Chapter 12

Chapter 12

Nonstationary Time Series Data and Cointegration

Prepared by Vera Tabakova, East Carolina University

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

Slide 12-2Principles of Econometrics, 3rd Edition

12.1 Stationary and Nonstationary Variables

Figure 12.1(a) US economic time series



Figure 12.1(b) US economic time series




(12.1a)

(12.1b)

(12.1c)

tE y

2var ty

cov , cov ,t t s t t s sy y y y

12.1.1 The First-Order Autoregressive Model


(12.2a)1 , 1t t ty y v

1 0 1

22 1 2 0 1 2 0 1 2

21 2 0

( )

..... tt t t t

y y v

y y v y v v y v v

y v v v y



(12.2b)

21 2[ ] [ .....] 0t t t tE y E v v v

1( ) ( )t t ty y v

1 , 1t t ty y v



(12.2c)

( ) / (1 ) 1/ (1 0.7) 3.33tE y

1( ) ( ( 1)) , 1 t t ty t y t v

1t t ty y t v


Figure 12.2 (a) Time Series Models



Figure 12.2 (b) Time Series Models



Figure 12.2 (c) Time Series Models


12.1.2 Random Walk Models


(12.3a)1t t ty y v

1 0 1

2

2 1 2 0 1 2 01

1 01

( )

ss

t

t t t ss

y y v

y y v y v v y v

y y v y v



(12.3b)

0 1 2 0( ) ( ... )t tE y y E v v v y

21 2var( ) var( ... )t t vy v v v t

1t t ty y v



1 0 1

2

2 1 2 0 1 2 01

1 01

( ) 2

ss

t

t t t ss

y y v

y y v y v v y v

y y v t y v



(12.3c)

0 1 2 3 0( ) ( ... )t tE y t y E v v v v t y

21 2 3var( ) var( ... )t t vy v v v v t

1t t ty t y v



1 0 1

2

2 1 2 0 1 2 01

1 01

; (since 1)

2 2 ( ) 2 3

( 1)2

ss

t

t t t ss

y y v t

y y v y v v y v

t ty t y v t y v

1 2 3 1 2t t t

12.2 Spurious Regressions


1 1 1

2 1 2

:

:

t t t

t t t

rw y y v

rw x x v

21 217.818 0.842 , .70

( ) (40.837)t trw rw R

t


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



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


12.3 Unit Root Test for Stationarity

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


(12.4)1t t ty y v

(12.5a)

1 1 1

1

1

1

t t t t t

t t t

t t

y y y y v

y y v

y v


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


0 0

1 1

: 1 : 0

: 1 : 0

H H

H H


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


(12.5b)1t t ty y v


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


(12.5c)1t t ty y t v

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).



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)11

m

t t s t s ts

y y a y v

1 1 2 2 2 3, ,t t t t t ty y y y y y

12.3.5 The Dickey-Fuller Tests: An Example


1 1

1 1

0.178 0.037 0.672

( ) ( 2.090)

0.285 0.056 0.315

( ) ( 1.976)

t t t

t t t

F F F

tau

B B B

tau

12.3.6 Order of Integration


1t t t ty y y v

10.340

( ) ( 4.007)

t tF F

tau

10.679

( ) ( 6.415)

t tB B

tau

12.4 Cointegration


(12.7)1ˆ ˆt t te e v

(12.8c)

(12.8b)

(12.8a)ˆ1: t t tCase e y bx

2 1ˆ2 : t t tCase e y b x b

2 1ˆˆ3: t t tCase e y b x b t

12.4 Cointegration


12.4.1 An Example of a Cointegration Test


(12.9)2ˆ 1.644 0.832 , 0.881

( ) (8.437) (24.147)t tB F R

t

1 1ˆ ˆ ˆ0.314 0.315( ) ( 4.543)

t t te e etau

12.4.1 An Example of a Cointegration Test

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


0

1

: the series are not cointegrated residuals are nonstationary

: the series are cointegrated residuals are stationary

H

H

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.


1t t ty y v

1t t t ty y y v

12.5.1 First Difference Stationary


(12.10a)1 0 1 1t t t t ty y x x e

1t t ty y v

t ty v

(12.10b)1 0 1 1t t t t ty y x x e

12.5.2 Trend Stationary

where

and


(12.11)

t ty t v

t ty t v

1 0 1 1t t t t ty y x x e

1 0 1 1t t t t ty t y x x e

1 1 2 0 1 1 1 1 2(1 ) ( )

1 1 2 0 1(1 ) ( )

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.


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

Chapter 12

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