chuỗi thời gian

7/29/2019 chui thi gian

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Chng trnh Ging dy Kinh t Fulbright Cc phng php nh lng II Kinh tlng c s- 3rd ed.Nin kha 2011-2013 Bi c Ch.21: Chui thi gian trong kinh tlng

Damodar N. Gujarati 1 Bin dch: Xun ThnhHiu nh: Cao Ho Thi

_________________

Phn V_________________CHUI THI GIAN

TRONG KINH T LNG

Cc d liu ca chui thi gian v ang c s dng mt cch thng xuyn v su rng,trong cc nghin cu thc nghim, ti mc cc nh kinh t lng gn y phi bt u ch mt cchk lng ti cc d liu ny. Trong Chng 1 chng ta nhn thy rng mt gi nhngm, to c s cho vic phn tch hi qui lin quan ti cc d liu ca chui thi gian, l cc dliu phi l dng. Nu khng nh vy th phng thc kim nh gi thuyt thng thngda trn t, F, cc kim nh khi bnh phng (X 2) v tng t c th tr nn khng ng tin cy.Trong cc Chng 21 v 22 chng ta s xem xt k hn cc d liu ca chui thi gian.

Trong Chng 21, u tin chng ta xc nh chui thi gian dng v sau pht trincc kim nh tm ra xem mt chui thi gian c l dng hay khng. V vn ny chng talm quen vi mt s khi nim lin quan, th d nh nghim n v, bc ngu nhin v chuithi gian kt hp. Sau chng ta s phn bit s khc nhau gia chui thi gian vi xu hngdng (TS) v chui dng vi sai phn (DS) v ch ra cc ng dng thc t ca chng. Mt vn thng gp trong lnh vc hi qui lin quan ti cc d liu ca chui thi gian l hin thngHi qui khng xc thcv chng ta s bn vcc ngha thc tin ca n. Tip theo, chng tas lm quen vi khi nim ng kt hp v ch ra tm quan trng ca n i vi nghin cuthc nghim. Tt c nhng khi nim ny s c minh ha mt cch r rng.

Chng 22 chng ta tp trung ch yu vo vic d bo s dng cc d liu ca chuithi gian. Vi gi nh rng mt chui thi gian l dng hoc c th tr nn dng bng cc

chuyn ha thch hp, chng ta s chng t qu trnh m hnh ha ARIMA, c bit tinh Box v Jenkins, c th c s dng cho vic d bo nh th no. Chng ny chng tacng bn ti mt phng php d bo khc, c bit n vi tn gi l t hi qui vector(VAR), v xem xt cc u im ca n so vi cc m hnh d bo kinh t lng truyn thngdng h phng trnh ng thi. Chng ta cng s th hin, vi cc th d thch hp, cc m hnhd bo ARIMA v VAR c thc hin nh th no.

Hai chng ny mi ch cp mt cch cn bn v chui thi gianca lnh vc kinh tlng. y l mt trong cc phm vi nng ng nht ca nghin cu kinh t lng v c mtlot cc cun sch chuyn ngnh vit v ti ny. Mc ch ca chng ta trong phm vi haichng ny l ch nhm gii thiu vi bn c th gii hp dn ca chui thi gian trong lnh

vc kinh t lng.


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Chng trnh Ging dy Kinh t Fulbright Cc phng php nh lng IIBi c

Kinh t lng c s- 3rd ed.Ch.21: Chui thi gian trong kinh tlng

Damodar N. Gujarati Bin dch: Xun ThnhHiu nh: Cao Ho Thi

2

_________________

CHNG

21_________________CHUI THI GIAN

TRONG KINH T LNG. PHN I:

TNH DNG (TNH TI),CC NGHIM N V,V TNH NG KT HP.

Nh nu Chng 1, mt trong hai d liu quan trng s dng trong nghin cu thc nghiml d liu ca chui thi gian. chng ny v chng tip theo chng ta s xem xt k hnnhng d liu v chng t ra mt lot cc thch thc i vi cc nh kinh t lng v ccnh thc nghim.

Th nht, cng tc thc nghim da vo d liu chui (thi gian) gi nh rng chui thigian c cp ti phi l dng. Mc d Chng I chng ta lm quen vi quan im trcgic ca tnh dng, chng ny chng ta s xem xt mtcch k lng hn. C th hn lchng ta s c gng xc nh tnh dngc ngha l g v ti sao ta li phi bi ri khi mtchui thi gian khng phi l chui dng.

Th hai, khi hi qui mt bin ca mt chui thi gian i vi mt bin ca chui thigian khc, ta thng thu c gi tr R2rt cao, mc d khng h c mi lin h c ngha nogia chng. Tnh hung ny l th d cho vn Hi qui khng xc thc (Hy xem mc 8.2).Vn ny xut hin bi v nu nh c hai chui thi gian c xt n u th hin cc xuhng mnh (xu hng ln hoc xung lin tc), th R2c gi tr cao l do s hin din ca xuhng loi ny, ch khng phi do mi quan h thc ca hai chui thi gian . Do , iuquan trng l tm ra c mi quan h gia cc bin s kinh t l thc hay gi.Chng ta s thy chng ny Hi qui khng xc thc c th xy ra nh th no nu cc chui thi gian khng

phi l dng.

Th ba, cc m hnh hi qui c cha cc d liu ca chui thi gian thng c dngcho cng tc d bo. T nhng lun im trn, ta cn phi bit xem liu vic d bo nh th cng tin cy hay khng khi m cc chui thi gian c s dng khng phi l chui dng.

Trong phn cn li ca chng ny chng ta s xem xt k hn v tnh dng ca mtchui thi gian.


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3

21.1 XEM XT MT VI CHUI THI GIAN C TRNG CA NN KINH THOA K

khi u, chng ta hy xem xt cc d liu ca chui thi gian nu trong Bng 21.1, ccd liu v 05 chui thi gian ca nn kinh t Hoa k c trnh by cho tng Qu ca cc nm

1970 n 1991. C 88 quan st c ghi nhn cho mi chui thi gian. Cc chui ny l TngSn phm X hi (GDP), Thu nhp Kh dng C nhn (PDI), Chi ph Tiu dng C nhn (PCE),Li nhun v C tc.

Hnh 21.1 th hin th c dng t cc d liu ca chui GDP, PCI v PCE rt ra tBng 21.1 v Hnh 21.2 th hin hai chui thi gian cn li.

Mt th c vch ra da vo cc d liu cho nh vy thng l bc u tintrong vic phn tch i vi bt k chui thi gian no. n tng u tin m chng ta c ct cc chui thi gian c v thnh th trong cc Hnh 21.1 v 21.2 l tt c cc chui dng nh u c xu hng tng, mc d xu hng ny khng phi l mt th duy tng, c

bit l i vi chui thi gian v Linhun. Cc chui thi gian ny thc cht l cc th d vcc chui thi gian khng dng. iu ny c ngha l g? Cu tr li c nu di y.

21.2. QU TRNH NGU NHIN DNG

D liu ca bt k chui thi gian no u c th c coi l c to ra nh mt qu trnhngu nhin v mt tp hp d liu c th, nh nu trong Bng 21.1, c th c coi l mtkt qu(c bit), tc l mt mu, ca qu trnh ngu nhin . S khc bit gia qu trnh ngunhin v kt qu ca n ging nh s khc bit gia tng th v mu trong d liu i chiu.Cng nh chng ta s dng cc d liu mu suy ra cc c lng v mt tp hp, th tronglnh vc chui thi gian, chng ta dng kt qu suy ra cc c lng v qu trnh ngu nhin. Mt dng ca qutrnh ngu nhin c cc nh phn tch v chui thi gian c bit quantm v xem xt k lng l ci c gi l Qu trnh ngu nhin dng.


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4

Ni chung, mt qu trnh ngu nhin c coi l dng nu nh trung bnh v phng sai ca nkhng i theo thi gian v gi tr ca ng phng sai gia hai thi on ch ph thuc vokhong cch v tr v thi gian gia hai thi on ny ch khng ph thuc vo thi imthc t m ng phng sai c tnh.1

Bng 21.1S liu Kinh t V m Hoa K, Qu I/1970 IV/1991

Qu GDP PDI PCE Li nhun C tc

1

Trong cc ti liu v chui thi gian, mt qu trnh ngu nhin nh vt c coi l mt qu trnh ngu nhin dngyu. i vi mc ch ca Chng ny, v trong hu ht cc tnh hung thc tin quan trng, dng dng ny s png.


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5

1970-I

1970-II

1970-III

1970-IV

1971-I

1971-II

1971-III1971-IV

1972-I

1972-II1972-III

1972-IV

1973-I

1973-II

1973-III

1973-IV1974-1

1974-Il

1974-III

1974-IV1975-1

1975-II

1975-III

1975-IV

1976-1

1976-II

1976-III

1976-IV

1977-1

1977-11

1977-III

1977-IV

1978-11978-lI

1978-III

1978-1'/

1979-1

1979-II

1 979-III1979-IV

1980-1

1980-II

1980-III

1980-IV

2,872.8

2,860.3

2,896.6

2,873.7

2,942.9

2,947.4

2,966.02,980.8

3,037.3

3,089.73,125.8

3,175.5

3,253.3

3,267.6

3,264.3

3,289.13,259.4

3,267.6

3,239.1

3,226.43,154.0

3,190.4

3,249.9

3,292.5

3,356.7

3,369.2

3,381.0

3,416.3

3,466.4

3,525.0

3,574.4

3,567.2

3,591.83,707.0

3,735.6

3,779.6

3,780.8

3,784.3

3,807.53,814.6

3,830.8

3,732.6

3,733.5

3,808.5

1 990.6

2,020.1

2,045.3

2,045.2

2,073.9

2,098.0

2,106.62,121.1

2,129.7

2,149.12,193.9

2,272.0

2,300.7

2,315.2

2,337.9

2,382.72,334.7

2,304.5

2,315.0

2,313.72,282.5

2,390.3

2,354.4

2,389.4

2,424.5

2,434.9

2,444.7

2,459.5

2,463.0

2,490.3

2,541.0

2,556.2

2,587.32,631.9

?,653.2

2,680.9

2,699.2

2,697.6

2,715.32,72'8.1

2,742.9

2 692.0

2,722.5

2,777.0

1,800.5

1,807.5

1,824.7

1,821.2

1,849.9

1,863.5

1,876.91,904.6

1,929.3

1,963.31,989.1

2,032.1

2,063.9

2,062.0

2,073.7

2,067.42,050.8

2,059.0

2,065.5

2,039.92,051.8

2,086.9

2,114.4

2,137.0

2,179.3

2,194.7

2,213.0

2,242.0

2,271.3

2,280.8

2,302.6

2,331.6

2,347,12,394.0

2,404.5

2,421.6

2,437.9

2,435.4

2,454.72,465.4

2,464.6

2,414.2

2,440.3

2,469.2

44.7

44.4

44.9

42.1

48.8

50.7

54.255.7

59.4

60.162.8

68.3

79.1

81.2

81.3

85.089.0

91.2

97.1

86.875.8

81.0

97.8

103.4

108.4

109.2

110.0

110.3

121.5

129.7

135.1

134.8

137.5154.0

158.0

167.8

168.2

174,1

178,1173A

174.3

144.5

151.0

154.6

24.5

23.9

23.3

23.1

23.8

23.7

23.823.7

25.0

25.526,1

26.5

27.0

27.8

28.3

29.429.8

30.4

30.9

30.530.0

29.7

30,1

30.6

32.6

35.0

36.6

38.3

39.2

40.0

41.4

42.4

43.544.5

46.6

48.9

50.5

51.8

52.754.5

57.6

58.7

59.3

60.5

Bng 21.1 (Tip theo)

Qu GDP PDI PCE Li nhun C tc1981-1

1981-II

1981-III

1981-IV

1982-1

1982-II

1982-Ill

1982-IV

1983-11983-II

3,860.5

3,844.4

3.864.5

3,803.1

3 756.1

3,771.1

3 754.4

3,759.6

3,783.53.886.5

2,783.7

2.776.7

2.814,1

2.808.8

2 795.0

2,824.8

2,829.0

2.832.6

2,843.62,867.0

2,475.5

2,476,1

2.487.4

2,468.6

2,484.0

2,488.9

2,502.5

2,539.3

2,556.52,604.0

159.5

143.7

147.6

140.3

114.4

114.0

114.6

109.9

113.6133.0

64.0

68.4

71.9

72.4

70.0

68.4

69.2

72.5

77.080.5


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6

1983-III

1983-IV

1984-1

1984-II

1984-III

1984-IV

1985-11985-II.

1985-III

1985-IV1936-1

1986-II

1936-III

1986-IV

1987-1

1987-II1987-III

1987-IV

1988-1

1988-II1988-III

1988-IV

1989-1

1989-Il

1989-III

1989-IV

1990-1

1990-II

1990-III

1990-Iil

1991-1

1991-II

1991-III1991-IV

3.944.4

4,012,1

4,089.5

4,144.0

4,166.4

4 194.2

4,221.84.254.8

4.309.0

4.333.54,390.5

4,387.7

4 412.6

4 427.1

4,460.0

4 515.34,559.3

4.625.5

4,655.3

4 704.84 734.5

4 779.7

4 809.8

4 832.4

4,845.6

4,859.7

4,880.8

4~900.3

4.903.3

4.855.1

4 824.0

4,840.?

4 862.74.868.0

2,903.0

2,960.6

3,033.2

3,065.9

3 102.7

3,l18.5

3 123.63,189.6

3,156.5

3,178.73,227.5

3.281.4

3,272.6

3,266.2

3,295.2

3 241.73,235.7

3,335.3

3 330.1

3,386.33 407.5

3.443.1

3,473.9

3 450.9

3.466.9

3 493.0

3,531.4

3.545.3

3 547.0

3 529.5

3.514.8

3 537.4

3 539.93.547.5

2,639.0

2,678.2

2,703.8

2,?41,1

2,754.6

2,784.8

2 824.92 849.7

2,893.3

2,895.12,922.4

2,947.9

2,993.7

3,012.5

3,011.5

3 046.83,075.8

3 074.6

3 123.2

3,147.83,170.6

3,202.9

3,200.9

3,203.6

3.241.1

3 241.6

3 258.8

3,253.6

3 231.2

3,251.8

3 241.1

3,252.4

3.271.23,271.1

145.7

141.6

155.1

152.6

141.8

136.3

125.2124.8

129.8

134.2109.2

106.0

111.0

119.2

140.2

157.9169,1

176.0

195.5

207.2213.4

226.0

221.3

206.2

195.7

203.0

199.1

193.7

196.3

199.0

189.7

182.7

189.6190.3

83.1

84.2

83.3

82.2

81.7

83.4

87.290.8

94,1

97.4105.I

110.7

112.3

111.0

108.0

105.5105.1

106.3

109.6

113.3117.5

121.0

124.6

127.1

129.1

130.7

132.3

132.5

133.8

136.2

137.8

136.7

138,1138.5

Ghi ch: GDP (Tng sn phm x hi), t la thi gi 1987, trang A-96.PDI (Thu nhp kh dng c nhn), t la thi gi 1987, trang A-112PCE (Chi ph tiu dng c nhn), t la thi gi 1987, trang A-96.Li nhun (Li nhun cng ty sau thu), t la thi gi 1987, trang A-110C tc (Cc khon chi tr c tc cng ty tnh), t la thi gi 1987, trang A-110.

Ngun:B Thng mi Hoa K, Cc Phn tch Kinh t,Bo co Thng k Kinh doanh,1963-1991, ra thng 6/1992.

minh gii cho lun im trn. Hy coi Yt l mt chui thi gian ngu nhin c cctnh cht sau:

Trung bnh: E (Yt) = (21.2.1)Phng sai: Var (Yt) = E (Yt - )

2 = 2 (21.2.2)ng phng sai: k= E [(Yt - ) Yt+k- )] (21.2.3)

y, k - ng phng sai (hoc s ng phng sai) ti tr k - l phng sai gia cc gitr Yt v Yt+k, tc l gia hai gi tr ca Y cc thi on cch qung k. Nu k=0, chng ta c o,n gin l phng sai ca Y (=2); nu k = 1, th 1l ng phng sai gia hai gi tr k cnnhau caY, tc l dng ng phng sai chng ta gp Chng 12 khi chng ta trnh by vch t tng quan.


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Gi s chng ta dch chuyn chui Y ban u t Y tn Yt+m. V nu Ytl dng, th trungbnh, phng sai v cc t ng phng sai ca Y t+mphing bng trung bnh, phng sai vcc t ng phng sai ca Yt. Tm li, nu mt chui thi gian l dng, th trung bnh, phngsai v t ng phng sai (ti cc tr khc nhau) s gi nguyn khng i d cho chng c

xc nh vo thi im no i na.

Nu mt chui thi gian khng phi l dng nh theo cch hiu va xc nh trn, thn c gi l chui thi gian khng dng(xin lu l rng, chng ta ang trnh by v tnhdng yu); i khi tnh khng dng c c l do s dch chuyn ca trung bnh.

th hin tt c iu ny, hy xem Hnh 21.3. Hnh 21.3a cho thy t sut sinh li thcca ch s c phiu S&P 500 i vi cc quan st hng nm t nm 1972 n 1986, v Hnh21.3b cho thy khong bin thin li sut Ngn hng ti Anh quc (s chnh lch gia li sutngn hn v di hn) hng qu ca giai on 1952-1988. Hnh u tin l mt th d v chuithi gian dng v hnh th 2 - chui thi gian khng dng.

Khi xt n cc chui thi gian ca nn kinh t Hoa k trong cc Hnh 21.1 v 21.2,chng ta c cm tng rng cc chui thi gian ny l khng dng v b ngoi, t nht trung

bnh, phng sai v cc t ng phng sai ca tng chui ring bit dng nh khng phi lbt bin theo thi gian. Lm sao m chng ta c th chc chn l Hnh 21.3a th hin mt chuithi gian dng v cc Hnh 21.3b, 21.1 v 21.2 li th hin cc chui thi gian khng dng?Chng ta s tho lun v cu hi ny phn tip theo.

21.3 KIM NH TNH DNG DA VO BIU TNG QUAN.

Mt cch kim nh n gin tnh dng l dng hm t tng quan (ACF). ACF vi tr k,k hiu bng k, c xc nh nh sau:

saiphng

treosaiphngong

k

k

k

0

(21.3.1)

Hy lu rng nu k = 0, th 0= 1 (ti sao?)

Do c ng phng sai ln phng sai c tnh bng cng mt n v o, nn kl mti lng khng c n v o, l trung tnh, l s. N nm trong khong t -1 n +1, ging nh

bt k mt h s tng quan no. Nu chng ta v th ktheo k, th th chng ta c c sl biu tng quan tng th.

V trong thc t chng ta ch c mt kt qu (tc l mu) ca mt qu trnh ngu nhin,nn chng ta ch c th tnh ton hm t tng quan mu, k. tnh hm ny, chng ta phitmng phng sai mu tr k, kv phng sai muotheo biu thc di y:

2

2Ni mt cch cht ch, chng ta phi chia ng phng sai mu tr k cho (n-k) v phng sai mu cho (n-1)

hn l chia cho n (ti sao?). Nhng cc mu ln iu ny khng gy ra khc bit my.


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8

n

YYYY kttk

))((

(21.3.2)

nYYt

2

0)( (21.3.3)

y, n l ln ca mu v Y l trung bnh mu.

Do hm t tng quan mu ti tr k s l:

0

k

k (21.3.4)

hm ny n gin l t l gia ng phng sai mu vi phng sai mu. th th hin k tr k c gi l Biu tng quan mu.

HNH 21.3


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9

Cc th d v chui thi gian tnh ti v khng tnh ti:(a) Ch s S&P 500 (t sut sinh li thc 1872-1986): l mt chui thi gian tnh ti.(b) Bin thin li sut ngn hng ti Anh quc (theo qu: I/1952IV/1988): mt chuithi gian khng tnh ti.(Ngun: Terence C. Mills.M hnh ho Kinh t lng ca cc Chui thi gian Ti chnh,NXB DHTH Cambridge, New York, 1993, trang 25 v 27.)

Hnh 21.4 th hin biu tng quan mu ca chui thi gian GDP cho trong Bng21.1, c c t chng trnh MICRO TSP phin bn 7.0. Chng ta trnh by biu tngquan ny vi 25 tr.3 Liu biu tng quan mu Hnh 21.4 c ch cho ta thy chui thigian GDP l dng hay khng? Mt c tnh ni bt ca biu tng quan mu ny l n c

bt u vi gi tr rt cao (khong 0,97 tr) v gim xung mt cch rt u n. Ngay c tr 14 (tc l tng quangia cc GDP cch nhau 14 qu) h s t tng quan vn cn ln -0,5. Dng tng quan kiu ny thng l mt du hiu cho thy rng chui thi gian l khngdng, ngc li nu nh mt qu trnh l hon ton ngu nhin th t tng quan ca n s bngkhng (zero) bt k tr ln hn khng no.

3 Mc d c cc kim nh v di ti a ca tr c s dng trong cc tnh ton, trong thc t cc tr ti

1/3 ln ca mu thng c s dng. Tuy nhin, vn ny rt thng xuyn l ch quan.


10/26




10

ngha thng k ca bt k k no u c th u c th c nh gi bi sai s chunca n. Bartlett ch ra rng nu mt chui thi gian l thun ty ngu nhin, tc l, n th hin(white noise)(Hy xem Phn 21.4), th cc h s t tng quan mu s c phn b gn nhchun vi trung bnh bng 0 v phng sai = 1/n, y n l ln ca mu.4i vi d liu cachng ta: n = 88, cng c ngha l l phng sai = 1/88 hoc sai s chun = 1/88 = 0,1066.Nh

vy, theo cc tnh cht ca mt phn b chun ha th khong tin cy 95% i vi kbt k sbng 1,96 (0,1066) = 0,2089 - gi tr hai pha ca 0 (zero). Do vy, nu gi tr c tnh caknm trong khong (-0,2089; 0,2089), chng ta s khng loi tr gi thuyt rng gi tr tht cak= 0. Tuy nhin, nu gi tr c tnh ny nm ngoi khong ni trn, chng ta c th loi tr githuyt cho rng gi tr tht ca k = 0. Kong tin cy 95% ny c th hinbng hai ngthng lin tc trong Hnh 21.4.

Bn c th thy rng tt c cc h s kca cc tr ti k = 23 trong Hnh 21.4 u c ngha thng k c bit, tc l khc 0 rt nhiu.

kim nh gi thuyt chungcho rng tt c cc h s t tng quan k ng thibng 0, ta c th s dng Tr thng k Q do Box v Pierce lp nn. Hm ny c xc nh nhsau:

m

k

knQ1

2 (21.3.5)

y, n = ln ca mum = thi lng ca tr

Tr thng k Q c phn b gn gingnh phn b khi bnh phng vi mdf. Trong khi pdng, nu nh gi tr tnh c ca Q vt qu gi tr gng/ti hn ca Q theo bng khi bnh

phng mt mc chn, ta c th loi tr gi thuyt khng - theo tt c k= 0; t nht phic mt vi k 0.

Mt bin th ca Tr thng k Q dng Box-Pierce ltr thng k Ljung-Box (LB) cxc nh di y:5

2

1

2

~

)2( m

m

k

k

knnnLB

(21.3.6)

Mc d trong cc mu ln, c tr thng k Q ln tr thng k LB u tun theo phn b khi bnhphng vi mdf, tr thng LB c coi l c cc tnh cht tt hn (mnh/hu hiu hn, v mtthng k) i vi cc mu nh so vi tr thng k Q.

i vi d liu GDP ca chng ta, tr thng k Q vi 25 tr l vo khong 793 v trthng k LB - 891, c hai gi tr ny l rt ng k; cc gi tr p rt ra cc gi tr khi bnh

4M.S. Barlett, V vic xc nh l thuyt cc tnh cht ca mu thuc chui thi gian t tng quan, Tp ch ca

Hi Thng k Hong gia, lot B, Quyn 27, 1946, trang 27 -41 (on the theoretical Specification of Samplingproperties of Autocorrelated Time Series, Journal of the Royal Statistical Society, series B, Vol.27, 1946, trang 27-

41.5 G.M. Ljung v G.P.E.Box, V mt cch o lng khng thch hp trong cc m hnh chui thi gian,

Biometrika, Quyn 66, 1978, trang 66-72.


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11

phng nh vy l gn nh bng 0. Do , c th kt lun rng hng phi tt c kca d liuGDP ca chng ta u bng 0.

Theo biu tng quan ny, chng ta i n kt lun tng th rng chui thi gian GDPnu Bng 21.1 khng phi l chui dng.

21.4 KIM NH NGHIM N V I VI TNH DNG.

Mt cch kim nh tnh dng khc c ph bin gn y l kim nh nghim n vcch ddng nht gii thiu v kim nh ny l xem xt m hnh sau:

Yt = Yt-1 + ut (21.4.1)

y Ut l s hng ch sai s ngu nhin xut pht t cc gi nh c in rng n c gi trtrung bnh bng 0, phng sai 2 l hng s v khng t tng quan. S hng sai s ny cnc bit ti di ci tn sai s nhiu ngu nhin(while noise error term) theo thut ng khoahc ng dng (engineering).6 T Chng 12 bn c s nhn ra rng Phng trnh (21.4.1) lmt hi qui bc mt, hoc AR(1), m chng ta hi qui gi tr ca Y ti thi im t da trngi tr ca n ti thi im (t-1). V nu h s ca Yt-1trong thc t bng 1, th chng ta ang

phi i mt vi ci gi l vn nghim n v, tc l tnh hung khng dng.7Do vy nuchng ta thc hin hi qui

Yt = Yt-1 + ut (21.4.2)

v tm ra rng = 1, th chng ta c th ni rng bin ngu nhin Y tc nghim n v. Trongkinh t lng (v chui thi gian), mt chui thi gian c nghim n v c gi lbc ngu

nhin(chui thi gian). V mt bc ngu nhin l mt th d ca chui thi gian khng dng.

8

V d, ta thy cc gi bn ti sn, nh cc gi c phiu chng hn, tun theo mt bc ngunhin, tc l, cc gi ny khng dng. Trong Ph lc ca chng ny chng ta thy rng mt

bc ngu nhin thc cht i din cho mt chui thi gian khng dng.

Phng trnh (24.1.2) thng c trnh by mt dng khc nhau sau:

Yt = ( - 1) Yt-1 + ut= Yt-1 + ut (21.4.3)

y = ( - 1) v , nh ta bit, l hmsai phn bc 1c gii thiu Chng 12. Hy

lu rng Yt = (Yt - Yt-1). Vi nh ngha ny, ngi c c th thy mt cch d dng l(21.4.2) v (21.4.3) l nh nhau. Tuy nhin, lc ny gi thuyt khng li l = 0 (ti sao?)

6Hy nh rng nu Ut khng ch l khng t tng quan m cn l c lp, th s hng sai s nh th c gi l

(strictly white noise). Xin cng lu rng nu s hng sai s l t tng quan, nh chng ta s ch ra sau y trongphn trnh by v Kim nh Dickey-Fuller gia tng (ADF), th chng ta c th d dng chp nhn c trng hpbt ng ny.7Mt quan im k thut: Ta c th vit (21.4.1) thnh Yt - Y t-1 = Ut. Tip n s dng ton t trn L sao cho

LYt = Yt-1. L2Yt = Yt-2v tng t, ta c th vit (21.4.1) thnh (1-L) Yt = Ut. Thut ng nghim n vl ni ti

nghim ca a thc trong ton t tr.8Bc ngu nhin thng c so snh vi bc i ca k say ru khi ra khi qun ru, ngi say bc mt

khong cch ngu nhin Ut vo thi im t, v nu ngi ny c tip tc bc i mi, th k cui cng s cngdt ra xa hn khi qun ru. iu tng t cng c dng ni v gi c phiu. Gi c phiu hm nay bnggi c phiu hm qua cng vi mt c sc ngu nhin.


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Nu thc s bng 0, ta c th vit (21.4.3) nh sau:

Yt = (Yt - Yt-1) = ut (21.4.4)

iu m phng trnh (21.4.4) ni ln l cc sai Phn bc 1 ca mt chui thi gian dng bcngu nhin (=ut) l mt chui thi gian dng do c gi nh rng utl thun ty ngu nhin.

Vy l, nu nh mt chui thi gian c ly sai phn mt ln v chui sai phn ldng, th ta c th ni rng chui ban u (dng bc ngu nhin) l mt chui kt hp bc1c k hiu l I(1). Tng t nh vy, nu nh chui ban u phi c ly sai phn hai ln(tc l ly sai phn bc 1 ca sai phn bc 1) tr thnh dng, th chui ban u c gi lchui kt hp bc 2,hoc I(2). Tm li, nu mt chui thi gian phi c ly sai phn d ln,th n s l chui kt hp bc d, hoc I(d). Do vy, bt k lc no nu ta c mt chui thi giankt hp bc 1 hoc ln hn, th c ngha l ta c mt chui thi gian khng dng. Theo qui c,nu d = 0 th qu trnh I(o) h qu s th hin mt chui thi gian dng. Chng ta s s dng ccthut ng qu trnh dngv qu trnh I(0) v[I(0) process] nh cc t ng ngha.

bit c liu chui thi gian Yt(chng hn nh GDP) c phi l chui khng dnghay khng, hy thc hin hi qui (21.4.2) v kim tra xem c bng 1 v mt thng k khng,hoc tng ng nh vy, hy c lng (21.4.3) v kim tra xem liu c phi =0 hay khngtrn c s tr thng k t. Tht khng may l gi tr t c c bng cch ny li khng tun theo

phn b students t ngay c i vi cc mu ln.

Theo gi thuyt khng rng = 1, tr thng k t c tnh theo qui c c bit ti nhl tr thng k (tau) [(tau statistic)], m cc gi tr ti hn ca n c sp thnh bng bi

Dickey v Fuller trn c s m phng Monte Carlo.9

Theo ti liu ny, kim nh Taucn cbit ti nh l kim nh Dickey-Fuller (DF), v s knh trng i vi nhng ngi phtminh ra n. Hy lu rng nu gi thuyt khng rng = 1 b bc b (tc l, chui thi gian ldng), th chng ta c th s dng kim nh t thng thng (students).

dng n gin nht ca n, chng ta c lng hi qui nh (21.4.2), sau chia h s c c lng cho sai s chun ca n tnh tr thng k Dickey-Fuller v i chiuvi cc bng Dickey-Fuller xem gi thuyt 0 = 1 c b bc b hay khng. Tuy nhin, cc

bng ny cha phi l hon ton y , chng c m rng mt cch ng k biMackinnon thng qua cc m phng Monte Carlo.10Trong s cc chng trnh phn mm thngk, ET, MICRO TSP v SHAZAMcho ra cc gi tr ti hn Dickey-Fuller v mackinnon ca

tr thng k DF.

Nu nh gi tr tuyt i tnh c ca tr thng k (tc l //) cao hn cc gi tr tihn tuyt i T hoc DF hoc Mackinnon DF, th chng ta s khng bc b gi thuyt cho rng

9 D.A. Dickey v W.A. Fuller, Phn b ca cc hm c lng i vi cc chui thi gian t hi qui vi nghim

n v, Tp ch ca Hip hi Thng k Hoa K, Quyn 74, 1979, trang 427-431. Hy xem thm Nhp mn vchui thi gian thng k ca W.A. Fuller, NXB John Wiley v cc con, New York, 1976.10

J.G. Mackinnon, Cc gi tr ti hn ca cc kim nh ng kt hp, trong R.F. Engle v C.W.J. Granger, Ccmi quan h kinh t di hn: Cc tho lun v ng kt hp, Chng 13, NXB Oxford University, New York1991.


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13

chui thi gian cho l dng. Nu mt khc, n thp hn gi tr ti hn, th chui thi gian sl khng dng.11

V nhng l do v mt l thuyt v thc tin, kim nh Dickey-Fuller c p dng ivi cc hi qui c thc hin cc dng sau:

Yt = Yt-1 + ut (21.4.3)Yt = 1 + Yt-1 + ut (21.4.5)Yt = 1 + 2

t +Yt-1 + ut (21.4.6)

y, t l binxu hng hoc bin thi gian. Trong mi trng hp gi thuyt khng s l =0, tc l c nghim n v. S khc bit gia (21.4.3) v hai hi qui khc l ch c s baogm c hng s (giao im vi trc ta ) v s hng xu hng.

Nu s hng sai s utl t tng quan, ta s bin i (21.4.6) thnh:

tit

m

i

itt YYtY

1

121 (21.4.7)

m , th d )(),( 322211 tttttt YYYYYY , tc l ta s dng cc s hng sai

phn ca tr. S lng cc s hng sai phn ca tr cn c thng c xc nh bngthc nghim - Khi nim v vic cn phi c bao nhiu s hng s hng sai s trong (21.4.7)l c lp vi chui. Gi thuyt khng vn l = 0 hoc = 1, c ngha l Y c nghim n v(y l khng dng).Khi kim nh DF c p dng cho cc m hnh nh (21.4.7), n c gi lkim nh gia tng Dickey-Fuller [Augmented Dickey-Fuller (ADF) test].Tr thng k cakim nh ADF c cng mt phn b tim cn ging nh ca tr thng k DF, do vy c th sdng cng cc gi tr ti hn ging nhau.

Chui Thi gian GDP ca Hoa k c phi l Dng hay khng?

Ta hy minh ha kim nh DF bng cch s dng cc d liu trong Bng 21.1. Theo phngphp MICRO TSP 7.0, cc hi qui tng ng vi (21.4.5) v (21.4.6) cho ra cc kt qu sau:

GDP t = 32,96930,0025 GDPt-1t = (1,3304) (0,3932) (21.4.8)

r2 = 0,0018 d = 1,3520

GDP t = 183,9751 + 1,3949t0,0579GDPt-1t = (1,7877) (1,5111) (1,5563) (21.4.9)

R2 = 0,0286 d = 1,3147

i vi mc ch ca chng ta,iu quan trng l tr thng k t (=tau) ca bin GDP t-1. Xin hynh rng gi thuyt khng ca ta l = 0, iu ny c ngha l = 1, hoc nghim n v. Vyi vi m hnh (21.4.8) cc tr thng k ti hn 1%, 5% v 10%, c tnh bi Mackinnon, l -3,5073; -2,8951; v -2,5844 tng ng. Do gi tr c tnh ca l - 0,3932, v mt gi tr tuyti l nh hn cc gi tr ti hn 1%, 5% v 10%, chng ta hng loi b gi thuyt 0 vi = 0,

11Nu hi qui c thc hin dng (21.4.3), th tr thng k T c c lng thng s mang du m (-). Do

vy, mt gi tr m ln ca T thng l mt du hiu ca tnh dng.


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tc l chui GDP th hin mt nghim n v, y l mt cch khc nirng chui GDP lkhng dng. Trn c s kim nh biu tng quan, chng ta phi thy trc c kt lunny.

i vi m hnh (21.4.9) cc gi tr ti hn 1%, 5% v 10% ca l - 4,0673; -3,4620; -3,1570

tng ng. Gi tr tnh c ca t (=tau) - 1,5563 i vi GDPt-1l khng ng k v mt thngk, cho thy mt ln na rng = 0 hoc c nghim n v trong d liu GNP. Nhn dy,xin lu

rng cc gi tr ti hn DF ph thuc vo liu c s hng khng i v/hoc xu hnghay khng.

bao gm c kh nng tng quan chui trong u t, chng ta c th s dng m hnh(21.4.7) v sau p dng kim nh ADF; xin nh rng tr thng k Durbin-Watson d, chotrong (21.4.8) v 21.4.9), m ch n mt kh nng nh th. Vi mt gi tr tr ca cc sai

phn bc 1 ca GDP, chng ta c c cc kt qu hi qui sau y:12

GDP t = 233,0806 + 1,8922 0,0787 GDPt-1 + GDPt-1t = (2,3848) (2,1522) (2,2153) (3,4647) (21.4.10)

R2 = 0,1526 d = 2,0858

Do Durbin Watson d tng ln, c kh nng y c tng quan chui. Tuy vy, hy lu rng = 2,2153 vn thp hn cc gi tr ti hn ADF : 4,0673 (1%); 3,4620 (5%); v 3,1570 (10%), iu ny ni ln rng chui thi gian GDP l khng dng.

Tm li, trn c s biu tng quan v cc kim nh nghim n v DF v ADF, ccd liu GDP ca Hoa K i vi thi on I-1970 n VI-1991 l khng dng.

Chui GDP Sai Phn Bc 1 c l Dng?

Chng ta hy cng lm li bi tp nu trn, nhng ln ny chng ta s tm xem liu GDP t =(GDP t - GDPt-1) c l dng hay khng? thun tin ta dng Dt ch GDPt.Ta c kt qusau:

D t = 15,53130,6748Dt-1t = (3,4830) (6,4956) (21.4.11)r2= 0,3436

Cc gi tr ti hn T 1%, 5% v 10%, c tnh bi Mackinnon, l 3,5082; 2,8955; v

2,5846 tng ng. V mt gi tr tuyt i, gi tr 6,4956 ln hn bt k mt trong ba gi trny, v th chng ta c th bc b gi thuyt rng (h s ca Dt-1) l bng 0. Tc l, cc d liuGDP sai phn bc 1 khng th hin nghim n v, hoc c th ni rng cc d liu ny l dng.iu c ngha: chng l I(0).13Hnh 21.5 cho thy cc d liu GDP sai phn bc 1 so vichui GDP ban u nh Hnh 21.1, th chui sai phn GDP Hnh 21.5 khng th hin bt kxu hng no.

12Vic b sung hai gi tr tr ca GDP khng lm cc kt qu thay i my

13Mt quan im k thut. C mt s tranh lun v vic liu ta c th p dng kim nh DF mt cch tun t, u

trn l vi d liu dng mc v sau l dng sai phn bc 1 hay khng. V vn ny, Hy tham kho D.A.Dickey v S.S. Pantula, Xc nh bc ca sai phn trong cc qu trnh t hi qui, Tp ch Thng mi v Thngk, Quyn 5, 1987, trang 455-461.


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Do GDPtl dng, nh nu trn, nn n l mt qu trnh ngu nhin I(0), c ngha lt bn thn GDPtl mt chui thi gian I(1); v bn cht, n l mt bc ngu nhin.

ng thi, Do DF hoc ADF ch cho ta thy mt chui thi gian c l kt hp hay khng,nn chng cn c bit n nh l kim nh s kt hp.

21.5 QU TRNH NGU NHIN C XU HNG DNG (TS) V SAI PHN DNG(DS).

Trong cc hi qui lin quan ti d liu chui thi gian, bin thi gian hoc xu hng t thngc dng nh mt trong cc bin hi qui c lp loi b vn tng quan khng xc thc(hy xem Chng 8). D liu thuc cc chui thi gian kinh t thng c xu hng chuyn dchtheo cng mt hng bi v mt xu hng thng l chung cho chng. Th d nh, trong hi quica PCE i vi PDI, nu nh chng ta quan st c R2 c gi tr rt cao, trng hp nythng xy ra, th iu c th phn nh khng phi mc thc ca mi lin h gia hai bin, m n gin ch l xu hng chung hin hu gia chng.

HNH 21.5 Cc sai phn bc I ca GDP, Hoa K, I/1970IV/1991.

trnh nhng mi lin h khng xc thc nh vy, thng thng phi thc hin hi quiPCE i vi PDI v t, bin xu hng. H s ca PDI c c t php hi qui ny s th hin

nh hng tnh ln PCE, sau khi loi b tc ng ca xu hng.

Ni mt cch khc, vic thhin bin xu hng trong hi qui c tc dng san bng xu hng (tc l loi b nh hng ca

xu hng ra khi) i vi c PCE v PDI.

Gn y, mt nhnh mi cc nh kinh t lng v chui thi gian chng li phngphp thc tiin chung ny. Theo h, phng php thc tin chun ny ch c th c chp nhnnu nh bin xu hng l xc nh(deterministic) ch khng phi lngu nhin.14

14Hy tham kho hai bi bit hp dn v qu gi: Charles R. Nelson v Charler I. Plosser, Cc xu hng v bc

ngu nhin trong chui thi gian kinh t v m: Mt s minh chng v cc ngha, Tp ch Kinh t Tin t, Quyn10, 1982, trang 139-162; James H. Stock v Mark W. Watson, Cc xu hng bin thin trong cc chui thi giankinh t, Tp ch cc trin vng kinh t, Quyn 2, No.3, Ma h 1988, trang 147-174.


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Ni rng ra, xu hng s l xc nh nu nh n l hon ton c th d on c vkhng bin thin. C th thy t Hnh 21.1 nhng g m iu ny th hin. Nu phi v mtng thng xu hng cho c chui thi gian GDP c minh ha , th ta s thy l ch mtng thng xu hng khng th hin y c cc d liu. C th c mt ng thng xuhng cho giai on I/1970-IV/1974, mt ng khc cho giai on I/1975-IV/1981 v mt

ng nacho giai on I/1982-IV/1991. Tm li, ng xu hng ny bn thn n thayi,tc l n l ngu nhin. Nu y l trng hp phi xt n, th phng php thc tin chung vsan bng xu hng s tr nn sai lch.

Bng cch no m ta c th bit c rng xu hng ca mt chui, chng hn nh GDP,l xc nh hay bin thin (tc l ngu nhin)? Ta c cu tr li trong hi qui (21.4.6). Khic lng hi qui ny, nu ta tm ra rng chui thi gian cho (th d: GDP) c nghim n v(tc l khng dng), th c th kt lun l chui thi gian th hin mt xu hng ngu nhin.

Nu khng c nghim n v, th chui thi gian th hin xu hng xc nh.

Tr li cc hi qui (21.4.9) v (21.4.10), chng ta kt lun rng chui thi gian GDP

ca Hoa K thuc giai on I/1970-IV/1991 l khng dng. Do , chui thi gian nh vythhin xu hng ngu nhin. Nh mt h qu, nu nh ta thc hin hi qui ny:

GDPt = 1 + 2t= ut (21.5.1)

vi gi thit sai lm l c xu hng xc nh, th GDP san bng xu hng thu c:

ut = (GDPt - 1 - 2t) (21.5.2)

s chng c gi tr g my khi trong thc t xu hng l ngu nhin. Do , cc d bo da vo

(21.5.1) s l khng ng tin cy. V l do ny chng ta cnh bo Chng 5 l khng cs dng ng hi qui c c lng theo cc d liu c trc a ra cc d bo quxa trong tng lai.

Tip theo chng ta gii htiu v hai khi nim c bn trong phn tch chui thi gian, cth l qu trnh xu hng dng(TSP) v qu trnh sai phn dng(DSP).Nu trong hi qui:

Yt = 1 + 2t + ut (21.5.3)

utthc ra l dng vi trung bnh bng 0, v phng sai 2, th (21.5.3) th hin mt TSP; nu ta

loi tr xu hng (tc l 1 + 2t) khi (21.5.3), th kt qu s l mt qu trnh dng. Tuy nhin,

nu Ytc pht sinh t

Yt - Yt-1 = + ut (21.5.4)

m l hng s v u tl dng vi trung bnh bng 0 v phng sai 2, th ta gi qu trnh nh

th l mt DSP. Hy nh rng (Yt - Yt-1) = Yt, tc l phng sai bc 1 ca Yt.

Quay li chui thi gian GDP ca chng ta, ta thy rng mc GDP khng phi l TSP,trong khi nh thy hi qui (21.4.11), GDP l mt DSP.


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Ni gn li, mt chui thi gian dng c th c m hnh ha nh mt qu trnh TS,trong khi mt chui thi gian khng dng li th hin mt qu trnh DS. Nh Holden, Peel vThomson nhn nh:

Vi mt xu hngxc nh, cc bin c th c lm thnh dng bng cch a mt xu hng

v thi gian vo bt c hi qui no hoc bng cch thc hin mt hi qui s b i vi thi gianv loi b xu hng c c lng [nh trong hi qui (21.5.2)]. Vi mt xu hngngunhin, cc kim nh i vi ng kt hp (s c bn ti sau y) v tnh ng kt hp l cnthit.15

Chng ta kt thc phnny bng cch nu ra cu hi: Tm quan trng thc tin ca TSP v SDPl g? Theo quan im d bo di hn, cc d bo c rt ra t TSP s l ng tin cy hn,trong khi nhng d bo rt ra t DST s l khng ng tin cy v i khi l rt nguy him. Nh

Nathan Balke nhn xt; s hin hu ca mt xu hng ngu nhin c ngha l cc bin ngtrong mt chui thi gian l kt qu ca cc c sc khng ch i vi thnh phn mang tnh nhtthi hay chu k m cn i vi thnh phn mang tnh xu hng. 16Tc l, nhng xo trn hoc

c sc i vi chui thi gian nh th s thay i hn mc ca chng mt cch vnh vin.

21.6 HI QUI KHNG XC THC

Cc hi qui lin quan ti cc d liu ca chui thi gian c cha kh nng thu c cc kt qu khng xc thchoc khng ng tin cy d rng b ngoi cc kt qu c v l tt nhng khixem xt k lng hn th chng tr nn ng ng. D thy tht r vn ny, chng ta hy cngquay tr li cc d liu PCE v PDI ca Ho K c nu Bng 21.1. Gi s chng ta thchin hi qui PCE i vi PDI tm mi quan h gia chng. Bng cch s dng cc d liu cho trong Bng 21.1, ta c c cc kt qu sau:

PCEt = -171,4412 + 0,9672 PDItt = (7,4809) (119,8711) (21.6.1)R2 = 0,9940 d = 0,5316

Cc kt qu hi qui ny c v ln khng tng ni: R2l qu cao, t l t ca PDI c gi tr quln, xu hng tiu dng bin t(MPC) so vi PDI l dng v c gi tr cao. Ch c mt iukhng nh l tr thng k Durbin-Watson d thp. Nh Granger v Newbold 17 xut,R2 >d l mt tin tt ng vc rng hi qui c c lng l hu qu ca mt hi quikhng xc thc.

Nu chng ta c lng phng trnh (21.4.9) i vi c PCE ln PDI, s c cc kt qusau:

PCEt = 91,7110 + 0,7704t 0,0432PCEt-1 (21.6.2)t= (1,6358) (1,2983) (1,3276)

PDIt = 326,2089 + 2,8834t 0,1579PDIt-1 (21.6.3)

15K.Holden, D.A.Peel v J.L. Thompson, Nhp mn v D bo Kinh t, NXB HTH Cambridge, New York,

1990, trang 81.16

Nathan S. Balke, Cc xu hng m hnh ha trong cc chui thi gian kinh t vi m, Tp ch Ph bnh v Kinht (Economic Review), Ngn hng D tr Lin bang ti Dallas, 05.1991, trang 21. y l mt bi bo rt ng c

c v cc qu trnh TS v DS.17C.W.J. Granger v P. Newbild, Cc hi qui khng xc thc trong kinh tlng, Tp ch Kinh t lng, Quyn2, 1974, trang 111-120.


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t= (2,7368) (2,5243) (2,5751)

i vi mc ch hin thi, chng ta quan tm ti gi tr T ca PCE v PDI theo cc tr. Ccgi tr ti hn T 1%, 5% v 10% hoc cc gi tr DF c tnh bi Mackinnon l 4,0673;3,4620; v3,1570 tng ng. Tip n, vmt gi tr tuyt i, cc gi tr =1,3276 v 2,5751

ca cc s hng tr PCE v PDI l nh hn cc gi tr ti hn T mc 10%. Do , kt lun y l PCE v PDI u c mt nghim n v, tc l chng khng phi l dng.

V th, khi tnh hi qui PCE i vi PDI trong (21.6.1), ta c th thc hin hi qui mtchui thi gian khng dng i vi mt chui thi gian khng dng khc. Trong trng hp nhvy, cc phng thc kim nh chun t v F khng c gi tr.18V mt ny, hi qui (21.6.1) lkhng xc thc. kt thc, hy nh rng trong (21.6.3) nu ta ly gi tr t = 2,5751 ca PDI t-1trn c s kim nh t thng thng, th n s l ng k v mt thng k mc 2%. Nhng trnc s kim nh T th n li khng ng k ngay c mc 10%, iu ny mt ln na cho thyti sao trong trng hp chui thi gian khng dng, ta khng c da vo cc gi tr cc lng t.

Chng ta ngi c, nh mt bi tp, t ch ra rng cc chui sai phn bc 1 PCE vPDI, PCEt v PDIt, l dng.

V PCE v PDI l dng, ti sao li khng hi qui chui trc i vi chui sau v loib vn v tnh khng dng, xu hng ngu nhin v cc vn c lin quan? Cu tr li l,gii quyt vn v tnh khng dng theo kiu ny c th cng ging nh tr nh i cng vinc tm, bi v khi ly sai phn bc 1 (hoc bc cao hn) ta c th nh mt mi quan h dihn qu gi gia PCE v PDI c cho trc bi cc mc (cng nh i vi saiphn bc 1) cahai bin ny. L thuyt kinh t hu nh c thhin nh mt mi quan h di hn gia cc bin

dng mc ca cc bin ch khng phi dng sai phn bc 1 ca chng. V th, gi thuyt thunhp c nh Milton Friedman mc nhin cng nhn rng mc tiu dng c nh l mt hm camc thu nhp c nh; mi quan h khng c th hin theo sai phn bc 1 ca cc bin ny.Cc mi quan h, chng hn, gia chi tiu ca chnh ph v cc khon thu v thu; cung ng tinmt v gi c (Hy nh li l thuyt s lng v Tin); li sut tri phiu kho bc 3 th ng v 6thng; v.v... cng c th hin tt nht dng mc.

thy iu ny, hy tham chiu Bng 21.1. Mc d c PCE ln PDI u c xu hngtng ln theo kiu ngu nhin, dng nh chng c xu hng cng nhau. Bin ng ny gingnh i bn nhy, mingi theo mt bc ngu nhin, m cc bc ngu nhin ca h dngnh li n khp vi nhau. S ng b, v mt trc gic l khi nhim ng sau chui thi gian

ng kt hp.Nh ta s ch ra phn tip theo, nu PCE v PDI l ng kt hp, th cc ktquhi qui cho (21.6.1) c th s khng phi l khng xc thc, v cc kim nh thng thngt v F s l ng. Nh Granger ghi nhn Mt kim nh v s ng kt hp c th c coinh mt tin kim nh loi b cc tnh hung hi qui khng xc thc.

18Vn ny c ghi nhn bi Dickey v Fuller, op.cit. v gn y hn bi P.C.B. Phillips trong Cc hi qui

chui thi gian vi nghim n v, Econometrica, Quyn 55, 1987, trang 277 -302, v Hng ti mt l thuyttim cn hp nht v t hi qui, Biometrica, Quyn 74, 1987, trang 535 -547. i vi cc tho lun thuc cc sch

gio khao c th tip cn c v mt ny v cc phng din khc ca chui thi gian, Hy c Terence C. Mills, Cc Phng php k thut v chui thi gian dnh cho cc nh kinh t, NXB HTH Cambridge, New York,1990.


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19

21.7 NG KT HP

Hy tr li cc hi qui (21.6.1), (21.62) v (21.6.3). Hai hi qui ny ch ra rng c PCE v PDI lcc qu trnh khng dng hoc bc ngu nhin [xin lu : mi qu trnh u l I(1)]. Mc dvy, s kt hp tuyn tnh ca hai bin ny c th s l dng. C th hn, nu ta vit (21.6.1)

nh:19

ut = PCFt + 1 - 2PDIt (21.7.1)

v tm ra rng ut[tc l kt hp tuyt tnh (PCEt - 1 - 2PDIt)] l I(0) hoc dng, th ta ni rngcc bin PCE v PDI l ng kt hp; Ni mt cch nm na, chng thuc cng mt bin song.V trc gic, ta thy rng khi u ttrong (21.7.1) l I(0), th cc xu hng trong PCE v PDI trnn cn bng vi nhau.20V chng s c cng mt bc sng nu chng c kt hp vi nhau cng mt bc.Nh vy, nu chui Y l I(1) v mt chui khc X cng l I(1), th chng c thl ng kt hp. Ni rng ra, nu Y l I(d) v X cng l I(d) m d c cng gi tr, th hai chuiny c th l ng kt hp. Nu ng l nh th, th hi qui i vi cc mc ca hai bin ,ging nh trong (21.6.1), s l c ngha (tc l khng phi l khng xc thc); v chng ta skhng mt bt k thng tin di hn qu gi no, ci m c th b mt i nu thay vo ta sdng cc sai phn bc 1 ca chng.

Tm li, min l chng ta chc chn c rng nhng g c c t cc hi qui nh(21.6.1) l I(0) hoc dng, th h phng php hi qui truyn thng (k c cc kim nh t v F)m chng ta bit cho n nay vn p dng c i vi cc d liu lin quan ti chui thigian. ng gp qu bu ca cc khi nim v nghim n v, ng kt hp, v.v... l ch bucta phi tm ra liu cc s d ca hi qui c l dng hay khng.Theo cch din t ca l thuytng kt hp, mt hi qui nh (21.6.1) c coi l hi qui ng kt hpv thng s 2c

gi lthng s ng kt hp.21

Mt lot cc phng php kim nh ng kt hp c xut trong cc bi vit trntp ch. Hai phng php n gin l (1) kim nh DF v ADF i vi Ut c c lng

bng hi quing kt hp v (2) kim nh ng kt hp Durbin-Watson (CRWD).22

Kim nh Engle-Granger (EG) hoc Kim nh EG Gia tng (AEG)

Chng ta bit cch p dng kim nh nghim n v DF hoc ADF. Tt c nhng g ta philm l c lng mt hi qui nh(21.6.1), thu nhn cc s d, v s dng kim nh DF hoc

19C.W.J. Granger, Cc pht trin trong nghin cu v cc bin kinh t ng kt hp, Bn tin Kinh t v Thng k

ca HTH Oxford, Quyn 48, 1986, trang 226.20

Cch xem xt ng kt hp bng trc gic ny c nu ra trong Tri thc v thc hnh kinh t lng caWilliam E. Griffiths, R. Carter Hill, v George G. Judge, NXB John Wiley v cc con, New York, 1993, trang 700-

702.21

Khi nim ng kphp c th c m rng i vi m hnh hi qui c cha k kbin hi qui c lp. Trongtrng hp ny, ta s c k thng s ng kt hp. tho lun chung v iu ny, hy tham kho cc tham chiunu trong cc ch thch khc nhau.22

S khc bit ny tn ti gia cc kim nh nghim n v v cc kim nh ng kt hp: Nh David A. Dickey,Dennis W. Jansen, v Daniel I. Thornton quan st thy, Cc kim nh nghim n v c thc hin i vi ccchui thi gian n bin (Univariate) [tc l duy nht]. Ngc li, ng kt hp li x l mi quan h gia mtnhm cc bin, m (mt cch v iu kin) mi bin c nghim n v. Hy c cc bi vit ca h,

Nhp mn v ng kt hp vi mt ng dng i vi tin t v thu nhp, Tp ch ph bnh kinh t, Ngn hng dtr Lin bang ti St. Lyis, 03-04.1991, trang 59. Nh tn gi ca bi vit nu, y l mt s nhp mn tuyt viv kim nh ng kt hp.


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ADF.23Tuy nhin, c mt iu cn phi phng khi thc hin. Do U c c lng theothng s ng kt hp 2 c c lng, nn cc gi tr ti hn quan trng DF hoc ADF skhng tht thch hp. Engle v Granger tnh ton cc gi tr ny, vic ny c th tm thytrong cc sch tham kho.24Do vy, kim nh DF hoc ADF trong bi cnh ny c gi lkim nh Engle-Granger (EG) vkim nh EG gia tng (AEG). Tuy nhin, hin nay TSP,

SHAZAM v cc phn mm khc li a ra cc gi tr ti hn ny cng vi cc thng tin khc.25

Chng ta hy cng quay li vi hi qui PCE PDI (21.6.1) v thc hin kim nhnghim n v DF i vi cc s d c c lng theo hi qui ny.

Ta c ccc kt qu sau:

ut = -0,2716 ut-1t = (-3,6725) (21.7.2)

r2 = 0,1422

Cc gi tr ti hn Engle-Granger (tr thng k t hi qui trc ) 1%, 5% v 10% tng ngl2,5899;1,9439; v1,6177. Do v mt gi tr tuyt i gi tr c c lng = 3,6725vt qu bt c mt trong cc gi tr ti hn ny, nn kt lun c th l u t c c lng sl dng (tc l, n khng c nghim n v), v v vy PCE v PDI, mc d l khng dng mtcch ring bit, li l ng kt hp.26

Kim nh Hi qui ng Kt hp Durbin-Watson (CRDW)

Mt cch khc, nhanh hn, tm ra liu PCE v PDI c l ng kt hp hay khng l kim nh

CRDW, m cc gi tr ti hn ca n u tin c cung cp bi Sargan v Bhargava.

27

TrongCRDW chng ta s dng gi tr d Durbin-Watson c c t hi qui ng kt hp, chng hn d= 0,5316 cho trong (21.6.1). Nhng n y gi thuyt khng s l d = 0 hn l d chun = 2.Da vo 10.000 m phng hnh thnh t 100 quan st cho mi m phng, cc gi tr ti hn 1%,5% v 10% kim nh gi thuyt khng vi gi tr tht ca d = 0 l 0,511; 0,386 v 0,322tng ng. Nh vy, nu gi tr d tnh c nh hn 0,511, th chng ta bc b gi thuyt ngkt hp o83 mc 1%. Trong th d ca chng ta, gi tr ca d = 0,5316 l cao hn mc ti hn

23

Nu PCE v PDI khng phi l ng kt hp, th bt k mt lin kt tuyn tnh no ca chng cng s l khngdng, v do , cc s d Ut cng s l khng dng.24

R.F. Engle v C.W.J. Granger. ng kt hp v hiu chnh sai s: Th hin, c lung v kim nh,Econometrica, Quyn 55, 1987, trang 251-276; R.F. Engle v B. Sam Yoo, Dbo v kim nh trong cc hthng ng kt hp, Tp ch Kinh t lng, Quyn 35, trang 143 -159; R.F. Engle, Chui thi gian kinh t ngkt hp: Mt tng quan vi cc kt qu mi, trong cun Cc mi quan h kinh t di hn : tri thc v ng kthp ca R.F. Engle v C.W.J. Granger, NXB HTH Oxford, 1991, Chng 12. Cun sch ny cng cha bi bit nu trn y ca Mackinnon, cng nh cc bi tham lun hi tho khc v lnh vc ny. 25

Mt s cc gi tr ti hn EG v AEG cng c sao chp litrong cun c lng v suy lun trong kinh tlng ca Russell Davidson v James G. Mackinnon, NXB HTH Oxford, New York 1993, Bng 20.2, trang 722.Chng cng c th c tm thy trong cc phin bn MICRO-TSP 7.0 v SHAZAM 7.0.26

Mt quan im k thut v hi qui ng kt hp c c lng (21.6.1): Do mi PCE v PDI l I(1), takhng th s dng cc sai s chun c c lng v cc gi tr lin quan t ca cc h s c c lng

nu trong hi qui ny cho mc ch a ra cc suy lun v cc thng s tp hp.27J.D. Sargan v A.S. Bhargava, Kim nh cc s d t hi qui bnh phng nh nht i vi bc ngu nhinc Gaussian to ra, Econometrica, Quyn 51, 1983, trang 153-174.


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ny, iu c th ng rng PCE v PDI l ng kt hp - mt kt lun tng t nh kt lunt c trn c s kim nh EG.28

Tm li, da trn cc kim nh EG v CRDW, ta i ti kt lun rng PCE v PDI lng kt hp.29 Mc d chng th hin cc bc ngu nhin mt cch ring bit, dng nh vn

c mt mi quan h di hn n nh gia hai bin ny;chng s khng ri xa nhau, iu ny cth thy r t Hnh 21.1.

21.8 NG KT HP V C CH HIU CHNH SAI S (ECM)

Chng ta va mi ch ra rng PCE v PDI l ng kt hp, tc l, c mt mi quan h cn bngdi hn gia hai bin ny. Tt nhin, trong bi cnh ngn hn c th c s mt cn bng. Do vy,ta c th coi s hng sai s trong (21.7.1) l sai s cn bng.30 V chng ta c th s dng shng sai s ny lin h ng thi ngn hn ca PCE vi gi tr di hn ca n. C ch hiu

chnh sai s (ECM), c s dng ln u tin bi Sargan.31v sau c ph bin rng ribi Engle v Granger, dng hiu chnh s mt cn bng. Khng i su vo cc chi tit lthuyt, y chng ta ch th hin, vi th d PCE/PDI, ECM vn hnh ra so; cc chi tit c thc tm thy trong bi hi tho ca Engle v Granger c nu phn trc (Hy xem chthch 24).

Hy xem xt m hnh sau nh mt th d n gin ca ECM:

PCEt = 0 + 1PDIt + 2ut-1 + t (21.8.1)32

y, nh thng l l k hiu sai phn bc 1; u t-1l gi tr ca s d t hi qui (21.6.1) tr mt thi on (one-period lagged value), c lng thc nghim ca s hng sai s cn bng;v l s hng sai s vi cc tnh cht thng thng.

Hi qui (21.8.1) lin h s thay i PCE vi s thay i PDI v sai s cn bng thion trc . Trong hi qui ny PDI th hin cc xo trn ngn hn ca PDI, trong khi shng hiu chnh sai s ut-1th hin s iu chnh hng ti cn bng di hn. Nu nh 2l ngk v mt thng k, th n cho ta thy t l mt cn i no trong PCE mt thi on c hiuchnh trong thi on tip theo. S dng cc d liu cho trong Bng 21.1, ta c cc kt qusau:

PCEt = 11,6918 + 0,2906PDIt 0,00867t-1 (21.8.2)t= (5,3249) (4,1717) (1,6003)

28C s tranh lun ng quan tm v tnh u vit ca CRDW so vi DF, iu ny c th tm thy trong sch ca

Engle-Granger, op.cit. S tranh lun xoay quanh sc mnh ca hai tr thng k, tc l xc sut khng phm phi sais Loi II (type II error), (c ngha l chp nhn gi thuyt khng trong khi iu l khng ng/sa).29

Cc kim nh EG v CRDW hin c b sung bi cc kim nh u vit hn do Johansen pht trin. Nhngtho lun vphng php Johansenl nm ngoi phm vi cun sch ny. Hy tham kho S.Johansen v K.Juseliuc lng v suy lun vi xc sut ti a v ng kt hp - vi cc ng dng i vi Nhu cu v Tin t, Bn tinKinh t v Thng k ca HTH Oxford, Quyn 52, 1990, trang 169-210.30

Thut ng ny l do Griffiths v cc tc gi khc, op.cit., trang 701.31

J.D. Sargan, Tin cng v gi c Vng Quc Anh: Mt nghin cu v h phng php lun Kinh t lng,

Xut bn ln u 1964 v c sao li trong cun Kinh t nh lng v phn tch kinh t lng ca K.F. Wallisv D.F. Hendry, NXB Basil Blackwell, Oxford, 1984.32

Cc s hng sai phn tr b sung ca PDI cng c th c cp ti.


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R2= 0,1717 d = 1,9233

Cc kt qu ny cho thy rng cc thay i ngn hn ca PDI c tc ng tch cc ng k lnPCE v rng khong 0,09 sai bit gia gi tr thc v gi tr di hn, hoc cn bng, ca PCEc loi tr hoc hiu chnh mi qu (ba thng). Tuy nhin, do gi tr p ca h s PCE l

khong 0,11, nn tm quan trng v mt thng k ca khm ph ny vn cn cha r.33

Khi xemxt hi qui ng kt hp (21.6.1), ta thy MPC di hn l khong 0,97, iu ny ni ln rngthc t c mi quan h 1/1 gia PCE v PDI v rng PCE iu chnh theo ng tng trng dihn ca n mt cch nhanh chng tip theo sau mi xo trn.

Trc khi kt thc phn ny, nn ghi nh li nhc nh sau ca S.G.Hall:

Trong khi khi nim ng kt hp r rng l mt s cng c vng chc quan trng v l thuytca m hnh hiu chnh sai s, vn cn c mt lot cc vn xung quanh vic ng dng thc tca n; cc gi tr ti hn v vic thc hin nhiu kim nh vi mu nh vn cn xa l i vicc m hnh a dng; vic kim tra mt cch chuyn su i vi biu tng quan c th vn l

mt cng c quan trng.34

21.9 TM TT V CC KT LUN

1. Phn tch hi qui da trn cc d liu chui thi gian ngm gi nh rng chui thi giani tng l dng. Cc kim nh t v F c in u da vo gi nh ny.

2. Trong thc t, hu ht cc chui thi gian l khng dng.

3. Mt qu trnh ngu nhin c coi l dng yunu nh trung bnh, phng sai v cc tng phng sai ca n l hng s theo thi gian (tc l chng bt bin theo thi gian).

4. mc khng chnh thc, tnh dng yu c th c kim nh bng biu tngquanca mt chui thi gian, l mt th t tng quan cc trn khc nhau. ivi cc chui thi gian dng, biu tng quan thu hp li rt nhanh, trong khi i vi ccchui thi gian khng dng th n bin mt mt cch t t. i vi mt chui thun tyngu nhin, t tng quan mi tr 1 v ln hn l bng 0.

5. mc chnh thc, tnh dng c th c thm nh bng cch tm xem liu chui thigian c cha nghim n v hay khng. Cc kimnh Dickey-Fuller (DF) v kim nhDF gia tng (ADF) c th c dng cho mc ch ny.

6. Mt chui thi gian kinh t c th l xu hng - dng (TS)hoc sai phn - dng (DS).Mt chui thi gian TS c mt xu hng xc nh, trong khi chui thi gian DS li c mtxu hng bin thin hoc ngu nhin. Thc tin chung v vic a cc bin thi gian hocxu hng v mt m hnh hi qui san bng xu hng cc d liu ch thch hp i vi

33Hy nh rng Hi qui ECM m ta s dng l rt n gin. Cc bin th khc ca hi qui ny c th c tm

thytrong cc ti liu tham kho. Cng xin lu rng chng ta khng cp thm cc bin tu khc thng quavic cp cc sai phn tr bc cao hn ca PDI ln PCE.34

S.G. Hall, Mt ng dng ca phng php c lng hai bc Granger v Engle i vi cc d liu tng tincng (Aggregate Wage) ca Vng Quc Anh,Bn tin kinh t v thng k ca HTH Oxford, Quyn48, s 3,

08.1986, trang 238. Hy c John Y. Campbell v Pierre Perron, Cm by v c hi: Nhng g cc nh kinh t vm phi bit v nghim n v, NBER (National, 1991, trang 141-219. Theo quan i ca ti, bi vit ny l loiphi ci vi cc nh kinh t ng dng.


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chui thi gian TS. Cc kim nh DF v ADF c th c s dng xc nh liu mtchui thi gian l TS v DS.

7. Hi qui mt bin ca chui thi gian i vi mt hoc nhiu bin khc ca chui thi gianthng c th cho ra cc kt qu khng c ngha hoc khng xc thc. Hin tng ny

c gi l hi qui khng xc thc. Mt cch trnh khi n l tm xem liu chui thigian c phi l ng kt hp hay khng.

8. ng kt hpc ngha l mc d l khng dng mt cch n l, nhng lin kt tuyn tnhca hai hoc nhiu chui thi gian c th s l dng. Cc kim nh EG, AEG v CRDW cth c dng tm ra liu hai hoc nhiu chui thi gian c l ng kt hp hay khng.

9. ng kt hp ca hai (hoc nhiu hn) chui thi gian ng rng c mt mi quan h dihn (hoc cn bng) gia chng.

10. C ch hiuchnh sai s (ECM), c Engle v Granger pht trin, l mt phng tin

iu ha ng thi ngn hn ca mt bin kinh t vi ng thi di hn ca n.

11. Lnh vc kinh t lng chui thi gian vn ang tin trin. Cc kt qu v cc kim nh c thit lp vn cn l, trong mt s trng hp, s b v cn nhiu cng vic phi lm.Mt cu hi quan trng ang cn c gii p l v sao mt s chui thi gian kinh t ldng v mt s khc li l khng dng.

CC BI TP

Cu hi:

21.1Tnh dng yu c ngha l g?

21.2Mt chui thi gian kt hp c ngha l g?

21.3 ngha ca mt nghim n v l g?

21.4 Nu mt chui thi gian l I(3), th ta phi ly sai phn bao nhiu ln n tr thnhdng?

21.5 Cc kim nh Dickey-Fuller (DF) v DF gia tng l g?

21.6 Cc kim nh Engle-Granger (EG) v EG gia tng l g?

21.7 ngha ca ng kt hp l g?

21.8 S khc bit, nu c, gia cc kim nh nghim n v v cc kim nh ng kt hp lg?

21.9 Hi qui khng xc thc l g?

21.10 Mi lin h gia ng kt hp v hi qui khng xc thc l g?


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21.11 S khc bit gia xu hng xc nh v xu hng ngu nhin l g?

21.12 Qu trnh xu hng - dng (TSP) v qu trnh sai phn - dng (DSP) c ngha l g?

21.13 Bc ngu nhin (M hnh) l g?

21.14 i vi mt qu trnh dng bc ngu nhin, phng sai l v hn, Bn c ng khng? V sao?

21.15 C ch hiu chnh sai s (ECM) l g? Mi quan h ca n vi ng kt hp l g?

Bi tp

21.16 S dng d liu cho trong Bng 21.1. Hy lp cc biu tng quan mu ti 25 tr i vi cc chui thi gian PCE, PDI, li nhun v c tc. Bn thy c trng thichung g? V trc gic, nhng chui thi gian no trong s ny c v l dng?

21.17 i vi mi trong s chuithi gian ca bi tp 21.16. Hy dng kim nh DF tmxem liu cc chui thi gian ny c cha nghim n v hay khng. Nu c nghim nv, Bn s c trng ha chui thi gian nh th no?

21.18 Tip tc bi tp 21.17, Bn s quyt nh ra sao nu nh kim nh ADF l thch hp hnso vi kim nh DF?

21.19 Hy xem xt cc chui thi gian Li nhunv C tc cho Bng 21.1. Do C tc phthuc vo Li nhun. Hy nghin cu m hnh n gin sau y:

C tc t = 1 + 2Li nhun + ut

(a) Bn c cho rng hi qui ny s chu hu qu ca hin tng hi qui khng xc thc?V sao?

(b) Cc chui thi gian C tc v Li nhunc l ng kt hp hay khng? Bn s kimnh mt cch chnh xc iu ny nh th no?Nu nh sau khi kim nh, bn thyrng chng l ng kt hp, liu cu tr li cho mc (a) trn ca bn c thay ikhng?

(c) Hy dng c ch hiu chnh sai s (ECM) nghin cu ng thi ngn hn v dihn ca C tc trong mi quan h vi Li nhun

(d) Nu bn xem xt cc chui C tc v Li nhun mt cch ring r, liu chng c thhin cc xu hng ngu nhin hoc xc nh khng? Bn s dng cc kim nh no?

*(e) Gi s C tcv Li nhun l ng kt hp. Sau thay v ly hi qui C tc theoLi nhun, bn hy ly hi qui Li nhun theo C tc. Liu mt hi qui nh th cgi tr hay khng?

*-Khng bt buc


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21.20 Hy ly cc sai phn bc 1 ca cc chui thi gian trong Bng 21.1 v v th cachng. Hy dng biu tng quan ca mi chui thi gian ti 25 tr. Bn c ntng g v cc biu tng quan ny?

21.21 Thay v ly hi qui C tc theo Li nhun dng mc, gi s bn ly hi qui ca sai

phn bc 1 ca C tc theosai phn bc 1 ca Li nhun. Bn c a giao im vi trcta vo hi qui ny khng? V sao c hoc v sao khng? Hy trnh by cc tnh ton.

21.22 Hy tip tc vi bi tp trn. Bn s kim nh tnh dng ca hi qui sai phn bc 1 rasao? Trong th d ny, Bn s hy vng nhn c mt kt cc chotrc (priori) no v vsao? Hy trnh by tt c cc php tnh.

21.23 Da vo s liu cc khu dn c mi (housing starts) - X thuc khu vc t nhn Vngquc Anh trong giai on 1948-1984, Terence Mills thu c kt qu hi qui sau:*

Xt = 31,030,188 Xt-1se = (12,50) (0,080)(t=) (2,35)

Ghi ch: Gi tr ti hn 5% l -2,95 v 10% l -2,60

(a) Trn c s cc kt qu ny, chui thi gian khu dn c mi l dng hay khng dng?Hoc, c nghim n v trong chui thi gian ny khng?Lm th no m bn bitc?

(b) Nu bn s dng kim nh t thng thng, liu gi tr t quan st c c l ng k

v mt thng k khng? Trn c s , liu bn c kt lun rng chui thi gian nyl dng?(c) By gi hy xem xt cc kt qu hi qui sau:

2Xt = 4,76 1,39Xt-1 + 0,3132Xt-1

se = (5,06) (0,236) (0,163)

(t=) (5,89)

y 2l ton t sai phn bc 2, tc l sai phn bc 1 ca sai phn bc 1. Gi tr c lng Tby gi l ng k v mt thng k. n y bn c th ni g v tnh dng ca chui thi gianang c xem xt ny?

Ghi chu: Mc ch ca hi qui nu trn l tm ra liu c mt nghim n v bc 2 trongchui thi gian hay khng.

*Terence C. Mills, op. cit., trang 127. Cc k hiu c hi khc i.


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PH LC 21A

21.A1 M HNH BC NGU NHIN

Gi s {ut} l mt chui ngu nhin vi trung bnh v phng sai (hng s) 2 v n l

khng tng quan v chui. (Ghi ch: {} l ch mt chui). Vy th chui {Y t} c gil bc ngu nhin nu nh:

Yt = Yt-1 + ut (1)

Trong m hnh bc ngu nhin, nh (1) th hin, gi tr Y thi im t tng ng vigi tr ca n thi im (t-1) cng vi mt c sc ngu nhin. Hy coi Y0= 0 thiim t = 0 sao cho:

Y1 = u1

Y2 = Y1 + u2 = u1 + u2

Y3 = Y2 + u3 = u1 + u2 + u3

v ni chung

Yt = ut

Do ,

E(Yt) = E (ut) = t (2)

Theo cch tng t, ta c th ch ra rng

Phng sai (Yt) = t 2 (3)

Nh (2) v (3) cho thy, do c trung bnh ln phng sai ca Ytthay i theo thi gian t(thc ra l tng ln trong trng hp ny), nn qu trnh l khng dng

Tuy nhin

Yt - Yt-1 = ut (4)

li l mt qu trnh hon ton ngu nhin. Tc l cc phng sai bc 1 ca mt chui thigian dng bc ngu nhin l dng.

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