非线性时间序列与马尔可夫链

非线性时间序列与马尔可夫链 . ……………………2 1. 2. . ……………………6 1. 2. 3. . ---AR…..12 1. 2. AR 3. . …………………………..29 1. 2. 3. AR 4. AR . …………………………….46 1. 2. …………………………………………50 １ . 1. {x}, t , . , . ---LAR(1): x=x+e, t=1,2,… (1.1) tt-1t 2{e}i.i.d.Ee=0, Ee=<, etttt {x,x,…}. (1.1), t-1t-1 x=x+e= e + x tt-1ttt-1 = e + { e + x} tt-1t-2 2= e + e + x =… tt-1t-2 2n-1n=e+e+e+…+e+x. (1.2) tt-1t-2t-n+1t-n||<1, n x 0, (1.3) t-n 2n-1j{e+e+e+…+e} e. (1.4) tt-1t-2t-n+1j=0t-j LAR(1), jx=e. (1.5) tj=0t-j , LAR(1). , p LAR(p). p LAR(p): ２ x=x+x+...+x+e, t=1,2,… (1.6) t1t-12t-2pt-pt 2{e}i.i.d.Ee=0, Ee=<, e{x, ttttt-1 x,…}. , (1.6), (1.2)t-1 , , AR(1) , LAR(1). ？,,,,,12px1,,,,,,t,,,,10？0x,,0,,,,t,1X=, U=, A=, (1.7) t,,,,,,？.........？？,,,,,,,,,,,,x0？000t,p，1,,,,,, (1.6): X=A X+ eU. (1.8) tt-1t (1.2), , X=AX+eU tt-1t 2= eU+ eAU+Ax= tt-1t-2 2n-1n=eU+eAU+eAU+…+eAU+Ax. (1.9) tt-1t-2t-n+1t-n A(A)(A), (A)<1, , (1.8): k X=AUe. (1.10) tk=0t-k Xx{x}, ttt (1.6). , x=e. (1.11) tk=0kt-k (1.6),, ... ,, , k12p ３ , (1.11), 2x=e, , (1.12) tk=0kt-kk=0k , (1.12)xkt. , {x}. t , , , ---NLAR(1), LAR(1) . NLAR(1) x=(x)+e, t=1,2,… (1.13) tt-1t 2{e}i.i.d.Ee=0, Ee=<, e{x, ttttt-1 x,…}, LAR(1), , t-1 2(x)x, (x)=x/{a+bx}. t-1t-1t-1t-1t-1 (1.13), , x= (x) +e= e+ (x) tt-1ttt-1 = e+ ( e+ (x)) tt-1t-2 = e+ ( e+ ( e+ (x)))=… tt-1t-2t-3 =e+ ( e+ ( e+ …+ (x))…). (1.14) tt-1t-2t-n , (x)t-1 , , . p ４ x=(x,x,…,x)+e, t=1,2,… (1.15) tt-1t-2t-pt (1.6)(1.9), ,(,,...,)xxx,,t,1t,2t,p,,x,,t,1 ( x,x,…,x), (1.16) t-1t-2t-p,,？,,,,xt,p，1,, (1.15) X=(X) +eU, t=1,2,… (1.17) tt-1t , (1.9)(1.11). , , , . , , . . 2. , . (1.12) , . , . k , (1.12){e}, t . . , ５ . . , , . , . . 1. (1.12){x}, {e}tt . i . ARMA. , , . , , . , . : {}i.i.d.E=0, ttt {x, x,…}. () t-1t-1 : x=(x,x,…,x)+, (2.1) tt-1t-2t-pt (…)p. , , (2.1) x=(x,x,…,x;)+, (2.2) tt-1t-2t-pt ６ , (2.1). : x=(x,x,…,x)+S(x,x,…,x),(2.3) tt-1t-2t-pt-1t-2t-qt (…)p, S(…)q. . (2.3). : x=(x,x,…,x; ,,…,),t=1,2,… (2.4) tt-1t-2t-pt t-1 t-q (…)p+q. , (2.4) ARMA, , , , , . , . pqPQx=x++x. (2.5) tj=1jt-jj=0jt-ji=1j=1ijt-it-j =1. 0 x=x+ x+. (2.6) tt-1t-1t-1t 2. (2.1)(2.2), . (2.2). : f(x)=f(x)+f(x)+...+f(x)+,(2.7) t1t-12t-2pt-pt f(.), , , =(,,…,)12p ７ . , {x}. t f(.), {f(x)}t , y=f(x)AR tt y=y+y+...+y+, (2.8) t1t-12t-2pt-pt . , , , . , , , . , Box-Cox, , ,. , f(.) , (2.7). f(.), {f(x)}, t (2.7). f(.), -1f(.), (2.7) -1x=f(f(x)+f(x)+...+f(x)+), (2.9) t1t-12t-2pt-pt (2.4), . x=f(x)+f(x)+...+f(x)+, (2.10) t1t-12t-2pt-pt f(.), , . (2.2), . ８ x=I(x<0)x+I(x0)x+, (2.11) t1t-1t-12t-1t-1t I(.). , TAR(). : ,x，,,x,0,,1t,1tt,1x= t=1,2,… (2.12) ,t,x,,x0.，,2t,1tt,1, , (2.10)(2.11), . . . , {x}, , t , , . : (2.2), x=(x,x,…,x;)+, (2.13) tt-1t-2t-pt (…)p, =(,,…,). , . 12p , ,x11t, x=+, (2.14) tt 21，,x21t, =(,), : 12 -<<, 0<. , (2.13), , , , (…) ９ . , (…) , . , (2.1) , . . , , . 3. (2.3). , (2.3). : x=(x,x,…,x)+S(x,x,…,x), (2.15) tt-1t-2t-pt-1t-2t-qt (…)p, S(…)q. , . : (2.15)S(…), 2E=1; , (2.15), (…)S(…)t . ARCH: x=(x,x,…,x)+e, (2.16) tt-1t-2t-pt e=S(e,e,…,e), tt-1t-2t-qt 1/222S(e,e,…,e)={+e+…+e}. (2.17) t-1t-2t-q01t-1pt-p １０ ARCH, , . GARCH, . (2.12): ,，,x，...，,x，,,x,c,,1011t,11pt,p1tt,dx= (2.18) ,t,,x...,x,,xc,，，，，,2021t,12qt,q2tt,d, {}{}i.i.d., 1t2t 22~N(0,), ~N(0,), , (2.18), d11t11t1 , c, , . . : x={+x+…+x+}I(x<0) t1011t-11pt-p1tt-d +{+x+…+x+}I(x0) 2021t-12qt-q2tt-d ={+x+…+x}I(x<0) 1011t-11pt-pt-d +{+x+…+x}{I(x0) 2021t-12qt-qt-d +{I(x<0)+I(x0)}. (2.19) 1tt-d2tt-d , {}={}={}, 1t2tt x={+x+…+x}I(x<0) t1011t-11pt-pt-d +{+x+…+x}{I(x0) +, (2.20) 2021t-12qt-qt-dt , (2.10). (2.20), (2.10)f(.)(k=1,2,…,p+q+2), k . , (2.19) １１ . , (2.19) ? , , , (2.15)(2.16). . : x=(x,x,…,x)+S(x,x,…,x), tt-1t-2t-p1t-1t-2t-q1t + S(x,x,…,x), (2.21) 2t-1t-2t-q2t {}{}i.i.d., E=E=0, 1t2t1t1t 2222E= E=. , 1t12t2 . (2.19) (x,x,…,x), S(x,x,…,x) S(x,x,…,x)t-1t-2t-p1t-1t-2t-q2t-1t-2t-q . , {}{}, 1t2t , . . . , . . ---AR 1. {x; t=1,2,…}, , t . {x(t); t=(0,)}, １２ , . , , . , , . , , {x; t=1,2,…}. t , xt . . : , () , , , (). x=1t t; x=2tt . {x, x, …}. 12 12, , . . , . t, , x. t x=1, , x=1tt+1 2. x=1, x=2t+1t+1 . x, t+1 . , x=2t, . , １３ x, x, …, x, x, 12t-11x, …, x , x, , x2t-1tt . (1906). : P(x=1x=1)=? P(x=1x=2)=? t+1tt+1t : P(x=kx=j,x=j,…,x=j) t+1tt-1t-111 =P(x=kx=j), (3.1) t+1t P(x=kx=j)t(), t+1t P(x=kx=j)=p, j, k=1,2. (3.2) t+1tjk pjk. , jk (x=12), t p+p=1, p=1- p, j=1,2. (3.3) j1j2j2j1 , ppp1,p,,,,1112,, P=,,, (3.4) ,,,,,pp1,qq,,2122,, P{x}, t . (3.3), p= p, 11p=q. p(2211 ), p. 22 , , P{x}. t １４ . . , , . , , , . , (). , , {x}, tt ; x()t . , , . : , : １５ , 20--- , 50--- , 60--- , 70--- , (1975). . . : {x} t, , P(x0, (3.11) k=1k, mxm(1)R, Am(1) , Lebesgue, . . (3.11) m, R, A, A. , xA, x A. . , , . , , . x=1t; t １８ x=2; x=3; x=4ttt . {x, x, …}. 12 . , x=12, x, 1tx=12, 34; , x=34t1, x=34, 12. , t (3.11)x=1, A={3,4}, (3.11)0. . , , , . , ----, . : A,A,…,A, {x} 12dt P(xA)=1, xA, k=2,3,…,d, , kk-1 P(xA)=1, xA. , 1d {x}d. t , , A1A, AA,…, , AA, 223d1 . , AA, k-1k A, , k . , , {x}, td １９ {x}(k=1,2,…,d-1){x}. , td+ktd , d=1. . : {x}, t mCR, q, , , m P(xA) (A), xC, AR, (3.12)q, C{x}. t , . , , . , , , , . . . x, , 0 , , F. xF, 011 , (m=1) F(x)=P(x1, FF, 101 PP, 01 ２０ P(A)=P(xA)=P(y, A)P(dy). (3.13) 110 P(A)=P(xA)=P(y, A)P(dy), t=1,2,… (3.14) ttt-1 xP, 00 . , P, (3.13) P(A)=P(xA)=P(y,A)P(dy)=P(A)=P(xA), (3.15) 110 , Px1 P, P, , P1 , P. (3.15)(3.14), P(A)=P(xA)=P(y, A)P(dy) ttt-1 =P(y, A)P(dy). t=1,2,… (3.16) , , xt . : {x; t t=0,1,2,…}, , , , , . , . , . . ２１ : {x}t mP, xR, (3.13) P(x=x)=1, x, 00 xP(3.14)x, nn x lim||P-P||=0, (3.17) nn {x}; >1 t nx lim||P-P||=0, (3.18) nn {x}, t xx. (3.18)||P-P||(P-P), nn xPP. n x(P-P). , n xx ||P-P||=|P(dy)-P(dy)|. nn , , , =1-1=0; , . , x>0, P=P. n , , , x, nxn xP, P, n . , . , . : {x}, t ２２ , , C, (m)g, c>0, c>0, 12 (i) E{g(x)| x =x}g(x)-c, xC, nn-11 (ii) E{g(x)| x =x}c, xC, nn-12 , . 0<<1, ’(i)(i), (ii), ’(i) E{g(x)| x =x}g(x)-c, xC, nn-11 , . , . 2. AR p x=(x,x,…,x)+, t=1,2,… (3.19) tt-1t-2t-pt (1.6)(1.9), ,(,,...,)xxx,,t,1t,2t,p1,,,,,,x,,0t,1,, (x,x,…,x), U=, (3.20) t-1t-2t-p,,,,？？,,,,,,,,0x,,t,p，1,, (3.19) X=(X) +U, t=1,2,… (3.21) tt-1t p=1, (3.19) x=(x)+, t=1,2,… tt-1t P( x0. wk=1jkjkjk , (3.22). , , . , . . , , (41998). . 3.3. : x=(x,x,…,x)+S(x,x,…,x), (3.24) tt-1t-2t-pt-1t-2t-qt ,(…)S(…) t 2 (i) , E=0, E=1, ttt ２５ (ii) ||.|| 0<<1, c0, w (3.22), (iii) S(…), lim S(x)/||x||=0, (3.25) ||x|| , (3.24){X}. t 2. {}3.2(i). t 3.1. . AR(p) x=(x,x,…,x)+, tt-1t-2t-pt (…), K<, |(x)| 0, W=I, (3.22), . 3.3. . AR(p) x=x+x+…+x+, t1t-12t-2pt-pt , ２６ 2p1-u-u-…-u0, |u|1. (3.27) 12p (1.6)(1.7)(1.8), X=A X+ eU. tt-1t (1.7)A, (3.27), ||.||(3.22), 0<(A)<<1, w (A)A. . 3.4. . x=x+x+…+x+f(x,x,…,x)+, t1t-12t-2pt-pt-1t-2t-qt (3.27), f(…)(3.26), . . 3.5. .(Threshold AR---TAR) (2.12)(2.18), ,,，x，,...，x,，,if,,,x,c,,1011t,11pt,ptt,d1,,,，x，...,，x,，,ifc,x,c,,2021t,12pt,pt1t,d2x=(3.28) t ,？？？, ,,，,x，...，,x，,,ifc,x,,,s0s1t,1spt,pts,1t,d, {}, , t . (3.28) p=max||<1, (3.29) 1ksj=1kj (3.23), . 3.6. -ARCH: 1/2x=h, t=1,2,… (3.30) ttt 22h=+|x|+…+|x|, (3.31) t01t-1pt-p ２７ >0, 0, i=1,2,…,p, 0i 0<<1, (3.25), , 3.3(i)(ii), . , . -GARCHi , . 3.7. ARCH: 1/2x=h, t=1,2,… (3.32) ttt 22h=+x+…+x, (3.33) t01t-1pt-p >0, 0, i=1,2,…,p, 0i p<1, j=1j (3.22), ARCH(3.32) . , (4,1998). 3.23.3. . , ? ? , , . , . ２８ i.i.d.. , , , , . , , . , 果. . . 1. , . , , , . : , ? , ? , --- . GARCH, . : ２９ 如(1.6) 如(2.1) -- 如AR-ARCH模 如(2.15) 型 , , . , : ? ? ? , ---ARMA, , . , AR--ARCH, , . , . 2. ３０ , , , . . , . , . 2.1. : , ARCH, . ---LM. ARCH, , 1/2x=h, t=1,2,… (4.1) tt {}i.i.d., ~N(0,1), tt 22h=+x+…+x, (4.2) t01t-1pt-p >0, 0, i=1,2,…,p, 0i ==…==0, (4.1). 12p , (2.12){x}t, {e}i.i.d., , t ==…==0, (4.1), . 12p ARCH, . , ARCH, . , ３１ H (): ==…==0; 012p H(): ++…+>0, 112p x,x,…,x, H. 12n0 , . : x,x,…,x, 12n (x,x,…,x), HH, 12n01 , H. 0 , (x,x,…,x), , 12n , . =(x,x,…,x). n12n =(,,…,)=(,,…,,), 12p+112p0 x,x,…,x, (4.1)(4.2), 12n L(). H, n1 , : L()/ =0. (4.3) n . H, L0 , ==, , p+10 L()(4.3), . n00L , , L()n , , H 0 ==…==0, (4.4) 12p , L(), n ３２ , : p+1{L()+}/ =0, j=1,2,…,p+1, (4.5) nk=1kkj p+1{L()+}/ =0, j=1,2,…,p, (4.6) nk=1kkj . , , , 0L0L p+1, , 00L ; , H, 0 . L0Ln , . D()=(L()/, L()/,…,L()/), (4.6) nn1n2np+1 2I()=-E(L()/). (4.7) nn , (4.6)(4.7) 0L D() I() . n0Ln0L -1LM={D()} {I()}{D()}. (4.8) n0Ln0Ln0L =(x,x,…,x)=LM, (4.9) n12n , . , . , 2 H, , n; (4.10) 0np H, , n; (4.11) 1n ３３ 2(4.10): p, np(4.11): . , Hn0 H, . , 1n 2(0.05), p 2 P(>)=, p >H, H; H. n010 (4.10)(4.11), , . , . L()n, I()n. H, n0L0 2D()/n0, ; , H, n0Lnp1 n, . nn , , (4.3)(4.8) L(), , n ; , LM. , L(), , n , , LM, . 2.2. : LM, . LM, . ３４ : , HH, 01 , , HH, 01 HH, H. 100 , . , , , , , . , x,x,…,x12n , , , : H (): ; 0 H(): . 1 . , , . , , . . , . {x} t E{x x,x,…}(x,x,…), (4.12) tt-1t-2t-1t-2 e x - (x,x,…). (4.13) ttt-1t-2 ３５ E{ex,x,…} tt-1t-2 =E{ x - (x,x,…) x,x,…} tt-1t-2t-1t-2 =E{ x x,x,…}-E{(x,x,…) x,x,…} tt-1t-2t-1t-2t-1t-2 =(x,x,…) - (x,x,…) t-1t-2t-1t-2 =0. (4.14) , (4.13) x =(x,x,…)+e. (4.15) tt-1t-2t , (1.15), , AR(), . , , , , , AR(p). ARMA(p,q)AR(), , . , (4.15). , p , AR(p). AR(p), , . AR(1), , . , , AR(p) . AR() . ３６ x,x,…,xAR(1) 12n x =(x)+e, (4.16) tt-1t , . : H (): (x), (x)=ax+b; 0t-1t-1t-1 H(): (x). 1t-1 H, (x), (4.16) 0t-1 x=ax+b+e. (4.17) tt-1t LM, , (4.16) . . H, ab. 0 (4.16), |a|<1. Ex=Ex=, tt-1 Ex=E(ax+b+e)= aEx+b+Ee=aEx+b, tt-1tt-1tt-1 =a+b, (4.18) , (x-)=a(x-)+e, tt-1t 2E(x-)(x-)=E[a(x-)+e(x-)] tt-1t-1tt-1 2=aE(x-). (4.19) t-1 (4.14), E[e| x]=0, tt-1 E[e(x-)]=E{E[e(x-)]| x} tt-1tt-1t-1 ３７ =E{(x-)E[e|x]}=0. (4.20) ttt-1 22=E(x-)(x-), =Ee, (4.18)(4.19) ktt-ket a=, b=(1-a)=(). (4.21) 1/01/0 , ab, . 10 H. 1 (.), (4.21)ab, (x)=ax+b, u=x-(x), ttt-1 u=x-(x)=(x)-(x) +e ttt-1t-1t-1t, H, (x)(x), (x)-(x) 0. 1t-1t-1 H, (x)=(x), (x)-(x) =0, u=e. 0t-1t-1tt, (x)=(x)(=ax+b), H! (4.22) 0 , H H. 01 (4.22)(4.14), , E{ux}=0, H! tt-10 , , E{ux}=0cE{uI(x)=, >, n H, H; H. 010 AR(1). AR(p). , AR(p). , AR(p). , . . . x,x,…,xAR(p)12n , , , . , AR(p), , u=x-(x, x ,…,x), t=p+1,p+2,…,n. ttt-1t-2t-p ˆˆˆˆaaa(x, x ,…,x)= (x+x+…+x+b), t-1t-2t-pt-1t-2t-pp12 2n2ˆ,=(1/n-1){x-(x, x ,…,x)}. t=2tt-1t-2t-pe , AR(1)(ii)(4.24)(4.25), 1/2 2ˆ,(c)=(1/m)nke ４０ m{x-(x, x ,…,x)}I(x k=1,2,…,pnk , H, H; H. 010 , AR(). , . , n, AR(p), pnnn . , , , pn, n p3.116)}, (5.2) 1tt-d2tt-d ( x, x,…, x)(2.19), t-1t-2t-12 . n=1920, , n+1, n+2,… *x, , n+k ****x=( x, x,…, x), k=1,2,… (5.3) n+kn+k-1n+k-2n+k-12 ** * x= x, x= x,…, x= x. nnn-1n-1n-11n-11 **x, x,…, , n+1n+2 , 9. , 910 . (5.1)xt-d d=2, , (), ４８ . TAR(5.1), , . . . . x t 3 4 10 9 2 5 6 1 8 7 11 xt-1 1,2,3,…(x,x), (x,x), (x,x),… 122334 ４９ (1983,1986). . (1998). . Brockwell, P.J. and Davis, R.A.(1987). Time Series: theory andmethods. Springer, New York. Engle,R.F.(1982). Autoregressive conditional hete- roskedasticity with estimates of the variance of he United Kingdom inflation. Econometrica, 50, 987-1007. Fan, J. and Yao, Q.(2003). Non-linear Time Series Analysis. Spriner, New York. Meyn, S.P. and R.L. Tweedie(1996). Markov Chains and Stochastics Stability. Springer-Verlag. Tong, H.(197). Some comments on the Canadian lynx data-with discussion.J.Roy.Stat.Soc.,A140,432-468. Tong, H.(1990). Non-linear Time Series. Oxford Science Publications. Tong, H. and Lim, K.S.(1980). Threshold auto- regression, limitcycles and cyclical data(with discussion).J.Roy.Stast.Soc.,B42,245-292. ５０ : : 40, 50, 60, 70, 80, 90, 00 () 3, “”() Kalman(13) : 80, 90, 00 () 21, “GARCH”() : 60, 70, 80, 90, 00 () 22, “”() : 50, 60, 70, 80, 90, 00 () 6(“”) : (15,16,17) : (18,19) : () ５１ (, 6) 1010, , 6:00-9:00 1012, , 1:30-4:00 1014, , 8:00-11:00 1016, , 1:30-4:00 1020, , 6:00-9:00 1022, , 8:00-11:00 ５２