Spurious Regression and Simple Cointegration
Gloria González-Rivera
University of California, Riverside
and
Jesús Gonzalo U. Carlos III de Madrid
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Spurious Regression
y t  y t 1  u t ; u t  iid (0,  2u )
Set-up:
x t  x t 1  v t ; v t  iid (0,  2v )
Regress
; E(u t , v s )  0 t, s
E(u t u t  k )  E( v t v t  k )  0 k
y t    x t   t
What do you expect to get?
p
ˆ 
0
p
2
R  0
t ˆ  t distributi on
Spurious Regression (cont)
What does it really happen?
ˆ  some distributi on
2
R  some distributi on
p
DW  0
T 1 / 2 t ˆ  some distributi on
Spurious Regression (cont)
Spurious Regression Problem (SPR):
Regression of an integrated series on another unrelated integrated
series produces t-ratios on the slope parameter which indicates a
relationship much more often than they should at the nominal test
level. This problem does not dissapear as the sample size is
increased.
The Spurious Regression Problem can appear with I(0) series too
(see Granger, Hyung and Jeon (1998)). This is telling us that the
problem is generated by using WRONG CRITICAL VALUES!!!!
In a Spurious Regression the errors would be correlated and the
standard t-statistic will be wrongly calculated because the variance
of the errors is not consistently estimated. In the I(0) case the
solution is:
ˆ ˆ
t  t - distributi on , where ˆ  (long - run varian ce of ˆ )1/2
ˆ
Maybe the same thing can be done to solve the SPR problem with
I(1) variables.
Spurious Regression (cont)
How do we detect a Spurious Regression (between I(1) series)?
Looking at the correlogram of the residuals and also by testing for a
unit root on them.
How do we convert a Spurious Regression into a valid regression?
By taking differences.
Does this solve the SPR problem?
It solves the statistical problems but not the economic interpretation of
the regression. Think that by taking differences we are loosing
information and also that it is not the same information contained in a
regression involving growth rates than in a regression involved the
levels of the variables.
Spurious Regression (cont)
Does it make sense a regression between two I(1) variables?
Yes if the regression errors are I(0).
Can this be possible?
The same question asked David Hendry to Clive Granger time ago.
Clive answered NO WAY!!!!! but he also said that he would think
about. In the plane trip back home to San Diego, Clive thought about
it and concluded that YES IT IS POSSIBLE. It is possible when both
variables share the same source of the I(1)’ness (co-I(1)), when both
variables move together in the long-run (co-move), ... when both
variables are
COINTEGRATED!!!!!!!!!!!!!!!!!!!!!!!
Some Cointegration Examples
Example 1: Theory of Purchasing Power Parity (PPP)
“Apart from transportation costs, good should sell for the same
effective price in two countries”
*
Pt  St Pt
An index of the price level
in the USA
In logs :
A weaker version
of the PPP:
$ per €
Price Index for
Spain
p t  s t  p*t
*
pt  st  pt  zt
If the three variables are I(1) and zt is I(0) then the PPP theory is
implying cointegrating between pt, st and p*t .
Some Cointegrating Examples (cont)
Example 2: Present Value Models (PVM)


Yt  (1  )
i E t ( y t  i )  c
i 0
Yt: Long-term yields
yt: short-term yields
Stock Prices
dividends
Consumption
labor income
If yt has a unit root and the PVM holds then Yt and yt will be
cointegrated (see Campbell and Shiller (1987)
Z t  Yt   y t is I(0)
Geometric Interpretation of Cointegration
What is an ATTRACTOR?
Consider the price (over time) of a commodity that is traded in
two different locations i and j.
.
pit  p jt
p jt
3
.
.
4
1 : ( pi1 , p j1 )
5
.
2 : ( pi 2 , p j 2 )
.
2
1

t : ( pit , p jt )
45
Suppose that pi1  p j1
pit
Demand will go to location j
 pi1 and  p j1
The adjustment does not have to be instantaneous but eventually
pit  p jt Long-run equilibrium: this is a linear attractor.
Shocks to the economy make us move out of the attractor.
Geometric Interpretation of Cointegration (cont)
The concept of attractor is the concept of long-run equilibrium
between two stochasic processes. We allow the two variables to
diverge in the short-run; but in the long-run they have to converge
to a common region denominated attractor region. In other words,
if from now on there are not any shocks in the system, the two
stochastic processes will converge together to a common attractor
set.
Question 1: Write in intuition terms two two economic examples
where cointegration can be present. Explain why?
Question 2: A drunk man leaving a bar follows a random walk. His
dog also follows a random walk on its own. Then they go into a
park where dogs are not allowed to be untied. Therefore the drunk
man puts a strap on his dog and both enter into the park Will their
paths be cointegrated? Why?
Definition of Cointegration
From an economic point of view we are interested on answering
(1) Can we test for the existence of this attractor?
(2) If it exists, how can be introduced into our econometric modelling?
Some rules on linear combinations of I(0) and I(1) processes
1. X t  I (0)  a  bX t  I (0)
X t  I (1)  a  bX t  I (1)
2. X t , Yt  I (0)  aX t  bYt  I (0)
3. X t  I (0), Yt  I (1)  aX t  bYt  I (1)
I (1) is dominant
4. X t , Yt  I (1)  aX t  bYt  I (1) in general
Definition
If X t and Yt are I(1) but there exists a linear
combinatio n, say
Z t  m  aX t  bYt
such Z t is I(0), then X t , Yt are said to be cointegrat ed.
Why Two Series Are Cointegrated?
Consider the following construction
~
X t  AWt  X t I (1)  I (1)  I (0) (rule 3)
~
Yt  Wt  Yt
The following linear combination
~
~
Z t  X t  AYt  AWt  X t  AWt  AYt
~
~
Z t  X t  AYt  I(0) (rule 2)
so X t , Yt have a " common I(1) factor".
Result 1.
If two I(1) series have a common I(1) factor and idiosincratic I(0)
components, then they are cointegrated.
It can be proved that Result 1 is an IF and ONLY IF result.
A Simple Test for Cointegration
• This test is due to Engle and Granger (1987)
• Estimate the following regression model in levels
y t    x t  z t ; y t , x t  I(1)
•Perform an ADF test on the residuals:
p
ẑ t  ẑ t 1 

 i ẑ t  i  error
i 1
• The null hypothesis
Ho :   0
• This means that the residuals have a unit root and therefore yt and xt are not cointegrated.
• If the residuals are I(0) then yt and xt are cointegrated
Error Correction Model
Vector Error Correction Model(VECM)
X t , Ytare I(1) and cointegrated,
X t  c1  1Z t 1  1X t 1  .....  1Yt 1  ....   xt
Yt  c2  2 Z t 1   1X t 1  .....   1Yt 1  ....   yt
For a bivariate VAR, where
where ( xt ,  yt )' is a bivariate white noise and
Z t  X t  AYt  I (0), and
at least one i  0
If X t , Yt are not cointegrat ed  Z t  I (1) (rule 4)
In the ECM, I(1) cannot explain I(0), i.e. X t , Yt  1  2  0
Result 2.
If X t , Yt are cointegrated, then exists an ECM representation.
Cointegration is a necessary condition for ECM and viceversa
(Granger Representation Theorem).
Geometric intuition of the Error Correction Model
Intuition on ECM
Zt  Yt  AX t : disequilib rium error
Yt
Zt  0
Yt  AX t

Xt
Wherever the system goes at time t+1 , depends on the magnitude
and sign of the disequilibrium error of the previous period t, at least.
Short-run dynamics: movements in the short run, modeled in the
ECM, that guide the economy towards the
Long-run equilibrium Yt  AX t
Cointegration and Econometric Modelling
1. Check the integration of X t , Yt  I (1) : use the Dickey-Fuller tests
2. Testing for cointegration between X t and Yt . Find the cointegrating
relation. OLS regression (minimize the variance of residuals).
Yt  cˆ  ˆX t  Zˆ t
H 0 : non - cointegrat ion Zˆ t  I (1)
H1 : cointegrat ion Zˆ t  I (0)
If H 0 is rejected  X t , Yt are cointegrat ed
Warning: we will be tempted to use the Dickey-Fuller tests but the
test is based in residuals . We need a different set of critical values,
as in Engle-Granger (89) or McKinnon (90).
P( w  1.65)  5% Normal distributi on
P( w  2.86)  5% Dickey - Fuller distributi on
P( w  3.34)  5% Engle - Granger/Mc Kinnon
Cointegration and Economic Modelling (cont)
(1, ˆ ) is the cointegrat ing vector
ˆ is super - consistent (T( ˆ  )  some distributi on)
3. Short-run dynamics: ECM
X t  c1  1 (Yt 1  ˆX t 1 )   x1X t 1  .....   y1Yt 1  ....   xt
Yt  c2  2 (Yt 1  ˆX t 1 )   x1X t 1  .....   y1Yt 1  ....   yt
Engle-Granger two-step estimation method:
(i) Estimate ˆ  Ẑt
(ii) Plug Ẑ t in the ECM (SURE estimation): estimators in the ECM
are consistent and efficient.
Cointegration with more than two variables
Example 1.
N  3, h  2
Yt  Wt  ut 

X t  Wt  vt Wt  I (1) ut , vt , st  I (0)
Z t  Wt  st 
1 common stochastic trend  Wt
2 cointegrat ing vectors : (1  1 0)' (0 1  1)'
Example 2.
N  3, h  1
Yt  Wt  ut


X t  Wt  Rt  vt Wt , Rt  I (1) ut , vt , st  I (0)

Z t  Rt  st

2 common stochastic trends  Wt , Rt
1 cointegrat ing vector : (1  1 1)'
Cointegration with more than two variables (cont)
Example 3.
N  2, h  1
Yt  AWt  ut 
Wt  I (1) ut , vt , st  I (0)
X t  Wt  vt 
1 common stochastic trend  Wt
Example 4.
1 cointegrat ing vector : (1  A)'
N  2, h  2; Yt and X t are stationary
2 cointegrat ing vectors : (1 0)' (0 1)'
Cointegration: Testing and Estimation with more than two variables
Johansen’s method:
Two major advantages with respect to Engle-Granger procedure:
(1) Testing for number of cointegrating vectors when N>2
(2) Joint procedure: testing and maximum likelihood estimation of
the vector error correction model and long run equilibrium relations.
Framework
Consider a VAR(p)
Yt  1Yt 1   2Yt 2  ....   pYt  p   t
We construct the vector error correction model transforming the VAR:
where  0    I  [ I  1   2  ....   p ]
Yt  1Yt 1   2 Yt 2  ....   p 1Yt  p 1   0Yt 1   t
subtractin g Yt 1 from both sides :
  [1   2  ....   p ]
where i  [ i 1   i 2  ....   p ] i  1,... p  1
Yt  1Yt 1   2 Yt 2  ....   p 1Yt  p 1  Yt 1   t
Vector error correction model:
Yt  1Yt 1   2 Yt 2  ....   p 1Yt  p 1   0Yt 1   t
If Yt is cointegrat ed with cointegrat ing rank h,
then  0   B A' (the matrix is of reduced rank) and
nxh hxn
 0Yt 1   B A' Yt 1   BZ t 1
nxh hxn
A contains the cointegrat ing vectors
B contains the coefficien ts of adjustment
Example 5: 2 variables, 1 cointegrating vector
Z t  X t  Z t
 X t 1 
 1 
0Yt 1    1    


Z
 2
 t 1 
H 0 : h cointegrat ing vectors at most (restricti ons on 0 )
H1 : n cointegrat ing vectors (no restrictio ns on 0 )
Objective:
Construct the likelihood function under the null and under
the alternative, and construct a likelihood ratio-type test.
max 
s.t.  0   BA'
Johansen’s algorithm to maximize the constrained likelihood is
based on canonical correlation analysis.
Likelihood ratio test has a non-standard distribution due to the
non-stationarity of the variables.
H 0 : h  n
Trace test

H1 : h  n
2(1   0 )  
H 0 : h
H : h  1 maximum eigenvalue test
 1