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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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SPECI$ICATION ISSUES
:i3erences (;yt# or levels (yt#6 5f all 5("#, then level.
5f some 5(1# but cointegrated,then level of -M.
5f 5(1# but not cointegrated, then
di3erence to 5("#.
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SPECI$ICATION ISSUES
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
JB
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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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