spurious reg and co integration

8/10/2019 Spurious Reg and Co Integration

1/22

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


2/22

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


3/22

Spurious Regression (cont)

What does it really happen?

ondistributisomet2/1T

0p

DW

ondistributisome2R

ondistributisome


4/22


5/22

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.


6/22


7/22

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.


8/22

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


9/22

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.


10/22

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?


11/22

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


12/22

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.


13/22

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


14/22

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 ,


15/22

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


16/22

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


17/22

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


18/22

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.


19/22

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.


20/22

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:


21/22

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


22/22

)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.

spurious reg and co integration

Documents