Lecture 7 VAR

8/17/2019 Lecture 7 VAR

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TOPIC 7:

VECTOR

AUTOREGRESSIVEMODELS AND ITSAPPLICATION

By:Assoc. Prof. Dr. Sallahu!" #assa"

SEEQ5133 Applied Econometrics

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INTRODUCTION

Some variables are not onlyexplanatory variables for a givendependent variable, but they arealso explained by the variable thatthey are used to determined.

Model of simultaneous equations –exogenous, endogenous andpredetermined.

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INTRODUCTION

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According to Sim (1!"#, if there issimultaneity among a number ofvariables, then all these variables should

be treated in the same $ay. %herefore, there should be no distinction

bet$een endogenous and exogenousvariables. All variables should betreated as endogenous variable.

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INTRODUCTION

%his situation leads to thedevelopment of the &A' model.

hy $e need &A') %he &A' model is a general frame$or*

to describe the dynamic interrelationshipbet$een stationary variables.

e are not really con+dent that avariable is actually exogenous.

erforming forecasting analysis.

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$EATURES O$ A VAR

MODEL -quations are identi+ed. an be estimated by /0S and get consistent

estimators.

All variables are endogenous. /nly laggedendogenous variables on 'S.

All variables are assumed stationary.

oe2cient in reduced form not structural

parameter. ontemporaneous e3ect captured by residuals.

are uncorrelated $hite4noise error terms.

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t t & 21 µ µ

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VAR MODEL

&A' is a multiple equation system. A set of k time series regressions.

%he regressors are lagged values ofall k series.

0et begin $ith t$o time4seriesvariables)

and

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t y t x

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VAR MODEL

%he dynamic relationship of t$ovariables yield a system of equations)

-ach variable is a function of its o$nlag and the lag of the other variable

in the system.

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y

t t t t x y y ε β β β +++=

−− 11211110

x

t t t t x y x ε β β β +++= −− 12212120

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VAR MODEL

5f and are stationaryvariables,

&A' model is)

%he above system can beestimated using /0S.

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t y t x ( )0 I

yt t t t x y y ε β β β +++= −− 11211110

x

t t t t x y x ε β β β +++= −− 12212120

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VAR MODEL

5f and are nonstationaryvariables, and not cointegrated,$e $or* $ith the +rst di3erence.

&A' model is)

All variables are no$ . %hesystem can be estimated using /0S

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( )1 I

t y t x ( )1 I

y

t t t t x y y ∆

−− +∆+∆=∆ ε β β

112111

x

t t t t x y x ∆

−− +∆+∆=∆ ε β β 112111

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VECM MODEL

5f and are nonstationaryvariables,

and cointegrated e need to modify the system of

equations to allo$ for the

cointegrating relationshipbet$een the nonstationaryvariables.

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t y t x

( )1 I

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VECM MODEL

hy $e do this6 %o retain and use valuable information

about the cointegrating relationship.

%o ensure the best technique that ta*einto account the properties of time seriesdata.

&- Model needs to be used. 5t is aspecial form of the &A' for nonstationaryvariables that are cointegrated.

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VECM MODEL

Model)

%he &-M model allo$s us to examineho$ much $ill change in response toa change in the explanatory variable (the

cointegration part, #, as $ell as the speedof the change (the error correction part,

#

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( ) t t t t

v X Y Y 11101111

+−−+=∆−−

β δ α α

( ) t t t t

v X Y X 21101212

+−−+=∆−−

β δ α α

t y

1−t ECT

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VECM MODEL

7eneral &-M speci+cation)

is the impact multiplier (the short4run

e3ect# that measures the immediateimpact that a current change in $ill haveon a change in .

is the feedbac* e3ect or the ad8ustmente3ect, and sho$s the speed of ad8ustmentor ho$ much of the disequilibrium is beingcorrected. %o ensure stability.

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t t t t X ECT Y ε β λ α +∆++=∆

−11

β

t X

t Y

λ

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SPECI$ICATION ISSUES

0ogs or no logs6 o$ many variables6

%he number of coe2cients in each

equation is proportional to the numberof variables.

9eep the number of variables small)

to ensure plausible relationship amongvariables. to avoid estimation error forecasting

accuracy.

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:i3erences (;yt# or levels (yt#6 5f all 5("#, then level.

5f some 5(1# but cointegrated,then level of -M.

5f 5(1# but not cointegrated, then

di3erence to 5("#.

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o$ many lags6 -nough to eliminate autocorrelation but

as fe$ as possible.

%oo many lags – consume degree offreedom and multicollinearity ( andare linearly dependent#

%oo fe$ lags – speci+cation error (omittingin


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SPECI$ICATION O$ VARMODEL1# %esting for stationarity

=ind out a given time series isstationary or non stationary.

erforming unit root tests – :=, A:=,or tests.

:ouble clic* on the series and

choose V!%&'U"!( Roo(T%s('P%rfor) (%s(s*%c!+ca(!o"'O,

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U"!( Roo( T%s(18


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SPECI$ICATION O$ VARMODEL

") >nit root?non stationary.

1) Stationary.

R%-%c( #. !f (h% AD$ s(a(!s(!cs /01 (h% cr!(!cal 2alu% ( # 5f stationary, then '5 @ 5("# if no

then '5 @5(n# nB". 5f non stationary, ta*e +rst

di3erences of '5 as 1−−=∆ t t PRI PRI PRI

τ

C τ

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SPECI$ICATION O$ VARMODELC# %esting for cointegration

>sing Dohansen test (multiple equation#. Steps)

%esting the order of integration of the variables. Setting the appropriate lag length of the model.

hoosing the appropriate model.

:etermining the number of cointegrating vector.

hoose variables then 3u!c4'Grou*s(a(!s(!cs'5oha"s%" Co!"(%6ra(!o"(%s('O,

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SPECI$ICATION O$ VARMODEL Step 1)%esting the order of integration

of the variables. Step C) Setting the appropriate lag

length of the model -stimate &A' model including all variables in

levels

5nspect the values of the A5, SE and dodiagnostic chec*ing (autocorrelation,heteroscedasticity, normality, possible A'e3ect#.

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SPECI$ICATION O$ VARMODEL &A' -stimation

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Nor)al!(y T%s(

>sing the Darque4Eera test fornormality.

5t based on t$o measures) S4%&"%ss – refers to ho$ symmetric

the residuals are around Fero. ,ur(os!s – refers to the Gpea*ednessH

of the distribution. =or a normaldistribution, the *urtosis value is I.

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Nor)al!(y T%s(

%he Darque4Eera statistics)

$here S J S*e$ness, 9 J 9urtosis, K JSample siFe

e re8ect the hypothesis of normally

distributed error if a calculated value of thestatistics exceeds a critical value selectedfrom the chi4squared distribution.

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( )

−+=

4

3

6

2

2 K S N

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SPECI$ICATION O$ VARMODEL Step I) hoosing the appropriate model

regarding the deterministic components inthe multivariate system.

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SPECI$ICATION O$ VARMODEL26

Information Criteria by Rank and Model

Data Trend: None None Linear Linear Quadrati

Rank or No Intere!t Intere!t Intere!t Intere!t Intere!t

No" of CE# No Trend No Trend No Trend Trend Trend

Lo$ Likeli%ood by Rank &ro'#( and Model &olumn#(

) *+,1+"))3 *+,1+"))3 *+,)-"5.3 *+,)-"5.3 *+,))"-./

1 *+,))")5, *+3//"5-. *+3/-")3/ *+3/0"-)3 *+3/1"-/5+ *+3/3",0- *+3/+"1-1 *+3/1"/0/ *+3/1"155 *+3-/".00

3 *+3/+"3)) *+3/)"/30 *+3/)"/30 *+3-/",). *+3-/",).

kaike Information Criteria by Rank &ro'#( and Model &olumn#(

) 1+1"5))1 1+1"5))1 1+1",0-+ 1+1",0-+ 1+1"+,3,

1 1+1"+)+0 1+1"++/3 1+1"+51/ 1+1"+/)1 1+1")/,-2

+ 1+1"103/ 1+1"+)/1 1+1"+,/) 1+1"3)00 1+1"+-3/

3 1+1",15) 1+1",/./ 1+1",/./ 1+1"50)3 1+1"50)3

S%'ar Criteria by Rank &ro'#( and Model &olumn#(

) 1++"+.)1 1++"+.)1 1++"3.,- 1++"3.,- 1++"+5.-

1 1++"+1.)2 1++"+-,- 1++"3/1/ 1++",0+, 1++"3.1,

+ 1++",,)5 1++"5.)+ 1++".,+3 1++"0-55 1++"-)3/

3 1++"/35) 1+3"1,35 1+3"1,35 1+3"3,3. 1+3"3,3.


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SPECI$ICATION O$ VARMODEL Step L) :etermining the ran* of or

the number of cointegrating vector.

>sing t$o tests) E!6%"2alu%s /charac(%r!s(!c roo(s0 (%s(

" ) 'an* ( # J r ($e have up to r

cointegrating relationship#

1) (r 1# vector.

Maximal eigenvalue statistic – to test ho$many of the number of the characteristic rootsare signi+cantly di3erent from Fero.

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Π

Π

( ) ( )1

11+

−−=+r maxˆ lnT r ,r λ λ


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SPECI$ICATION O$ VARMODEL >sing t$o tests)

L!4%l!hoo Ra(!o (%s( " ) %he number of cointegrating vectors is less

than of equal tor .

1) %he number of cointegrating vectors is more

than r .

%race statistic)

5f trace statistic is smaller than the NO criticalvalue so the model does not sho$ cointegration.

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( ) ( )∑+=

+−−=n

r i

r trace

ˆ lnT r

1

11 λ λ


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SPECI$ICATION O$ VARMODEL

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4nre#trited Cointe$ration Rank Te#t &Trae(

y!ot%e#ied Trae )")5

No" of CE( Ei$en6alue Stati#ti Critial 7alue 8rob"22

None 2 )",,/00/ 3/",).+/ +,"+05/. )")))3

t mo#t 1 2 )"+-)+35 15"5)--- 1+"3+)/) )")1,1

t mo#t + )")501/1 +"355.55 ,"1+//). )"1,0,

Trae te#t indiate# + ointe$ratin$ e9n( at t%e )")5 le6el

2 denote# re:etion of t%e %y!ot%e#i# at t%e )")5 le6el

22Ma;innon*au$*Mi%eli# &1///( !*6alue#

4nre#trited Cointe$ration Rank Te#t &Ma


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SPECI$ICATION O$ VARMODEL Eoth the trace and the maximal

eigenvalue statistics suggest theexistence of t$o cointegrating

vectors. -vie$s then reports results regarding

the coe2cients of the speed of

ad8ustment coe2cients ( # and thematrix of the long4run coe2cients( #.

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α β


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SPECI$ICATION O$ VARMODEL After establishing the number of

cointegrating vectors, $e

proceed $ith the estimation ofthe -M.

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VECM MODELESTIMATION 5f there is cointegration, $e can

estimate the &-M.

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VECM MODELESTIMATION

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Cointe$ratin$ E9: CointE91

RE=8>M&*1( 1"))))))

R?DI>M&*1( *1)"5+/-/

&1)"3-0,(

@*1")130+A

RBD8>M&*1( *)")1/+5+

&)"))35+(

@*5",0.+-A

Error Corretion: D&RE=8>M( D&R?DI>M( D&RBD8>M(

CointE91 *)")3/)30 *)")11++/ *,",50555

&)")1../( &)"))5+/( &)"/5+.,(

@*+"33/,1A @*+"1+3).A @*,".0/10A

D&RE=8>M&*1(( )",50/.5 *)")+.,1) 0"/+,..)

&)"1-5/)( &)")5-/+( &1)".1+/(

@ +",.3,/A @*)",,-+3A @ )"0,.0)A

D&R?DI>M&*1(( *)"-31,-- *)"),3..1 *3"31+5+1

&)".3.--( &)"+)1-.( &3."35-/(

@*1"3)550A @*)"+1.+/A @*)")/111A

D&RBD8>M&*1(( *)"))5.03 *)"))+03) *)"+,+33)

&)")),1)( &)"))13)( &)"+3,).(

@*1"3-3.3A @*+"1)11+A @*1")353+A


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VAR MODEL ESTIMATION

5f there is no evidence of cointegration, $ecan estimate the unrestricted &A'.

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Lecture 7 VAR

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