day 4 regression with distributed lag

Economics 20 - Prof. Anderson 1

D bo s dng m hnh chui thigian(Time Series Models for Forecasting)

Hi qui vi bin trRegression with distributed lags

Nguyn Ngc AnhTrung tm Nghin cu Chnh sch v Pht trin

Nguyn Vit Cngi hc Kinh t Quc dn


Gii thiu

Chng ta trong cc bi trc, xem xt m hnh hi qui, s dng cho c d liu cho, ln d liu chui thi gian.Tuy nhin, chng ta y li thng quan tm n nhngbin s thay i theo thi gian, ch khng phi l nhngbin thay i theo cc c nhnM hnh hi qui tnh cho ta bit quan h gia cc chuithi gian. y, tc ng ca mt bin X ln mt bin Y c githit l ch c tc ng trong cng thi k.


M hnh ng

Tc ng mang tnh ng (Dynamic effects) Chnh sch cn c thi gian mi c tc dng Mc cng nh tnh cht ca tc ng c th

thay i theo thi gian Tc ng thng xuyn (Permanent) v tcng tm thi (Temporary effects.)


Trong kinh t hc v m Tc ng ca tin t M i vi Y (GDP) trong

ngn hn c th khc vi trong di hn

Ngi ta thng gi l hm phn ng (impulse response function) Tng cung tin trong mt nm nm th Sau s quay tr li bnh thng, khng tng M

na

iu g s xy ra vi Y

time

Y


Phn b tr (Distributed Lag)Tc ng c phn b theo thi gian(Effect is distributed through time) Hm tiu dng : Tc ng ca thu nhp cng

thay i theo thi gian Tc ng ca thu thu nhp i vi GDP s c tr

Tc ng ca chnh sch tin t vi SX cngqua thi gian

yt = + 0 xt + 1 xt-1 + 2 xt-2 + etit

ti x

yE

=

)(


Tc ng phn b tr

Hot ng kinh t ti cc thi im t

Tc ng tiThi im t

Tc ng tiThi im

t+1

Tc ng tiThi im t+2


Tc ng phn b trTc ng ti thi im t

Hot ng kinh t tithi im t Hot ng kinh t ti

thi im t-1 Hot ng kinh t tithi im t-2


Hai cu hi

1. Tr bao lu (How far back)? - tr l bao lu ?- Tr hu hn hay v hn

2. Liu cc h s c nn b hn ch hay khng(restricted)?

- iu chnh (smooth adjustment)- Hay s liu quyt nh (let the data

decide)


1. Phn b tr hu hn khng hnch (Unrestricted Finite DL)

Hu hn: bin ng ca mt bin s ch ctc ng ln mt bin khc trong mtkhong thi gian c nh V d: Tc ngc ca CS tin t thng c tcng ln GDP khong 18 thng

tr c gi thit l bit mt cch chc chnKhng hn ch (Unrestricted - unstructured) Tc ng giai on t+1 khng c quan h vi

tc ng giai on t


yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . + n xt-n + et

C n tr khng hn ch (unstructured lags)

Khng c mt dng cu trc (systematic structure)

no i vi cc sCc tham s s khng b hn ch (rng buc - restricted)

C th s dng OLS: s cho ta cc clng nht qun (consistent) v khngtrch


Nhng vn ny sinh1. Ta s mt n quan st khi tr l n

S liu t nm 1960, gi s c tr l 5 thi k, tc lthi im sm nht c th s dng trong m hnh hi qui l nm 1965 Mt t do (a thm bin tr mt t do)

2. Vn a cng tuyn gia cc bin tr xt-j xt rt ging vi xt-1 t thng tin c lpc lng khng chnh xc (xem bi trc) lch chun ca c lng l ln, kim nh t c gi

tr thpKim nh gi thuyt l kh khn (uncertain)


3. C th c nhiu bin tr th sao? Mt nhiu t do

4. C th c lng chnh xc hn nu xydng mt s cu trc trong m hnh

Nhng vn ny sinh


Tr s hc tr vn hu hn : Tc ng ca X cuicng s bng 0Cc h s khng c lp vi nhau Tc ng ca mi bc tr s nh dn i VD: Chnh sch tin t ca nm 1995 s tcng ti GDP ca nm 1998 t hn chnh schtin t ca nm 1996


2. Tr s hc

i

i0 = (n+1)

1 = n2 = (n-1)

n =

.

.

.

0 1 2 . . . . . n n+1

..

.

.

linearlag

structure


Tr s hc

p t quan h:

i = (n - i+ 1)

0 = (n+1) 1 = n 2 = (n-1) 3 = (n-2) . .n-2 = 3 n-1 = 2 n =

Ch cn c lng 1 tham s , ,Thay v n+1 tham s , 0 , ... , n .

yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . + n xt-n + et


Gi s X l cung tin ( dng log) v Y lGDP (dng log), n=12 v c c lngc gi tr l 0.1Tc ng ca x thay i i vi GDP tronggiai on hin ti s l 0=(n+1)=1.3Tc ng ca CS tin t mt nm sau sl 1=n=1.2n nm sau ,tc ng s l n= =0.1Sau n+1 nm, tc ng s l 0

it

ti x

yE

=

)(


c lng

c lng s dng OLSCh cn c lng mt tham s : Phi bin i mt cht vit m hnh didng c th c lng c


yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . + n xt-n + et

yt = + (n+1) xt + n xt-1 + (n-1) xt-2 + . . . + xt-n + et

Bc 1: p t rng buc: = (n - i+ 1)

Bc 2: Bc tc tham s, .yt = + [(n+1)xt + nxt-1 + (n-1)xt-2 + . . . + xt-n] + et


Bc 3: Xc nh zt .

zt = [(n+1)xt + nxt-1 + (n-1)xt-2 + . . . + xt-n]

Bc 5: Chy OLS :

yt = + zt + et

Bc 4: Xc nh tr , n.

Vi n = 4: zt = [ 5xt + 4xt-1 + 3xt-2 + 2xt-3 + xt-4]


u/ nhc imt tham s phi c lng (ch mt tham s) hnso vi m hnh khng hn ch/rng buc Sai s chun thp T cao Kim nh tt

Nhng nu cc rng buc khng ng th sao? c lng s b trch

Rng buc tuyn tnh c thc t khng? Xem xt m hnh khng hn ch nh gi Tin hnh kim nh F


2

1

//)(

dfSSEdfSSESSEF

U

UR =

Kim nh Fc lng m hnh khng rng buc (unrestricted model)c lng m hnh c rng buc (arithmetic lag) Tnh ton ch s F


So snh vi gi tr F ti hn F(df1,df2) df1=n s rng buc/hn ch (number of

restrictions) S b ta tr i s gamma = (n+1)-1

df2=s quan st s bin trong m hnh khngb rng buc (k c intercept)

df2=(T-n)-(n+2)


3. Phn b tr s m

Rng buc tuyn tnh c th l qu cngnhcMun c dng liRng buc s m (Polynomial quadratic hoc cao hn)

i = 0 + 1i + 2i2iit

t

xyE

=

)(


Phn b tr s m (Polynomial Lag)

.. . .

.

0 1 2 3 4 i

01 2 3

4

i


Tng t nh m hnh tr s hc Ch c hnh dng ca dng hm phn ng l

khc (impulse response function)Vn hu hn : Tc ng ca X cui cng sbng 0 Cc h s c quan h vi nhau Tc ng ca mi bc tr khng nht thit s

nh hn bc tr trc (not uniform decline)


c lngS dng OLSCh cn c lng mt tham s : S lng tham s bng vi bc s m (number

of parameters is equal to degree of polynomial)Phi thc hin mt s bin i m hnhc dng c lng c . M hnh ny tr thnh dng m hnh s hc

(arithmetic) nu bc m l 1 OLS vi m hnh bin i


i = 1, . . . , np = 2 v n = 4

V d: M hnh m bc 2 0 = 01 = 0 + 1 + 22 = 0 + 21 + 423 = 0 + 31 + 924 = 0 + 41 + 162

n = trp = Bc m

Trong i = 1, . . . , ni = 0 + 1i + 2i +...+ pi2 p

i = 0 + 1i + 2i2


yt = + 0 xt + 1 xt-1 + 2 xt-2 + 3 xt-3 + 4 xt-4 + et

yt = + 0 xt + (0 + 1 + 2)xt-1 + (0 + 21 + 42)xt-2+ (0 + 31 + 92)xt-3+ (0 + 41 + 162)xt-4 + et

Bc 2: Bc tc cc tham s: 0, 1, 2.yt = + 0 [xt + xt-1 + xt-2 + xt-3 + xt-4]

+ 1 [xt-1 + 2xt-2 + 3xt-3 + 4xt-4]+ 2 [xt-1 + 4xt-2 + 9xt-3 + 16xt-4] + et

Bc 1: p t rng buc: i = 0 + 1i + 2i 2


Bc 3: xc nh zt0 , zt1 and zt2 for 0 , 1 , and 2.

yt = + 0 [xt + xt-1 + xt-2 + xt-3 + xt-4]+ 1 [xt-1 + 2xt-2 + 3xt-3 + 4xt-4]+ 2 [xt-1 + 4xt-2 + 9xt-3 + 16xt-4] + et

zt0 = [xt + xt-1 + xt-2 + xt-3 + xt-4]

zt1 = [xt-1 + 2xt-2 + 3xt-3 + 4xt- 4 ]

zt2 = [xt-1 + 4xt-2 + 9xt-3 + 16xt- 4]


yt = + 0 zt0 + 1 zt1 + 2 zt2 + et

Bc 5: Biu din i di dng ca 0 , 1 , v 2.^ ^ ^ ^

0 = 01 = 0 + 1 + 22 = 0 + 21 + 423 = 0 + 31 + 924 = 0 + 41 + 162

^^^^^^

^ ^ ^^ ^ ^^ ^ ^^ ^ ^

Bc 4: OLS


u nhc imt tham s c lng Chnh xc hn

Nhng nu rng buc khng chnh xc thsao? c lng trch

Liu c lng m c ng khng? Linh hot hn tr s hc

Nu ch xp x ng ? Kim nh F test


Kim nh Fc lng m hnh khng hn chc lng m hnh hn ch (polynomial lag model)

Tnh ton con s kim nh thng k nh trc

So snh vi gi tr ti hn F(df1,df2) df1=s rng buc = s cc tr i s =(n+1)

(p+1) df2=s quan st s lng cc bin trong m hnh

khng rng buc (k c intercept) df2=(T-n)-(n+2)


trVi c 03 m hnh va nu, ta cn phichn tr (lag length) C th coi nh l chn im ct m Sau bin s khng cn tc ng VD: CS tin t khng cn tc ng ti GDP sau

2 nmKhng c tiu ch tha ng chn laiu ny

C nn cho tr n l v hn hay khng?


Tiu ch chn tr (Lag-Length Criteria)

Tiu ch Akaikes AIC criterion

Tiu ch Schwarzs SC criterion

Vi mi tiu ch trn, ta chn bc tr sao chocc tiu ch trn l nh nht. V khi a thm bintr vo s lm gim SSE, nn phn th 2 cami tiu ch l mt penalty function i vi vica thm bin tr vo m hnh

2 ( 2 )ln nS S E nA ICT N T N

+= +

( )2 ln( )( ) ln n n T NSSESC nT N T N

+ = +


Tm tt

1. Tr bao lu?? - tr l khong bao lu th ph ?- Khng c cu tr li (no good answer)

2. Liu cc tham s c nn b rng buc khng? - Th hin qua s liu- S hc hay s m-Bc ca s m


4. M hnh tr Geometric

C tr di v hnNhng chng ta khng th c lng mts lng v hn cc tham sBuc cc h s ca bin tr phi tun thmt trt t nht nh c lng cc tham s cho trt t/cu trc ny. i vi dng m hnh tr geometric th cutrc ca tr s c dng gim lin tc vitc gim dn.


Cu trc tr ca m hnh geometric

i.

.. . .

0 1 2 3 4 i

1 = 2 = 23 = 34 = 4

0 = geometrically

decliningweights


c lngKhng th c lng dng OLS Ch cn c lng hai tham s : ,Phi bin i biu din m hnh didng thc c th c lng cSau s dng bin i Koyck (Koycktransformation)Sau s dng bnh phng cc tiu haibc (2SLS)


yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . + etM hnh phn b tr v hn:

yt = + i xt-i + eti=0

Cu trc tr c dng geometric:

i = i where 0 0 .


yt = + 0 xt + 1 xt-1 + 2 xt-2 + 3 xt-3 + . . . + etTr v hn khng cu trc :

yt = + (xt + xt-1 + 2 xt-2 + 3 xt-3 + . . .) + etTr geometric v hn (infinite geometric lag):

thay th i = i 0 = 1 = 2 = 2 3 = 3. ..


S nhn gia k (v d 3 k) (interim multiplier) :

S nhn tc ng (impact multiplier) :

S nhn di hn :

(1 + + 2 + 3 + . . . ) =

yt = + (xt + xt-1 + 2 xt-2 + 3 xt-3 + . . .) + et

+ + 2

1

Phn ng ng (Dynamic Response):


yt = + (xt + xt-1 + 2 xt-2 + 3 xt-3 + . . .) + et

yt yt-1 = (1 ) + xt + (et et-1)

Tr tt c cc ton t mt bc, nhn vi , v sau ly m hnhgc tr i

yt-1 = + ( xt-1 + 2 xt-2 + 3 xt-3 + . . .) + et-1

Bin i Koyck (Koyck Transformation)

yt = (1 ) + yt-1 + xt + (et et-1)yt = 1 + 2 yt-1 + 3xt + t


Cn s dng 2SLS

yt-1 c lp vi et-1 (xem m hnh)Nhng yt-1 li c tng quan vi vt-1Nh vy OLS s khng ph hp OLS khng th phn bit gia nhng thay i

ca yt do yt-1 gy ra vi nhng thay i do vtgy ra

OLS s coi nhng thay i ca vt nh lnhng thay i ca yt-1


S dng 2SLS

1. Hi qui yt-1 ln xt-1 v tnh gi tr clng ca yt-1 (fitted value)

2. S dng gi tr c lng ca yt-1trong m hnh hi qui Koyck regression

tttt vxyy +++= 3121


Sao li th nh? T m hnh hi qui bc 1, gi tr c lng

yt-1 khng cn tng quan vi et-1 trong khi yt-1 th c tng quan

Nh vy gi tr c lng (fitted value) yt-1khng cn tng quan vivt =(et -et-1 )

2SLS s cho kt qu c lng nht qun(consistent) ca m hnh phn b trGeometric (Geometric Lag Model)


M hnh k vng iu chnh dn(Adaptive Expectations Model)

Mt dng m hnh ca m hnh bin tr geometric Nu chng ta gi thit rng cc c nhn c kvng dng iu chnh dn (adaptive expectation) th m hnh bin tr geometric l ph hpGi thit v k vng K vng c xc lp trn kinh nghim qu kh K vng c iu chnh da trn nhng sai lm ca

qu khK vng iu chnh ny khng ph hp vi githuyt v k vng hp l (rational expectations)


yt = + x*t + etyt = Cu tin tx*t = li sut k vng

(x*t khng quan st c)

V d: Cu tin t

x*t - x*t-1 = (xt-1 - x*t-1) iu chnh k vng da trn cc sai lm ca qu kh:


Bin i mt cht c th tin hnh clng

x*t - x*t-1 = (xt-1 - x*t-1)

x*t = xt-1 + (1- ) x*t-1

Cho x*t v mt pha

xt-1 = [x*t - (1- ) x*t-1]or


Ly m hnh ban u, tr mt bc v nhn vi(1 )

yt = + x*t + et (1)

yt = - (1 )yt-1+ [x*t - (1 )x*t-1]+ et - (1 )et-1

Tr i, ta c

(1 )yt-1 = (1 ) + (1 ) x*t-1 + (1 )et-1 (2)


Thay xt-1 = [x*t - (1- ) x*t-1] vo ta c

yt = - (1 )yt-1+ xt-1 + utTrong ut = et - (1 )et-1

y chnh l m hnh phn b tr m=(1)Chng ta c th c lng m hnh nybng 2SLS


V d: hm tiu dng

C l tiu dng, Y* l thu nhp k vngtrong tng lai quyt nh mc tiu dng, cc c nhn phi

d on v thu nhp trong tng lai ca mnhNu ngi ta iu chnh k vng theo githuyt iu chnh dn

ttt eyc ++= *

)( * 11*

1*

= tttt yyyy


Thay vo ta s c dng

S dng 2SLS c lng bn OLS:

S dng thay cho

tttt vycc +++= 13121

1

3

2

1

1

==

==

ttt eev

ttt eyaac ++= 110

1 tc 1tc


M hnh iu chnh dn

Mt dng khc ca m hnh iu chnh dnGi thit rng cc c nhn iu chnh mith dn dn Vic iu chnh c th tn km, nn khng iu

chnh ngayV d : Hn trong kho ca cc cng ty

y*t = + xt + et


Hng trong kho s c iu chnh dn timc ti u

Tham s cho bit t l chnh lch giacon s thc t v con s mong mun iuchnhVic iu chnh ngay lp tc c th c tnkmM hnh trn rt ging, nhng khng gingtuyt i m hnh k vng iu chnh dn(AE model)

yt - yt-1 = (y*t - yt-1)


Bin i mt chtyt - yt-1 = (y*t - yt-1)

= ( + xt + et - yt-1)= + xt - yt-1+ et

yt = + (1 - )yt-1 + xt + etTm yt :


Kt lunTrong bi ging ny ta xem xt m hnhphn b trMt bc tin so vi m hnh tnhNhng ni chung, m hnh vn gi thitrng chng ta vn c s liu l cn bng(stationary processes.)Vic dy s khng cn bng s c xemxt tip trong cc phn tip sau

D bo s dng m hnh chui thi gian (Time Series Models for Forecasting) Hi qui vi bin tr Regression with distributGii thiuM hnh ngPhn b tr (Distributed Lag)Tc ng phn b tr Tc ng phn b trHai cu hi1. Phn b tr hu hn khng hn ch (Unrestricted Finite DL)Nhng vn ny sinhNhng vn ny sinhTr s hc 2. Tr s hc Tr s hc c lngu/ nhc imKim nh F3. Phn b tr s mPhn b tr s m (Polynomial Lag) c lng u nhc im Kim nh F tr Tiu ch chn tr (Lag-Length Criteria)Tm tt4. M hnh tr Geometric Cu trc tr ca m hnh geometric c lngPhn ng ng (Dynamic Response):Bin i Koyck (Koyck Transformation)Cn s dng 2SLSM hnh k vng iu chnh dn (Adaptive Expectations Model)V d: Cu tin t Bin i mt cht c th tin hnh c lng Ly m hnh ban u, tr mt bc v nhn vi (1)V d: hm tiu dng M hnh iu chnh dn Kt lun

day 4 regression with distributed lag

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