Spurious Reg and Co Integration
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Transcript of Spurious Reg and Co Integration
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You are free to use and modify these slides for educational purposes, but please if you improvethis material send us your new version.
Spurious Regression and Simple Cointegration
Gloria Gonzlez-Rivera
University of California, Riverside
and
Jess Gonzalo U. Carlos III de Madrid
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Spurious Regression
Set-up:
k0)ktvtv(E)ktutE(ust,0)sv,tE(u;)2v,0(iidtv;tv1txtx
)2u,0(iidtu;tu1tyty
Regress ttxty What do you expect to get?
ondistributitt
0p2R
0p
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Spurious Regression (cont)
What does it really happen?
ondistributisomet2/1T
0p
DW
ondistributisome2R
ondistributisome
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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 thelevels of the variables.
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Some Cointegration Examples
Example 1:Theory of Purchasing Power Par ity (PPP)
Apart from transportation costs, good should sell for the sameeffective price in two countries
*tPtStP
An index of the price level
in the USA$ per Price Index for
Spain
*tptstp
In logs :
A weaker version
of the PPP: tz*tptstp
If the three variables are I(1) and ztis I(0) then the PPP theory is
implying cointegrating between pt, stand p*t.
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Some Cointegrating Examples (cont)
Example 2:Present Value Models (PVM )
c
0i
)ity(tEi)1(tY
Yt: Long-term yields yt: short-term yields
Stock Prices dividends
Consumption labor income
If ythas a unit root and the PVM holds then Ytand ytwill becointegrated(see Campbell and Shiller (1987)
I(0)istytYtZ
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Geometric Interpretation of Cointegration
What is an ATTRACTOR?
Consider the price (over time) of a commodity that is traded in
two different locations i andj.
.
1
.2
.
3 .
4 .
5
jtp
itp
45
jtit pp
),(:
),(:2
),(:1
22
11
jtit
ji
ji
ppt
pp
pp
Suppose that 11 ji pp Demand will go to locationj11 and ji pp
The adjustment does not have to be instantaneous but eventually
jtit pp Long-run equilibrium: this is a linear attractor.
Shocks to the economy make us move out of the attractor.
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Geometric Interpretation of Cointegration (cont)
The concept of attractoris the concept of long-run equi li br ium
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?
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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
generalin)1()1(,4.
dominantis)1(
)1()1(),0(.3
)0()0(,.2
)1()1(
)0()0(.1
IbYaXIYX
I
IbYaXIYIX
IbYaXIYX
IbXaIX
IbXaIX
tttt
tttt
tttt
tt
tt
Definition
ed.cointegratbetosaidare,thenI(0),issuch
Z
sayn,combinatio
linearaexistsbut thereI(1)areandIf
t
ttt
tt
tt
YXZ
bYaXm
YX
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Why Two Series Are Cointegrated?
Consider the following construction
ttt
ttt
YWY
IIIXAWX~
3)(rule)0()1()1(~
The following linear combination
factor".I(1)common"ahavetY,tXso
2)(rule)0(ItY~
AtX~
tZt
Y~
At
AWt
X~
tAW
tAY
tX
tZ
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.
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A Simple Test for Cointegration
This test is due to Engle and Granger (1987)
Estimate the following regression model in levels
Perform an ADF test on the residuals:
I(1)tx,ty;tztxty
p
1i
erroritzi1tztz
The null hypothesis 0:oH
This means that the residuals have a unit root and therefore ytand xtare not cointegrated.
If the residuals are I(0) then yt and xtare cointegrated
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Error Correction Model
Vector Error Correction Model(VECM)
For a bivariate VAR, where are I(1) and cointegrated,
yttttt
xttttt
YXZcY
YXZcX
.........
.........
1111122
1111111
tt YX ,
0oneleastat
and),0(
andnoisewhitebivariateais)',(where
i
ttt
ytxt
IAYXZ
0,i.e.I(0),explaincannotI(1)ECM,theIn
4)(rule)1(edcointegratnotare,If
21
tt
ttt
YX
IZYX
Result 2.
If are cointegrated, then exists an ECM representation.
Cointegration is a necessary condition for ECM and viceversa
(Granger Representation Theorem).
tt YX ,
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Geometric intuition of the Error Correction Model
Intuition on ECMerrorriumdisequilib:ttt AXYZ
tY
tX
tt AXY
0tZ
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 equilibriumtt AXY
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Cointegration and Econometric Modelling
1.Check the integration of : use the Dickey-Fuller tests
2.Testing for cointegration between . Find the cointegrating
relation. OLS regression (minimize the variance of residuals).
)1(, IYX tt
tt YX and
)0(ioncointegrat:)1(
ioncointegrat-non:
1
0
IZHIZH
ZXcY
t
t
ttt
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).
KinnonGranger/Mc-Engle%5)34.3(
ondistributiFuller-Dickey%5)86.2(
ondistributiNormal%5)65.1(
wP
wP
wP
edcointegratare,rejectedisIf 0 ttYXH
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Cointegration and Economic Modelling (cont)
on)distributisome)(T(consistent-superis
vectoringcointegrattheis),1(
3. Short-run dynamics: ECM
yttytxttt
xttytxttt
YXXYcY
YXXYcX
.........)(
.........)(
11111122
11111111
Engle-Granger two-step estimation method:
(i) Estimate
(ii) Plug in the ECM (SURE estimation): estimators in the ECM
are consistent and efficient.
tZ
tZ
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Cointegration with more than two variables
Example 1.
)'111(:vectoringcointegrat1
,trendsstochasticcommon2
)0(,,)1(,
1,3
)'110()'011(:vectorsingcointegrat2trendstochasticcommon1
)0(,,)1(
2,3
tt
ttttt
ttt
tttt
ttt
t
tttt
ttt
ttt
ttt
RW
IsvuIRW
sRZ
vRWX
uWY
hN
W
IsvuIW
sWZ
vWX
uWY
hN
Example 2.
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Cointegration with more than two variables (cont)
1)'(00)'(1:vectorsingcointegrat2
stationaryareand;2,2)'1(:vectoringcointegrat1
trendstochasticcommon1
)0(,,)1(
1,2
tt
t
tttt
ttt
ttt
XYhNA
W
IsvuIWvWX
uAWY
hN
Example 3.
Example 4.
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Cointegration: Testing and Estimation with more than two variables
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)tptpttt YYYY ....2211
We construct the vector error correction model transforming the VAR:
]....[where
....
:sidesbothfromgsubtractin
]....[
1,...1]....[where
....
210
10112211
1
21
21
1112211
p
ttptpttt
t
p
piii
ttptpttt
II
YYYYY
Y
pi
YYYYY
Johansens method:
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adjustmentoftscoefficienthecontains
vectorsingcointegratthecontains
'
andrank)reducedofismatrix(the'then
,rankingcointegratwithedcointegratisIf
....
11xx
10
xx0
10112211
B
A
BZYABY
AB
hY
YYYYY
ttnhhn
t
nhhn
t
ttptpttt
Vector error correction model:
Example 5: 2 variables, 1 cointegrating vector
1
1
2
1
10 1t
t
t
ttt
Z
XY
ZXZ
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)onnsrestrictio(novectorsingcointegrat:
)onons(restrictimostatvectorsingcointegrat:
01
00
nH
hH
Objective:
Construct the likelihood function under the null and under
the alternative, and construct a likelihood ratio-type test.
Likelihood ratio test has a non-standard distribution due to the
non-stationarity of the variables.
testeigenvaluemaximum
1h:1H
h:0H
testTracenh:1H
nh:0H
)01(2
'..
max
0 BAts
Johansens algorithm to maximize the constrained likelihood is
based on canonical correlation analysis.