Kiem Dinh Nghiem Don Vi

Economics 20 - Prof. Anderson 1

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

Nguyn Ngc AnhTrung tm Nghin cu Chnh sch v Pht trin

Nguyn Vit Cngi hc Kinh t Quc dn

Kim nh nghim n v: Phngphp v vn Unit Root Tests: Methods and Problems


n tp bui trc

Chui cn bng >< chui khng cn bngHm s (autocorrelation fucntion) v tht tng quan (correlogram)Kim nh Q v kim nh Ljung-BoxHi qui khng gi trXu hng: Xc nh (Deterministic) hay ngu nhin (Stochastic)?Gii thiu qua v ARMA


n nh

Tnh cht ca cc c lng (VD OLS) sph thuc vo vic dy s c n nh/cnbng hay khngDy s yt l n nh nu hm xc sutkhng ph thuc vo thi gianC ngha l:

E(yt) khng i theo tVar(yt) khng ph thuc vo t Cov(yt,yt+s) ph thuc vo s v khng vo t


n nh yu

Cn bng yu nu mt dy s c m-men bc nht v bc 2 khng ph thuc vo t Cn bng yu s l nhng trng hp ta giiquyt v gp phi


Qua trnh ngu nhin gin n nht

Trong t l nhiu trng (white noise) l binphn phi iid c trung bnh l 0 v phng sai l 2

Kim tra v thy rng :E(yt)=0Var(yt)= 2

Cov(yt,yt-s)=0

Y cng l nhiu trng rt t gp trong cc dys thi gian trong kinh t

Bui trc Xt = ut ut ~ IID(0, 2 )

0t ty = +

White Noise

-0.6

-0.4

-0.2

0

0.2

0.4

0.6


Dy s t qui bc nht - AR(1)

0 1 1t t ty y = + +

( )1 0 0 1 1,...,t t tE y y y y = +

Gi tr ca k hin ti ch ph thuc vo k trc


Khi no th dy s AR(1) c tnh n nh (stationary)?

Ly k vng ton ca biu thc ta c :

Nu cn bng, ta c th vit nh sau :

( ) ( )0 1 1t tE y E y = +

( ) ( ) 0111

t tE y E y

= =

Khng ph thuc vo thi gian


Xt ti phng sai

Nu dy s cn bng th :

( ) ( )2 21 1t tVar y Var y = +

( )2

211

tVar y

=

Ch c ngha nu nh |1 |


Dy s t qui ph qut(General Auto-Regressive Processes)

Dy s t qui bc p AR(p) c dng nh sau :0 1

pt i t i ti

y y == + +Dy s ny s n nh, nu nghim ca dy s m bc p nmtrong vng trn nghim n v

1

pp p iii

z z =

iu kin cn l : (xem phn ly k vng ton ca phng sai trn)

11 1p ii = <


Dy s trung bnh trt(Moving-Average Processes)

Mt dy s ht sc ph bin khc l dy s trungbnh trt bc 1 MA(1), c dng sau :

0 1t t ty = + +

Dy s trung bnh trt lun l dy s cnbng:

( ) 0tE y =


Tnh n nh ca dy s MA

Mi ng phng sai xa hn u bng khng

( ) ( ) ( )2 2 21 1t t tVar y Var Var = + = +

( ) ( )( ) ( ) 21 1 1 2 1,t t t t t t tCov y y E Var = + + = =

( ) ( )( )2 1 2 3, 0t t t t t tCov y y E = + + =


Dy s MA(q)

Dy s ny lun cn bngng phng sai gia hai quan st s lzero nu nh khong cch gia hai quan stl ln hn q thi k

0 1

qt t i t ii

y == + +


Quan h gia dy s AR v MA

Trng hai dy s c v khng quan h, nhng thc ra c quan h gia hai dy sXt dy s AR(1) vi 0=0:

1 1t t ty y = +Thay yt-1 ta c:

[ ] 21 1 2 1 1 2 1 1t t t t t t ty y y = + + = + +


Tip tc thay ta c

Nh vy, dy s AR(1) c th c biu din di dngdy s MA() v c trng s ngy cng gim dnCn c tnh cn bng, m bo rng ton t cui cngs bng 0

3 21 3 1 1 1 2t t t t ty y = + + +

1 11i

t t t iiy y == + +


S dng php ton tr (lag operator) C th b qua

Php ton tr :1t tLy y =

st t sL y y =

C th vit dy s AR(1) nh sau:

[ ]1 11t t t t ty Ly L y = + =


[ ]11t

ty L

=

( )111 1

1 i it t t t ii iy L

= = = + = +

Tng t nh vic thay th dn


Vi dy s AR(p) - B qua

Nu (L) l c th nghich o, ta s c (invertible) :

( )0 011p i

i t t t tiL y L y

= = + = +

( )( )1 0t ty L = +

Nh vy dy s AR(p) c th c vit didng mt dy s MA() nht nh no


T dy MA thnh dy AR

S dng php tr ta c th vit dy s MA(q) nh sau:

( )t ty L =

Nu (L) l c th nghch o:

( )1 t tL y =

Nh vy dy s MA(q) c th c biu dinthnh dy s AR() c th no


Chui ARMA

Cc dy s thi gian c th c c phn AR vphn MA Mt dy s ARMA(p,q) c th c vit nh sau

0 1 1

p qt i t i t i t ii i

y y = == + + +


Kim nh nghim n v

Lm th no bit mt dy s c cn bng hay khng? Do phn MA lun cn bng, nn s ch tp trungvo phn AR


Xu hng: Xc nh (Deterministic) hay ngu nhin (Stochastic)?

0 50 100 150 200 250 300 350 400 450 500

100

200

0 50 100 150 200 250 300 350 400 450 500

100

200

0 5 10 15 20 25

.25

.5

.75

1

0 5 10 15 20 25

.25

.5

.75

1

M hnh 1

M hnh 2

Y a Yt t t= + +1 1

Y a a Y a tt t= + + +1 2 1 3

(a2 0)


Y a a Y a tt t= + + +1 2 1 3

Chui s ny c xu hng xc nh nu (if a3 > 0)

Cc suy din thng k s c gi tr(vi iu kin l a2 < 1).

Chui ny c th c chuyn sang chui cn bngbng cch loi b xu hng xc nh

Y a t a a Yt t = + +3 1 2 1


Y a Yt t t= + +1 1

Chui ny khng cn bng Xu hng l ngu nhin

Suy din thng k s khng c gi tr

C th cn bng thng qua ly sai phn (difference stationary)

Y Y at t t = +1 1


Y b0 Y I

Y Y Y b0 It t t

t t t t

= + +

= = +

1

1

(1)

(0)

Bc ng nht (tch hp) ca mt dy s-Order of Integration of a Series

Mt dy s sau khi ly sai phn (difference) tr thnh dy s cn bngc gi l dy s c tch hp bc 1 , v k hiu l I(1).

Ni chung, mt dy s thi gian tr thnh cn bng sau khi csai phn d ln c gi l c bc tch hp d, k hiu l I(d).

Mt dy s khng cn ly sai phn m vn l dy s cnbng c gi l c tch hp 0, v k hiu l I(0)


Cc xc nh cc chui khng cn bng cch khng chnh thng

(1) th s liu (a) Trung bnh c thay i?

(b) Phng sai c thay i ?

0 50 100 150 200 250 300 350 400 450 500

-200

-150

-100

-50

0

50

100

150

200 var

0 50 100 150 200 250 300 350 400 450 500

0

2

4

6

8

10

12RW2


(2) S dung CorrelogramVi dy s cn bng, th tim cn 0 rt nhanh. Chui

s khng c b nh (no memory)

0 50 100 150 200 250 300 350 400 450 500

-0.25

0.00

0.25

0.50whitenoise

0 5 10

-0.5

0.0

0.5

1.0ACF-whitenoise



(2) S dng CorrelogramVi dy s c dng bc ngu nhin, thcorrelogram khng tim cn 0. C tng quan rt caogia cc k (High autocorrelation for large values of k)

0 50 100 150 200 250 300 350 400 450 500

0.0

2.5

5.0

7.5

10.0

12.5randomwalk

0 5 10

0.25

0.50

0.75

1.00ACF-randomwalk



Kim nh thng k s dung t-test

Xy dng m hnh AR(1) c trt (b0) Yt = b0 + b1Yt-1 + t t ~ iid(0,2) (1)

Phng php gin n l c lng phng trnh (1) s dngOLS v xem xt cc con s c lng b1

S dng t-test vi gi thuyt trng Ho: b1 = 1 (khng cn bng)vi gi thuyt thay th Ha: b1 < 1 (cn bng).

Kim nh : TS = (b1 1) / (Std. Err.(b1)) Bc b gi thuyt trng khi gi tr t ln v c du m

gi tr ti hn (critical value) mc - 5% l -1.65


Kim ng thng k chui cn cng: kim nh t

Kim nh t i vi dy s AR(1) c trt (b0)

Yt = b0 + b1Yt-1 + t t ~ iid(0,2) (1)

Mt s vn vi phng php ny(1) Bin tr ph thuc => b1 s b c lng trch xung , c bit l nhng mu nh

(2) Khi b1 =1, chng ta s c chui khng cn bng, vvic s dng phng php hi qui l khng c gi tr


Kim nh nghim n v - Nhngvn c bn

Mun kim nh H0:1=1 so vi H1:1


Kim nh Dickey Fuller (DF)

Dickey v Fuller (1979): Tr Yt-1 t 2 v ca phng trnh

t ~ iid(0,2) = 1 1 (2)

Mun kim nh H0:1=0 vi H1: 1


Kim nh Dickey Fuller (DF)

S dng kim nh t vi gi thuyt trng lHo: 1 = 0 (khng cn bng hay c nghim n

v - Unit Root) v gi thuyt thay th - Ha: 1 < 0 (cn bng).

- Khi kim nh c gi tr ln v c du mbc b gi thuyt chui cn bng (reject non-stationarity)

- y chnh l kim nh nghim n v (unit root test) v phng trnh slide trc Ho: b1 =1.


Mt s dng kim nh DF (Variants of DF test)

C 3 m hnh hi qui c th s dng kim nh nghim n v

Y YY b YY b Y b t

t t

t t

t t

= += + += + + +

1

0 1

0 1 2

S khc bit gia cc m hnh ny l s hin din ca cc biu thc b0 vb2t. 1 kim nh xem Y c phi l mt bc ngu nhin(Random Walk) hay khng2 kim nh xem Y c phi l mt bc ngu nhin c trt hay khng(Random Walk with Drift)3 kim nh xem Y c phi l mt bc ngu nhin c h s trt vc xu hng hay khng (Random walk with Drift and Deterministic Trend)


Y Yt t= + 1

M hnh n gin nht (ch thch hp khi ta cho rng khng c ccYu t khc trong m hnh (true regression model))

S dng kim nh t v so snh vi gi tr ti hn do Dickey vFuller tnh ton. Nu gi tr t nm ngoi khong tin cy, bc bgi thuyt trng l c nghim n v (unit root)

Statistic


Y b Yt t= + +0 1

M hnh c trt (drift)

1

Kim nh s dng kim nh F xem = b0 = 0 , s dng bng phi chnh thng

S dng kim nh t xem beta c bng khng khng? =0 , s dng bng phi chnh thng(non-standard tables)


V d

Dy s c s quan st n = 25 vi mc ngha 5% cho phng trnh

-critical value = -3.00 t-test critical value = -1.65

pt-1 = -0.007 - 0.190pt-1 (-1.05) (-1.49)

= -0.190 = -1.49 > -3.00

Do khng th bc b H0 c unit root.


Kim nh DF c tnh ti yu t xu hng ca dy s

i khi dy s thi gian c xu hng i ln hoc i xung (khngcn bng v trung bnh ca dy s - non-stationary mean).

V th nn ua xu hng vo m hnh v s dng kim nh DF.

Yt = b0 + Yt-1 + b2 trend + t (4)

Hon ton c kh nng l dy s Yt s cn bng xung quanh mt xuhng no . Nu khng a yu t xu hng vo m hnh th dys s khng cn bng/n nh ( non-stationary.)


Cc kim nh DF khc nhau Tm tt cc loi kim nh t

Yt = b0 + Yt-1 + b2 trend + t(a) Ho: = 0 Ha: < 0

Yt = b0 + Yt-1 + t(b) Ho: = 0 Ha: < 0

Yt = Yt-1 + t(c) Ho: = 0 Ha: < 0

Gi tr ti hn c th xem trong cc sch hoc Fuller (1976)


Kim nh DF S dng kim nh loi F (F-type test)

3 Yt = b0 + Yt-1 + b2 trend + t(a) Ho: = b2 = 0 Ha: 0 v/hoc b2 0

1 Yt = b0 + Yt-1 + t (b) Ho: = b0 = 0 Ha: 0 v/hoc b0 0

Cc gi tr ti hn c th xem bi nghin cu ca Dickey v Fuller (1981)


Tm tt Dickey-Fuller Tests

M hnh Gi thuyt Kim nh

Gi tr ti hn cho khong tiin cy 95% v 99%

Y b Y b tt t= + + +0 1 2

= 0 -3.45 and -4.04 b0 = 0 given = 0 3.11 and 3.78 b2 = 0 given = 0 2.79 and 3.53 = b2 = 0 3 6.49 and 8.73 = b0 = b2 = 0 2 4.88 and 6.50 Y b Yt t= + +0 1 =0 -2.89 and -3.51 b0 = 0 given = 0 2.54 and 3.22 = b0 = 0 1 4.71 and 6.70 Y Yt t= + 1 =0 -1.95 and -2.60 (Gi tr ti hn cho n = 100)


Kim inh Dickey Fuller b xung (Augmented Dickey Fuller) Kim nh Dickey Fuller gi thit rng cc residuals t trong m hnh hi qui DF l khng t tng quan

Gii php: a cc bin tr ca bin ph thuc vo m hnh

Vi s liu theo qu, c th tr 4 bc ta cYt = b0 + Yt-1 + 1Yt-1 + 2Yt-2 + 3Yt-3 + 4Yt-4 + t (3)

Lc ny c vn pht sinh khi cn phn bit cc m hnhS dng phng php t chung ti ring (general to specific) loi b

cc bin khng c nghaKim tra m hnh cui cng (parsimonious model) xem c t tng

quan hay khng

S dng kim nh F-test i vi cc bin c nghaS dng h s thng tin. Cn nhc gia m hnh parsimony vi phng sai caphn d (residual)


Xem xt chui s v Correlogram

0 50 100 150 200 250 300 350 400 450 500

100

200

Y

0 5 10 15 20 25 30

.25

.5

.75

1ACF-Y

Bin Y ny r rng l c xu hng, v chng ta phi xem xt xem xu hng ny l xc nh(deterministic) hay ngu nghin (stochastic). Sau khi to ra bin sai phn Y ,

ta c lng m hnh c tr ca Y. S lng tr nhiu n mc ta ngh l ph hp.(trong v d trn, tr ca bin sai phn Y l 4)


Kim nh nghim n v (Unit Root Testing)

Y b b t Y Yt t t= + + + + 0 2 1 1 1

Sau khi c lng xong m hnh

Cc gi thuyt c th kim nh l

H b b bv

H b b b

0 0 2 0

1 0 2 0

0 0

0 0

: , , , ,

: , , , ,

=

kim nh, s dng F-Test v tham s phi


V d - Real GDP (2000 Prices) Seasonally Adjusted(1) V th theo thi gian khng cn bng

(trung bnh thay i theo thi gian, v correlogramkhng bng khng)

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

50

75

100Y

0 5 10

0.25

0.50

0.75

1.00ACF-Y

k

Time

GDP

r


Kim nh nghim n v(1) Ly sai phn cn bng

(Trung bnh khng i v correlogram bng khng)

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

-1

0

1

2

3DY

0 5 10

-0.5

0.0

0.5

1.0ACF-DY

Timer

k


Kim nh nghim n v(3) Xc nh s bc tr - s dng ADF test

c lng m hnh chung v kim nh serial correlation

EQ ( 1) Yt = b0 +b2 trend+ Yt-1 + 1Yt-1 + 2Yt-2 + 3Yt-3 + 4Yt-4 + t

Coefficient Std.Error t-value t-prob Part.R^2

Constant 0.538887 0.3597 1.50 0.136 0.0121Trend 0.00701814 0.004836 1.45 0.148 0.0114Y_1 -0.0156708 0.01330 -1.18 0.240 0.0075DY_1 -0.0191048 0.07395 -0.258 0.796 0.0004DY_2 0.137352 0.07297 1.88 0.061 0.0190DY_3 0.188071 0.07354 2.56 0.011 0.0345DY_4 0.0474897 0.07473 0.635 0.526 0.0022

AR 1-5 test: F(5,178) = 1.7263 [0.1308] Kim nh chp nhn gi thuyt rng khng c tng quanVn tip tc s dng F-test v Schwarz Criteria kim tra m hnh


Kim nh nghim n v(3) Xc nh s bc tr s dng kim nh ADF test

ModelEQ ( 1) Yt = b0+b2 trend+ Yt-1 + 1Yt-1 + 2Yt-2 + 3Yt-3 + 4Yt-4 + tEQ ( 2) Yt = b0+b2 trend+ Yt-1 + 1Yt-1 + 2Yt-2 + 3Yt-3 + tEQ ( 3) Yt = b0+b2 trend+ Yt-1 + 1Yt-1 + 2Yt-2 + tEQ ( 4) Yt = b0+b2 trend+ Yt-1 + 1Yt-1 + tEQ ( 5) Yt = b0+b2 trend+ Yt-1 + t

S dng c F-test v Schwarz information Criteria (SC).

Gim s bc tr (number of lags) khi F-test chp nhn gi thuyt

Chn m hnh (phng trnh) c SC l nh nhttc l chn m hnh c phng sai ca phn d (residual) v s cc

tham s nh nht


(3) Xc nh bc tr s dng ADF testProgress to dateModel T p log-likelihood Schwarz Criteria EQ( 1) 190 7 OLS -156.91128 1.8450EQ( 2) 190 6 OLS -157.12068 1.8196EQ( 3) 190 5 OLS -160.37203 1.8262EQ( 4) 190 4 OLS -162.16872 1.8175EQ( 5) 190 3 OLS -162.17130 1.7899

Tests of model reduction EQ( 1) --> EQ( 2): F(1,183) = 0.40382 [0.5259] Accept model reductionEQ( 1) --> EQ( 3): F(2,183) = 3.3947 [0.0357]* Reject model reductionEQ( 1) --> EQ( 4): F(3,183) = 3.4710 [0.0173]* EQ( 1) --> EQ( 5): F(4,183) = 2.6046 [0.0374]*

Mt s kt qu mu thun nhau. F-tests cho rng phng trnh (2) tthn phng trnh s (1) v phng trnh (3) th khng tt hn phng trnh (2)

Kim nh nghim n v


Kim nh nghim n v

(B) Tin hnh mt cch chnh thc

Coefficient Std.Error t-value t-prob Part.R^2

Constant 0.505231 0.3552 1.42 0.157 0.0109Trend 0.00655304 0.004772 1.37 0.171 0.0101Y_1 -0.0141798 0.01307 -1.08 0.279 0.0064DY_1 -0.0119522 0.07297 -0.164 0.870 0.0001DY_2 0.142437 0.07241 1.97 0.051 0.0206DY_3 0.185573 0.07332 2.53 0.012 0.0336

AR 1-5 test: F(5,179) = 0.68451 [0.6357]

Vn chnh l kim nh gi thuyt serial correlation assumption. CHng tac chp nhn gi thuyt trng la khng c serial correlation khng? Chngta chp nhn!


Mt s vn i vi kim nhnghim n v


Perron (1989) cho rng cc chui thi gian khng phi l cc chuc nghim n v, m l cc chui cn bng c xu hng v c bini v cu trc (Structural Breaks)

V d Khng hong nm 1929 Cn sc gi du Thay i cng ngh

Nhng s kin ny s lm thay i trung bnh (mean) ca cc dyS nh GDP. Nu ta khng nhn ra cc structural break, th sLun tm thy nghim n v cho d khng c nghim

Khi c bin i v cu trc th mi kim nh nghim n v u b trch.C xu hng khng bc b gi thuyt c nghim n v

Vn th : Structural Breaks


Vn 2 : Lc kim nh thp (Low Power)

Lc kim nh ca mt php kim nh l xc sut bc b gi thuyttrng khi gi thuyt ny sai (reject a false Null Hypothesis)

Kh kim nh gia 2 dy s (1) c nghim n v; (2) gn nghim n v Kh kim nh gia xu hng v trt (Trend and Drift)

Kim nh nghim n v

0 10 20 30 40 50 60 70 80 90 100

-8

-6

-4

-2

0

2

4

Y1 Z1

Y l dy s nghim n v

Z l dy s xp x nghimn v


Kim dnh = 0 trong m hnh Yt = b0 + Yt-1 + t

Kt qu kim nh da vo sai s chun (standard error) ca - Sai s chun cho bit c lng ca chng ta chnh xc n u- cng nhiu quan st, sai s chun cng nh

Trong trng hp ny, lc kim nh ca mt kim nh l kh nng bc b githuyt trng v vic dy s khng cn bng khi gi thuyt ny sai. (ni mt cchkhc, l kh nng chp nhn gi thuyt thay th l chui cn bng).

Lc kim nh thp c ngha l mt dy s c th l cn bng, nhng kim nhDF li cho rng dy s c nghim n vLc kim nh thp s gy ra vn nghim trng khi dy s l cn bng, nhngli xp x dy s c nghim n v. Gii php l tng s quan st ca dy s.

Kiem Dinh Nghiem Don Vi

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