(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
1
Pongsa PornchaiwiseskulFaculty of Economics
Chulalongkorn University
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
2
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
3
–Koyck–Polynomial DL (PDL)
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
4
• 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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(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
5
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
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
6
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 +++=
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
7
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.
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
8
• 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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(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
9
Lag (j)
βkj
βk0
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
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
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
11
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
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
12
• 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
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(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
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
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
14
( )( )( )( )
( )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
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
15
t2211000 ... εγγγγφ ++++++= mtmtttt ZZZZY
==> Unrestricted OLS is BLUE
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
16
0)1(... nRestrictio
0 Case
210
1
=−+++−
=−
mmγγγγ
β
Effect starts from zero==> Restricted LS is BLUE
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(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
17
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
(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University
18
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
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