2b) Principles of Econometrics, 3rd Edition

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**12.1.1 The First-Order Autoregressive Model**

(12.2c) Principles of Econometrics, 3rd Edition

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**12.1.1 The First-Order Autoregressive Model**

Figure 12.2 (a) Time Series Models Principles of Econometrics, 3rd Edition

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**12.1.1 The First-Order Autoregressive Model**

Figure 12.2 (b) Time Series Models Principles of Econometrics, 3rd Edition

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**12.1.1 The First-Order Autoregressive Model**

Figure 12.2 (c) Time Series Models Principles of Econometrics, 3rd Edition

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Random Walk Models (12.3a) Principles of Econometrics, 3rd Edition

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Random Walk Models (12.3b) Principles of Econometrics, 3rd Edition

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Random Walk Models Principles of Econometrics, 3rd Edition

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Random Walk Models (12.3c) Principles of Econometrics, 3rd Edition

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Random Walk Models Principles of Econometrics, 3rd Edition

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12.2 Spurious Regressions Principles of Econometrics, 3rd Edition

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**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 Principles of Econometrics, 3rd Edition

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**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 Principles of Econometrics, 3rd Edition

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**12.3 Unit Root Test for Stationarity**

Dickey-Fuller Test 1 (no constant and no trend) (12.4) (12.5a) Principles of Econometrics, 3rd Edition

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**12.3 Unit Root Test for Stationarity**

Dickey-Fuller Test 1 (no constant and no trend) Principles of Econometrics, 3rd Edition

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**12.3 Unit Root Test for Stationarity**

Dickey-Fuller Test 2 (with constant but no trend) (12.5b) Principles of Econometrics, 3rd Edition

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**12.3 Unit Root Test for Stationarity**

Dickey-Fuller Test 3 (with constant and with trend) (12.5c) Principles of Econometrics, 3rd Edition

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**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). Principles of Econometrics, 3rd Edition

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**12.3.4 The Dickey-Fuller Testing Procedure**

Principles of Econometrics, 3rd Edition

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**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) Principles of Econometrics, 3rd Edition

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**12.3.5 The Dickey-Fuller Tests: An Example**

Principles of Econometrics, 3rd Edition

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Order of Integration Principles of Econometrics, 3rd Edition

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**12.4 Cointegration (12.7) (12.8a) (12.8b) (12.8c)**

Principles of Econometrics, 3rd Edition

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12.4 Cointegration Principles of Econometrics, 3rd Edition

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**12.4.1 An Example of a Cointegration Test**

(12.9) Principles of Econometrics, 3rd Edition

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**12.4.1 An Example of a Cointegration Test**

The null and alternative hypotheses in the test for cointegration are: Principles of Econometrics, 3rd Edition

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**12.5 Regression When There Is No Cointegration**

First Difference Stationary The variable yt is said to be a first difference stationary series. Principles of Econometrics, 3rd Edition

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**12.5.1 First Difference Stationary**

(12.10b) Principles of Econometrics, 3rd Edition

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**12.5.2 Trend Stationary where and (12.11)**

Principles of Econometrics, 3rd Edition

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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. Principles of Econometrics, 3rd Edition

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**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 Principles of Econometrics, 3rd Edition

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