Nonstationary Time Series Data and
Cointegration
Prepared by Vera Tabakova, East Carolina University
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12.1 Stationary and Nonstationary Variables
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12.2 Spurious Regressions
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12.3 Unit Root Tests for Stationarity
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12.4 Cointegration
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12.5 Regression When There is No Cointegration
Principles of Econometrics, 3rd Edition
Slide 12-2
Figure 12.1(a) US economic time series
Principles of Econometrics, 3rd Edition
Slide 12-3
Figure 12.1(b) US economic time series
Principles of Econometrics, 3rd Edition
Slide 12-4
E  yt   
(12.1a)
var  yt   2
(12.1b)
cov  yt , yt s   cov  yt , yt s    s
Principles of Econometrics, 3rd Edition
(12.1c)
Slide 12-5
Principles of Econometrics, 3rd Edition
Slide 12-6
yt  yt 1  vt ,
 1
(12.2a)
y1  y0  v1
y2  y1  v2  (y0  v1 )  v2  2 y0  v1  v2
yt  vt  vt 1  2vt 2  .....  t y0
Principles of Econometrics, 3rd Edition
Slide 12-7
E[ yt ]  E[vt  vt 1  2vt 2  .....]  0
( yt  )  ( yt 1  )  vt
yt     yt 1  vt ,
Principles of Econometrics, 3rd Edition
 1
(12.2b)
Slide 12-8
E ( yt )     / (1  )  1/ (1  0.7)  3.33
( yt    t )  ( yt 1    (t  1))  vt ,
yt    yt 1  t  vt
Principles of Econometrics, 3rd Edition
 1
(12.2c)
Slide 12-9
Figure 12.2 (a) Time Series Models
Principles of Econometrics, 3rd Edition
Slide 12-10
Figure 12.2 (b) Time Series Models
Principles of Econometrics, 3rd Edition
Slide 12-11
Figure 12.2 (c) Time Series Models
Principles of Econometrics, 3rd Edition
Slide 12-12
yt  yt 1  vt
(12.3a)
y1  y0  v1
2
y2  y1  v2  ( y0  v1 )  v2  y0   vs
s 1
t
yt  yt 1  vt  y0   vs
s 1
Principles of Econometrics, 3rd Edition
Slide 12-13
E ( yt )  y0  E (v1  v2  ...  vt )  y0
var( yt )  var(v1  v2  ...  vt )  tv2
yt    yt 1  vt
Principles of Econometrics, 3rd Edition
(12.3b)
Slide 12-14
y1    y0  v1
2
y2    y1  v2    (  y0  v1 )  v2  2  y0   vs
s 1
t
yt    yt 1  vt  t  y0   vs
s 1
Principles of Econometrics, 3rd Edition
Slide 12-15
E ( yt )  t  y0  E (v1  v2  v3  ...  vt )  t  y0
var( yt )  var(v1  v2  v3  ...  vt )  tv2
yt    t  yt 1  vt
Principles of Econometrics, 3rd Edition
(12.3c)
Slide 12-16
y1      y0  v1;
(since t  1)
2
y2    2  y1  v2    2  (    y0  v1 )  v2  2  3  y0   vs
s 1
t
 t (t  1) 
yt    t  yt 1  vt  t  
   y0   vs
s 1
 2 
1 2  3 
t  t  t  1 2
Principles of Econometrics, 3rd Edition
Slide 12-17
rw1 : yt  yt 1  v1t
rw2 : xt  xt 1  v2t
rw1t  17.818  0.842 rw2t ,
(t )
Principles of Econometrics, 3rd Edition
R 2  .70
(40.837)
Slide 12-18
Figure 12.3 (a) Time Series of Two Random Walk Variables
Principles of Econometrics, 3rd Edition
Slide 12-19
Figure 12.3 (b) Scatter Plot of Two Random Walk Variables
Principles of Econometrics, 3rd Edition
Slide 12-20

12.3.1 Dickey-Fuller Test 1 (no constant and no trend)
yt  yt 1  vt
(12.4)
yt  yt 1  yt 1  yt 1  vt
yt     1 yt 1  vt
(12.5a)
  yt 1  vt
Principles of Econometrics, 3rd Edition
Slide 12-21

12.3.1 Dickey-Fuller Test 1 (no constant and no trend)
H0 :   1  H0 :   0
H1 :   1  H1 :   0
Principles of Econometrics, 3rd Edition
Slide 12-22
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12.3.2 Dickey-Fuller Test 2 (with constant but no trend)
yt     yt 1  vt
Principles of Econometrics, 3rd Edition
(12.5b)
Slide 12-23
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12.3.3 Dickey-Fuller Test 3 (with constant and with
trend)
yt    yt 1  t  vt
Principles of Econometrics, 3rd Edition
(12.5c)
Slide 12-24
First step: plot the time series of the original observations on the
variable.
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If the series appears to be wandering or fluctuating around a sample
average of zero, use test equation (12.5a).
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If the series appears to be wandering or fluctuating around a sample
average which is non-zero, use test equation (12.5b).
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If the series appears to be wandering or fluctuating around a linear
trend, use test equation (12.5c).
Principles of Econometrics, 3rd Edition
Slide 12-25
Principles of Econometrics, 3rd Edition
Slide 12-26

An important extension of the Dickey-Fuller test allows for the
possibility that the error term is autocorrelated.
m
yt     yt 1   as yt s  vt
(12.6)
s 1
yt 1   yt 1  yt 2  , yt 2   yt 2  yt 3  ,
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The unit root tests based on (12.6) and its variants (intercept excluded
or trend included) are referred to as augmented Dickey-Fuller tests.
Principles of Econometrics, 3rd Edition
Slide 12-27
Ft  0.178  0.037 Ft 1  0.672Ft 1
(tau )
(  2.090)
Bt  0.285  0.056 Bt 1  0.315Bt 1
(tau )
Principles of Econometrics, 3rd Edition
(  1.976)
Slide 12-28
yt  yt  yt 1  vt
  F t   0.340  F t 1
(tau )
(  4.007)
  B t   0.679  B t 1
(tau )
Principles of Econometrics, 3rd Edition
(  6.415)
Slide 12-29
eˆt  eˆt 1  vt
(12.7)
Case 1: eˆt  yt  bxt
(12.8a)
Case 2 : eˆt  yt  b2 xt  b1
(12.8b)
Case 3: eˆt  yt  b2 xt  b1  ˆ t
(12.8c)
Principles of Econometrics, 3rd Edition
Slide 12-30
Principles of Econometrics, 3rd Edition
Slide 12-31
Bˆt  1.644  0.832 Ft , R 2  0.881
(12.9)
(t ) (8.437) (24.147)
eˆt  0.314eˆt 1  0.315eˆt 1
(tau ) (4.543)
Principles of Econometrics, 3rd Edition
Slide 12-32
The null and alternative hypotheses in the test for cointegration are:
H 0 : the series are not cointegrated  residuals are nonstationary
H1 : the series are cointegrated  residuals are stationary
Principles of Econometrics, 3rd Edition
Slide 12-33

12.5.1 First Difference Stationary
yt  yt 1  vt
yt  yt  yt 1  vt
The variable yt is said to be a first difference stationary series.
Principles of Econometrics, 3rd Edition
Slide 12-34
yt  yt 1  0 xt  1xt 1  et
(12.10a)
yt    yt 1  vt
yt    vt
yt    yt 1  0 xt  1xt 1  et
Principles of Econometrics, 3rd Edition
(12.10b)
Slide 12-35
yt    t  vt
yt    t  vt
yt   yt1 0 xt 1xt1  et
(12.11)
yt     t   yt 1  0 xt  1 xt 1  et
where   1 (1  1 )   2 (0  1 )  11  1 2
and
  1 (1  1 )  2 (0  1 )
Principles of Econometrics, 3rd Edition
Slide 12-36
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
Slide 12-37
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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
Slide 12-38