ardl-kkOYCKtRANSFORMATION

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

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Pongsa PornchaiwiseskulFaculty of Economics

Chulalongkorn University


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–Koyck–Polynomial DL (PDL)


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• A form of short-term model• Explanatory variables

– lagged values of dependent variable– current and lagged values of exogenous variables

• Three components– Auto-Regressive part– Distributed-Lag part– random part

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t

pKtKpKKtKKKtK

ptptt

ptptt

ptpttt

XXX

XXXXXX

YYYY

ε

βββ

βββ

βββ

φφφφ

+

++++

++++

++++

++++=

−−

−−

−−

−−−

...

...

...

...

,,1,10

2,22,21,221220

1,11,11,111110

22110

M


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Effect of current Xk at time t on Y is distributed over pk periods.

βk0 = Current or short-run effect of Xk on Yβkj = j-period delayed effect of Xk on Yβk = total or long-run effect of Xk on Ywhere

pkkkkk ,10 ... ββββ +++=


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Note that OLS is valid as lagged Y can beuncorrelated with the error term. If the random part is auto-correlated error terms, OLS becomes invalid. No problem with DL models (without AR part).

Without restriction on parameters, there will be too many parameters (β), especially for very long lag.


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• No lagged Y on the right-hand side.• Only one X• error terms could be ARMA• Declining effect over infinite lag or• Restriction on βkj

( )

,...1,0 ,10 , 0 =<<= jjj λβλβ

∞=pk



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Lag (j)

βkj

βk0


10

1010

1001

00

100

2200

202

1000

1100

)1()1(

)1()1()1()1(

...)1(

...

...

−−

−−

−

−−

−−

−+++−=−++−=−−++−=−

+−+=

+++++=

+++++=

+++++=

ttttt

ttttt

ttt

tt

tt

tttt

tptpttt

XYYXYY

LXYLXL

XLLXXX

XXXY

λεεβλλφλεεβλφλ

ελβλφλελβφ

ελλβφ

εβλλββφ

εβββφ


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EstimationStep 0 GuessStep 1 Estimate for

Step 2 Use new and go back to Step 1 until convergence

1001 )1(ˆ−− −++−=− ttttt XYY λεεβλφλ

λ̂),,( 00 λβφ

λ̂


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• No lagged Y on the right-hand side.• Allow more than one X• error terms could be ARMA• polynomial effect over finite lag • Restriction on βkj

,...,1,0 ,

...2210

pjpmjjj m

mj

=<<

++++= γγγγβ



13t

22

10

3210

2210

1210

00

1100

)...(

)3...93(

)2...42(

)...(

...

ε

γγγγ

γγγγ

γγγγ

γγγγγφ

εβββφ

+

+++++

+++++

+++++

++++++=

+++++=

−

−

−

−

−−

ptmm

tmm

tmmtm

t

tptpttt

Xppp

XX

XX

XXXY

M


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( )( )( )( )

( )t

21

3213

2212

211

10

0

...2

...8

...4

...2

...

ε

γ

γ

γ

γ

γφ

+

++++

++++

++++

++++

++++=

−−−

−−−

−−−

−−−

−−

ptm

tm

tm

pttt

pttt

pttt

pttt

t

XpXX

XpXX

XpXX

pXXXXXX

Y

M

tZ0

mtZ


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t2211000 ... εγγγγφ ++++++= mtmtttt ZZZZY

==> Unrestricted OLS is BLUE


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0)1(... nRestrictio

0 Case

210

1

=−+++−

=−

mmγγγγ

β

Effect starts from zero==> Restricted LS is BLUE



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0)1(...

)1()1( nRestrictio

0 Case

22

10

1

=+++

++++

=+

mm

p

ppp

γ

γγγ

β

Effect dies down to zero.==> Restricted LS is BLUE


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0)1(...

)1()1(

0)1(... nsRestrictio

0,0 Case

22

10

210

11

=+++

++++

=−+++−

== +−

mm

mm

p

ppp

γ

γγγ

γγγγ

ββ

Effect starts from zero and finally back to zero. ==> Restricted LS is BLUEPrinted with FinePrint - purchase at www.fineprint.com

ardl-kkOYCKtRANSFORMATION

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