### ch12unitroots - Memorial University of Newfoundland

```ECON 6002
Econometrics
Memorial University of Newfoundland
Nonstationary Time Series Data and
Cointegration
Adapted from Vera Tabakova’s notes

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
Fluctuates about a rising trend
Yt-Y t-1
On the right
hand side
Fluctuates about a zero mean
“Differenced
series”
Figure 12.1(a) US economic time series
Principles of Econometrics, 3rd Edition
Slide 12-3
Yt-Y t-1
On the right
hand side
“Differenced
series”
Figure 12.1(b) US economic time series
Principles of Econometrics, 3rd Edition
Slide 12-4
Stationary if:
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)
Each realization of the process has a proportion rho of the previous one
plus an error drawn from a distribution with mean zero and variance
sigma squared
It can be generalised to a higher autocorrelation order
We just show AR(1)
Principles of Econometrics, 3rd Edition
Slide 12-7
yt  yt 1  vt ,
 1
(12.2a)
We can show that the constant mean of this series is zero
y1  y0  v1
y2  y1  v2  (y0  v1 )  v2  2 y0  v1  v2
yt  vt  vt 1  2vt 2  .....  t y0
E[ yt ]  E[vt  vt 1  2vt 2  .....]  0
Principles of Econometrics, 3rd Edition
Slide 12-8
We can also allow for a non-zero mean, by replacing yt with yt-mu
( yt  )  ( yt 1  )  vt
Which boils down to (using alpha = mu(1-rho))
yt     yt 1  vt ,
 1
(12.2b)
E ( yt )     / (1  )  1/ (1  0.7)  3.33
Principles of Econometrics, 3rd Edition
Slide 12-9
Or we can allow for a AR(1) with a fluctuation around a linear trend
(mu+delta times t)
The “de-trended” model , which is now stationary, behaves like
an autoregressive model:
( yt   t )  ( yt 1   (t 1))  vt ,
yt    yt 1  t  vt
 1
(12.2c)
With alpha =(mu(1-rho)+rho times delta)
And lambda = delta(1-rho)
Principles of Econometrics, 3rd Edition
Slide 12-10
Figure 12.2 (a) Time Series Models
Principles of Econometrics, 3rd Edition
Slide 12-11
Figure 12.2 (b) Time Series Models
Principles of Econometrics, 3rd Edition
Slide 12-12
Figure 12.2 (c) Time Series Models
Principles of Econometrics, 3rd Edition
Slide 12-13
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
The first component is usually
just zero, since it is so far in the
past that it has a negligible
effect now
The second one is the stochastic
trend
Slide 12-14

A random walk is non-stationary, although the mean is constant:
E ( yt )  y0  E (v1  v2  ...  vt )  y0
var( yt )  var(v1  v2  ...  vt )  tv2
Principles of Econometrics, 3rd Edition
Slide 12-15
yt    yt 1  vt
(12.3b)
A random walk with a drift both wanders and trends:
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 12-16
E ( yt )  t  y0  E (v1  v2  v3  ...  vt )  t  y0
var( yt )  var(v1  v2  v3  ...  vt )  tv2
Principles of Econometrics, 3rd Edition
Slide 12-17
rw1 : yt  yt 1  v1t
rw2 : xt  xt 1  v2t
Both independent and artificially generated, but…
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

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

Dickey-Fuller Test 1 (no constant and no trend)
H0 :   1  H0 :   0
H1 :   1  H1 :   0
Easier way to test the hypothesis about rho
Remember that the null is a unit root: nonstationarity!
Principles of Econometrics, 3rd Edition
Slide 12-22

Dickey-Fuller Test 2 (with constant but no trend)
yt     yt 1  vt
Principles of Econometrics, 3rd Edition
(12.5b)
Slide 12-23

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.

If the series appears to be wandering or fluctuating around a sample
average of zero, use Version 1

If the series appears to be wandering or fluctuating around a sample
average which is non-zero, use Version 2

If the series appears to be wandering or fluctuating around a linear
trend, use Version 3
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  ,

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
F = US Federal funds interest rate
Ft  0.178  0.037 Ft 1  0.672Ft 1
(tau )
(  2.090)
B = 3-year bonds interest rate
Bt  0.285  0.056 Bt 1  0.315Bt 1
(tau )
Principles of Econometrics, 3rd Edition
(  1.976)
Slide 12-28
In STATA:
use usa, clear
gen date = q(1985q1) + _n - 1
format %tq date
tsset date
TESTING UNIT ROOTS “BY HAND”:
* Augmented Dickey Fuller Regressions
regress D.F L1.F L1.D.F
regress D.B L1.B L1.D.B
Principles of Econometrics, 3rd Edition
Slide 12-29
In STATA:
. regress D.F
L1.F ROOTS
L1.D.F
TESTING
UNIT
“BY HAND”:
Source
SS Fuller
dfRegressions
MS
* Augmented
Dickey
7.99989546
regressModel
D.F L1.F
L1.D.F 2 3.99994773
Residual
9.54348876
76
.12557222
regress D.B L1.B L1.D.B
Total
17.5433842
78
Number of obs
F( 2,
76)
Prob > F
R-squared
Root MSE
.224915182
t
P>|t|
79
31.85
0.0000
0.4560
0.4417
.35436
D.F
Coef.
F
L1.
LD.
-.0370668
.6724777
.0177327
.0853664
-2.09
7.88
0.040
0.000
-.0723847
.5024559
-.001749
.8424996
_cons
.1778617
.1007511
1.77
0.082
-.0228016
.378525
Principles of Econometrics, 3rd Edition
Std. Err.
=
=
=
=
=
=
[95% Conf. Interval]
Slide 12-30
In STATA:
Augmented Dickey Fuller Regressions with built in functions
dfuller F, regress lags(1)
dfuller B, regress lags(1)
Choice of lags if we want to allow
For more than a AR(1) order
Principles of Econometrics, 3rd Edition
Slide 12-31
In STATA:
Augmented Dickey Fuller Regressions with built in functions
dfuller F, regress lags(1)
. dfuller F, regress lags(1)
. dfuller F, regress lags(1)
Augmented Dickey-Fuller test for unit root
Augmented Dickey-Fuller test for unit root
TestTest
Statistic
Statistic
Z(t) Z(t)
Number of obs
Number of obs
=
=
79
79
Interpolated Dickey-Fuller
Interpolated Dickey-Fuller
1% 1%
Critical
5%
10%Critical
Critical
Critical
5% Critical
Critical
10%
Value
Value
Value
Value
Value
Value
-2.090
-2.090
-3.539
-3.539
-2.907
-2.907
-2.588
-2.588
MacKinnon
approximate
p-value
Z(t)= =0.2484
0.2484
MacKinnon
approximate
p-value
forfor
Z(t)
D.F D.F
Coef. Std.
Std.
Err.
Coef.
Err.
tt
P>|t|
P>|t|
[95%
[95% Conf.
Conf.Interval]
Interval]
F
F L1.
-.0370668
.0177327
L1. LD.-.0370668
.6724777 .0177327
.0853664
LD.
.6724777
.0853664
_cons
_cons
.1778617
.1778617
Principles of Econometrics, 3rd Edition
.1007511
.1007511
-2.09
7.88
7.88
-2.09
0.040
0.000
0.000
0.040
-.0723847
-.0723847
.5024559
1.77
0.082
-.0228016
1.77
0.082
.5024559
-.0228016
-.001749
-.001749
.8424996
.8424996
.378525
.378525
Slide 12-32
In STATA:
Augmented Dickey Fuller Regressions with built in functions
dfuller F, regress lags(1)
Alternative: pperron uses Newey-West standard errors to account for
serial correlation, whereas the augmented Dickey-Fuller test implemented in
dfuller uses additional lags of the first-difference variable.
Also consider now using DFGLS (Elliot Rothenberg and Stock, 1996) to
counteract problems of lack of power in small samples. It also has in STATA
a lag selection procedure based on a sequential t test suggested by Ng and
Perron (1995)
Principles of Econometrics, 3rd Edition
Slide 12-33
In STATA:
Augmented Dickey Fuller Regressions with built in functions
dfuller F, regress lags(1)
Alternatives: use tests with stationarity as the null
KPSS (Kwiatowski, Phillips, Schmidt and Shin. 1992) which also has an
automatic bandwidth selection tool or the Leybourne & McCabe test .
Principles of Econometrics, 3rd Edition
Slide 12-34
yt  yt  yt 1  vt
The first difference of the random walk is stationary
It is an example of a I(1) series (“integrated of order 1”
First-differencing it would turn it into I(0) (stationary)
In general, the order of integration is the minimum number of times a
series must be differenced to make it stationarity
Principles of Econometrics, 3rd Edition
Slide 12-35
yt  yt  yt 1  vt
  F t   0.340  F t 1
(tau )
(  4.007)
So now we reject the
Unit root after differencing
once:
We have a I(1) series
  B t   0.679  B t 1
(tau )
Principles of Econometrics, 3rd Edition
(  6.415)
Slide 12-36
In STATA:
dfuller D.F, noconstant lags(0)
dfuller D.B, noconstant lags(0)
. dfuller F, regress lags(1)
Augmented Dickey-Fuller test for unit root
Number of obs
=
79
Interpolated Dickey-Fuller
5% Critical
10% Critical
Value
Value
Number of obs
=
79
-3.539
-2.907
-2.588
Interpolated Dickey-Fuller
Test
1% Critical
5% Critical
10% Critical
MacKinnon approximate
p-value
for Z(t) = 0.2484
Statistic
Value
Value
Value
. dfuller D.F, noconstantTest
lags(0)
Statistic
Dickey-Fuller test for unit root
Z(t)
-2.090
Z(t)
-4.007
D.F
Coef.
1% Critical
Value
-2.608
Std.
Err.
t
-1.950
P>|t|
-1.610
[95% Conf.
Interval]
F
L1.
LD.
-.0370668
.6724777
.0177327
.0853664
-2.09
7.88
0.040
0.000
-.0723847
.5024559
-.001749
.8424996
_cons
.1778617
.1007511
1.77
0.082
-.0228016
.378525
Slide 12-37
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 12-38

If you have unit roots in the time series in your model, you risk the
problem of spurious regressions

However, spuriousness will not arise if those series are cointegrated,
so that determining whether cointegration exists is also key

The series are cointegrated if they follow the same stochastic trend or
share an underlying common factor

In that case you can find a linear combination of your nonstationary
variables that is itself stationary
Principles of Econometrics, 3rd Edition
Slide 12-39

You must make sure that you have a balanced (potentially)
cointegrating regression, so you want to find out the level of
integration of your series (usually they are all I(1))

The coefficients in that linear combination form the cointegrating
vector, which should have one of its elements normalized to one,
because the cointegrating vector is only defined up to a factor of
proportionality

The cointegrating vector may include a constant, in order to allow for
unequal means of the two series
Principles of Econometrics, 3rd Edition
Slide 12-40

The estimator from a cointegrating regression is superconsistent
Principles of Econometrics, 3rd Edition
Slide 12-41
Two main approaches can be used to check if there is cointegration:

The residual approach

The error correction approach

Principles of Econometrics, 3rd Edition
Slide 12-42
Two main approaches can be used to check if there is cointegration:

The residual approach. The classic Engle-Granger approach, based
on testing whether the error of the (potentially) cointengrating
regression is itself stationary

The error correction approach, which test whether the error
correction term is significant
Principles of Econometrics, 3rd Edition
Slide 12-43
Not the same as for dfuller, since the residuals are estimated errors no actual
Ones (also no constant!)
Note: unfortunately STATA dfuller as such will not notice and give you erroneous
critical values! They would lead to an overoptimistic conclusion 
Principles of Econometrics, 3rd Edition
Slide 12-44
Bˆt  1.644  0.832Ft , R 2  0.881
(12.9)
(t ) (8.437) (24.147)
eˆt  0.314eˆt 1  0.315eˆt 1
(tau ) (4.543)
Check: These are wrong!
. dfuller ehat, noconstant lags(1)
Augmented Dickey-Fuller test for unit root
Test
Statistic
Z(t)
-4.543
Number of obs
=
79
Interpolated Dickey-Fuller
1% Critical
5% Critical
10% Critical
Value
Value
Value
-2.608
-1.950
-1.610
Slide 12-45
Using egranger with option regress 
Engle-Granger test regression
D._egresid
Coef.
_egresid
L1.
LD.
-.3143204
.3147476
Std. Err.
.0691906
.1021565
t
-4.54
3.08
P>|t|
0.000
0.003
[95% Conf. Interval]
-.4520965
.1113281
-.1765443
.5181671
eˆt  0.314eˆt 1  0.315eˆt 1
(tau ) (4.543)
Now these are right!
. egranger B F, regress lags(1)
Augmented
Engle-Granger test for cointegration
Replacing variable _egresid...
Number of lags
= 1
Augmented Engle-Granger test for cointegration
Number of lags
= 1
Test1% Critical
StatisticValue
Test
Statistic
Z(t)
Z(t)
-4.543
-4.543
-3.515
Critical values from MacKinnon (1990, 2010)
N (1st step)
N (test)
1%5% Critical
Critical
Value
Value
=
=
81
79
N (1st step)
N (test)
5% Critical
Value
10% Critical
Value
-2.898
-3.515
-2.586
-2.898
=
=
81
79
10% Critical
Value
-2.586
Critical values from MacKinnon (1990, 2010)
Slide 12-46
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 12-47

Let us consider the simple form of a dynamic model:

Here the SR and LR effects are measured respectively by:

Rearranging terms, we obtain the usual ECM:
Slide 12-48
Where the LR effect will be given by:
And
is a partial correction term for the extent to which Yt-1 deviated from its
Equilibrium value associated with Xt-1
Slide 12-49
This representation assumes that any short-run shock to Y that pushes it off
the long-run equilibrium growth rate will gradually be corrected, and
an equilibrium rate will be restored
is the residual of the long-run equilibrium relationship between X and Y and its
Coefficient can be seen as the “speed of adjustment”
Slide 12-50
Tthis representation assumes that any short-run shock to Y that pushes it off
the long-run equilibrium growth rate will gradually be corrected, and
an equilibrium rate will be restored
Usually
So the SR effect is weaker than the LR effect
Slide 12-51
. egranger B F, ecm
Replacing variable _egresid...
Engle-Granger 2-step ECM estimation
N (1st step)
N (2nd step)
=
=
81
80
Engle-Granger 2-step ECM
D.B
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
_egresid
L1.
-.2357887
.0723679
-3.26
0.002
-.3798917
-.0916858
F
D1.
.7529687
.1080218
6.97
0.000
.5378699
.9680675
_cons
-.0315012
.0477661
-0.66
0.512
-.1266156
.0636133
Slide 12-52
If you have cointegration, you can run an Error Correction Model, so you can
estimate both the long run and the short run relationship between the
relevant variables
The integration of the variables suggests that we should not use them in a
regression, but rather only their differences. We may obtain inconsistent
estimates (the spurious regression problem)
However, the fact that they are cointegrated (a weighted average of the
variables is stationary, I(0)) means that you can include linear combinations
of the variables in regressions of their differences in and Error Correction
Model (ECM)
Slide 12-53

By having already concluding that the variables are
cointegrated, we have implicitly decided that there is a
long-run causal relation between them.

Then the causality being tested for in a VECM is sometimes
called “short-run Granger causality”
Slide 12-54

The ECM analysis can show (by the magnitude and
significance of the EC terms) that when values of the
relevant variables move away from the equilibrium
relationship implied by the contegrating vector, there was a
strong tendency for the variable(s) to change so that the
equilibrium would be restored

The ECM analysis under cointegration allows us not to
throw away the information on the LR effect behind the
relationship
Slide 12-55
The EC term will be significant if there is a cointegrating
relationship
 Therefore, you can test the existence of cointegration by
looking at the significance of that coefficient

Slide 12-56

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.
Then we revert to the techniques we saw in Ch. 9
Principles of Econometrics, 3rd Edition
Slide 12-57
Manipulating this one you can construct and Error Correction Model
to investigate the SR dynamics of the relationship between y and x
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 12-58
yt    t  vt
yt    t  vt
yt   yt1  0 xt  1xt1  et
(12.11)
yt     t   yt 1  0 xt  1xt 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 12-59
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.

Then we can use the lagged residuals from the cointegrating
regression in an ECM model

This is the best case scenario, since if we had to first-differentiate the
variables, we would be throwing away the long-run variation

Additionally, the cointegrated regression yields a “superconsistent”
estimator in large samples
Principles of Econometrics, 3rd Edition
Slide 12-60
To summarize:

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 12-61
.
Principles of Econometrics, 3rd Edition
Slide 12-62















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 12-63
Further issues
Kit Baum has really good notes on these topics that can be used to learn
also about extra STATA commands to handle the analysis:
http://fmwww.bc.edu/EC-C/S2003/821/EC821.sect05.nn1.pdf
http://fmwww.bc.edu/EC-C/S2003/821/EC821.sect06.nn1.pdf
For example, some of you should look at (quarterly) seasonal unit root analysis
(command hegy4 in STATA implements the test suggested by Hylleberg et al. 1990)
Panel unit roots would be here
http://fmwww.bc.edu/EC-C/S2003/821/EC821.sect09.nn1.pdf
Principles of Econometrics, 3rd Edition
Slide 12-64
Further issues: more powerful tests
A host of new tests have been developed to try and overcome the shortcomings of the
first Dickey-Fuller ones
Alternative: pperron uses Newey-West standard errors to account for serial
correlation, whereas the augmented Dickey-Fuller test implemented in dfuller uses
additional lags of the first-difference variable.
Also consider now using DF-GLS (Elliot Rothenberg and Stock, 1996) to counteract
problems of lack of power in small samples. It also has in STATA a lag selection
procedure based on a sequential t test suggested by Ng and Perron (1995) that uses a
Modified AIC (command dfgls)
Principles of Econometrics, 3rd Edition
Slide 12-65
Further issues: reverting the null
Some tests use stationarity as the null hypothesis: See kpss which implements
the test suggested by Kwiatowski etal. (1992)
Kwiatowski, D., Phillips, P. C. B., Schmidt, P. and Shin, Y. (1992). `Testing the Null Hypothesis of Stationarity against the Alternative of a
Unit Root', Journal of Econometrics, 54, 91-115.
Leybourne, S.J. and B.P.M. McCabe. A consistent test for a unit root. Journal of Business and Economic Statistics, 12,
1994, 157-166.
This type of test is complementary to the dfuller-type ones, so if they do not
give you consistent results, you might be facing fractional unit roots or long
range dependence
[See lomodrs in Stata to learn more and Baum et al. 1999]
Principles of Econometrics, 3rd Edition
Slide 12-66
In STATA:
Principles of Econometrics, 3rd Edition
Slide 12-67
Further issues: unit root tests and structural breaks
You may want to consider unit root tests that allow for structural
Breaks, otherwise with a more basic test you might think you are detecting
a unit root, while all you have is a structural break
See
Perron, Pierre. 1989. The Great Crash, The Oil Price Shock and the Unit Root Hypothesis. Econometrica, 57,
1361–1401.
Perron, P. 1990. Testing for a unit root in a time series with a changing mean, Journal of Business and
Economic Statistics, 8:2, 153-162.
Perron, Pierre. 1997. Further Evidence on Breaking Trend Functions in Macroeconomic Variables. Journal of
Econometrics, 80, 355–385.
Perron, P. and T. Vogelsang. 1992. Nonstationarity and level shifts with an application to purchasing power
parity, Journal of Business and Economic Statistics, 10:3, 301-320.
You can also take a look at the literature review in this working paper:
http://ideas.repec.org/p/wpa/wuwpot/0410002.html
Principles of Econometrics, 3rd Edition
Slide 12-68
Further issues: unit root tests and structural breaks
You may want to consider unit root tests that allow for structural
Breaks:
Stata has zandrews
Zivot, E. & Andrews, W. K. Further Evidence on the Great Crash, the Oil Price Shock, and the Unit-Root Hypothesis
Journal of Business and Economic Statistics, 1992, 10, 251-270
And
Cleamo1 Cleamao2 Clemio1 Clemio2
Clemente, J., Montañes, A. and M. Reyes. 1998. Testing for a unit root in variables with a double change in the
mean, Economics Letters, 59, 175-182
Principles of Econometrics, 3rd Edition
Slide 12-69
Further issues
Apart from the fact that in your cointegration relationship you must choose one
variable to be the regressand (giving it a coefficient of one)
When you deal with more than 2 regressors you should consider the
Johansen’s method to examine the cointegration relationships
This is because when there are more than 2 variables involved, there can
be multiple cointegrating relationships!!!
In this case, you we exploit the notion of Vector Autoregression (VAR) Models
that involve a structural view of the dynamics of several variables
The generalization of these VAR techniques in this case resulted in the
Vector Error Correction Models (VECM)
Principles of Econometrics, 3rd Edition
Slide 12-70
Further issues
Apart from the fact that in your cointegration relationship you must
choose one
variable to be the regressand (giving it a coefficient of one)
When you deal with more than 2 regressors you should consider the
Johansen’s method to examine the cointegration relationships
You can use vecrank in Stata to run this test
Johansen, S. and K. Juselius. 1990. Maximum likelihood estimation and inference on cointegration with applications to
the demand for money. Oxford Bulletin of Economics and Statistics, 522, 169–210.
Principles of Econometrics, 3rd Edition
Slide 12-71
Further issues: unit root tests for panels
Since:
• many interesting relations involve relatively short
time–series and
• unit root tests are infamous when applied to a
single time series for their low power
there may be hope from tests that can be used on
short series but available across a cross–section of
countries, regions, firms, or industries
Principles of Econometrics, 3rd Edition
Slide 12-72
Further issues: unit root tests for panels
We need to logically extend the unit roots testing machinery
for univariate time series to the panel setting
• We can choose the null
• We need to consider how stationary (or nonstationary) a panel has to
be for us to deem it all stationary (or nonstationary)
• We can use a logic of pooling the series and finding one indicator or
averaging the indicators we find in each series instead
• We can use the residual approach or the ECM approach
Principles of Econometrics, 3rd Edition
Slide 12-73
Further issues: unit root tests for panels
One key issue with panel unit root tests is that they
should try and consider cross-sectional dependence
Only the second-generation tests can account for it,
the first-generation tests assume cross-sectional
independence
Principles of Econometrics, 3rd Edition
Slide 12-74
Further issues: unit root tests for panels
STATA offers:
• MADFULLER for MADF test, which is an extension of the ADF test (not
good for longitudinal panels)
• The test's null hypothesis should be carefully considered will be violated if
even only one of the series in the panel is stationary
• A rejection should thus not be taken to indicate that each of the series is
stationary
Sarno, L. and M. Taylor, 1998. Real exchange rates under the current float: Unequivocal evidence of mean reversion.
Economics Letters 60, 131–137.
Taylor, M. and L. Sarno, 1998. The behavior of real exchange rates during the post–Bretton Woods period. Journal of
9 International Economics, 46, 281–312.
Principles of Econometrics, 3rd Edition
Slide 12-75
Further issues: unit root tests for panels
STATA offers:
Levin Lin Chu (old levinlin now xtunitroot llc)
One of the first unit root tests for panel data, originally circulated in working
paper form in 1992 and 1993, published, with Chu as a coauthor, in 2002
This model allows for two–way fixed effects and unit–specific time trends
This test is a pooled Dickey–Fuller (or ADF) test, potentially with differing lag
lengths across the units of the panel
Unlike the MADF test, it works with short wide panels
Assumes that the autoregressive parameter rho is identical for all cross section
units (homogeneous alternatives)
Principles of Econometrics, 3rd Edition
Slide 12-76
Further issues: unit root tests for panels
STATA offers: ipshin now xtunitroot ips
The Im–Pesaran–Shin test extends the LLC to allow for heterogeneity in the
value of rho (heterogeneous alternatives)
Under the null, all series nonstationary; under the alternative, a fraction of the
series are assumed to be stationary in contrast to the LLC test, which presumes
that all series are stationary under the alternative hypothesis
IPS use a group–mean Lagrange multiplier statistic to test the null hypothesis.
The ADF regressions (which may be of differing lag
lengths) are calculated for each series, and a standardized statistic
computed as the average of the LM tests for each equation
Im, K., Pesaran, M., and Y. Shin, 1997. Testing for unit roots in heterogeneous panels. Mimeo, Department of Applied
Economics, University of Cambridge.
Principles of Econometrics, 3rd Edition
Slide 12-77
Further issues: unit root tests for panels
Hadri et al. LM test whose null hypothesis is that all series in the panel
are stationary, just like the KPSS test differs from that of Dickey–Fuller
style tests in assuming stationarity rather that nonstationarity
Hadri, K., 2000. Testing for stationarity in heterogeneous panel data. Econometrics Journal, 3, 148–161.
Principles of Econometrics, 3rd Edition
Slide 12-78
Further issues: unit root tests for panels
Hadri et al. LM test whose null hypothesis is that all series in the panel
are stationary, just like the KPSS test differs from that of Dickey–Fuller
style tests in assuming stationarity rather that nonstationarity
Hadri, K., 2000. Testing for stationarity in heterogeneous panel data. Econometrics Journal, 3, 148–161.
Principles of Econometrics, 3rd Edition
Slide 12-79
Further issues: unit root tests for panels
STATA offers: nharvey
The Nyblom–Harvey Test of Common Stochastic Trends
Nyblom, J. and A. Harvey. Tests of common stochastic trends. Econometric Theory, 16, 2000, 176-199.
Nyblom, J. and A. Harvey. Testing against smooth stochastic trends. Journal of Applied Econometrics, 16, 415–429.
Nyblom, J. and T. Makelainen, 1983. Comparison of tests for the presence of random walk components in a simple
linear model. Journal of the American Statistical Association, 78, 856–864.
Principles of Econometrics, 3rd Edition
Slide 12-80
Further issues: unit root tests for panels
STATA offers:
Breitung test
Breitung, J. 2000. The local power of some unit root tests for panel data. In Advances in Econometrics, Volume 15: Nonstationary Panels,
Panel Cointegration, and Dynamic Panels,
ed. B. H. Baltagi, 161-178. Amsterdam: JAI Press.
Breitung, J., and S. Das. 2005. Panel unit root tests under cross-sectional dependence. Statistica Neerlandica 59: 414-433.
Harris-Tzavalis test
Harris, D. and Inder, B. (1994). `ATest of the Null Hypothesis of Cointegration', in Non-Stationary Time Series Analysis
and Cointegration, ed. C. Hargreaves, Oxford University Press, New York.
Harris, R. D. F. and Tzavalis, E. (1999). `Inference for Unit Roots in Dynamic Panels where the Time Dimension is Fixed',
Journal of Econometrics, 91, 201-226
Fisher-type tests (combining p-values)
Maddala, G.S. andWu, S. (1999), A Comparative Study of Unit Root Tests with Panel Data and a new simple test, Oxford
Bulletin of Economics and Statistics, 61, 631-652.
Principles of Econometrics, 3rd Edition
Slide 12-81
Further issues: unit root tests for panels
Other tests
Pedroni, P.L., 1999. Critical values for cointegration tests in heterogeneous panels with multiple regressors.
Oxford Bulletin of Economics and Statistics 61 (4), 653–670. Pedroni, P.L., 2004.
Panel cointegration; asymptotic and finite sample properties of pooled time series tests with an application to
the purchasing power parity hypothesis. Econometric Theory 20 (3), 597–625.
Principles of Econometrics, 3rd Edition
Slide 12-82
Further issues: unit root tests for panels
STATA offers:
Pesaran’s pescadf (Lewandowski, 2007) and multipurt -- Running 1st
and 2nd generation panel unit root tests for multiple variables and lags
To detect cross-sectional dependence:
xtcds and xtcd
Pesaran, M. Hashem (2004) General Diagnostic Tests for Cross Section Dependence in Panels' IZA Discussion Paper No. 1240.
Sarafidis, V. & De Hoyos, R. E. On Testing for Cross Sectional Dependence in Panel Data Models The Stata Journal,
2006, 6, StataCorp LP, vol. 6(4), pages 482-496
Principles of Econometrics, 3rd Edition
Slide 12-83
Further issues: cointegration tests for panels
We also need tests for cointegration in panels
Both residual-based and ECM-based (see Breitung and Pesaran, 2005, for
a review)
Most residual-based cointegration tests, both in time series and in panels,
require that the long-run parameters for the variables in their levels are
equal to the short-run parameters for the variables in their differences
The failure to meet this common-factor restriction can lead to a
significant loss of power for residual-based cointegration tests
Principles of Econometrics, 3rd Edition
Slide 12-84
Further issues: unit root tests for panels, another issue
STATA offers:
xtwest a ECM-based cointegration test Persyn and Westerlund
(2008)
Westerlund, J. 2007. Testing for error correction in panel data. Oxford Bulletin of Economics and Statistics 69:
709–748.
Principles of Econometrics, 3rd Edition
Slide 12-85
Further issues: unit root tests for panels, more issues
• We need to worry now also about the heterogeneity of panels (e.g. see
xtpmg)
•
Blackburne, E. F. & Frank, M. W. Estimation of Nonstationary Heterogeneous Panels
The Stata Journal, 2001, 7, 197-208
• Also, it is also now necessary to consider the possibility of
cointegration between the variables across the groups (cross
section cointegration) as well as within group cointegration
Principles of Econometrics, 3rd Edition
Slide 12-86
```