BAY ES IAN INFERENCE FO R CO INTEGRATED SY STEM S

997 2 3 JOU RNAL O F SYST EM S EN G IN EER IN G 997 V o l 2 N o 3 (, 300072), :,, : F224 BAY ES IAN INFERENCE FO R CO INTEGRATED SY STEM S Zhag Sh iy ig Zhu Hui Zhag X ibi (Schoo l of M aagem e t T ia ji U iversity, T ia ji 300072) Abstract I th is paper, the p rob lem of testig of co i tegrated system s is discu ssed A Bayesia co i tegratio test based o errio r2co rrectio rep rese tatio of co i tegrated system s is p ropo sed T he efficiecy of the Bayesia m ethod is show by M o to Carlo experim e tṡ Key words: Co i tegratio system s, erro r2co rrectio m odel, co i tegratio test 0, D ickey Fu ller D F AD F [, 2 ] D F AD F, ph illp s D F Z [ 3 ],,, Z D F ( ), Joha se, L R [ 4 ], [5 7 ], AD F Joha se L R,, Joha se, ( 950562), 59,, 59,m ale, p roḟ 996 8 28 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved

24 2 3, M o to Carlo, ( ) k VA R y t = Υy t- + Υ2y t- 2 + + Υky t- k + Εt () y t m, y 0 = y - = = y - k = 0, Εt iidn (0, ) z m m Υ (Z ) = I - Υz - Υ2z 2 - - Υkz k,, y t I (), () Y = X B + Ε Y = (Y T, Y T 2,, Y T t,, Y T T) T Y t = (y t,, y ti,, y tm ) T t m B = (Υ T, Υ T 2,, Υ T k ) T Ε= (Ε T, Ε T 2,, Ε T T ) T Y T 0 Y T - i Y T - k X = Y T t- Y T t- i Y T t- k Y T T - Y T T - i Y T T - k () (ECM ) Y t = ΠY t- + Y t- + 2 Y t- 2 + + k- Y t- k+ + Εt (3) 0 = - (I - Υ - Υ2 - - Υk) i = - (Υi+ + + Υk) (i =, 2,, k - ) 0, Grager [ 8 ], r, rak (0 ) = r r 3 ) r = m, y t, y t I () 2) r = 0, y t, 3) 0 < r < m, m =, (3) D ickey2fu ller (AD F ) [ 2 ] y t, 0 = 0 rak (0 ) = 0, 0, 0, 0 Κj (j =, 2,, m ) (2) 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved

997 : 25 2 Κ M o to Carlo B T, (2) L (B, gx, Y ) = ( = ( Tm 2 g g - T 2 exp (- Tm 2 g g - T 2 exp {- B δ = (X T X ) - X T Y, Τ= T - m k ΤS = (Y - X B δ ) T (Y - X B δ ) T 2 t= Ε T t - Εt) 2 tr - [ΤS + (B - B δ ) T X T X (B - B δ ) ]} (B, ), B ) B, p (B ) C ( ),, p ( ) g g - 2 (m + ) (B, ) p (B, gx, Y ) L (B, gx, Y ) p (B ) p ( ) = g g - 2 (T + m + ) exp {- B p ( gx, Y ) g g - 2 (Τ+ m + ) exp {-, B p (B g X Y ) g g - 2 (m k) exp {-, p (B, ) = p (B ) p ( 2 tr - [Τs + (B - B δ ) T X T X (B - B δ ) ]} (4) 2 Τtr - S } (5) 2 (B - Bδ ) T ( - g X T X ) (B - B δ ) } (6), p ( gx Y ) (W ishart), B Κi (i =, 2,, m ) 0, B, Κ B,, VA R Κ M o to Carlo Κ [ 9 ], (B, ) p (B, gx Y ) = p ( gx Y ) p (B g X Y ), ( W ishart ) (i), B ( ) (i) B (i) B (i) 0 (i), 0 (i) (0, [4 ]), Κ (i) = (Κ (i), Κ 2 (i),, Κ (i) m ), Κ M o to Carlo, (i =, 2,, ) 3 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved

26 2 3 0 r, H 0: r < r0, H : r r0 (7) r0 {Κj} (j =, 2,, m ) w r 0, w (i) = Κ (i) + Κ 2 (i) + + Κ (i) r 0 r 0 = u r 0 + Ε (i) r 0 (r0 =, 2,, m ; i =, 2,, ) u r 0 w r 0, Ε (i) r 0 iid (0, Ρ 2 r 0 ) w r 0, u r 0 Ρ 2 r 0 r < r0, r0 {Κr 0 } 0, u r 0 = u r 0 - (r0 =, 2,, m ; u 0 = 0), (7) H 0: u r 0 = u r 0 -, H : u r 0 u r 0 - (8) w m w,, r0 0 r, w r 0 x, u r 0 Η, Η0 = u r 0 - (u 0 = 0), Ρ = Ρr 0 H 0: Η= Η0, H : Η Η0 K 0, K 0 P (H 0gx ) K 0 = P (H gx ) = P (H 0 ) P (H ) g P (H 0) > 0 = P (H 0 ) P (x gη= Η0) P (H ) P (x gη Η0) 0 + + - 0 P (x gη0, Ρ) P (Ρ) dρ P (x gη, Ρ) P (Η, Ρ) dηdρ Η 8 0, P (H ) = - P (H 0) > 0 Η 8 P (H 0 ) P (H ),, H 0 H K 0 >, H 0 H, H 0 (9) ) P (ΗgΡ) N (Η0, ΑΡ 2 ), Α Ρ P (Ρ) Ρ = K g Ρ K 0 = P (H 0 ) P (H ) g (K ), (9) 0 - P (x gη0) = 0 = K ( 2 P (x gη0, Ρ) P (Ρ) dρ 0 P (x gη, Ρ) P (ΗgΡ) p (Ρ) dηdρ (0) P (x gη0, Ρ) P (Ρ) dρ = K ( Τ= -, Τs = i= P (x gη0) = K g ( (0) 0 Ρ- exp {- 2 g ( 2 ) - 2 [Τs + (x - Η0 ) 2 ]gρ 2 } Ρ dρ 2 + (Τs) - 2 ( 2 ) [ + (x - Η0 ) Τs (x i - X ) 2 t = x - Η0 sg, (), 2 ( 2 ) - ] - v+ 2 () 2 + ( 2 ) (Τs) - t 2 2 ( + Τ ) - v+ 2 (2) 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved

997 : 27 P (x ) = + - 0 t 2 Τ( + Α) ]- P (x gη, Ρ) P (ΗgΡ) P (Ρ) dηdρ = K ( + Α) - 2 g ( 2 ( 2 ) - 2 + (Τs) - 2 ( 2 ) [ + v+ 2 (3) (2) (3) (0) K 0 = P (H 0 ) P (H ) g ( + t 2 gτ) - (Τ+ ) g2 (4) ( + Α) - g2 [ + t 2 gτ( + Α) ] (Τ+ ) g2, (4) K 0 P (H 0 ) P (H ) ( + Α) g2 exp [- 2 t2 Αg( + Α) ] (5) P (x gη0), Α Α, P (x ) 2) (Cauchy) P (ΗgΡ) = Ρ Π Ρ 2 + (Η- Η0) 2, P (Ρ) = K g Ρ [0 ] K 0 = P (H 0 ) P (H ) ( ΠΤ 2 ) g2 ( + t2 Τ ) - (Τ- ) g2 (6) t Τ K 0 P (H 0 ) P (H ) ( ΠΤ 2 ) g2 exp (- 2 t2 ) (7), r0 m (r0 = m, m -,, ) {w r 0 }, H 0: r < r0, H : r r0, K 0, H 0 H, H 0, r0 4 ) (D GP) D GP (a) : y t = Βy t- + Εt Εt iidn (0, ) D GP (b) : y t = Βy t- + Β2y t- 2 + Εt Εt iidn (0, ) D GP (a), 0 = - ( - Β) 3, i) Β= 0 90, ii) Β= 0 98, iii) Β= 3, D GP (b), 0 = - ( - Β - Β2), : iv) Β + Β2 =, Β = 0 3, Β2 = 0 7, v) Β + Β2 <, Β = 0 3, Β2 = 0 2 D GP (a) D GP (b) 5, 50 00 200 5 5, M o to Carlo 500, p (Η, Ρ) (Α=, Α= ) (4) - (7), K 0,D GP (a) (i),, H 0: 0 = 0 (Β = ) H : 0 0 (Β ), (i) I () D GP (a) (ii) (i), 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved

28 2 3, iii), H 0: 0 = 0 (Β= ) H, I () (iv) (v) H 0: 0 = 0 (Β + Β2 = ), H : 0 0 (Β + Β2 ) (iv), (v) 5, K 0 D GP (a) D GP (b) i) 0 9 ii) 0 98 iii) 00 iv) 0 3, 0 7 v) 0 3, 0 2 50 K c 7 7E - 7 8 32E - 2 6 004 553 5 485 779 2 39E - K c 8 08E - 88 3 50E - 4 5 936 56 5 45 52 0 K 9 93E - 7 5 86E - 2 4 788 776 4 373 058 6 68E - 0 K 5 67E - 87 3 36E - 4 4 786 34 4 368 258 0 K 2 64E - 70 6 9E - 67 89 9 6 329 07 2 02E - K 2 9 24E - 87 3 96E - 3 67 57 7 6 263 54 0 00 K c 6 87E - 69 9 37E - 53 4 975 78 7 854 48 4 54E - 5 K c 66E - 77 85E - 33 4 96 4 7 799 926 0 K 36E - 68 22E - 33 998 74 6 299 74 2 08E - 3 K 03E - 76 40E - 52 2 007 07 6 35 26 0 K 2 53E - 68 9 29E - 33 69 26 7 62 397 38 3 53E - 2 K 2 89E - 76 06E - 5 69 244 8 62 553 99 0 200 K c 2 04E - 92 04E - 42 6 968 5 494 964 3 69E - 63 K c 0 3 70E - 76 6 964 43 5 46 223 0 K 8 66E - 92 7 24E - 43 3 603 6 4 429 02 3 93E - 56 K 0 7 23E - 76 3 606 3 4 450 725 0 K 2 2 65E - 92 4 2E - 42 9 867 9 30 724 35 0E - 63 K 2 0 4 2E - 75 9 906 30 880 4 0 K c K c K K (Α= ) K 2 K 2 (Α= ) 2) [ ] x t - 0 4 x t- ( - z ) = (, - 2) + Εt y t 0 y t- 0 0, x 0 = y 0 = 0, V ar (Εt) = 50, 00, 200,M o to Co rlo 500,, ) Α=, 2) Α= 2 2 K 0 50 00 200 r < 2 r < r < 2 r < r < 2 r < K c 2 356 52 82E - 54 453 63 4 89E - 84 2 8 88 2 08E - 95 K c 2 355 35 0 442 84 0 2 5 24 0 K 9 957 223 4 3E - 53 9 222 443 58E - 76 9 763 774 2 95E - 83 K 9 958 36 0 9 229 682 0 9 766 52 0 K 2 442 3 4 6E - 3 350 6 3 08E - 4 389 82 2 3E - 4 K 2 442 83 0 35 07 0 389 375 0 K c K c K K (Α= ) K 2 K 2 (Α= ) 2 H 0: r < 2 K 0 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved

997 : 29, H 0: r <,,, r = 5, (y t) (ct) 952-993 ( 978 00) 994 ) AD F {ct} {y t}, 3 AD F 5 ct y t, ct y t, {ct} {y t} I () ct y t K 0,, ct y t K 0,, {ct} {y t} 3 A D F (I) (k = 2) K 0 ct 252 2 2 47 y t 0 430 3 9 42 ct - 3 259 6 3 0 53E - 0 y t - 5 23 9 3 0 72E - 32 3 5-2 93 [2 ] I: A D F, k = 2: A D F 2 2) Joha se L R 0, VA R k k Schw arz, k = 2 4 H 0: rak = Joha se 95 95 r Κm ax H 0: rak = r K 0 r = 0 30 47 3 5 4 27 56 3 4 4 r < 2 0 085 r 2 907 3 8 2 907 3 8 r < 7 9E - 0 3 95 [4 ] 4 H 0: r = 0, Joha se 95, Κm ax 95, r = 0, r, Κm ax 95, r = Joha se L R 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved

30 2 3 H 0: r < 2, K 0 >, H 0, H 0: r <, K 0, H 0, r < 2, r =, 6,,, Α, I (), ECM,, D ickey D A, Fuller W A D istributio of the E stim ato rs fo r A uto regressive T im e series W ith a U it Roo ṫ Jo rual of the Am erica Statistical A ssoc, 979, 74: 427 43 2 D ickey D A, Fuller W A T he L ikelihood R atio Statistics fo r A uto regressive T im e Series w ith a U it Roo ṫ Ecoom etrica, 98, 49: 057 072 3 Ph illip s P C B T im e Series R egressio w ith a U it Roo ṫ Ecoom etrica, 987, 55: 277 30 4 Johase S Statistical A alysis of Co itegratio V ecto rṡ Joural of Ecoom ic D yam ics ad Cotro l, 988, 2: 23 254 5 Sim C A Bayesia Skep ticism o U it Roo t Ecoom etricṡ Joural of Ecoom ic D yam ics ad Cotro l, 988, 2: 463 474 6 Do rfm a J H A N um erical Bayesia T est fo r Co itegratio of A R P rocesseṡ Joural of Ecoom etrics, 955, 66: 289 324 7 D ejog D N Co itegratio ad T red2statioarity i M acroecoom ic T im e Series: Evidece F rom the L ikelihood Fuctio Joural of Ecoom etrics, 992, 52: 347 370 8 Egle R F Grager C W J Co itegratio ad E rro r Co rrectio: R ep resetatio, E stim atio ad T est2 ig Ecoom etrica, 987, 55: 25 276 9 Gew eke J Bayesia Iferece i Ecoom etric M odelṡ U sig M oto Carlo Itegratio Ecoom etrica, 989, 57: 37 340 0 Box G E P, T iao G C Bayesia Iferece i StatisticalA alysiṡ N ew Yo rk: Joh W iley ad Sos, 992 Egle R F, Yoo B S Fo recastig ad T estig i Co itegrated System ṡ Joural of Ecoom etrics, 987, 35: 43 59 2 Baerjee A, Do lado J J, Galbraith J W, et al Co itegratio, E rro r Co rrectio ad the Ecoom etric A alysis of N o2statioary D ata O xfo rd uiversity P ress,o xfo rd, 993 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved