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

 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 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

  t
  var
  t
  2 cov
 t
, t s
  cov
 t
, t s
   s
Principles of Econometrics, 3rd Edition
(12.1a)
(12.1b)
(12.1c)
Slide 12-5

Principles of Econometrics, 3rd Edition Slide 12-6

y t
  y t

1
 v t
,
 
1 y
1
  
0 v
1 y
2 y
1 v
2
( y
0 v
1
)
 v
2
  2 y
0 v
1 v
2
(12.2a) y t v t v t

1
 
2 v t

2

.....
  t y
0
Principles of Econometrics, 3rd Edition Slide 12-7

[ ] t

[ t
 v t

1
 2 v t

2

.....] 0
( y t
) ( y t

1
) v t y t
    y t

1
 v t
,
 
1 (12.2b)
Principles of Econometrics, 3rd Edition Slide 12-8

( y t
( ) t
   
/ (1 ) 1 / (1 0.7)

3.33
t ) ( y t

1
   
1))
 v t
,
 
1 y t
    y t

1 t v t
(12.2c)
Principles of Econometrics, 3rd Edition Slide 12-9

Principles of Econometrics, 3rd Edition
Figure 12.2 (a) Time Series Models
Slide 12-10

Principles of Econometrics, 3rd Edition
Figure 12.2 (b) Time Series Models
Slide 12-11

Principles of Econometrics, 3rd Edition
Figure 12.2 (c) Time Series Models
Slide 12-12

y t
 y t

1
 v t y
1
 y
0
 v
1 y
2 y
1 v
2
( y
0
 v
1
) v
2 y
0
 s
2 

1 v s y t
 y t

1 v t y
0
 s t 

1 v s
(12.3a)
Principles of Econometrics, 3rd Edition Slide 12-13

( ) t
 y
0

(
1
  
2 v t
)
 y
0 y t
 var( v v
1 2
...
v t
) t
2 v y t
   y t

1
 v t
(12.3b)
Principles of Econometrics, 3rd Edition Slide 12-14

y
1
   y
0
 v
1 y
2 y v
1 2
( y
0
 v
1
) v
2
2 y
0
 s
2 

1 v s y t
   y t

1 v t t y
0
 s t 

1 v s
Principles of Econometrics, 3rd Edition Slide 12-15

( ) t
   y
0

(
1
   
2 3 v t
) t y
0 y t
 var( v v v
1 2 3
...
v t
) t
2 v y t t y t

1
 v t
(12.3c)
Principles of Econometrics, 3rd Edition Slide 12-16

y
1
     y
0
 v
1
; (since t

1) y
2
2 y
1 v
2
2 ( y
0
 v
1
) v
2
2 3 y
0

2  s

1 v s y t t y t

1 v t t
(

1)
2
  y
0
 t  s

1 v s
1 2 3 t
   
Slide 12-17

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

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

rw
1 t
 rw
1
: y t
 y t

1
 v
1 t rw
2
: x t
 x t

1
 v
2 t
 rw
2 t
, R
2 
.70
Principles of Econometrics, 3rd Edition Slide 12-20

Figure 12.3 (a) Time Series of Two Random Walk Variables
Principles of Econometrics, 3rd Edition Slide 12-21

Figure 12.3 (b) Scatter Plot of Two Random Walk Variables
Principles of Econometrics, 3rd Edition Slide 12-22

 12.3.1 Dickey-Fuller Test 1 (no constant and no trend) y t
  y t

1
 v t
(12.4) y t
 y t

1
  y t

1
 y t

1
 v t y t

1
 y t

1
 v t
  y t

1
 v t
Principles of Econometrics, 3rd Edition
(12.5a)
Slide 12-23

 12.3.1 Dickey-Fuller Test 1 (no constant and no trend)
H
0
:
 
1

H
0
:
 
0
H
1
:
 
1

H
1
:
 
0
Principles of Econometrics, 3rd Edition Slide 12-24

 12.3.2 Dickey-Fuller Test 2 (with constant but no trend) y t y t

1
 v t
(12.5b)
Principles of Econometrics, 3rd Edition Slide 12-25

 12.3.3 Dickey-Fuller Test 3 (with constant and with trend) y t y t

1 t v t
(12.5c)
Principles of Econometrics, 3rd Edition Slide 12-26

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 Slide 12-27

 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

Principles of Econometrics, 3rd Edition
Th
Slide 12-29

 An important extension of the Dickey-Fuller test allows for testing for unit root in AR(p) processes: AUGMENTED Dickey-Fuller. y t y t

1
 s m 

1 a y t s
 v t
 y t

1
  y t

1
 y t

2

,
 y t

2
  y t

2
 y t

3

,
(12.6)
 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)
Principles of Econometrics, 3rd Edition Slide 12-30

  t
( tau

F t

1

0.672

F t

1
  t
( tau

B t

1

0.315

B t

1
Principles of Econometrics, 3rd Edition Slide 12-31

Principles of Econometrics, 3rd Edition y t y t
 y t

1
 v t
  t
 
0.340
( tau
 t

1
  t
 
0.679
  t

1
( tau

Slide 12-32





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

   t e
ˆ t

1
 v t
Case 1: e
ˆ t
  t bx t
Case 2 : e
ˆ t
 y t
 b x
2 t
 b
1
Case 3: e
ˆ t
  t b x t
  
1
ˆ t
Principles of Econometrics, 3rd Edition
(12.7)
(12.8a)
(12.8b)
(12.8c)
Slide 12-34

Principles of Econometrics, 3rd Edition Slide 12-35

B t
 
F t
R
2 
0.881
ˆ t
0.314
( tau
 e
ˆ t

1

0.315
 e
ˆ t

1
(12.9)
Principles of Econometrics, 3rd Edition Slide 12-36

The null and alternative hypotheses in the test for cointegration are:
H
0
: the series are not cointegrated
 residuals are nonstationary
H
1
: the series are cointegrated
 residuals are stationary
Principles of Econometrics, 3rd Edition Slide 12-37

 12.5.1 First Difference Stationary y t
 y t

1
 v t y t y t
 y t

1
 v t
The variable y t is said to be a first difference stationary series.
Principles of Econometrics, 3rd Edition Slide 12-38

y t y t

1 0 x t 1 x t

1
 e t y t
   y t

1
 v t
    t v t y t y t

1 0 x t 1 x t

1
 e t
Principles of Econometrics, 3rd Edition
(12.10a)
(12.10b)
Slide 12-39

y t
     v t y t
     v t y t
   y t


1

0 x t
 
1 x t


1
 e t y t t y t

1
 
0 x t
 
1 x t

1
 e t where and
1
(1
1
) (
2 0 1
)
1 1 1 2
         
1
(1
1
) (
2 0 1
)
Principles of Econometrics, 3rd Edition
(12.11)
Slide 12-40

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 12-41
















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 12-42