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

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Presentation on theme: "Nonstationary Time Series Data and Cointegration Prepared by Vera Tabakova, East Carolina University."— Presentation transcript:

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

2  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

3 Figure 12.1(a) US economic time series

4 Figure 12.1(b) US economic time series

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10 Figure 12.2 (a) Time Series Models

11 Figure 12.2 (b) Time Series Models

12 Figure 12.2 (c) Time Series Models

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18 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 Principles of Econometrics, 3rd Edition

19 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 Principles of Econometrics, 3rd Edition

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21 Figure 12.3 (a) Time Series of Two Random Walk Variables

22 Figure 12.3 (b) Scatter Plot of Two Random Walk Variables

23  Dickey-Fuller Test 1 (no constant and no trend)

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25  Dickey-Fuller Test 2 (with constant but no trend)

26  Dickey-Fuller Test 3 (with constant and with trend)

27 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).

28  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: Principles of Econometrics, 3rd Edition

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30  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)

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33  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 Principles of Econometrics, 3rd Edition

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37 The null and alternative hypotheses in the test for cointegration are:

38  First Difference Stationary The variable y t is said to be a first difference stationary series.

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40 where and

41 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.

42 Slide Principles of Econometrics, 3rd Edition  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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