Time Series Econometrics Time Time SeriesSeries 10110010 · 11/7/2013 7 Thursday, November 07, 2013...

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7 November 2013 Vijayamohan: CDS MPhil: Time Series 1

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Time Series EconometricsTime Series EconometricsTime Series EconometricsTime Series Econometrics

10101010

VijayamohananVijayamohananVijayamohananVijayamohanan PillaiPillaiPillaiPillai NNNN


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MethodologyMethodologyMethodologyMethodology

Time Time Time Time SeriesSeriesSeriesSeries

Thursday, November 07, 2013

Vijayamohan: CDS M Phil 103: Time Series

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Time Series: Methodology

Three alternative approaches:

(1): general to specific model (GETS):LSE Approach

Hendry, Pagan and Sargan (1984) and Hendry (1987).

(2): vector autoregressions model (VAR):Sims (1980): dominant approach in the USA ; and

(3): vector error correction model (VECM):Follows the Granger Representation theorem (Engle and Granger 1987)

11/7/2013 10:19:14 AM Vijayamohan: M Phil: Time Series 4

GETS Approach

Suppose, the theory implies that there is a relationship between consumption ( Ct) and income ( Yt ) i.e., say

Ct = αααα0 + αααα1Yt (1)

Since this is an equilibrium relationship,

a dynamic adjustment equation can be searched by starting first with a very general and elaborate specification.

This initial general specification is termed the general unrestricted model (GUM).


CDS MPhil FQ Vijayamohan

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A good GUM for the consumption equation:

CCCCtttt ==== ααααCCCCtttt––––1111+ + + + ββββ0000YYYYtttt + + + + ββββ1111YYYYtttt––––1 1 1 1 ++++ εtttt ;;;;

where Yt’= (Y1t, Y2t,…., Ykt),

∆∆∆∆CCCCtttt ==== ((((αααα−−−−1)1)1)1)CCCCtttt––––1111+ + + + ββββ0 0 0 0 ∆∆∆∆YYYYtttt + (+ (+ (+ (ββββ0 0 0 0 ++++ββββ1111))))YYYYtttt––––1 1 1 1 ++++ εtttt ; or; or; or; or

∆∆∆∆CCCCtttt ==== ββββ0 0 0 0 ∆∆∆∆YYYYtttt + + + + λλλλ [[[[CCCCtttt––––1111−−−− kkkk YYYYtttt––––1111] +] +] +] + εtttt ;;;;

where where where where kkkk = = = = ((((ββββ0 0 0 0 ++++ββββ1111)/)/)/)/ ((((1111−−−− αααα) and ) and ) and ) and λλλλ = = = = ((((αααα−−−−1)1)1)1)

GETS Approach



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GETS Approach

A good GUM for the consumption equation in general:

(((( )))) t

n

itt

m

iityiitcit kYCYCC εεεελλλλ∆∆∆∆ββββ∆∆∆∆ββββ∆∆∆∆ ++++−−−−++++++++====∑∑∑∑ ∑∑∑∑ −−−−−−−−−−−−−−−− 11

Include enough lagged variables Include enough lagged variables Include enough lagged variables Include enough lagged variables so that there is so that there is so that there is so that there is no no no no

serial correlation in the residuals of the GUM. serial correlation in the residuals of the GUM. serial correlation in the residuals of the GUM. serial correlation in the residuals of the GUM.

Finally, Finally, Finally, Finally, a parsimonious version of equation (2) is a parsimonious version of equation (2) is a parsimonious version of equation (2) is a parsimonious version of equation (2) is

developeddevelopeddevelopeddeveloped, , , , by deleting the insignificant variables and by deleting the insignificant variables and by deleting the insignificant variables and by deleting the insignificant variables and

imposing constraints on the estimated coefficientsimposing constraints on the estimated coefficientsimposing constraints on the estimated coefficientsimposing constraints on the estimated coefficients. . . .

GETS is thus a highly empirical approach.GETS is thus a highly empirical approach.GETS is thus a highly empirical approach.GETS is thus a highly empirical approach.

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ConsumptionConsumptionConsumptionConsumption IncomeIncomeIncomeIncome

InflationInflationInflationInflation



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ConsumptionConsumptionConsumptionConsumption

IncomeIncomeIncomeIncome



9 Thursday, November 07, 2013


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Residual Residual Residual Residual

AnalysisAnalysisAnalysisAnalysis

ACF PACF



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ConsumptionConsumptionConsumptionConsumption IncomeIncomeIncomeIncome

InflationInflationInflationInflation






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PcGive: Test: Dynamic Analysis



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PcGive: Test: Dynamic Analysis

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Residual Residual Residual Residual

AnalysisAnalysisAnalysisAnalysis

ACF PACF


Note: the equilibrium theoretical consumption

relationship can be recovered from equation(2)

by imposing the equilibrium condition that all the

changes in the variables are zero , i.e., from the term in

the last part in (2), since in equilibrium, equatio n (2) will

be

0 = 0 + 0 + λλλλ(Ct – 1 – kYt – 1) + 0

∴∴∴∴ C* = kY*.

GETS Approach

(2)(((( )))) t

n

itt

m

iityiitcit kYCYCC εεεελλλλ∆∆∆∆ββββ∆∆∆∆ββββ∆∆∆∆ ++++−−−−++++++++====∑∑∑∑ ∑∑∑∑ −−−−−−−−−−−−−−−− 11


The expression in the lagged level variables,

λλλλ(Ct – 1 – kYt – 1) in equation (2)

is known as the error correction term and models that

include it are known as the error correction models

(ECM).

It implies that departures from the equilibrium pos ition in

the immediate past period will be offset in the cur rent

period by λλλλ proportion.

Note that λλλλ should be negative.

GETS Approach

(((( )))) t

n

itt

m

iityiitcit kYCYCC εεεελλλλ∆∆∆∆ββββ∆∆∆∆ββββ∆∆∆∆ ++++−−−−++++++++==== ∑∑∑∑ ∑∑∑∑ −−−−−−−−−−−−−−−− 11



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Developed as an alternative to the large scale econometric models based on the Cowles Commission approach .

Sims (1980) argued:

• The classification of variables into endogenous and exogenous ,

• The constraints implied by the traditional theory on the structural parameters , and

• The dynamic adjustment mechanisms used in the large

scale models, are all arbitrary and too restrictive .

∴∴∴∴ Include all variables as endogenous .

VAR Models



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For a set of n time series variables

a VAR model of order p (VAR(p)) can be written as:

where the Ai’s are (n x n) coefficient matrices and

is an unobservable i.i.d. zero mean error term.

)'y...,y,y(y kt,ttt 21====

tptpttt yA...yAyAy εεεε++++++++++++++++==== −−−−−−−−−−−− 2211

)',...,,( ktttt εεεεεεεεεεεεεεεε 21====

VAR Models



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Consider a two-variable VAR(1) with k =2.

yttttt zcyczby εεεε++++++++++++==== −−−−−−−− 11211112

zttttt zcycybz εεεε++++++++++++==== −−−−−−−− 12212121

),(d.i.i~ iit20 εεεεσσσσεεεε 0====),cov( zy εεεεεεεεwith and

In matrix form:

++++

====

−−−−−−−−

−−−−

−−−−

zt

yt

t

t

t

t

z

y

cc

cc

z

y

b

b

εεεεεεεε

1

1

2221

1211

21

12

1

1

VAR Models

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ttt XBX εεεεΓΓΓΓ ++++==== −−−−11

Structural VAR (SVAR) or the Primitive System

To normalize the LHS vector, we need to multiply the equation by inverse B:

ttt BXBBXB εεεεΓΓΓΓ 111

11 −−−−−−−−

−−−−−−−− ++++====

ttt eXAX ++++==== −−−−11

VAR in standard form (unstructured VAR: UVAR).

VAR Models

++++

====

−−−−−−−−

−−−−

−−−−

zt

yt

t

t

t

t

z

y

cc

cc

z

y

b

b

εεεεεεεε

1

1

2221

1211

21

12

1

1



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++++

====

−−−−

−−−−

t

t

t

t

t

t

e

e

z

y

aa

aa

z

y

2

1

1

1

2221

1211

These error terms are composites of the structural innovations from the primitive system.

tt Be εεεε1−−−−====

−−−−====

zt

yt

t

t

b

b

)bb(e

e

εεεεεεεε

1

1

1

1

21

12

12212

1

−−−−====−−−−

1

1

1

1

21

12

1221

1

b

b

)bb(B

VAR Models



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∆∆∆∆εεεεεεεε ztyt

t

be

12

1

++++====

12211 bb−−−−====∆∆∆∆

∆∆∆∆εεεεεεεε ztyt

t

be

++++==== 21

2

0====)e(E it

2

22

12

2

2

22

12

2

2

11 ∆∆∆∆σσσσσσσσ

∆∆∆∆εεεεεεεε zyztyt

tt

b)b(E)e(E)e(Var

++++====

++++========

time independent, and the same is true for the other one.

VAR Models



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But covariances are not zero:

======== )ee(E)e,e(Cov tttt 2121

====++++++++

2

2112

∆∆∆∆εεεεεεεεεεεεεεεε )]b)(b[(E ytztztyt

02

221

212 ≠≠≠≠

++++∆∆∆∆

σσσσσσσσ )bb( yz

So the shocks in a standard VAR are correlated. The only way to remove the correlation and make the covar = 0 is to assume that the contemporaneous effects are zero: 02112 ======== bb

VAR Models

yttttt zcyczby εεεε++++++++++++==== −−−−−−−− 11211112zttttt zcycybz εεεε++++++++++++==== −−−−−−−− 12212121



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VAR Models

Same Same Same Same regressorsregressorsregressorsregressors for all equationsfor all equationsfor all equationsfor all equations; ; ; ;

so so so so model estimation is straightforwardmodel estimation is straightforwardmodel estimation is straightforwardmodel estimation is straightforward: : : :

ML ML ML ML estimatorestimatorestimatorestimator →→→→ OLS estimatorOLS estimatorOLS estimatorOLS estimator for each equation. for each equation. for each equation. for each equation.

This This This This propertypropertypropertyproperty →→→→ popularity of VAR modelspopularity of VAR modelspopularity of VAR modelspopularity of VAR models....

Note: Note: Note: Note: All variables in the reduced form equations are All variables in the reduced form equations are All variables in the reduced form equations are All variables in the reduced form equations are

endogenousendogenousendogenousendogenous, and hence the equations can be seen as , and hence the equations can be seen as , and hence the equations can be seen as , and hence the equations can be seen as

the the the the basic ARDL formulationbasic ARDL formulationbasic ARDL formulationbasic ARDL formulation of a simple VAR modelof a simple VAR modelof a simple VAR modelof a simple VAR model....



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VAR / GETS / VECM

The simple VAR models do not identify structural coefficients (hence ‘Structural VAR models’)

nor do they take seriously the relevance of unit root tests .

In GETS, although there is some awareness of the unit root characteristics of the variables, the crucial theoretical relationship , in the error correction part, is specified in the levels of the variables.

In contrast, VECM, like VAR, treats all variables as endogenous ,

but limits the number of variables to those relevant fo r

a particular theory .

++++

====

−−−−

−−−−

t

t

t

t

t

t

e

e

z

y

aa

aa

z

y

2

1

1

1

2221

1211

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VECM

This method developed by Johansen (1988) is undoubtedly the most widely used method in applied work.

Models using this approach are also known as

Cointegrating VAR (CIVAR) models.

VECM can be seen as scaled down (reduced form) VAR model in which the structural coefficients are identified .

The theoretical basis of VECM:

Granger Representation Theorem .



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VECM Granger Representation TheoremGranger Representation TheoremGranger Representation TheoremGranger Representation Theorem (Engle and Granger (Engle and Granger (Engle and Granger (Engle and Granger

1987):1987):1987):1987):

If a set of variables are If a set of variables are If a set of variables are If a set of variables are cointegratedcointegratedcointegratedcointegrated, then there , then there , then there , then there

exists a exists a exists a exists a VARMA representation for them and an VARMA representation for them and an VARMA representation for them and an VARMA representation for them and an

‘‘‘‘errorerrorerrorerror----correcting’ mechanismcorrecting’ mechanismcorrecting’ mechanismcorrecting’ mechanism (ECM); for example,(ECM); for example,(ECM); for example,(ECM); for example,

If If If If YYYYtttt and and and and XXXXtttt are both I(1)are both I(1)are both I(1)are both I(1) and and and and have constant meanshave constant meanshave constant meanshave constant means and and and and

are are are are cointegratedcointegratedcointegratedcointegrated, then , then , then , then there exists an ECMthere exists an ECMthere exists an ECMthere exists an ECM, (with the , (with the , (with the , (with the

equilibrium error equilibrium error equilibrium error equilibrium error UUUUtttt = = = = YYYYtttt –––– ββββ XXXXtttt ), ), ), ), of the formof the formof the formof the form::::

∆∆∆∆YYYYtttt = = = = ––––λλλλ1 1 1 1 UUUUtttt––––1111+ + + + lagged{lagged{lagged{lagged{∆∆∆∆YYYYtttt, , , , ∆∆∆∆XXXXtttt}}}} + + + + θθθθ (L)(L)(L)(L)εεεε1111tttt,,,,

Where Where Where Where θθθθ (L) is a finite polynomial in lag operator L and (L) is a finite polynomial in lag operator L and (L) is a finite polynomial in lag operator L and (L) is a finite polynomial in lag operator L and

εεεε1111tttt is a white noise.is a white noise.is a white noise.is a white noise.



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VECM

In In In In ∆∆∆∆YYYYtttt = = = = ––––λλλλ1111UUUUtttt––––1111+ + + + lagged{lagged{lagged{lagged{∆∆∆∆YYYYtttt, , , , ∆∆∆∆XXXXtttt} + } + } + } + θθθθ (L)(L)(L)(L)εεεε1111tttt,,,,

The The The The ‘equilibrating’ error‘equilibrating’ error‘equilibrating’ error‘equilibrating’ error in the in the in the in the previous periodprevious periodprevious periodprevious period, , , , UUUUtttt––––1111, , , ,

captures the captures the captures the captures the adjustment towards longadjustment towards longadjustment towards longadjustment towards long----run equilibriumrun equilibriumrun equilibriumrun equilibrium, , , ,

and the and the and the and the expected expected expected expected ––––veveveve signsignsignsign⇒⇒⇒⇒ error would correct in the error would correct in the error would correct in the error would correct in the

longlonglonglong----runrunrunrun....

Note: No feedback assumed from Note: No feedback assumed from Note: No feedback assumed from Note: No feedback assumed from YYYYtttt to to to to XXXXtttt....

If If If If feedback assumed from feedback assumed from feedback assumed from feedback assumed from YYYYtttt to to to to XXXXtttt, an additional , an additional , an additional , an additional

equation:equation:equation:equation:

∆∆∆∆XXXXtttt = = = = ––––λλλλ2222UUUUtttt––––1111+ + + + lagged{lagged{lagged{lagged{∆∆∆∆YYYYtttt, , , , ∆∆∆∆XXXXtttt}}}} + + + + θθθθ (L)(L)(L)(L)εεεε2222tttt,,,,

wherewherewherewhere εεεε2222tttt is is is is white white white white noisenoisenoisenoise....

7 November 2013 Vijayamohan: CDS M Phil: Time Series 7

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Level variables

Statistics > Multivariate time series > Vector error-correction model (VECM)

VECM in Stata



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provides information about the sample, the model fi t, and the identification of the parameters in the cointegrating equation.

VECM in Stata



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The main estimation table the estimates of the short-run parameters, + their standard errors and confidence intervals.

VECM in Stata

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The second estimation table the estimates of the parameters in the cointegratin gequation, + their standard errors and confidence intervals.

VECM in Stata



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-15

-10

-50

510

Pre

dict

ed c

oint

egra

ted

equa

tion

1950q1 1960q1 1970q1 1980q1 1990q1time

. predict ce, ce

. line ce time

VECM in Stata



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VECM

Remember: VECM, like VAR, treats Remember: VECM, like VAR, treats Remember: VECM, like VAR, treats Remember: VECM, like VAR, treats all variables as all variables as all variables as all variables as

endogenousendogenousendogenousendogenous,,,,

but but but but limits the number of variableslimits the number of variableslimits the number of variableslimits the number of variables to those relevant for a to those relevant for a to those relevant for a to those relevant for a

particular theory. particular theory. particular theory. particular theory.

For example, For example, For example, For example, with two variables: with two variables: with two variables: with two variables: YYYYtttt and and and and XXXXtttt,,,,

two endogenous variablestwo endogenous variablestwo endogenous variablestwo endogenous variables, , , , two possible CVs;two possible CVs;two possible CVs;two possible CVs;

If If If If our equation of interestour equation of interestour equation of interestour equation of interest is is is is YYYYtttt = = = = ffff((((XXXXtttt) only) only) only) only, that is,, that is,, that is,, that is,

If No If No If No If No feedback assumed from feedback assumed from feedback assumed from feedback assumed from YYYYtttt to to to to XXXXtttt, or , or , or , or XXXXtttt = = = = ffff((((YYYYtttt), ), ), ), we we we we

need to consider and estimate only one, first, equationneed to consider and estimate only one, first, equationneed to consider and estimate only one, first, equationneed to consider and estimate only one, first, equation, , , ,

unlike in VAR.unlike in VAR.unlike in VAR.unlike in VAR.



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VECM

Thus we need to test for direction of feedback:

From Xt to Yt: whether Yt is exogenous to Xt;

From Yt to Xt: whether Xt is exogenous to Yt;

That is, we need to test for ‘exogeneity’ of variables:

This exogeneity test is:

Granger non-‘causality’ test (Granger 1969):



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VECM: Granger non- ‘causality’ test: an Exogeneity test

ConsiderConsiderConsiderConsider thethethethe followingfollowingfollowingfollowing equationsequationsequationsequations::::

YYYYtttt==== ΣΣΣΣααααiiiiYYYYtttt−−−−iiii ++++ ΣΣΣΣββββ iiiiXXXXtttt−−−−iiii ++++ eeee1111tttt ,,,, ((((1111))))

XXXXtttt==== ΣΣΣΣγγγγiiiiYYYYtttt−−−−iiii ++++ ΣΣΣΣδδδδiiiiXXXXtttt−−−−iiii ++++ eeee2222tttt ,,,, ((((2222))))

wherewherewherewhere thethethethe summationssummationssummationssummations areareareare forforforfor somesomesomesome laglaglaglag lengthlengthlengthlength kkkk,,,,andandandand eeee1111tttt andandandand eeee2222tttt areareareare independentlyindependentlyindependentlyindependently distributeddistributeddistributeddistributed

whitewhitewhitewhite noisesnoisesnoisesnoises....

((((1111)))) hypothesiseshypothesiseshypothesiseshypothesises thatthatthatthat thethethethe currentcurrentcurrentcurrent valuevaluevaluevalue ofofofof YYYY isisisis

relatedrelatedrelatedrelated totototo pastpastpastpast valuesvaluesvaluesvalues ofofofof YYYY itselfitselfitselfitself andandandand thosethosethosethose ofofofof XXXX,,,,

whilewhilewhilewhile ((((2222)))) postulatespostulatespostulatespostulates aaaa similarsimilarsimilarsimilar behaviourbehaviourbehaviourbehaviour forforforfor XXXX....



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WeWeWeWe havehavehavehave thethethethe followingfollowingfollowingfollowing implicationimplicationimplicationimplication::::

XXXX doesdoesdoesdoes notnotnotnot ‘Granger‘Granger‘Granger‘Granger----cause’cause’cause’cause’ YYYY ifififif,,,, andandandand onlyonlyonlyonly if,if,if,if, ββββiiii ≡≡≡≡ 0000,,,,

forforforfor allallallall iiii,,,, asasasas aaaa groupgroupgroupgroup;;;;

ThusThusThusThus thethethethe measuremeasuremeasuremeasure ofofofof linearlinearlinearlinear feedbackfeedbackfeedbackfeedback fromfromfromfrom XXXX totototo YYYY isisisis zerozerozerozero

((((GewekeGewekeGewekeGeweke 1982198219821982))))....

ThatThatThatThat is,is,is,is, thethethethe pastpastpastpast valuesvaluesvaluesvalues ofofofof XXXX dodododo notnotnotnot helphelphelphelp totototo predictpredictpredictpredict YYYY....

InInInIn thisthisthisthis case,case,case,case, YYYY isisisis exogenousexogenousexogenousexogenous withwithwithwith respectrespectrespectrespect totototo XXXX

((((EngleEngleEngleEngle etetetet alalalal.... 1983198319831983))))....

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Similarly,Similarly,Similarly,Similarly, wewewewe havehavehavehave thethethethe followingfollowingfollowingfollowing implicationimplicationimplicationimplication::::

YYYY doesdoesdoesdoes notnotnotnot ‘Granger‘Granger‘Granger‘Granger----cause’cause’cause’cause’ XXXX,,,, if,if,if,if, andandandand onlyonlyonlyonly if,if,if,if, γγγγiiii ≡≡≡≡ 0000 forforforfor allallallall

iiii asasasas aaaa groupgroupgroupgroup;;;; thethethethe measuremeasuremeasuremeasure ofofofof linearlinearlinearlinear feedbackfeedbackfeedbackfeedback fromfromfromfrom YYYY totototo XXXX

isisisis zerozerozerozero....

ThatThatThatThat is,is,is,is, thethethethe pastpastpastpast valuesvaluesvaluesvalues ofofofof YYYY failfailfailfail totototo helphelphelphelp predictpredictpredictpredict XXXX....

HereHereHereHere XXXX isisisis exogenousexogenousexogenousexogenouswithwithwithwith respectrespectrespectrespect totototo YYYY....




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IfIfIfIf thethethethe laggedlaggedlaggedlagged termstermstermsterms havehavehavehave significantsignificantsignificantsignificant nonnonnonnon----zerozerozerozero

coefficientscoefficientscoefficientscoefficients,,,,

thenthenthenthen theretheretherethere isisisis ‘causality’‘causality’‘causality’‘causality’ orororor feedbackfeedbackfeedbackfeedback inininin bothbothbothboth

directionsdirectionsdirectionsdirections....


7 November 2013 Vijayamohan: CDS M Phil: Time Series 7

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Level variables

Granger non-‘causality’ test in STATA



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In Stata, first run a VAR model:

Statistics > Multivariate time series > Vector autoregression(VAR)



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In Stata

Statistics > Multivariate time series > VAR diagnostics and tests > Granger causality tests

H0: x does not Granger-cause y.




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H0: x does not Granger-cause y.

XY




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‘Granger‘Granger‘Granger‘Granger causality’causality’causality’causality’:::: concernedconcernedconcernedconcerned withwithwithwith onlyonlyonlyonly shortshortshortshort runrunrunrun

forecastabilityforecastabilityforecastabilityforecastability,,,, whilewhilewhilewhile

CointegrationCointegrationCointegrationCointegration:::: concernedconcernedconcernedconcerned withwithwithwith longlonglonglong runrunrunrun equilibriumequilibriumequilibriumequilibrium;;;;

ErrorErrorErrorError correctioncorrectioncorrectioncorrection modelmodelmodelmodel (ECM)(ECM)(ECM)(ECM) bringsbringsbringsbrings thethethethe twotwotwotwo

(different)(different)(different)(different) conceptsconceptsconceptsconcepts togethertogethertogethertogether....

SupposeSupposeSupposeSuppose YYYYtttt andandandand XXXXtttt areareareare bothbothbothboth IIII((((1111)))) seriesseriesseriesseries andandandand theytheytheythey areareareare

cointegratedcointegratedcointegratedcointegrated suchsuchsuchsuch thatthatthatthat uuuutttt ==== YYYYtttt −−−− ββββXXXXtttt isisisis IIII((((0000))))....



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ThisThisThisThis cointegrtaedcointegrtaedcointegrtaedcointegrtaed systemsystemsystemsystem cancancancan bebebebe writtenwrittenwrittenwritten inininin termstermstermsterms ofofofof ECMECMECMECM

asasasas::::

∆∆∆∆yyyytttt==== −−−−λλλλ1111uuuutttt−−−−1111++++ lagged{lagged{lagged{lagged{∆∆∆∆yyyytttt ,,,, ∆∆∆∆xxxxtttt}}}} ++++ θθθθ1111((((LLLL))))εεεε1111tttt,,,, ((((1111))))

∆∆∆∆xxxxtttt==== −−−−λλλλ2222uuuutttt−−−−1111++++ lagged{lagged{lagged{lagged{∆∆∆∆yyyytttt ,,,, ∆∆∆∆xxxxtttt}}}} ++++ θθθθ2222((((LLLL))))εεεε2222tttt,,,, ((((2222))))

where where where where θθθθ1111((((LLLL))))εεεε1111tttt and and and and θθθθ2222((((LLLL))))εεεε2222tttt are are are are finite order moving finite order moving finite order moving finite order moving

averagesaveragesaveragesaverages and one of and one of and one of and one of λλλλ1111, , , , λλλλ2 2 2 2

≠≠≠≠ 0. 0. 0. 0.



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In In In In the ECM, the the ECM, the the ECM, the the ECM, the error correction termerror correction termerror correction termerror correction term, , , , UUUUtttt−−−−1111, , , ,

‘‘‘‘Granger causes’Granger causes’Granger causes’Granger causes’ ∆∆∆∆YYYYtttt or or or or ∆∆∆∆XXXXtttt (or both). (or both). (or both). (or both).

As As As As UUUUtttt−−−−1111 itself is a itself is a itself is a itself is a functionfunctionfunctionfunction of of of of YYYYtttt−−−−1111and and and and XXXXtttt−−−−1111, , , ,

either either either either XXXXtttt is ‘is ‘is ‘is ‘Granger caused’Granger caused’Granger caused’Granger caused’ by by by by YYYYtttt−−−−1111 or or or or YYYYtttt by by by by XXXXtttt−−−−1111. . . .



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That That That That is, is, is, is, the coefficientthe coefficientthe coefficientthe coefficient of of of of ECECECEC contains contains contains contains information on information on information on information on

whether the past values of the variableswhether the past values of the variableswhether the past values of the variableswhether the past values of the variables ‘‘‘‘affectaffectaffectaffect’ the ’ the ’ the ’ the

current values of the variable under consideration. current values of the variable under consideration. current values of the variable under consideration. current values of the variable under consideration.

This then implies that there must be some This then implies that there must be some This then implies that there must be some This then implies that there must be some ‘Granger ‘Granger ‘Granger ‘Granger

causality’causality’causality’causality’ between the two series in order to induce them between the two series in order to induce them between the two series in order to induce them between the two series in order to induce them

towards towards towards towards equilibriumequilibriumequilibriumequilibrium....

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Granger non-‘causality’ test: ‘Causality’- test ?

First suggested by Wiener (1956):

More properly called Wiener-Granger non-‘causality’ test.

Economists (e.g., Zellner 1979) and even philosophers(e.g., Holland 1986) question the very term ‘causality’:

To mean ‘cause-effect’ relationship, when there is only temporal lead-lag relationship?

Not ‘causality’ but ‘precedence’ as suggested by Edward Leamer.

Unfortunately several studies to infer ‘cause-effect’ relationship!



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Granger non-‘causality’ test: ‘Causality’- test ?

Adrian Pagan (1989) on Granger causality:

“There was a lot of high powered analysis of this

topic, but I came away from a reading of it with the

feeling that it was one of the most unfortunate

turnings for econometrics in the last two decades,

and it has probably generated more nonsense results

than anything else during that time.”

Pagan, A.R. (1989), '20 Years After: Econometrics 1966-1986,' in B. Cornet and H. Tulkens (eds)., Contributions to Operations Research and Econometrics, The XXth Anniversary of CORE, (Cambridge, Ma., MIT Press).


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Consider a (linear) filter –

The impact multiplier is:

∑∑∑∑ −−−−====k

ktkt XY ΦΦΦΦ

t

t

X

Y

∂∂∂∂∂∂∂∂


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Consider a (linear) filter –

��k = response in the output at time t to a unit pulse in the input at time t – k:

one-period multiplier or transient response or impulse response:

∑∑∑∑ −−−−====k

ktkt XY ΦΦΦΦ

t

kt

kt

t

X

Y

X

Y

∂∂∂∂∂∂∂∂====

∂∂∂∂∂∂∂∂ ++++

−−−−


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Given the linear filter

Impulse response function =

The time path of all the periodical impulse responses:

∑∑∑∑ −−−−====k

ktkt XY ΦΦΦΦ

t

kt

kt

t

X

Y

X

Y

∂∂∂∂∂∂∂∂====

∂∂∂∂∂∂∂∂ ++++

−−−−


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The first order Autorregresive process AR(1):

ACF = ρρρρk = γγγγk / γγγγ0 = Φk, k = 0, 1, 2, …..

Impulse responses

ACF ↓↓↓↓, as k ↑↑↑↑, since |Φ| < 1: Sign of stationarity.

+Φ: direct convergence;

negative Φ: oscillatory convergence.

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In PcGiveFirst run a VAR



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SYS( 1) Estimating the system by OLS (using Data.in 7)The estimation sample is: 1953 (3) to 1992 (3)

URF equation for: CONSCoefficient Std.Error t-value t-prob

CONS_1 0.720835 0.1408 5.12 0.000CONS_2 0.114970 0.1219 0.943 0.347INC_1 0.119269 0.08579 1.39 0.166INC_2 0.028640 0.07645 0.375 0.708INFLAT_1 -1.14639 0.3395 -3.38 0.001INFLAT_2 -0.222212 0.3554 -0.625 0.533Constant U 14.1709 15.12 0.937 0.350

sigma = 1.87171 RSS = 525.4948372

URF equation for: INCCoefficient Std.Error t-value t-prob

CONS_1 -0.201124 0.2311 -0.870 0.386CONS_2 0.294033 0.2001 1.47 0.144INC_1 0.847601 0.1408 6.02 0.000INC_2 -0.0203814 0.1255 -0.162 0.871INFLAT_1 0.470490 0.5574 0.844 0.400INFLAT_2 -1.26264 0.5836 -2.16 0.032Constant U 73.8946 24.83 2.98 0.003

sigma = 3.0729 RSS = 1416.406296

URF equation for: INFLATCoefficient Std.Error t-value t-prob

CONS_1 0.00227184 0.02679 0.0848 0.933CONS_2 0.000632921 0.02319 0.0273 0.978INC_1 0.0103773 0.01632 0.636 0.526INC_2 -0.0157283 0.01455 -1.08 0.281INFLAT_1 1.54190 0.06460 23.9 0.000INFLAT_2 -0.648024 0.06764 -9.58 0.000Constant U 2.41866 2.877 0.841 0.402

sigma = 0.356165 RSS = 19.02800862

Vector Portmanteau(12): 78.1333Vector AR 1-5 test: F(45,395)= 0.84719 [0.7481] Vector Normality test: Chi^2(6) = 9.1045 [0.1678] Vector hetero test: F(72,723)= 1.2828 [0.0642]

Vector hetero-X test: F(162,695)= 1.1138 [0.1819]



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0 50 100 150

0.0

0.5

1.0CONS (CONS eqn)

0 50 100 150

0.0

0.2

0.4INC (CONS eqn)

0 50 100 150

0.000

0.025

0.050INFLAT (CONS eqn)

0 50 100 150

0.0

0.2

0.4 CONS (INC eqn)

0 50 100 150

0.0

0.5

1.0INC (INC eqn)

0 50 100 150

-0.02

0.00

INFLAT (INC eqn)

0 50 100 150

-5.0

-2.5

0.0CONS (INFLAT eqn)

0 50 100 150

-4

-2

0 INC (INFLAT eqn)

0 50 100 150

0.0

0.5

1.0INFLAT (INFLAT eqn)

Response ofConsumption Income Inflation

Shock fromConsumption

Shock fromIncome

Shock fromInflation

Quarterly data 1953 (1) to 1992 (3)



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0 50 100 150

2

4

6 cum CONS (CONS eqn)

0 50 100 150

1

2

3

4cum INC (CONS eqn)

0 50 100 150

0.25

0.50

0.75cum INFLAT (CONS eqn)

0 50 100 150

5

10 cum CONS (INC eqn)

0 50 100 150

5

10 cum INC (INC eqn)

0 50 100 150

-0.3

-0.2

-0.1

0.0 cum INFLAT (INC eqn)

0 50 100 150

-150

-100

-50

0cum CONS (INFLAT eqn)

0 50 100 150

-100

-50

0cum INC (INFLAT eqn)

0 50 100 150

2.5

5.0

7.5

10.0cum INFLAT (INFLAT eqn)

Accumulated Response of

Consumption Income Inflation

Shock fromConsumption

Shock fromIncome

Shock fromInflation

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In StataFirst run a VAR or VECM

. var cons inc inflat, noconstant lags(1/2)



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In StataFirst run a VAR or VECM

. var cons inc inflat, noconstant lags(1/2)

. irf create order1, step(10) set(myirf1)(file myirf1.irf created)(file myirf1.irf now active)(file myirf1.irf updated)

. irf graph oirf, impulse(inc) response(cons)



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-2

-1.5

-1

-.5

0

0 5 10

order1, inc, cons

95% CI orthogonalized irf

step

Graphs by irfname, impulse variable, and response variable

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1. All the series are I(0)

Simply model the data in their levels, using OLS estimation.

2. All the series are integrated of the same order (e.g., I(1)), but not cointegrated.

Just difference (appropriately)each series, and estimate a standard regression model using OLS.



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3. All the series are integrated of the same order, and they are cointegrated.

estimate two types of models: (i) An OLS regression model using the levels of the data.

→→→→ the long-run equilibrating relationship between the variables.

(ii) An error-correction model (ECM), estimated by O LS. →→→→ the short-run dynamics of the relationship between

the variables.



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4. Finally,

Some of the variables in question may be stationary,

some may be I(1)

and there may be cointegration among some of the I(1) variables.



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ARDL= Autoregressive-Distributed Lag.

in use for decades, but in more recent times provide a very valuable vehicle for testing for the presence of long-run relationships between economic time-series.



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In its basic form, an ARDL regression model : (MARMA model)

yt = β0 + β1yt-1 + .......+ βkyt-p+ α0x t + α1x t-1 + α2x t-2 + ......... + αqx t-q + εt

where εt is a random "disturbance" term.

"autoregressive“ = y t is "explained (in part) by lagged values of itself.

"distributed lag" = successive lags of the explanat ory variable "x" .

Sometimes, the current value of x t is excludedThursday, November 07, 2013


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Pesaran MH and Shin Y. 1999. “An autoregressive distributed lag modelling

approach to cointegration analysis.”

Chapter 11 in Econometrics and Economic Theory in the 20th

Century: The Ragnar Frisch Centennial Symposium,

Strom S (ed.). Cambridge University Press: Cambridge.

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“… a new approach to testing for the existence of a relationship between variables in levels which is applicable irrespective of whether the underlying regressors are purely I(0), purely I(1) or mutually cointegrated.”

“Bounds Testing Approaches to the Analysisof Level Relationships”

M. Hashem Pesaran, Yongcheol Shin and Richard J. Sm ithJournal of Applied Econometrics

16: 289–326 (2001)



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Two sets of asymptotic critical values : one when all regressors are purely I(1) and the other if they are all purely I(0).

These two sets of critical values provide critical value bounds for all classifications of the regressors int o purely I(1), purely I(0) or mutually cointegrated.

Accordingly, various bounds testing procedures are proposed.



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The ARDL / Bounds Testing methodology of Pesaran and Shin (1999) and Pesaran et al. (2001) has a number of features that many researchers feel give it some advantages over conventional cointegration testing.



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For instance:

It can be used with a mixture of I(0) and I(1) data.

It involves just a single-equation set-up, making it simple to implement and interpret.

Different variables can be assigned different lag-lengths as they enter the model.



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A conventional ECM for cointegrated data :

∆yt = β0 + Σ βi∆yt-i + Σγj∆x1t-j + Σδk∆x2t-k + φzt-1 + et ;

The ranges of summation : from 1 to p , 0 to q 1, and 0 to q 2respectively.

z, the "error-correction term", is the OLS residuals series from the long-run "cointegrating regression",

yt = α0 + α1x1t + α2x2t + vt



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Step 1:

use the ADF/PP/KPSS tests to check that none of the series are I(2).

Step 2:Formulate the following model:

∆yt = β0 + Σ βi∆yt-i + Σγj∆x1t-j + Σδk∆x2t-k

+ θ0yt-1 + θ1x1t-1 + θ2 x2t-1 + et ;

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∆yt = β0 + Σ βi∆yt-i + Σγj∆x1t-j + Σδk∆x2t-k+ θ0yt-1 + θ1x1t-1 + θ2 x2t-1 + et ;

almost like a traditional ECM. The error-correction term, z t-1 replaced with the terms

yt-1, x1t-1, and x2t-1 from y t = α0 + α1x1t + α2x2t + vt

the lagged residuals series would be zt-1 = (yt-1 - a0 - a1x1t-1 - a2x2t-1),

where the a's are the OLS estimates of the α's.

"unrestricted ECM", or an "unconstrained ECM". Pesaran et al. (2001) call this a "conditional ECM".



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Step 3:

The ranges of summation :from 1 to p , 0 to q 1, and 0 to q 2 respectively

Maximum lags are determined by using one or more of the "information criteria" : AIC, SC (BIC), HQ, etc.

Remember: Schwarz (Bayes) criterion (SC) is a consistent model-selector.



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Step 4:A key assumption in the ARDL / Bounds Testing methodology of Pesaran et al. (2001) :

The errors of the equation must be serially independent .

Once an apparently suitable version of the equation has been estimated, use the LM test to test the null hypothesis that

the errors are serially independent, against the alternative hypothesis that the errors are (either) AR(m) or MA(m), for m = 1, 2, 3, ...



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Step 5:We have a model with an autoregressive structure, so we have to be sure that the model is " dynamically stable ".

Check that all the associated with the model

.



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Step 6:Now perform the " "


Do a "F-test" of the hypothesis, H0: θ0 = θ1 = θ2 = 0 ;

against the alternative that H 0 is not true.



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Step 6: "Bounds Testing"

yt = β0 + Σ βi∆yt-i + Σγj∆x1t-j + Σδk∆x2t-k+ θ0yt-1 + θ1x1t-1 + θ2 x2t-1 + et ;

As in conventional cointegration testing, we're testing for

between the variables.

This absence coincides with zero coefficients for yt-1, x1t-1 and x 2t-1 in the equation:

H0: θ0 = θ1 = θ2 = 0

A rejection of H0 implies that we have a long-run relationship.

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Exact critical values for the F-test not available for an arbitrary mix of I(0) and I(1) variables.

Pesaran et al. (2001) supply bounds on the critical values for the asymptotic distribution of the F-statistic.

lower and upper bounds on the critical values .

the lower bound is based on the assumption that all of the variables are I(0) , and the upper bound : all the variables are I(1) .



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If the computed F-statistic falls below the lower bound

conclude : the variables are I(0) , so no cointegration possible, by definition.

If the F-statistic exceeds the upper bound , conclude : we have cointegration .

Finally, if the F-statistic falls between the bounds , the test is inconclusive .



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Step 7:If the bounds test proves cointegration, estimate the long-run equilibrium relationship betw een the variables:

yt = α0 + α1x1t + α2x2t + vt ;

as well as the usual ECM:

∆yt = β0 + Σ βi∆yt-i + Σγj∆x1t-j + Σδk∆x2t-k + φzt-1 + et ;

where zt-1 = (yt-1 -a0 - a1x1t-1 - a2x2t-1), and the a's are the OLS estimates of the α's .



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Step 8:

“Extract" long-run effects from the unrestricted ECM .


at a long-run equilibrium, ∆yt = 0, ∆x1t = ∆x2t = 0,

the long-run coefficients for x 1 = -(θ1 / θ0)

and x 2 = -(θ2 / θ1).





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