Chapter 12 modified JJ

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Chapter 12 modified JJ

• Nonstationary Time Series Data and Cointegration

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

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12.1 Stationary and Nonstationary Variables

• Figure 12.1(a) US economic time series

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12.1 Stationary and Nonstationary Variables

• Figure 12.1(b) US economic time series

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12.1 Stationary and Nonstationary Variables

(12.1a)
(12.1b)
(12.1c)
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12.1 Stationary and Nonstationary Variables

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12.1.1 The First-Order Autoregressive Model

(12.2a)
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12.1.1 The First-Order Autoregressive Model

(12.2b)
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12.1.1 The First-Order Autoregressive Model

(12.2c)
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12.1.1 The First-Order Autoregressive Model

• Figure 12.2 (a) Time Series Models

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12.1.1 The First-Order Autoregressive Model

• Figure 12.2 (b) Time Series Models

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12.1.1 The First-Order Autoregressive Model

• Figure 12.2 (c) Time Series Models

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12.1.2 Random Walk Models

(12.3a)
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12.1.2 Random Walk Models

(12.3b)
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12.1.2 Random Walk Models

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12.1.2 Random Walk Models

(12.3c)
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12.1.2 Random Walk Models
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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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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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12.2 Spurious Regressions

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12.2 Spurious Regressions

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

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12.2 Spurious Regressions

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

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12.3 Unit Root Test for Stationarity

• 12.3.1 Dickey-Fuller Test 1 (no constant and no
  trend)

(12.4)
(12.5a)
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12.3 Unit Root Test for Stationarity

• 12.3.1 Dickey-Fuller Test 1 (no constant and no
  trend)

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12.3 Unit Root Test for Stationarity

• 12.3.2 Dickey-Fuller Test 2 (with constant but no
  trend)

(12.5b)
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12.3 Unit Root Test for Stationarity

• 12.3.3 Dickey-Fuller Test 3 (with constant and
  with trend)

(12.5c)
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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).

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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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12.3.4 The Dickey-Fuller Testing Procedure

• Th

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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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12.3.5 The Dickey-Fuller Tests An Example

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12.3.6 Order of Integration

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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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12.4 Cointegration
(12.7)
(12.8a)
(12.8b)
(12.8c)
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12.4 Cointegration
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12.4.1 An Example of a Cointegration Test
(12.9)
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12.4.1 An Example of a Cointegration Test

• The null and alternative hypotheses in the test
  for cointegration are

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12.5 Regression When There Is No Cointegration

• 12.5.1 First Difference Stationary
• The variable yt is said to be a first difference
  stationary series.

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12.5.1 First Difference Stationary
(12.10a)
(12.10b)
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12.5.2 Trend Stationary

• where
• and

(12.11)
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12.5.2 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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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