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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
Economics 20 - Prof. Anderson 2
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
Economics 20 - Prof. Anderson 3
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
Economics 20 - Prof. Anderson 4
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
Economics 20 - Prof. Anderson 5
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
Economics 20 - Prof. Anderson 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
Economics 20 - Prof. Anderson 7
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
Economics 20 - Prof. Anderson 8
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 |
Economics 20 - Prof. Anderson 9
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 = <
Economics 20 - Prof. Anderson 10
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 =
Economics 20 - Prof. Anderson 11
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 = + + =
Economics 20 - Prof. Anderson 12
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 == + +
Economics 20 - Prof. Anderson 13
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 = + + = + +
Economics 20 - Prof. Anderson 14
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 == + +
Economics 20 - Prof. Anderson 15
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 = + =
Economics 20 - Prof. Anderson 16
[ ]11t
ty L
=
( )111 1
1 i it t t t ii iy L
= = = + = +
Tng t nh vic thay th dn
Economics 20 - Prof. Anderson 17
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
Economics 20 - Prof. Anderson 18
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
Economics 20 - Prof. Anderson 19
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 = == + + +
Economics 20 - Prof. Anderson 20
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
Economics 20 - Prof. Anderson 21
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)
Economics 20 - Prof. Anderson 22
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
Economics 20 - Prof. Anderson 23
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
Economics 20 - Prof. Anderson 24
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)
Economics 20 - Prof. Anderson 25
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
Economics 20 - Prof. Anderson 26
(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
Cc xc nh cc chui khng cn bng cch khng chnh thng
Economics 20 - Prof. Anderson 27
(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
Cc xc nh cc chui khng cn bng cch khng chnh thng
Economics 20 - Prof. Anderson 28
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
Economics 20 - Prof. Anderson 29
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
Economics 20 - Prof. Anderson 30
Kim nh nghim n v - Nhngvn c bn
Mun kim nh H0:1=1 so vi H1:1
Economics 20 - Prof. Anderson 31
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
Economics 20 - Prof. Anderson 32
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.
Economics 20 - Prof. Anderson 33
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)
Economics 20 - Prof. Anderson 34
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
Economics 20 - Prof. Anderson 35
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)
Economics 20 - Prof. Anderson 36
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.
Economics 20 - Prof. Anderson 37
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.)
Economics 20 - Prof. Anderson 38
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)
Economics 20 - Prof. Anderson 39
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)
Economics 20 - Prof. Anderson 40
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)
Economics 20 - Prof. Anderson 41
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)
Economics 20 - Prof. Anderson 42
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)
Economics 20 - Prof. Anderson 43
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
Economics 20 - Prof. Anderson 44
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
Economics 20 - Prof. Anderson 45
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
Economics 20 - Prof. Anderson 46
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
Economics 20 - Prof. Anderson 47
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
Economics 20 - Prof. Anderson 48
(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
Economics 20 - Prof. Anderson 49
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!
Economics 20 - Prof. Anderson 50
Mt s vn i vi kim nhnghim n v
Economics 20 - Prof. Anderson 51
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
Economics 20 - Prof. Anderson 52
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
Economics 20 - Prof. Anderson 53
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.