Носко В.П. - Эконометрика. Введение в регрессионный анализ рядов - 2002

.. 2002 1. 2. . ARMA 2.1. . 2.2. 2.3. 2.4. 2.5. ( ) 2.6. ARMA, 3. ARMA 3.1. ARMA 3.2. 3.3. 4. 4.1. 4.2. 4.3. 4.4. 5. 5.1. ARMA 5.2. TS DS 5.3. TS DS ARMA. . 6. TS DS 6.1. 6.2. 6.3. - 6.4. 6.5. DGP SM 6.6. . 6.7. 6.8. 6.8.1. 6.8.2. 6.8.3. . 6.8.4. DF-GLS 6.8.5. (KPSS) 6.8.6. ( ) 6.9. , TS DS 6.9.1. 6.9.2. 6.9.3. TS DS 6.9.4. 6.10. 6.10.1. 6.10.2. 7. 7.1. 7.2. . 7.3. . 7.4. 8. 8.1. 8.2. 1 , [(2000)], yt = 1 xt1+ 2 xt2+ + p xtp + t , t = 1, 2, , n , , xt1, xt2, , xtp , t =1,2,,n,, 1, 2,, n() , ( ). . ( ); ; . . : ,, , , , , t - ; , t - ; , F- ; , . , n , xt1, xt2, , xtp ,t = 1, 2, , n ; ( ) t , t =1,2,,n,, y1, y2, , yn , . , 1 , .,, n . , , , , , . ., () . , ., ,.. (,,,..). ,, , . yt = yt 1 + t , t = 1, 2, , n ,y 0 = 0, 1, 2,, n, 2., xtt , ..xt = yt 1. ,( ), , = ==ntttnttyy y22112. ,, t: x1 = 0 , x2 = y1 = 1 , x3 = y2 = y1 + 2 = 1 + 2 , x4 = y3 = y2 + 3 = ( 1 + 2 ) + 3 = 2 1 + 2 + 3, xn = yn 1 = n 2 1 + n 3 2 + + n 1 . , , t-F- , , t-F-, .1 , = 1, ( ), yt =yt 1 + t , t = 1, 2, , n ,y 0 = 0, () . , . [White(1958)], , . TS(trendstationary) ( )DS(differencestationary), . DS, ., ,, DickeyD.A.,FullerW.A,GrangerC.W.J.,HansenB.E.,JohansenS., JuseliusK.,PerronP.,PhillipsP.S.B.,SimsC.A.,StockJ.H.,WatsonM.W.J. [Maddala G.S., Kim In-Moo (1998)]; . [Hatanaka M. (1996)]. , , . , [ (2000)], ., -. , ,, ,, (,), . . , . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm1 1. , yt = 1xt1+ 2 xt2+ + p xtp + t , t = 1, 2, , n , , xt1, xt2, , xtp , t=1,2,,n,, 1, 2,, n() , ( ). . :, , , , , , t - ; , t - ; F- ; , . , n, xt1, xt2, , xtp ,t = 1, 2, , n ; ( ) t,t=1,2,,n,, y1, y2, , yn , ., , . . ,( ). , .. . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm2,- p : y= X+ , y= (y1, ..., yn)T - n, X(np)- n, n > p, = ( 1, 2, , p)T - , = ( 1, 2, , n)T - () n. Xp,XTX , (XTX) 1 , = (XTX) 1XTy . E() = E ((XTX)1XT(X + )) = E ((XTX)1XTX ) + E ((XTX)1XT ) = + E (XTX) 1XT ). X, E ((XTX) 1XT ) = (XTX) 1XT E ( ) = 0, E() = , .. . (, ) , E ((XTX) 1XT ) 0, E() , ,,, . , , , , ( ). A s () xt1, xt2, , xtp ts , 1, 2,, n, 2 > 0. ( t ~ i.i.d. N (0, 2). i.i.d. independent, identically distributed.) E ((XTX) 1XT ) = E ((XTX) 1XT) E( ) = 0, . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm3. () . X; ( ). , XX . , : | X~( ) ( )12,X X NT . , ( ) ( )12,X X NT X . , , . , j- : j | X~( ) ( )1 2,j jTjX X N , ( ) 1 jjTX X j- ( ) 1 X XT, 1 ) (j jTj jX X X~N (0, 1) . S2/2, S2 = RSS/(n p), RSS , - (n p) , S2/2| X~2(n p) . , t- H0:j= *j t = ( )2 21 *1 *) () ( SX XX X Sj jTj jj jTj j= . ,H0, t-t-(np) , t | X~t(n p) . X . ,X,t-H0: j = *j t(n p) . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm4 F- . A . A| X~ N(0, 2In) , In ( n n) . xt =(xt1,xt2,,xtp)Tpt- . B s () xt1, xt2, , xtp ts ; t , t ~ i.i.d., E( t) = 0, D( t) = 2 > 0E( t4) = 4< ; E(xt xtT) = Qt ,(1/n)( Q1 + + Qn) Qn , Q ; E(xit xjt xkt xst) < i, j, k, s ; (1/n)( x1 x1T + + xn xnT) Q . , ,A. n-() t

S2,t-F-, ,. , ,, , [Hamilton (1994)]. (n)n,Xn n, 2nS ,tn ,Fn-S2,t ,F,n. , B, n n ((n) ) N (0 , 2 Q 1), n(2nS 2 ) N (0, 4 4), tnN (0 , 1), qFn 2(q) , q . ,.. , , n , . (n) N ( , 2 Q 1 n),(n) N ( , 2 (XnT Xn) 1) , ( ), . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm5

2nS N (2 , (4 4) n),tnN (0, 1), qFn 2(q). Bn , t(n p) t- (N(0,1)) F(q, n p) F- ( 2(q) qFn), ( ,). , ,n tn Fn . C , , , t | X~ i.i.d. , X n- N (0 , 2 V) ; V nn. V,; V1 .(nn)-P,V1 =PTP.P, * = P . E(*)=0(X) * Cov (*| X ) = E (**T | X ) = E (P (P )T | X ) =P E ( T | X ) PT = P 2 V PT . V = (V 1) 1 = (PTP) 1 , Cov (*| X ) = P 2 V PT = 2 P(PTP) 1PT = 2 In . P y= X+ , : Py = PX+ P, y* = X*+ *, y* = Py ,X*= PX ,* = P . * | X~N (0 , 2 In) , , A. , , A, y* = X*+ * . , * = (X*TX*) 1 X*T y* = (XTPTPX) 1 XTPTPy = (XT V 1X) 1 XT V 1y . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm6 , ..E(*) = , ( X) Cov ( * | X ) = 2(X*TX*) 1 = 2(XT V 1X) 1. (GLS generalized least squares). y*=X*+* , t-F-. V, ,,V=V(), ,. , GLS *=(XTV1X)1XTV1y V=V(0)(V(0))V(n), n0., . ,, . , , . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm1 2. . ARMA 2.1. . (timeseries),. (, , ..), , x1, ..., xn t = 1, , n . ,x1,...,xn ,, X1,...,Xn, F(v1, v2, , vn) = P{ X1 < v1, X2 < v2, ... , Xn < vn }. , X1,...,Xn p( x1, x2, , xn). , , . . xt,t=1,,n,( ),m(m 0 () = 0 0. , t s Xt Xs , ts, . , Xt , X 1, ..., X n N(0, 2), , ..Xt ~ i.i.d. N(0, 2). .,,X1,..., Xn ,, ,.. Xt /( ). ,, - t s Xt Xs . (NOISE) D(Xt) 0.04. -0.6-0.4-0.20.00.20.40.60.850 100 150 200 250 300 350 400 450NOISE . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm4 , . ,, , . , , t . , ,,,, ()-1984(). -3-2-10123450 100 150 200 250DOW-JONES_TEMP ,, xt ( ), . 2.3. (). : Xt = a Xt 1 + t ,t = 1, , n, t , 2,X0,a0 . E(Xt) = a E(X t 1), E(Xt) = 0 t = 0, 1, , n. ,Xt = a X t 1 + t = a (a Xt 2 + t1) + t = a2 Xt2 + a t1 + t = = = a t X0 + a t 1 1 + a t2 2 + + t , Xt1 = a Xt2 + t1 = a t1 X0 + a t2 1 + a t3 2 + + t1 , Xt2 = a Xt3 + t2= a t2 X0 + a t3 1 + a t4 2 + + t2, X1 = a X0 + 1. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm5 X0 1, 2, , n, , Cov(X0, 1) = 0,Cov(X1, 2) = 0, , Cov(Xt2, t1) = 0, Cov(Xt1, t) = 0 D(Xt) = D(aXt1 + t) = a2D(Xt1) + D(t),t = 1, , n. , , D(X0) = D(Xt) = X2 t = 1, , n, : X2 = a2X2+ 2. a2 < 1, .. a < 1. X2 X2 = 2 (1 a2) . , Cov(Xt , Xt +) =Cov(a t X0 + a t1 1 + a t2 2 + + t ,a t+X0 + a t+1 1 + a t+2 2 + + t+) = = a2t+D(X0) + a (1 + a2 + + a2(t1))2 = = a [a2t2 (1 a2) + (1 a2t)2 (1 a2)] = [a (1 a2)]2, Corr(Xt , Xt +) = a , .. , . ,, Xt = a Xt1 + t ,t = 1, , n, , a < 1 ; X0 1, 2, , n ; E(X0) = 0 ; D(X0) = 2 (1 a2) . Corr(Xt , Xt +) = () = a . (), . yt E(Yt) = , , Xt =Yt : Yt = a (Yt1 ) + t ,t = 1, , n, Yt =aYt1 + + t ,t = 1, , n, = (1 a) . , . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm6Xt( ), , (1) = E(Xt Xt1) = E [(a Xt1 + t) Xt1] = a (0), (1) = (1) (0) = a , a>0,1, , . a < 0 ,,, . , Xt = aXt1 + t 2 = 0.2a = 0.8 ( ) a = 0.8 ( ). -1.0-0.50.00.51.01.550 100 150 200 250 300 350 400 450a = 0.8 -1.5-1.0-0.50.00.51.01.550 100 150 200 250 300 350 400 450a = - 0.8 . X0 =x0, Xt = a Xt1 + t Xt ,, , , Xt = a Xt1 + t a 2. ,t=0 ,x0 . Xt a < 1. . , t=1,x0 . Xt = a t x0 + a t1 1 + a t2 2 + + t , E(Xt) = a t x0 + a t1E(1) + a t2E(2) + + E(t) = a t x0 , D(Xt) = (a2(t1) + a2(t2) + + 1) 2 = [(1 a2t) (1 a2)]2 = 2 (1 a2) [a2t (1 a2)]2, Cov(Xt , Xt +) = Cov(Xt a t x0 , Xt + a t+ x0) = a (1 + a2 + + a2(t1))2

= a (1 a2t)2 (1 a2) , Xt , Cov(Xt , Xt +) t . , a < 1, t E(Xt) 0 ,D(Xt) 2 (1 a2), Cov(Xt , Xt +) a [2 (1 a2)], . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm7..t Xt ,Cov(Xt,Xt +), . , a < 1 , Xt = aXt1 + t X0 = x0 . Xt ,

100=+ =tkk tk tta x a X , a < 1 , , .~0==kk tkta X : ;~0 k tt kk tt ta x a X X=+ = t at x0 0. 0 2 22 = ==t kkt kk tka a E , tX~Xt;Xt tX~t.Xt tX~ , X0a . tX~

( ) ( ) , 0~0 0= = |.|

\|= ==kk tkkk tktE a a E X E D(tX~) = ( ) ,12202 20 0aa D a a Dkkkk tkkk tk= = = |.|

\| === Cov(tX~,~+ tX ) = ( ) ,12202 20 0aa E a a a a Ekk tkkk tkkk tk= |.|

\|=((

|.|

\||.|

\| == += tX~ ( ). , ,1 ~11 = =kk tktaaX ,~ ~01 tkk tkt tX a X a = = += .. tX~ .~ ~1 t t tX a X + = t 1~ tX , 2~ tX ,..., t 1~ tX ,2~ tX , , .. t . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm8()., tX~ ,, AR(1). xt,Xt=aXt1+t =0.2 a x0. -0.50.00.51.01.52.02.510 20 30 40 50 60 70 80 90 100Xx0=2; a=0.5-0.50.00.51.01.52.02.510 20 30 40 50 60 70 80 90 100Xx0=0;a1=0.5 -0.50.00.51.01.52.02.510 20 30 40 50 60 70 80 90 100Xx0=2; a=0.9-0.50.00.51.01.52.02.510 20 30 40 50 60 70 80 90 100Xx0=0;a=0.9

Xt=aXt1+t .p( AR(p)) Xt = a1 Xt1 + a2 Xt2 + + ap Xtp + t , ap 0, t D(t) = 2 . ,Cov(Xts,t)=0s>0;, t(), tt. , t p t , , , , t , Xt . L(lagoperator), L Xt = Xt1 ; . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm9 B (backshift operator). k,Lk, Lk Xt = Xtk . a1 Xt1 + a2 Xt2 + + ap Xtp (a1L + a2 L2 + + ap Lp) Xt , , p- , a(L) Xt = t , a(L) = 1 (a1L + a2 L2 + + ap Lp). ,, a(z) = 0 ()z1.( , AR(1) a(z) = 1 a z , a(z) = 0 z = 1/a , z > 1 a < 1. ) a(L) Xt = t Xt = ) (1L at= =0 jj t jb , ,0 1, ()a(z)=0. k(k) Ak,=1/zminzmin a(z)=0,, A k.A>0 , a1 , , ap . Xtk (k>0), AR(p),, (k) =a1 (k1)+ a2 (k2)+ + ap (kp) , k > 0 . (0), (k) =a1 (k1)+ a2 (k2)+ + ap (kp) , k > 0 . ,p,aj p, (.3.1 3.2). . AR(2)Xt = 4.375 + 0.25Xt1 0.125Xt2 + t . a(z) = 0 1 0.25z + 0.125z2 = 0,z2 2z + 8 = 0,z1,2 =1i7. ,. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm11 = / (1 a1 a2) = 4.375/(1 0.25 + 0.125) = 5, 5. .p = 2, (k) =0.25 (k1) 0.125 (k2), k > 0 . ,(0) = 1. (1) (1) = 0.25 (0) 0.125 (1) = 0.25 0.125 (1), :(1) = 0.25 / (1 + 0.125) = 2/9 = 0.222. : (2) = 0.25 (1) 0.125 (0) = 0.25 * 0.222 0.125 = 0.069, (3) = 0.045,(4) = 0.003,(5) = 0.005 .. , , . . -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0123456789 RHO 024681050 100 150 200 250 300 350 400 450AR(2) 2.4. q(MA(q)). , Xt =t + b1 t1 + b2 t2 + + bq tq , bq 0 , t . . , , Xt =t + b1 t1 + b2 t2 + + bq tq , .. Xt= + t + b1 t1 + b2 t2 + + bq tq . qMA(q) ( moving average). q = 0 = 0 . q = 1 , Xt = t + bt1 . D(Xt) = (1 + b2)2 , E [(Xt )(Xt1 )] = b2 , E [(Xt )(Xtk )] = 0, k > 1, . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm12 Xt E(Xt) = 0 , D(Xt) = (1 + b2)2 , (k) =( ) . 1 , 0, 1 ,, 0 , 122 2>== +kk bk b (k) = , 1 , 0, 1, ) 1 (, 0 , 12>= +=kk b bk ... . ,b>0,b0,,, MA(1)b ||.|

\|+=, , 0 , 0,2 0q kq k b bk jk qjj MA(q) , X2 = (1 + b12 + + bq2)2 k =+ + ==||.|

\|||.|

\| =+=., 2 1, , 0 , , 1, , 0,02 0KKq q kq k b b bqjj k jk qjj q (.. q). , , . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm13, , . , bj = ja ,0 < a < 1, , t

, MA() Xt = = 0 jja t j = =0 jj t jb , .0 p1 + q2 . , p q .(,aX(z)aY(z), Xt Yt , .) , AR(1), , ARMA(2, 1). . ,AR, ARMA . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm16,AR, MA., MA MA . ,,AR(p) ,, (.. MA(0)). ARMA(p, p). ,ARMA(p,q)Xt , AR(). , , AR(p), , . , ARMA, AR MA . , ,,, ( ) .ARMA . 2.6. ARMA, , ARMA,,, . (SAR(1)) Xt = a4 Xt4+ t , a4 < 1 (SMA(1)) Xt = t + b4 t4. (k) = a4k/4 k = 4m,m = 0, 1, 2, , (k) = 0 k > 0. (0) = 1,(4) = b4 ,(k) = 0 k > 0.SAR(1)a4=0.8 SMA(1) b4 = 0.8.-4-2024610 20 30 40 50 60 70 80 90 100X_SAR -4-2024610 20 30 40 50 60 70 80 90 100X_SMA

. . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm17 , , ARMA((1, 4), 1)

Xt = a1 Xt1+ a4 Xt4+ t + b1 t1

ARMA(1, (1,4)) Xt = a1 Xt1+ t + b1 t1+ b4 t4. C a1 =2/3,a4 = 1/48, b4 = 1/5 a1 = 0.4,b1 = 0.3, b4 = 0.8 . -4-202410 20 30 40 50 60 70 80 90 100ARMA((1, 4), 1)-4-202410 20 30 40 50 60 70 80 90 100ARMA(1, (1,4)) , a(z) = 0 1 2/3z + 1/48 z4 = 0, ..z4 32z + 48 = 0; z1 = 2,z2 = 2,z3 = 2 + i8,z4 = 2 i8, .a(z)=010.4z=0; z = 2.5 > 1, . , , , (1 a1L)Xt = (1+ b1L)(1+ b4L4) t , (1 a1L)(1 a4L4) Xt = (1+ b1L) t . Xt = a1 Xt1+ t + b1 t1+ b4 t4+ b1b4 t5 ,

Xt = a1 Xt1+ a4 Xt4 a1a4 Xt5+ t + b1 t1 . 14(..t1 t4), 14(..Xt1 Xt4 )., Xt = a1 Xt1+ t + b1 t1+ b4 t4+ b5 t5 , Xt = a1 Xt1+ a4 Xt4+ a5 Xt5+ t + b1 t1

cb5 = b1b4 , a5 = a1a4 . (,), , (parsimonymodel).,-, ( ), .ARMA,, [Enders (1995)]. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm18 . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm1 3. ARMA ,x1,x2,,xT

ARMA, , : 1. ; 2. ; 3. . ARMA, .. p q. , . a1, a2, , ap, b1, b2, , bq . . ,, X1,X2,, . (misspecificationtests). ,, ,, .. , . ,, , . ARMA(p,q), Xt ~ ARMA(p, q). , AR(p), Xt ~ AR(p), MA(q), Xt ~ MA(q). 3.1. ARMA ARMA (ACF) (PACF) (ACF autocorrelation function, PACF partial autocorrelation function) , ARMA. ARMA . (MA) . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm2(PACF)Xt.part(k)k Xt Xt+k , Xt+1 , , Xt+k1 . , part(k) Xtk Xt1,,Xtk , Xt .,(.,, [Hamilton(1994)]),part(k)ak k (s) =a1 (s1)+ a2 (s2)+ + ak (sk) , s = 1, 2, , k , (s1) a1+ (s2) a2 + + (sk) ak = (s) , s = 1, 2, , k , , a1 , a2 , , ak , (1k), , (k1) . kk,, PACF part(0) = 1,part(1) = (1), part(2) = 1 ) 1 () 1 ( 1) 2 ( ) 1 () 1 ( 1 = ) 1 ( 1) 1 ( ) 2 (22 , part(3) =1 ) 1 ( ) 2 () 1 ( 1 ) 1 () 2 ( ) 1 ( 1) 3 ( ) 1 ( ) 2 () 2 ( 1 ) 1 () 1 ( ) 1 ( 1 , K part(k) =1 ) 3 ( ) 2 ( ) 1 () 3 ( 1 ) 1 ( ) 2 () 2 ( ) 1 ( 1 ) 1 () 1 ( ) 2 ( ) 1 ( 1) ( ) 3 ( ) 2 ( ) 1 () 3 ( 1 ) 1 ( ) 2 () 2 ( ) 1 ( 1 ) 1 () 1 ( ) 2 ( ) 1 ( 1KM O M M MKKKKM O M M MKKK k k kkkkk k k k . part(k) , (1), (2), ...,(k). . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm3 , Xt AR(p), part(p) 0 , part(k) = 0k > p. PACF ARMA(p, q) q > 0. , ACF q MA(q). ,PACFpAR(p). ACFPACF ARMA(p, q) p 0,q 0. , (k)part(k) ACF, ( )( )( )( )( )( ), 1 ,..., 1 ,0 1 1121 = = ===+T kkxTx xk Tk rTttk Ttk t t

== =TttxTx11 = E(Xt) , ( ) ( )( )=+ =k Ttk t tx xk Tk1 1 (k),PACF, rpart(k)., part(k) (s) r(s). ,,part(k)Xtk Xt1,,Xtk , Xt., Xt = a1 Xt1 + a2 Xt2 + + ak Xtk + ut

( ut ut = Xt (a1 Xt1 + a2 Xt2 + + ak Xtk) , -). ak rpart(k) . XtARMA(p,q)( ) 0 (k)= a1k part(1) = a1 part(k) = 0, k 2 AR(1),a1 < 0 (k)= a1k part(1) = a1 part(k) = 0, k 2 AR(p) k p MA(1),b1 > 0 k = 1; k> 1 ; part(1) > 0 MA(1), b1 < 0 k = 1; k> 1 ; part(k) 0 1; (1) (a1+ b1) 1; part(1) = (1) ARMA(1, 1) a1 < 0 1; (1) (a1+ b1) 1; part(1) = (1); part(k) (1), k > 1 ARMA(p, q) , , . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm5 q p SAR(1) , ; , ; SMA(1) , ; , ; AR(p)MA(q) r(k)rpart(k), (k)part(k) r(k) rpart(k). , , (.., t ). Xt MA(q), n, ) ( 2 11)) ( (12q k jTk r Dqj>+ = , (k), k > q . , limT E(r(k)) = (k) . ,Tk>qr(k) ., H0 : Xt MA(q) , , ) ( 12) (12j rTk rqj=+ > k > q . 0.05. ,q=0,Xt~MA(0),H0:Xt . 0,2) ( > > kTk r (2) Xt AR(p), Tk > prpart(k) rpart(k) N (0, T 1)( D(rpart(k)) T 1 ). , H0: Xt ~ AR(p) ,,2) ( p kTk rpart> > , 0.05. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm6 , ACF PACF, 2/ T. , 0.95,r(k),Xt,rpart(k),Xt~AR(p).. , , 0.05, H0 k .,,? ,. T = 499 x1, x2, , x499. () k = 1, 2, , 36. ACF PACF ACF PACF ACF PACF -0.019 -0.0193 0.102 0.1265 -0.053 -0.031 -0.013 -0.0144 -0.071 -0.0516 -0.015 -0.018 -0.083 -0.0835 -0.044 -0.0367 -0.064 -0.035 0.038 0.0356 0.017 0.0348 0.032 0.042 -0.047 -0.0497 -0.083 -0.1159 -0.057 -0.075 0.017 0.0098 0.035 0.0280 -0.053 -0.044 -0.024 -0.0199 -0.049 -0.0851 0.011 -0.006 0.062 0.0530 0.069 0.0322 0.034 0.021 0.061 0.0691 0.041 0.0223 0.029 0.034 0 0.074 0.0732 -0.014 -0.0574 -0.042 -0.057 1 0.079 0.0993 -0.035 -0.0185 0.013 0.064 2 0.021 0.0344 0.034 0.0126 0.046 0.055 ACF,,2/T=0.0895 r(13)=0.102., H0:Xt?PACF, , , . (T=499), rpart(k), k = 1, 2, . Bk , , rpart(k) 2/T . 0.05. ( 36)rpart(k), k = 1, 2, , 36, P2 = C362 (0.05)2(1 0.05)362 = (3635/2) 0.0520.9534,. . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm7 lg(P2) = lg(630) + 2lg(0.05) + 34lg(0.95) = 0.560. :P2=0.275, PACF 36 . ACF, r(k), k = 1, 2, , Xt ( MA(q) q 1 ). ACF P1 = C361 (0.05)(1 0.05)35 , lg(P1) = 0.780, : P1 = 0.166. , ACFPACF. ,,r(k) EVIEWS, : ) ( k T k , T . , (k) . ACF PACFQ-, , .Q-. ( ) [Box, Pearce (1970)] ==Mkk r T Q12) ( . r(1), r(2), , r(M) , Xt , , TTr(k)N(0,1),Tr2(k)[N(0,1)]2=2(1). (,Xt.[(1974)].) , T Q ~2(M). H0. 0.05, Q > 20.95(M). P-Q M = 1, 2, . M H0, P- 0.05. ,, 2(M)T. [Ljung, Box (1979)] =+ =Mkk Tk rT T Q12) () () 2 (, . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm8(T)2(M), T , .EVIEWS(EconometricViews) P-, 2(M).Q- .P-(Prob)Q- . Prob Prob Prob 1 0.670 13 0.064 25 0.061 0.8734 0.0456 0.077 0.2925 0.0497 0.063 0.3486 0.0668 0.072 0.3497 0.0379 0.065 0.4558 0.0440 0.061 0.5399 0.0441 0.076 0.4380 0.0332 0.084 0.3601 0.0373 0.096 0 0.2432 0.0494 0.099 1 0.1463 0.0565 0.119 2 0.1874 0.0646 0.119 P-,M=14,15,1722,0.05, MH0:Xt, MP- , 0.05, H0 M . -,, ,M,. [Kwan (1996a,b)]. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm9, ARMA . , . , . . AR,.., Xt AR(k) =kjkja1(Xt j ) = t ,ak0 = 1, k , k [Akaike(1973)]. ,k, ) 2 ( ln ) AIC(2T k kk + = T , 2k t AR k- . 2k k- j skjj k sa 1== , j ka , j = 1, , k , j ka , x , t , ( ) x x aj tkjj k t ==0 , 2k 2k = =TttT121 . , ()k0 . , , ln 2k + k cT , cT = O(T 1 ln T)(..cT T , T 1 ln T ). SIC [Schwarz (1978)],TTkkln ln SIC2+ = . [Hannan, Quinn (1979)], cT = 2ck T 1lnlnT , c > 1, . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm10TT ckkln ln 2 ln HQ2+ = , k0T . T . AR(2) Xt = 1.2 Xt-1 0.36 Xt-2 + t . a(z) = 0 1 1.2 z + 0.36 z2 = 0z=5/3>1,, .C t = 1, 2, , 500 . -8-404850 100 150 200 250 300 350 400 450 500PRIM1 : ACFPACFAC PAC Q-StatProb .|******* .|******* 0.899 0.899 406.25.000 .|*********|.0.732 -0.396675.97.000 .|**** .|.0.561 -0.005834.98.000 .|*** .|.0.409 -0.027919.59.000 .|**.|.0.277 -0.048958.40.000 .|*.|.0.167 -0.015972.59.000 .|*.|* 0.095 0.071 977.15.000 .|. .|.0.045 -0.045978.19.000 .|. .|.0 0 9. . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm11.014.01178.29.000 .|. .|. 0 -0.001 0.020 978.29.000 .|. .|. 1 0.003 0.055 978.30.000 .|. .|. 2 0.019 0.001 978.49.000 2/T=0.089PACF, k=1,2., , H0: Xt ~ AR(2). ,AR4-,3-,2-1- ,AR .

p = 1 p = 2 p = 3 p = 4 IC 3.083264 2.91800 2.919244 2.924441 IC 3.100148 2.94336 2.953116 2.966846 AR(2). AR,, , ARMA(p0, q0) ( p0, q0) a(L) Xt = b(L) t , (p0,q0), (. [Kavalieris (1991)]). AR(k) , 1 ,00 = == k t j tkjj ka X a j ka , j = 1, , k, . 1 , ) ( 00 = == k j tkjj k ka x a t k ., .(, , .) Xt Xt j , j = 1, , p, Xt k (t j),j=1,,q. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm12 aj , .. j ka ,j = 1, , p, jb bj , j = 1, , q. , a(z), b(z) ,) ( , ) (0 0jqjjjpjjz b z b z a z a = == = t tx L a L b ) ( ) (1 = , ==Ttt q pT12 2,~1~ . , , ARMA , t tx L a L b ) ( ) (1 = . p0,q0 ( ) q p~,~, ( )TTq pq p q pln) (~ln~SIC2,2,+ + = . , ( )2,~SICq p pq , p p0 , q q0 , ( ) q p~,~. 3.2. ()ARMA,.. p,q ARMA(p,q),, ., . (,AR(1)), , () , . ,., (k)r(k) (k)., AR(p),a1,,app , , , 1), ( ) (1p k j k a kpjjK = == . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm13(1),,(p) ( ) r(1), , r(p) . MA(q) (q > 0) ,.2.5.MA(1) Xt = t + bt1 ,t = 1, , T . x1, x2, , xT , 1, 2, , T () 0 : 1 = X1 b0 ,2 = X2 b1 = X2 b(X1 b0) = (X2 ) b(X1 ) + b2 0, T = XT bT 1 = (XT ) b(XT 1 ) + b2(XT 2 ) +(1)T 1 b T 1 (X1 ) + (1)Tb T 0 . (b), x1, x2, , xT 0 , Q(b) = 12 + 22 + + T2, b . , ,,() b . , . b0, . , b < 1, b 1.,0=0. 0 , ,0=0, ,0, . MA(q) q >1:0=1== 1 + q =0. , ( EVIEWS) (backcasting), 0 , 1 , , 1 + q . , , . ARMA ,,[Hamilton (1994)]. , , . . AR(p) a(L) Xt = + t. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm14 , Xt .) 1 (2 1 pa a a =Ka1,,ap, Xt = + a1 Xt1 + a2 Xt2 + + ap Xtp + t ,, , ... , 1 pa a , .) 1 (2 1 pa a a =K,,, EVIEWS(EconometricViews),., Xt = (1 a1 a2 ap) + a1 Xt1 + a2 Xt2 + + ap Xtp + t a1, , ap . ., ,MA, (NLLSnonlinearleastsquares) . ( , ).9460.89560.4 9470.39570.2 9481.19580.6 9490.99590.9 9501.89600.3 9511.29610.7 9521.29620.6 953 11.4 9630.7 9541.29640.5 . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm159550.59650.9 :

10.010.511.011.512.046 48 50 52 54 56 58 60 62 64X , , Autocorrelation Partial Correlation AC PAC Q-StatProb .|*** .|***0.429 0.429 4.2694 0.039 .|*** .|**0.366 0.222 7.5350 0.023 .|. **|.0.059 -0.204 7.6255 0.054 .|. .|.0.016 -0.034 7.6324 0.106

*|. *|.-0.156 -0.129 8.3498 0.138 **|. **|.-0.255 -0.195 10.393 0.109

***|. *|.-0.321 -0.123 13.879 0.053 . *|. .|* -0.133 0.175 14.523 0.069 , 2/T = 2/20 = 0.447. 0.447 ., , MA(0) X0 = + t . , , ACF 1., AR(1)MA(1)., : , , .- , , , , .. . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm16, , ,- , . , 19401945(). 1940 1965 : 78910111240 42 44 46 48 50 52 54 56 58 60 62 64X 19421944.. 19401965. 1946 1965 .. AR(1),,,(1)=a1. (1)r(1)=0.429, a1., AR(1) , . Dependent Variable: X Method: Least Squares Sample(adjusted): 1947 1965 Included observations: 19 after adjusting endpoints Convergence achieved after 3 iterations VariableCoef. Std. Error t-Statistic Prob. C10.81451 0.159261 67.90445 0.0000 AR(1)0.430515 0.219257 1.963522 0.0662 R-squared0.184864 Meandependent var 10.81053 AdjustedR-squared 0.136915 S.D.dependent var 0.425434 S.E.of regression 0.395238 Akaikeinfo criterion 1.080645 Sumsquared resid 2.655627 Schwarz criterion 1.180060 a1 .MA(1),(1)=b1/(1+b12). (1)r(1)=0.429 b1/(1+b12)=0.429.0.5671.704. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm17MA(1); MA(1).MA(1) (backcasting) . Dependent Variable: X Method: Least Squares Sample: 1946 1965 Included observations: 20 Convergence achieved after 13 iterations Backcast: 1945 VariableCoef. Std. Error t-Statistic Prob. C10.81379 0.113355 95.39726 0.0000 MA(1)0.280610 0.228102 1.230195 0.2345 R-squared0.117961 Meandependent var 10.81000 AdjustedR-squared 0.068959 S.D.dependent var 0.414094 S.E.of regression 0.399561 Akaikeinfo criterion 1.097739 Sumsquared resid 2.873684 Schwarz criterion 1.197313 Log likelihood-8.977395 F-statistic2.407257 Durbin-Watson stat 1.789895 Prob(F-statistic)0.138178 b1 .P-t-F-b1. , ,20 . 1945 , : Dependent Variable: X Method: Least Squares Sample: 1946 1965 Included observations: 20 Convergence achieved after 24 iterations Backcast: OFF VariableCoefficient Std. Error t-Statistic Prob. C10.81515 0.112582 96.06431 0.0000 MA(1)0.274024 0.229231 1.195405 0.2474 R-squared0.116800 Meandependent var 10.81000 AdjustedR- 0.06 S.D.dependent0.4. . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm18squared7734var14094 S.E.of regression 0.399824 Akaikeinfo criterion 1.099054 Sumsquared resid 2.877464 Schwarz criterion 1.198628 Log likelihood-8.990543 F-statistic2.380443 Durbin-Watson stat 1.778286 Prob(F-statistic)0.140261 3.3. ,.., () . ,H0 , ,,t . ARMA(p, q) a(L) Xt = b(L) t , .. Xt = a1 Xt1 + + ap Xtp + t + b1 t1 + + bq tq , t tL b X L a ) () ( =, ppL a L a L a 1 ) ( 1 = K , qqL b L b L b 1 ) (1+ + + = K . MA ARMA(p, q) , ) () (L bL at = Xt , t a(L)b(L) ) ( L a) (L b , : t tXL bL a ) () ( = . ,,, - . t t .C,t , . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm19,[Bartlett(1946)][Box, Pierce(1970)] t ==+=Tttk Ttk t tk r12 1 ) ( ) (12k r T QMkBP == (Q- ). , QBP , 2(M p q), , TM, (M/T) . , QBP > 20.95(M p q).,T QBP2(Mpq). [Ljung, Box (1978)], , ( ) .2 2) 5 () (++ TM Mq p M Q EBP ,M(M+5)/(2T+2), 2(M p q) . . QBP2(E(QBP)) ,E(QBP)( ). D(r(k))(1/T) (T k)/(T2 + 2T). Q- ( )=+ =MkLBk TrT T Q12) 2 (, 2(M p q), QBP , T QLB 2(M p q), QBP . , ; M . , , t , , t . ,D(t).( , ,,. t,. t . .) . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm20EVIEWSARMA P- Q .

AR(1)A(1) . AR(1): ACF PACF AC PAC Q-Stat Prob

*|. *|. -0.096 -0.096 0.2033

.|** .|** 0.271 0.265 1.9334.164

*|. *|. -0.116 -0.078 2.2687.322

.|* .|. 0.076 -0.008 2.4232.489

*|. .|. -0.099 -0.049 2.7024.609

*|. *|. -0.125 -0.175 3.1852.671

**|. **|. -0.257 -0.260 5.3801.496

.|. .|. 0.019 0.051 5.3928.612

.|. |*0.047 0.189 5.4826.705

*|. **|.0 -0.178 -0.259 6.8945.648

.|. .|.1 0.040 -0.050 6.9748.728

*|. *|.2 -0.187 -0.132 8.9579.626 MA(1): ACF PACF AC PAC Q-StatProb.|* .|*0.100 0.100 0.2306 .|*** .|***0.365 0.358 3.4838.062 .|.

*|. -0.050 -0.127 3.5491.170 .|.

*|. 0.058 -0.068 3.6417.303 . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm21

*|.

*|. -0.139 -0.088 4.2051.379

*|.

*|. -0.169 -0.182 5.1071.403

*|.

**|. -0.277 -0.199 7.6991.261

*|. .|*-0.066 0.094 7.8603.345 .|. .|*-0.021 0.159 7.8772.446 . *|.

**|.0 -0.187 -0.302 9.4191.400 .|. .|. 1 0.002 -0.042 9.4192.493 . *|.

*|.2 -0.187 -0.104 11.338.415 P-QLB 0.05, , t ,., t :P-JarqueBera, , 0.480 0.608. . ,, (). ,t1,,tn nt tX X , ,1 K ., ., () . [Lomnicki (1961)]. ( )= =Ttkt kX XTm11, 3,2242 2 / 3231 = =mmGmmG . ,Xt ,TG1G2 . ) (24) ( , ) (6) ( 42 31kTG D kTG Dk k = == = . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm22, (k)r(k). ), (1G D ), (2G D

) (,) (222111G DGGG DGG = = , , . , T ( ) ( ) . ) 2 (22221 + G G,,T2G . ,( )., ( ) ( )2221 + G G ,AR MA,..Xt, . , Xt,, Xt.ARMA t,,, ,, N(0,2). , : ,24) ( ,6) (2 1TG DTG D = =(.. (k) = 0 k 0), ,24,62211GT GGT G = = ( ) ( )+ = + 24 622212221G GT G G,, Jarque Bera [Jarque, Bera (1980)]. ,JarqueBera ( ), , , , , . -,EVIEWS T(TK),K , . ,. , . ?. . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm23T(p,q) ARMA,, , . ,, ., , . MA(1)AR(1) ,,H0 :a1=0AR(1) H0 : b1 = 0 A(1) . , MA(0) Xt = + t . , : VariableCoef. Std. Error t-Statistic Prob. C10.81000 0.092594 116.7460 0.0000 R-squared0.000000 Meandependent var 10.81000 AdjustedR-squared 0.000000 S.D.dependent var 0.414094 S.E.of regression 0.414094 Akaikeinfo criterion 1.123258 Sumsquared resid 3.258000 Schwarz criterion 1.173045 Log likelihood-10.23258 Durbin-Watson stat 1.138735 , , AR(1)MA(1)., (1)part(1), ,, . ,, . , ,, . [Sargan,Bhargava(1983)]., 0.05 T= 21 1.069. , , MA(0). MA(0), MA(1) AR(1) . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm24MA(0) MA(1) AR(1) IC 1.123 1.098 1.081 IC 1.173 1.197 1.180 AR(1), MA(0). : , . .3.1 Xt=1.2Xt10.36Xt2+t., ,() .AR4,3,21AICSIC . . Dependent Variable: X Sample(adjusted): 3 500 Included observations: 498 after adjusting endpoints Convergence achieved after 3 iterations VariableCoef. Std. Error t-Statistic Prob. C0.001015 0.330958 0.003067 0.9976 AR(1)1.256580 0.041257 30.45723 0.0000 AR(2)-0.397095 0.041290 -9.617188 0.0000 ACF PACF AC PAC Q-StatProb .|.| .|.| -0.003 -0.0030.0042 .|.| .|.| -0.005 -0.0050.0165 .|.| .|.| 0.000 0.000 0.0165.898 .|.| .|.| 0.036 0.036 0.6866.709 . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm25 .|.| .|.| 0.037 0.037 1.3675.713

*|.|

*|.| -0.086 -0.0855.0736.280 .|.| .|.| 0.005 0.005 5.0882.405 .|.| .|.| -0.004 -0.0065.0977.531 .|.| .|.| -0.002 -0.0045.0993.648 .|.| .|.|0 -0.054 -0.0506.5887.582 .|.| .|.|1 -0.014 -0.0086.6897.669 .|.| .|.|2 0.019 0.011 6.8676.738 P- QLB 0.05, , t , . t(P- JarqueBera0.616).,Xt , AR(2). , Dependent Variable: X Sample(adjusted): 3 500 Included observations: 498 after adjusting endpoints Convergence achieved after 2 iterations VariableCoef. Std. Error t-Statistic Prob. AR(1)1.256581 0.041215 30.48807 0.0000 AR(2)-0.397096 0.041248 -9.627056 0.0000 S.E.of regression 1.036707 Akaikeinfo criterion 2.91398 Sumsquared resid 533.0816 Schwarz criterion 2.93089 InvertedAR Roots .63 -.05i .63+.05i . AR(2),AR(3)AR(4)P-, AR(2). Xt 1 Xt 2Xt 3Xt 4 AR(2)1.26 0.40 . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm26AR(3)1.25 0.39P = 0.87 AR(4)1.25 0.40P = 0.72P = 0.56 , , ,, .AR(2) . ,,. ,AR(2), , a(z) = 0 , .. 1 1.2 z + 0.36 z2 = 0 , z=5/31.67., ,.,, ,, , .0.630.05i, z=1.580.125i.,, ()a(z)=0, ,,AR(2) . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm1 4. ,,, . 4.1. 1,, , (A,A,B,C), . , . AR(p)yt = + a1 yt 1 + a2 yt 2 + + ap yt p + t , t ,D(t)=2. yt = xtT + t , xt = (1, yt 1, yt 2, , yt p)T , = ( , a1 , a2 , , ap)T . ,A B.s xt st, s = t 1, t 1 yt 1, xt . , A B.,AC. , XN(0, 2V)() V(A). xt+1=(1,yt,yt1,,ytp+1)T xt t . , AR(p) , : D yt yt = + a1 yt 1+ a2 yt 2 + + ap yt p + t (t ); . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm21a1za2 z2ap z p=0 ; t ~ i.i.d., E( t) = 0, D( t) = 2 > 0, E( t4) = 4 < . , n =(,a1,a2,,ap),n, n 1/2(n ) N(0, 2Q 1 ) , Q, yt. 2Q1 Sn2(XnTXn n)1 , , , nN(, Sn2(XnTXn )1).(Xnn.) , , . , ,, . . yt yt = + t + t , t~i.i.d.,E( t)=0,D( t)=2>0,E( t4)=4 0, E( t4) = 4 < . t-qF (q,FF-) N(0,1)2(q). , . , , , .

y= X+ ,X = Xn , , , yt = xtT + t ,t = 1, 2, , n , xt = (xt1, xt2, , xtp)T pt-, n =(1,2,,p)T , n.(.,,[Green(1997)]), n n : 01plim1= =tnttxn

[ : plim (n 1 XnT ) = 0]; Q x xnTtntt= =11plim [ :( ) Q X X nTn=1plim ] , Q ; ( ) Q N xntntt21, 01 = [ : (n 1/2 XnT ) N(0, 2Q)]. (plim; .),n , D,n ( n) N(0, 2Q 1 ). [Mann,Wald(1943)](-). . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm4 Q x xnTtntt= =11plim [:plim(n1 XnTX)=Q],Q , t ~ i.i.d., E( t) = 0, D( t) = 2 > 0, E t m < m = 1, 2, , E(xt t) = 0,t = 1, 2, , n , , n n . , E(xt t) = 0,t = 1, 2, , n , E( t) = 0, , Cov(xt k , t) = 0k= 1, 2, , p , ..t .E t m 0, E t m < m = 1, 2, , Cov(xt k , t) = 0k= 1, 2, , K . n n (n ) N(0, 2Q 1 ). ,xt (K-), E(xt) = = const ,Cov(xt) = Q ,Cov(xt k , xt +s, l) = kl (s) t, s k, l = 1, 2, , K. (kl (s) - k- l- xt , s . s, kl (s)s,kl (s)- k- l- xt .) ,F, . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm5Cov(xt)=Q xt xtT t = 1, 2, , n . F ARX : yt = a1 yt 1 + a2 yt 2 + + ap yt p + ztT + t , zt = (zt1, zt2, , ztM)T , = (1 , 2 , , M)T. F , xt = (yt 1, yt 2 , ,y t p, z1 , z2 , , z M)T , =(a1 , a2 , , ap , 1 , 2 , , M)T . : zt (M-) ; ,1plim1ZTtnttnQ z zn= = ZQ ; t ~ i.i.d., E( t) = 0, D( t) = 2 > 0, E t m < m = 1, 2, ; Cov(zt m , t) = 0m= 1, 2, , M ; Cov(yt j , t) = 0j= 1, 2, , p ; a(z) = 1 a1 z a2 z2 ap zp = 0 . (. [Green (1993)]) F, n n 1/2(n ) N(0, 2Q 1). ,a(z)=0, ARX., (..t) (long-run)yt,zt1,zt2,,ztM , . 4.2. ARX, ( ADL ) yt = 0 + a1 yt 1 + a2 yt 2 + + ap yt p ++ (10 x1, t + 11 x1, t 1 + + 1r x1, t r ) + + ++ (s 0 xs, t + s 1 xs, t 1 + + s r xs, t r) + t . ADL(p,r;s),p yt ,r x1, t , x2, t , ,xs, t ,yt,s .,. . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm6ij , xi, t . ADL(p,r; s) a(L) yt = + b1(L) x1, t + + bs(L) xs, t + t , a(L) = 1 a1 L a2 L 2 ap L p, bi(L) = i 0 + i 1L + + i r L r ,i = 1, , s . , yt ,) (1) () (1) () (1) (1, , 1 1 t t s s t tL ax L bL ax L bL a L ay + + + + = K ,) (1) ( ) () (1, , 1 1 t t s s t tL ax L c x L cL ay + + + + = K .) () () (L aL bL cii=, yt

L = 1, t 0. ; ) 1 ( ) 1 () 1 (1, , 1 1 t s s t tx c x cay + + + = K , t : . ) 1 ( ) 1 () 1 (11 1 s sx c x cay + + + = K 1(1),,s(1) (long-runmultipliers). ADL(1, 1; 1), (1 1L) yt = + 0 xt + 1 xt 1 + t . 1 < 1 ( ) ( )( ),1 11 11 1 01 1t t t tx xL Ly + ++= .. yt = (1 + 1 + 12 + ) + (1 + 1 L + 12 L2 + )(0 xt + 1 xt 1 + t), : yt xt = 0 ,yt +1 xt = yt xt 1 = 1 + 1 0 , yt +2 xt = yt xt 2 = 1 1 + 120 , yt +3 xt = yt xt 3 = 121 + 130 , . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm7 .. , ()xt yt., : yt xt + yt xt 1 + yt xt 2 + yt xt 3 + = = 0 (1 + 1 + 12 + ) + 1(1 + 1 + 12 + ) = = (1 1L) 1(0 + 1). ,, ADL(1,1;1). ,yt xt . ,ADL , . , ( . . 4.1, F): t- N(0,1). FF-q,qF 2 q . tqF ()(t-, F-). . t ,. , . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm8dd=2, ( t ). ADL(3, 2; 1) (1 0.5L 0.1L 2 0.05L 3) yt = 0.7 + (0.2 + 0.1 L + 0.05L 2) xt + t . y xL = 1 t 0: (1 0.5 0.1 0.05) y = 0.7 + (0.2 + 0.1 + 0.05) x , .. 0.35 y = 0.7 + 0.35 x , y = 2 + x . xt =0.7xt1 +xt ,xt ~i.i.d.N(0,1),yt, ADL(3,2;1),t~i.i.d.N(0,1),t xt . :x1 = 0, y1 = y2 = y3 = 0.-4-2024610 20 30 40 50 60 70 80 90 100Y X ,, .yt=+xt +t ; yt = 1.789 + 0.577xt + et , et . : -6-4-202410 20 30 40 50 60 70 80 90 100DELTA , ACFPACFC PAC Q-Stat Prob. . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm9 . |***** . |*****.696 0.696 49.981 .000 . |****. |* .536 0.099 79.868 .000 . |*** *| . .364 -0.081 93.801 .000 . |**.| . .227 -0.056 99.279 .000 . |*.| . .130 -0.015 101.10 .000 .| ..| . .057 -0.020 101.46 .000 ,P- 0.0000. , , . . xt ACFPACFACPAC Q-Stat Prob . |*****. |*****0.6860.68648.4680.000 . |****| . 0.429-0.07967.5940.000 . |* *| . 0.193-0.13271.5270.000 . | . *| . 0.024-0.06671.5910.000 *| . | . -0.0580.00371.9580.000 *| . *| . -0.140-0.10774.0900.000 xt AR(1). yt ACFPACFACPACQ-Stat Prob . |******. |******0.7670.767 60.58 0.000 . |***** . |* 0.6290.100 101.75 0.000

|****. | . 0.494-0.042 127.37 0.000 . |*** . | . 0.3990.019 144.32 0.000 . |**. | . 0.318-0.003 155.21 0.000 . |**. | . 0.2570.004 162.38 0.000 AR(1). ADL , : yt = + a1 yt 1 + 0 xt + 1 x t 1 + t . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm10 ADL(1, 1; 1), : Dependent Variable: Y Variable Coefficient Std. Error t-StatisticProb. C 0.5585880.1572763.55163 0.0006Y(-1)0.6952040.06609510.51828 0.0000X0.2089710.1261351.65673 0.1009X(-1)0.1616900.1323521.22166 0.2249(P- AR(1)0.164), t (P-JarqueBera= 0.267),(P-=0.159), , , , t-F-. H0:0 =1=0F-P-0.0032; 2(2)P-0.0022. .xt1, P-, xt , : Dependent Variable: Y VariableCoefficient Std. Errort-StatisticProb. C0.5178680.1540983.3606480.0011 Y(-1)0.7197380.06313111.400640.0000 X0.3103430.0952413.2585110.0015 R-squared0.637523Mean dependent var1.844751 Adjusted R-squared 0.629971S.D. dependent var1.709520 S.E. of regression1.039901Akaike info criterion2.945962 Sum squared resid103.8138Schwarz criterion3.024602 Log likelihood-142.8251 F-statistic84.42207 Durbin-Watson stat2.256404Prob(F-statistic)0.000000 , . x t 1, x t , Dependent Variable: Y VariableCoefficient Std. Errort-StatisticProb. C0.5678210.1586003.5802150.0005 Y(-1)0.6925230.06667310.386900.0000 X(-1)0.3059390.1005823.0416980.0030 R-squared0.632818Mean dependent var1.844751 Adjusted R-squared 0.625169S.D. dependent var1.709520 S.E. of regression1.046627Akaike info criterion2.958857 Sum squared resid105.1611Schwarz criterion3.037497 Log likelihood-143.4634 F-statistic82.72550 Durbin-Watson stat2.221594Prob(F-statistic)0.000000 . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm11 xt1,., . , (1 0.720 L) yt = 0.518 + 0.310 xt + et . L=1et 0,:0.28yt=0.518+0.310xt, y = 1.839 + 1.107x . , , . , ,ADL(3,2;1). , : Dependent Variable: Y VariableCoefficient Std. Errort-StatisticProb. C0.5396170.1740573.1002390.0026 Y(-1)0.5902930.1055835.5907950.0000 Y(-2)0.1539360.1205171.2773030.2048 Y(-3)-0.031297 0.099814-0.3135550.7546 X0.2055700.1294831.5876260.1159 X(-1)0.1919590.1536081.2496660.2147 X(-2)-0.024779 0.138389-0.1790530.8583 R-squared0.643818Mean dependent var1.882787 Adjusted R-squared 0.620072S.D. dependent var1.706160 S.E. of regression1.051648Akaike info criterion3.008022 Sum squared resid99.53667Schwarz criterion3.193826 Log likelihood-138.8891 F-statistic27.11327 Durbin-Watson stat1.999213Prob(F-statistic)0.000000 yx , , : y = 1.882 +1.300 x , ,, ., . 4.3. (VAR vector autoregression). y1ty2t : y1t = 1 + 11.1 y1, t 1 + 12.1 y2, t 1 + 1t , y2t = 2 + 21.1 y1, t 1 + 22.1 y2, t 1 + 2t , .. , , y1t y1,t1 ,y2,t1 y2t. 1t2t : Cov(jt , ls) = 0t s j , l = 1, 2 ; Cov(jt , yl, t r) = 0r 1 j , l = 1, 2 . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm12 , 1t2t

. y1t y2t . , , . p , VAR(p). ky1t,y2t,,ykt. p, y1t = 1 + 11.1 y1, t 1 + 11.2 y1, t 2+ + 11.p y1, t p + + 12.1 y2, t 1 + 12.2 y2, t 2 + + 12.p y2, t p + + + + 1k. 1 yk, t 1 + 1k. 2 yk, t 2+ + 1k.p yk, t p + 1t , y2t = 2 +21.1 y1, t 1 + 21.2 y1, t 2 + + 21.p y1, t p + + 22.1 y2, t 1 + 22.2 y2, t 2 + + 22.p y2, t p + + + + 2k.1 yk, t 1 + 2k.2 yk, t 2 + + 2k.p yk, t p + 2t , . . . yk t = k +k1.1 y1, t 1 + k1.2 y1, t 2+ + k1.p y1, t p + + k2.1 y2, t 1 + k2.2 y2, t 2 + + k2.p y2, t p + + + + kk. 1 yk, t 1 + kk. 2 yk, t 2+ + kk.p yk, t p + kt , ij.r - yj, t r yi t . 1t , 2t , , kt , Cov(jt , ls) = 0t s j , l = 1, , k ; Cov(jt , yl, t r) = 0r 1 j , l = 1, , k ; Cov(jt , lt) .C 1t , 2t , , kt t = (1t , 2t , , kt)T, y1t,y2t,,ykt . () Yt 1 = ( y1, t 1 , y1, t 2 , , y1, t p , , yk, t 1 , yk, t 2, , yk, t p) . VAR(1) (k = 2, p = 1): y1t = 0.6 + 0.7 y1, t 1 + 0.2 y2, t 1 + 1t , y2t = 0.4 + 0.2 y1, t 1 + 0.7 y2, t 1 + 2t . y1t, y2tt=2,3,,100. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm13y11 =y21=0;1t2t , N(0, 0.12).012345610 20 30 40 50 60 70 80 90 100Y1 Y2 (y1t y2t). 0.00.20.40.60.810 20 30 40 50 60 70 80 90 100Y1-Y2 ,: ., y1t

y2t 0.4 ( y1t y2t 0.403). y1t , y2t VAR . ,VAR ,,. VAR(p) k yt = + 1 yt 1 + 2 yt 2 + + p yt p + t . yt = (y1t , y2t , , ykt )T , = ( 1, 2, , k )T , t = (1t , 2t , , kt)T, r = ( ij.r ) k k y1, t r , y2, t r , ,yk, t r k. yt 1 yt 1 2 yt 2 p yt p = + t , (Ik 1 L 2 L2 p Lp) yt = + t , A(L) yt = + t , A(L) = Ik 1 L 2 L2 p Lp . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm14 VAR : k det(Ik z 1 z2 2 zp p) = 0 (.. det A(z) = 0) (.. k ). , , y1 , y2 , , yp. L = 1 0 =t . A(1) yt= , yt = A 1(1) . VAR(1) y1t = 0.6 + 0.7 y1, t 1 + 0.2 y2, t 1 + 1t , y2t = 0.4 + 0.2 y1, t 1 + 0.7 y2, t 1 + 2t . yt = + 1 yt 1 + t , =tttyyy21,=4 . 06 . 0 ,= 7 . 0 2 . 02 . 0 7 . 01,=ttt21 , A(L) yt = + t , 7 . 0 1 2 . 02 . 0 7 . 0 17 . 0 2 . 02 . 0 7 . 01 00 1) (1 2 == =L LL LL LL LL I L A , .6 44 6) 1 ( ,3 . 0 2 . 02 . 0 3 . 0) 1 (1==A A det A(z) = 0 , 07 . 0 1 2 . 02 . 0 7 . 0 1) ( det= =z zz zz A .. (1 0.7 z)2 (0.2 z)2 = 0 ,(1 0.9 z)(1 0.5 z) = 0 . z = 1/0.9z = 1/0.5 1, .. . () == =8 . 42 . 54 . 06 . 06 44 6) 1 (1 A yt. , y1t = 5.2,y2t = 4.8, . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm15 y1t y2t y1 y2 = 0.4 . ,,,y1t 5.2, y2t 4.8;(y1t y2t)0.4. VAR(1) . ,,, VAR, VAR, , (), (). ,-, , . , . , VAR . VAR A(L) yt = + B(L) xt + t , A(L) =I 1 L 2 L2 p Lp

B(L) .detA(z)=0 ,, yt = A 1(L) + (L) xt + A 1(L) t , (L)=A1(L)B(L)(transferfunction).(L); . (, , long-run ) , L = 1 t 0. : yt = A 1(1) + (1) xt . (1) . ( i , j)-cij(1)xjt yit (.. 4.2). VAR(1) VAR y1t = 0.6 + 0.7 y1, t 1 + 0.2 y2, t 1 + 0.1 x1, t 1 + 0.2 x2, t + 1t , y2t = 0.4 + 0.2 y1, t 1 + 0.7 y2, t 1 + 0.2 x1, t + 0.4 x2, t 1 + 2t . A(L) , , . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm16=+= + =LLL L B B L B4 . 0 2 . 02 . 0 0.1 4 . 0 00 1 . 00 2 . 02 . 0 0) (1 0, .4 . 0 2 . 02 . 0 1 . 0) 1 (1 0= + = B B B == =2 . 3 6 . 18 . 2 4 . 14 . 0 2 . 02 . 0 1 . 06 44 6) 1 ( ) 1 ( ) 1 (1B A C , +=21212 . 3 6 . 18 . 2 4 . 18 . 42 . 5xxyy, .. . 2 . 3 6 . 1 8 . 4, 8 . 2 4 . 1 2 . 52 1 22 1 1x x yx x y+ + =+ + = , x1, t x2, t AR(1) , x1, t = 0.7 x1, t 1 + 1t , x2, t = 0.5 x2, t 1 + 2t ; 1t2t~ i.i.d. N(0, 1). x11 =x21 = 0,y11 =y21 = 0 ( 1) x11 =x21 = 0,y11 =5.2,y21 = 4.8 ( 2). :0246810 20 30 40 50 60 70 80 90 100Y1 Y2Variant 1 123456710 20 30 40 50 60 70 80 90 100Y1 Y2Variant 2 ,- ,, . . VAR(1) : y1t = 0.8 y1, t 1 + 0.2 y2, t 1 + 1t , y2t = 0.2 y1, t 1 + 0.8 y2, t 1 + 2t . ,8 . 0 2 . 02 . 0 8 . 0212121+=ttttttyyL LL Lyy . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm17 =L LL LL A8 . 0 1 2 . 02 . 0 8 . 0 1) ( . =2 . 0 2 . 02 . 0 2 . 0) 1 ( A , , A 1(1) .

det A(z) = 0 (1 0.8 z)2 (0.2 z)2 = 0,..(1 z)(1 0.6 z) = 0.(1/0.6)1.,1, .? . -8-6-4-20210 20 30 40 50 60 70 80 90 100Y1 Y2 ., -y11 =y21 =0. , :y11 =y21 = 5. -20246810 20 30 40 50 60 70 80 90 100Y1 Y2 -, VAR . 4.4. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm18 , , ADL(1,1;1) yt = + a1 yt 1 + 0 xt + 1 x t 1 + t . ,, . =(a1 ,0 ,1). , .ADL(1,1;1) . 9 . (1) (a1 = 1 = 0): yt = + 0 xt + t . yt xt ; yt 1 x t 1 yt ., ,,,t . (2) (0 = 1 = 0): yt = + a1 yt 1 + t . yt yt1 ; xt t(t 1) yt . -, , yt . (3) (a1 = 0 = 0): yt = + 1 x t 1 + t . , yx .,, 1 . , , .,- y . (4) (a1 = 1, 1 = 0) yt = + 0 xt + t , (=1L,yt=ytyt 1,xt=xtxt 1), ,,( ). . . , . (5) (a1 = 0) yt = + 0 xt + 1 x t 1 + t

y. , , xtx t 1 . (6) (1 = 0) . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm19yt = + a1 yt 1 + 0 xt + t

x. , , . y*t = + xt y , yt =yt yt 1 yt yt 1 =(1 )( y*t yt 1) + t , 0 1, .. yt=(1 ) y*t + yt 1 + t , ,t ,yt y*ty. (, yt ,xt .) yt = yt 1 + (1 )( + xt yt 1) + t = (1 ) + yt 1 + (1 ) xt + t , yt = + a1 yt 1 + 0 xt + t , = (1 ) , a1 = ,0 = (1 ) . ,x t1 a1, a1 . (7) , (0 = 0):yt = + a1 yt 1 + 1 x t 1 + t . ,,xt=x t1 +ut. xt ADL(1,1;1) : yt = + a1 yt 1 + 0 xt + 1 x t 1 + t = + a1 yt 1 + (0 + 1) x t 1 + (t + 0 ut), yt= + a1 yt 1 + *1 x t 1 + *t . () 0 1,... , ( ADL(1,1;1)). (8) (1 = a10): yt = + a1 yt 1 + 0 xt a10 x t 1 + t . yt a1 yt 1 = (1 a1) + 0 (xt a1 x t 1) + t . , yt = + 0 xt + ut , ut = a1 ut 1 + t , a1 < 1. (9) ( a1 < 1,0 +1 = b(1 a1), b 0): yt = + 0 xt (1 a1)( yt 1 b x t 1) + t , yt = 0 xt (1 a1)( yt 1 a b x t 1) + t , a = (1 a1), b = ( 0 +1) (1 a1). . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm20 . y = a + b x ytxt yt 1 a b x t 1 . () ADL(1,1;1) yt = + a1 yt 1 + 0 xt + 1 x t 1 + t

yt yt 1 = (1 a1) yt 1 + 0 (xt x t 1) + (0 + 1) x t 1 + t . a1 < 1 ( ), yt = + 0 xt (1 a1)( yt 1 ((0 + 1)/(1 a1)) x t 1) + t , 0 + 1 0 . ,a1 0, Xt ()Xt1 = xt1 ,E( XtXt1 = xt1) = xt1,,xt1.,xt1>0, Xt = xt xt1 , xt1 r ; W(r) 1. , , , W(1) = W(1) W(0) ~ N(0, 1), [W(1)]2 ~ 2(1). , T { } { } { } { } 68 . 0 1 ) 1 ( 0 1 )] 1 ( [ 0 1 1 2 21 1= < = < < = < P W P a P a P , DGP ( ), DGP:xt = xt1 + t , SM: xt = a1 xt 1 + t 1 1 < a 2/3 . T(1 a 1)a1=1 T(); [Fuller(1976)]. , t~ N(0, 2),1 , , T x0 =0.,x0 T(1 a 1) , T(1 a 1) . , T(1 a 1), H0: a1 = 1 HA: a1 < 1 5% T(1 a 1), T(1 a 1), 1 a , 1 a , TT(1 a 1) 1 a 25 7.30.708 50 7.70.846 100 7.90.921 250 8.00.968 500 8.00.998 8.1 , T H0 1 a ,1. . , TH0:a1=1HA:a1

1 a = 0.604,t = 2.027 > t = 3.50, . ST_3 1 a = 0.733 >1 a = 0.604,t = 2.687 > t = 3.50. 1 a t,WALK_2, , . ,,ST_3,,DGP, 1 a t SM: xt = + t + a1 xt1 + t , DGP:xt = xt1 + t. , 1 a t , DGP:xt = + xt1 + t, 0, SM: xt = + t + a1 xt1 + t 1 a t DGP:xt = xt1 + t DGP:xt = + xt1 + t, 0. SM:xt =+ t+a1xt1 +tDGP:xt =xt1 +t ,WALK_1. ,(DGP:xt =0.2+xt1 +t ): 1 a =0.858>1 a =0.604,t=2.027>t=3.50, . , , [MacKinnon (1991)], t- . , , t(p, T) t-,p T , t(p, T) + 1T 1 + 2T 2 , ,1 ,2,p, . p = 0.01, 0.05, 0.10.. . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm8 5 1974.1985. : Xt 47962.75 315.1909 t = 0.884803 (Xt1 47962.75 315.1909 (t1)) + t, Xt = 5804.037 + 36.30898 t + 0.884803 Xt1 + t. ,,0.884803Xt1 (.. AR(1)), , . . , SM: xt = + t + a1 xt 1 + t H0:xt = +xt1 +t , EVIEWS, : Test Statistic -1.4252771% Critical Value*-4.1630 5% Critical Value-3.5066 10% Critical Value-3.1828 *MacKinnon critical values for rejection of hypothesis of a unit root. ,H0., , xt = + t + a1 xt 1 + t, xt = + xt 1 + t. xt = + t : Variable Coefficient Std. Errort-StatisticProb. 236.895880.809982.9315170.0052 Durbin-Watson stat 2.135959 xt = 236.8958 + xt 1 + t. SM: xt = a1 xt 1 + t SM: xt = + a1 x t 1 + t

SM: xt = + t + a1 xt 1 + t a1 t-a1=1 : DGP:xt = xt 1 + t ( ), . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm9DGP:xt = + xt 1 + t ( ). , , SM: xt = + a1 xt 1 + t , xt = xt 1 + t H0: = 0, a1 = 1. , F- ) 2 ) 1 ((2 ) (01 = T RSSRSS RSS, H0F- F(2, T3)T3.,, H0,,,, 1 H0 ( ) F(2,T3).[Dickey, Fuller(1981)].1 H0: = 0, a1 = 1. 5% 1 , ,()5% F , F(2, n 3) (., [Hamilton (1994), .7 Case 2] [Enders (1995), ]): T1 F 255.183.44 504.863.20 1004.713.10 2504.633.00 5004.613.00 4.593.00 WALK_1, ST_2, ST_1 ( 50 ). SM:xt = + a1 xt1 + t WALK_1, = 0.579, 1 a = 0.850, RSS = 48.0335. = 0, a1 = 1tx = xt1 , ( ) , 7939 . 52221 0= ==Ttt tx x RSS86 . 4 329 . 2) 2 1 50 /( 0335 . 482 / ) 0335 . 48 7939 . 52 (1< = = H0 . ST_2 : = 0.181, 1 a = 0.777, RSS = 52.6618. = 0, a1 = 1RSS0 = 59.0547, 86 . 4 853 . 2) 2 1 50 /( 6618 . 522 / ) 6618 . 52 0547 . 59 (1< = = H0 . ST_1 : = 0.042, 1 a = 0.785, RSS = 52.7007. = 0, a1 = 1RSS0 = 58.0671, 1 = 2.662 < 4.86 H0 . ,,t- a1 , , F-. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm10SM: xt = + t + a1 xt1 + t DGP: xt = xt1 + t H0: = = 0, a1 = 1 , DGP: xt = + xt1 + t H0: = 0, a1 = 1 . F-2;5% (. [Enders (1995), ]) T1 255.68 505.13 1004.88 2504.75 5004.71 4.68 F-3;5% (. [Hamilton (1994), .7 Case 4] [Enders (1995), ]): T1 257.24 506.73 1006.49 2506.34 5006.30 6.25

WALK_1, WALK_2, ST_3. SM: xt = + t + a1 xt1 + t H0: = = 0, a1 = 1 , 2; H0: = 0,a1 = 1 , 3 . H0: = = 0, a1 = 1 WALK_1: =0.854,=0.012, 1 a =0.858,RSS=46.7158. RSS0 = 52.7939, 2 = 1.995 < 5.13 H0 . WALK_2: =0.711,=0.040, 1 a =0.858,RSS=46.7158. RSS0 = 52.7939, 2 1.995 < 5.13 H0 . ST_3: = 0.345, = 0.070,1 a = 0.733,RSS = 50.3928, 2 = 1.207 < 5.13 H0 . H0: = 0, a1 = 1 WALK_1:RSS = 46.7158. RSS0 = 52.7282, 3 = 1.973 < 5.13 H0 . WALK_2:3 1.973 H0 . ST_3:RSS = 50.3928, 2 = 0.711 H0 . , . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm11 . ., 1 a1 < 0.8, 1 >1.23. , t- T(1 a 1). , H0 . , () . ,, ., . , () . :50 ST_1 ST_2 DGP:xt =xt1 + t, SM:xt = + a1 xt1 + t ; ST_3 DGP:xt = +xt1 + t(DGP:xt =xt1 + t) ,

SM:xt = + t + a1 xt1 + t . ,, ;T=100. (10% ): n = 50n = 100 ST_1 t = 2.298 > t = 2.92, t = 3.238 < t = 2.89, ST_2 t = 2.245 > t = 2.92, t = 3.217 < t = 2.89, ST_3 t = 2.687 > t = 3.18. 10% t = 3.207 < t = 3.15, 10% , . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm12 6.3. - (GNP) 1947 . 1961 . 5 Xt 217.740 5.222 t = 1.380 (Xt1 217.740 5.222(t1)) 0.630 (Xt2 217.740 5.222(t2)) + t, Xt= 55.017 + 1.304 t + 1.380 Xt1 0.630Xt2 + t . , ? . . SM:xt = + t + a1 xt1 + a2 xt2 + + ap xtp + t . (#)xt = + t + xt1 + (1 xt1 + + p1 xt p+1 ) + t , = a1 + a2 + + ap , j= (aj + 1 + + ap) . ( GNP xt = 55.017 + 1.304 t + 1.380 Xt1 0.630Xt1 + 0.630Xt1 0.630Xt2 + t = = 55.017 + 1.304 t + 0.750 Xt1 + 0.630 Xt1 + t .) , a(z) = 0 z= 1, p1, ,a1 +a2++ap=1,..=1.(., ,[Hamilton(1994)].), AR(p)H0:=1 (#). ,T( 1)t-=1, (augmented)(#)(, ). t- ADF (augmented Dickey Fuller), DF, AR(1). , t- = 1, (#) (# #) xt = + t + xt1 + (1 xt1 + + p-1 xt p+1 ) + t , = 1, H0: = 1 (#) H0: = 0 (# #). H0:=1(#)HA: 3.1744, 10%. , . P- F- 0.44. , , . -, (, ). , xt ARMA(p, q) q > 0 ? xt~ ARMA(p, q) , a(L) xt = b(L) t , a(L),b(L)pq,b(L), c(L) xt = t , c(L) xt = a(L) b(L) = 1 + 1L + 2L2 + . (##) (# # #)xt = + t + xt 1 + (1 xt 1 + 2 xt 2 + ) + t . ? [Said,Dickey(1984)],ARIMA(p,1,q) pq ARI(p*, 1) p* < 3T . (# # #) . 6.4. , ,[Dickey(1976)], [Fuller(1976)],[Dickey,Fuller(1979)],[Dickey,Fuller(1981)]. (),xt DS(DS-); TS(TS-). , (, ). , . (SM,statisticalmodel;DGP , data generating process). 1) tx ( ), SM: , , , 2 ,1T t t x xt t t = + + + = . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm15DGP:. 0 , , , 2 , = + = T t xt t t , .. SM t-tH0:=0. tcrit ,, DGP ( ). DS- , t < tcrit. ,,, [Fuller(1976)],[Fuller(1996)], T = 25, 50, 100, 250, 500. T, ,, [MacKinnon (1991)]. 2)xt( ) , SM: , , , 2 ,1T t x xt t t = + + = DGP: . , , 2 , T t xt t = = SM t-tH0:=0.tcrit ,, DGP( ).DS-,t 0) 25 2.972.61 50 2.892.56 100 2.862.54 250 2.842.53 500 2.832.52 2.832.52 =0,, = 0 DGP = 0. =0, H0: = 0 SM:xt = + xt1 + t, DGP:xt = + t 0 . t- = 0 , ,H0:=0 xt =+t 0.,T ,, . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm23,=0, . 5 ,4H0:=0, SM:xt = xt1 + t . t- H0: = 0 ( 1). , : H0: = 0 ; H0: = 0 xt = t. (, xt = 1 xt1 + + p1 xt p+1 + t.) ., , , . ., . . 1:tpjj t j t tx x t x +||.|

\| + + + = = 111 2:tpjj t j t tx x x +||.|

\| + + = = 111 3:tpjj t j tx x +||.|

\| + = =11 . 1959 1985 ( , 1982 .). . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm248001200160020002400280060 62 64 66 68 70 72 74 76 78 80 82 84DPI , , . . 1 SM:xt = + t + xt1 + 1 xt1 + + p1 xt p+1 + t. , SM:xt = + t + xt1 + t. (T = 26): xt = 461.338 + 25.857 t 0.448 xt1 + et ; t-H0:=0t=2.640.(5%) DGP:xt = + t , 0 , t=3.59(T=26).tt . 2 , DGP DGP:xt = + t + t. H0: = 0. = 2.680. (5%) (HA: > 0) 2.85 H0: = 0 . 3 SM:xt = + xt1 + t; DGP DGP:xt = t. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm25 H0: = 0. :xt = 47.069 + 0.00522 xt1 + et;t-t=0.335.(5%)t=2.98 . 4 SM:xt = + xt1 + t,DGP:xt = t,H0: = 0. :xt = 47.069 + 0.00522xt1 +et ;t-t=1.682 t = 1.95 H0: = 0 . : xt = xt1 + t . (,), : . H0: = 0 2. (, t- : = 2.680,t = 2.85.) 2 H0: = 0 , 0, H0: = 0 SM:xt = + t + xt1 + t,DGP:xt = + t + t, 0. t N(0, 1) , 5% t = 1.645. t = 2.640 (. 1), ,, : xt = 461.338 + 25.857 t 0.448 xt1 + t. 6.8. 6.8.1. ,[Phillips,Perron(1988)], xt DS H0 : = 0 SM: , , , 2 ,1T t u x t xt t t = + + + = ,, .,, ut ( ), . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm26()( , < C u Et>2)., , . t-H0 :=0, Zt , ut . Zt (long-run)ut , ( ) . ... lim211T2Tu u E T + + = tu () , , , 2 ,1T t u t x xt t t = + + + = ( )2 2 [Newey, West (1987)] ( ) ,11 2102 =((

+ + =jljlj + = =j tlj tt ju u T11j- ut . lT , , ( ) 04 / 1 T l , ( )2 2 (. [Phillips (1987)]) Zt t. , lNeweyWest(l windowsize).,.. ()., . ,- . (,, [Phillips,Perron(1988)],[Schwert(1989)])- l.[Schwert(1989)], l = [K(T/100)1/4], [a] a, K 4 12 . l,,EVIEWS,l= [4(T/100)2/9]([Newey,West(1994)]). ,l . Zt[Fuller(1976)] [acKinnon(1991)]. ,xtIMA(1,q), q l Newey-West. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm27q=1, 1 1 + = t t tb x ,b1>0- ,, .,b1 3.1744, 10%. , . P- F- 0.44. , , . , . [Newey,West(1994)]l= [4(T/100)2/9] = 3 ; PP Test Statistic-2.8711781% Critical Value*-4.1190 5% Critical Value-3.4862 10% Critical Value-3.1711 Lag truncation for Bartlett kernel: 3 ( Newey-West suggests: 3 ) . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm28Residual variance with no correction29.28903Residual variance with correction54.87482Phillips-Perron Test Equation Dependent Variable: D(GNP) Method: Least Squares Sample(adjusted): 1947:2 1961:4 Included observations: 59 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. GNP(-1)-0.1530240.072723 -2.1042120.0399 C38.3321115.97824 2.3990200.0198 @TREND(1947:1)0.8063260.378145 2.1323220.0374 , ,[Newey,West(1994)], , . ST_1, ST_2, ST_3: ST_1 ST_2 DGP:xt =xt1 + t, SM:xt = + a1 xt1 + t ; ST_3 DGP:xt = + xt1 + t(DGP:xt =xt1 + t) ,

SM: xt = + t + a1 xt1 + t . DF PP(l), , l, [Newey, West (1994)]. n = 50n = 100 ST_1 DF = 2.298 > t10% = 2.60, PP(3)= 2.394 > t10% = 2.60, 10% DF = 3.238 < t5% = 2.89, PP(4)= 3.399 < t5% = 2.89, 5% ST_2 DF = 2.387 > t10% = 2.60, PP(3)= 2.322 > t10% = 2.60, 10% DF = 3.217 < t5% = 2.89, PP(4) = 3.364 < t5% = 2.89, 5% ST_3DF = 2.687 > t10% = 3.18, PP(3) = 2.755 > t10% = 3.18, 10% DF = 3.207 < t10% = 3.15, PP(4) = 3.368 < t10% = 3.15, 10% ,DFPP, . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm29 6.8.2. [Leybourne(1995)] DFxt, ,DFmax . DFmax T = 25, 50, 100, 200, 400 () . (). , . . ST_3100 DF = 3.207 . 3.352.,3.207, 5% 3.45, . 5%() 3.15,ST_3 5% . 6.8.3. . [Schmidt, Phillips (1992)] DS ( ) t tw t x + + = , t t tw w1 + = , T t ,..., 2 = . ,( = 1 1) , . (DS)(TS) , . ,,lT1/2. . [Maddala,Kim(1998),.85]. . ST_3100, :3.12.,5% 3.06. 5% . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm306.8.4. DF-GLS,,, [Elliott, Rothenberg, Stock (1996)]. DF-GLS (. [Maddala, Kim (1998)]) a0 = 0 ydt= a0 ydt1+a1ydt1++ apydtp+error, ydt - ( . ). ,DF-GLS. 3.246,5%2.89. 5%,, -. 6.8.5. (KPSS) , [Kwiatkowski, Phillips, Schmidt, Shin (1992)], TS. = + + . , ,,, ., , DS . LM . -, , -. lNewey-West, l. , (. [Schwert (1989)]). KPSS [Maddala, Kim (1998), .120-122]. ST_3 100 KPSS l = 30.157., TS,, .,, :5%0.146,TS DS . , ,-DF-GLS, TS DS , . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm316.8.6. ( ) ,[Cochrane(1998)], 1VVVRkk =(VR variance ratio),( )k t t kx x DkV = 1 . xt,VRk= 1 , xt , ( ), VRk 0 k . T/(Tk+1) VRk . VRk k = 1,, K TSDS, . VRk : jkjkrkjVR11 2 11=|.|

\|+ + = , rj j xt = xt xt1 . ST_3, , TS DS , . .0.20.40.60.81.01.25 10 15 20 25 30 35 40VARRATIO TS . WALK_2 : . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm320.80.91.01.11.21.35 10 15 20 25 30 35 40VARRATIO ,WALK_2DS . 6.9. , TS DS 6.9.1. , .,, , .,( )., , , DS(., ,[Ghysels,Perron(1993)]),. -([Davidson,MacKinnon(1993)]). (dummy) D1,, D12()D1,,D4(). , . -[Dickey,Bell,Miller(1986)],, t . 6.9.2. ,, ,,, ., ,,, . , , [Shiller, Perron (1985)] [Perron (1989b)]. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm336.9.3. TS DS TSDS , TS, DS, : H0: TS H0: TS H0: DS 1 2 H0: DS 3 4 : 2 DS . 3 TS . 1 - . 4 (DGP) TS DS . 6.9.4. [Fuller(1976)][Dickey,Fuller(1981)] , ., . , , . , ,1. . 2. , , . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm34. :, ,. , , . ,, ,>1,[Dickey, Pantula(1987)] , ., ; 1 ... AR(2) a(L) xt = t , .. (1 a1L a2L2) xt = t , (1 aL)( 1 bL) xt = t ,a = 1/z1 , b = 1/z2 , z1, z2 a(z) = 0. , , | z1|, |z2| 1, , | a|, |b| 1. , xt , , : xt = (a + b) xt 1 abxt 2 + t . xt 1 : xt = (a + b 1) xt 1 abxt 2 + t . xt 1 : 2xt = xt xt 1 = xt 1 + (a + b 1) xt 1 abxt 2 + t = = (a + b 2) xt 1 + (1 ab) xt 2 + t . : 2xt = (a + b 2) xt 1 + [ (ab 1) xt 2 + (ab 1) xt 1] (ab 1) xt 1 + t = (a + b ab 1) xt 1 + (ab 1) xt 1+ t , 2xt = (a 1)(1 b) xt 1 + (ab 1) xt 1+ t . ,2 ,1. , a = b = 1( ), 2xt = t . a = 1, |b| < 1( ), 2xt = (b 1) xt 1+ t , 2xt = xt 1+ t < 0 . |a| < 1 |b| < 1( ), 2xt = xt 1 + xt 1+ t < 0 < 0. , , , . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm35, 2xt = + xt 1 + ut

t- (12,, =00).ut 2xt 1 , ... , 2xt p + 1 . (=0), 2xt = xt 1 + xt 1+ ut = 0 < 0. , xt , xt ~ I(1). AR(2) : -8-6-4-202410 20 30 40 50 60 70 80 90 100ROOT0 -16-12-8-40410 20 30 40 50 60 70 80 90 100ROOT1 . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm36-8-6-4-20210 20 30 40 50 60 70 80 90 100ROOT2

, . SM: 2xt = + xt 1+ t

= 0 < 0. ( .) ROOT2, 2(0), . T = 100 5% 2.89.t-1.64; .ROOT0ROOT1 ,1(a=0), . T=1005%1.95. t- 7.83 ROOT0 5.50 ROOT1; . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm37ROOT0 ROOT1. SM:2xt = xt 1 + xt 1+ t =0= 0 1BBtT tT tDMU; > = 0 BB BtT tT t T tDTS ; >= 0BBtT tT t tDT . : A. = 1, = = 0,d 0. B. = 1, = = 0, 0. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm40 C. = 1, = = 0,d 0, 0. A. < 1, 0, 0, d = 0 B. < 1, 0, 0, = 0 C. < 1, 0, 0,d = 0, 0 . . t- , =T/TB , (AO), ,(IO), .[Perron(1989a)] 2. ,[Perron, Vogelsang (1993)]. ,, ., ,([Perron (1989a)],, . zt , zt = a1 zt 1 + t , yt yt = f(t) + zt , f(t) = 0t TBf(t) = 0t > TB . E(zt) = 0, E(yt) = 0t TBE(yt) = t > TB . , TByt ( ). , AR(1) yt . , yt = f(t) + a1 yt 1 + t , a1 < 1, f(t) = 0t TBf(t) = (1 a1)t > TB , 0 . 2, . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm41TByt. ytTB? : yt = a1 yt 1 + (f(t) + t) = a1 yt 1 + t . t = TB + h k th TBkk h TBh TBa y y +=++ + = 101 0 = = ( )k th TBkk h TBk t f a y +=++ + ) (101 0 = = ( ) = +=+ +((

+101 1101 01 `hkkh TBkk tk h TBa a a y . () E(yt) = 0. h ( ) = = 101 11 limhkkha a . t = TByt , . f(t) t

(), ., . ,(), . , . :

-4048121610 20 30 40 50 60 70 80 90 100Y1_ADD Y1_INNOV : . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm42 02040608010010 20 30 40 50 60 70 80 90 100Y2_ADD Y2_INNOV : 02040608010012014010 20 30 40 50 60 70 80 90 100Y3_ADD Y3_INNOV : 02040608010012010 20 30 40 50 60 70 80 90 100Y4_ADD [Perron (1989a)]. -14 , . 1114,.. -. , . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm43,xt=M1,1, , . 1995:062000:07.Xt=M1 : 1000002000003000004000005000006000007000001996 1997 1998 1999 2000M1 (.[ (2001)]), . , 19981999.,-1998 ., 1998., , ()(AO ). ,TB, xt = + t + DTSt + ut , DTSt t TB t > TB 0 t.et. etet 1 e t 1 ,, et p : et = et 1 + =pjjc1 et j + t ; t-H0:=1 ,[Perron, Vogelsgang(1993),.249]). , .. . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm44TB=42,1998:08. 12 , .. (11)P--(AR(1) ), 0.0002 . ,GS() SIC,, (10% ) . SICP-val LM-.P-val White P-val J-B t- ( 12 ) 22.2361 0.983 2 0.967 0.7010.281-1.92 822.157-2.27 1122.089-2.60 1022.018-2.90 9*21.9861 0.590 2 0.844 3 0.954 0.3720.223-3.27 421.9740.040-2.78 521.9350.035-2.59 321.8980.016-2.22 1** ( GS)21.8370.0060.518-2.04 7 21.8340.0020.184-1.37 621.7930.008-1.31 2 ( SIC)21.7820.006-0.92 , . , 10% . (SIC), . P-(P-values)LM- -.,P-,() . P-(White) . P-Jarque-Bera . t-() -, ( ) . ( 10% ) 12 ,8,11,10,9(). ,,1712; , ., , 10% , , , . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm45, . , ,, , . = TB/T, TB ,,T .=42/62=0.667.5% () 3.94(=0.6)3.89(=0.7). . , 1998:08 1998. 1. , t- . 6.10.2. 14 , .[Zivot,Andrews(1992)] ,GNP 1973.( -). , (). ,(1964.),, 80-., 1973.,GNP., , (), ., , [Perron (1989a)], , , . ,ZivotAndrews (), , t-t H0: = 1; (tmin),t . ? ,,,(18901970). [Zivot, Andrews (1992)] (A) (. A, B, C),DTBt. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm46, GNP, 1929 . ().t t = 4.95, TB = 40, = 40/81= 0.49. (5%) t 3.76,. ,,ZivotAndrews (1929.),tmin=4.95.t-. tmin t :5%tmin5.26. tmin=4.95>5.26,(H0:=1) . .1114 .GNP() (19861970). (): , . , , , GNP, (kurtosis):5.68,4.658,4.324, , tmin.( . 3.3) () 5% :5.86,5.815.86(5.38,5.335.63, ). tmin 5.82, 5.30 5.61, , , . ,, ,D(t)=,tmin ,5% . [Perron (1997)], Zivot, Andrews, tmin, . (A)(C),(Zivot,Andrews) DTBt . 3 = ( 3), 0 . - , , ECONOMETRICVIEWS,() . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm47,[Perron(1997)], PERRON97 RATS. IO1 ,IO2, , AO , . : UR t- = 1; STUDABS t- , ( IO1) ( IO2); STUD t- ,(IO1) ( IO2); , . ( 1) , ,PERRON97 RATS,, [Perron(1997)]., , , . , (IO). PERRON97 : break date TB = 1999:07;statistic t(alpha=1) =-3.34124 critical values at1%5%10%for 70obs. -6.32 -5.59 -5.29 number of lag retained : 12 explained variable :M1 coefficient student CONSTANT124786.795613.33345 DU-2506239.31872 -3.77751 D(Tb)40455.794422.72347 TIME9769.037083.44839 DT23866.026863.78217 M1{1}-0.91050-1.59235 DUt =1 t>TBDUt = 0 t ; D(Tb)t =1 t=TB+1D(Tb)t = 0 t ; DT= tt>TBDTt=0 t ; (M1{1})t =M1t1. (, 6 . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm48: CONST, DU, D(Tb), TIME, DT M1{1}.) PERRON971999:07, t- t=1, . t=1 = 3.341, 5%5.59,. ,, 12 GS 10% . t-dDTt, ,1998:04.t=1=0.547,5% 5.33; . ( 11)., t- DT, ,1998:04 (UR-). ,( )(AO). PERRON97 : break date TB = 1999:02;statistic t(alpha=1) =-3.59417 critical values at1% 5%10% for 100 obs.-5.45 -4.83-4.48number of lag retained : 12 explained variable :M1 coefficientstudent CONSTANT104939.65455 20.48279 TIME4832.56930 26.73200 DT14335.07564 21.11189 M1{1}-0.75752 -1.54915 (, . CONST,TIME,DT; et . et et1 et 1,, et p ). t=1 1et1 . 1999:02,t=1=3.594(12 ), 5% 4.83, UR- . ,, : 1.626 ,3.[Zivot,Andrews(1992)]( ), , , UR- . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm4911995:062000:07, [(2001)]. . () () DSTS - () - DF-GLS KPSS DS ( ) ( ) , ,:DS-, TS-; DS-. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm1 7. 7.1. (,spurious) ., [ (2000)]. 19571966 :E- ( -),C - ( . )H- ( ). : . - .( ) 195734.9 716478 195835.9 724478 195937.9 797478 196041.1 844481 196143.5 881483 196246.7 946493 196348.91011520 196452.01083528 196556.11157528 196662.61249534 : 02004006008001000120014001 3 5 7 9 . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm2,, . , , E H RC H R= + == + =26255 7131 0 900129 30 0 350 087122. . . ;. . , . ; .,

. 993 . 0, 0498 . 0 860 . 0; 993 . 0 , 950 . 19 90 . 23 22= + == + =R E CR C E (, , , , ,19 950 0 0498 0 993 . . . = R2 .) , ,, ,., , ,0.9 . ,, (,-spurious) ., ( )- .,,, R2 yx : R ryx2 2=. , y x . rCov y xVar y Var xyx =( , )( ) ( ) , ( )( ), ) , (111= =nii i nx x y y x y Cov ( ) , ) (1211= =nii ny y y Var( ) . ) (1211= =nii nx x x Var Var x ( ) Var y ( ) , yxr 1, Cov y x ( , ) > 0 . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm3y yi x xi y x ,, . ( ,, .,, ,,,, .) , yxr 1, Cov y x ( , ) < 0 . y yi x xi y x ,, ,.(, ,,,, , .) , , . 1t 2t, ,N(0,1). -3-2-101235 10 15 20 25 30 35 40 45 50EPS_1 -3-2-101235 10 15 20 25 30 35 40 45 50EPS_2 DGPDGP : xt = 1 + 0.2 t + 1t , yt = 2 + 0.4 t + 2t , SM:yt = + xt + t . xt yt . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm405101520255 10 15 20 25 30 35 40 45 50X Y . : Dependent Variable: Y Sample: 1 50 Included observations: 50 VariableCoefficientStd. Errort-StatisticProb. C1.5538660.6857712.2658680.0280 X1.8002550.10299717.478780.0000 R-squared0.864218Mean dependent var12.22809 Adjusted R-squared0.861389S.D. dependent var5.925326 S.E. of regression2.206028Akaike info criterion4.459442 Sum squared resid233.5948Schwarz criterion4.535923 Log likelihood-109.4860F-statistic305.5076 Durbin-Watson stat2.150060Prob(F-statistic)0.000000 , , . . : Dependent Variable: Y VariableCoefficientStd. Errort-StatisticProb. C2.0374500.2948616.9098790.0000 T0.4122320.02805514.693940.0000 X-0.0541860.133658-0.4054100.6870 R-squared0.975727Mean dependent var12.22809 Adjusted R-squared0.974694S.D. dependent var5.925326 S.E. of regression0.942598Akaike info criterion2.777771 Sum squared resid41.75908Schwarz criterion2.892492 Log likelihood-66.44428F-statistic944.6386 Durbin-Watson stat2.249075Prob(F-statistic)0.000000 , . ,xt, . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm5-, yt .xt Dependent Variable: Y VariableCoefficientStd. Errort-StatisticProb. C1.9900200.2682917.4174030.0000 T0.4014930.00915743.847270.0000 R-squared0.975642Mean dependent var12.22809 Adjusted R-squared0.975134S.D. dependent var5.925326 S.E. of regression0.934357Akaike info criterion2.741262 Sum squared resid41.90511Schwarz criterion2.817743 Log likelihood-66.53155F-statistic1922.583 Durbin-Watson stat2.249658Prob(F-statistic)0.000000 ., yt = + xt + t , , SM233.59, 41.91. ,yt xt .

,DGP . () . ,, . . DGP: xt = xt 1 + 1t , yt= yt 1 + 2t , 1t 2t , . -20-15-10-5051010 20 30 40 50 60 70 80 90 100X Y . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm6, , 50(51100). SM:yt = + xt + t : Dependent Variable: Y Sample: 51 100 Included observations: 50 VariableCoefficientStd. Errort-StatisticProb. C8.6164960.74827711.515120.0000 X0.5975130.0775207.7078730.0000 R-squared0.553120Mean dependent var3.404232 Adjusted R-squared0.543810S.D. dependent var3.354003 S.E. of regression2.265356Akaike info criterion4.512519 Sum squared resid246.3283Schwarz criterion4.589000 Log likelihood-110.8130F-statistic59.41131 Durbin-Watson stat0.213611Prob(F-statistic)0.000000 , DGP yt xt , 0.553.,, : -20-15-10-5051055 60 65 70 75 80 85 90 95 100X Y ,,100, : Dependent Variable: Y Sample: 1 100 Included observations: 100 VariableCoefficientStd. Errort-StatisticProb. C1.4902060.6645382.2424700.0272 X0.0550970.0839780.6560860.5133 R-squared0.004373Mean dependent var1.120548 Adjusted R-squared-0.005786S.D. dependent var3.513463 S.E. of regression3.523613Akaike info criterion5.376648 Sum squared resid1216.753Schwarz criterion5.428752 Log likelihood-266.8324F-statistic0.430449 Durbin-Watson stat0.061638Prob(F-statistic)0.513306 . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm7, xt0.05510.5975, 51100. DGP , , xt ,. (0.214 0.062 ). , , ( !) . . DGP:xt =xt1 +1t,yt=yt1 +2t,x0 =0,y0=0,1t2t ,1t~ N(0, 12),2t~ N(0, 22), Cov(xt , yt) = 0.,T(xt ,yt),t=1,2,,T, SM: yt= xt + ut , ut~ i.i.d. N(0, u2),Cov(xt , ut) = 0. |.|

\||.|

\|= = =TttTtt t Tx x y121. DGP, T T, . ,SM, ( DGP !) : Cov(xt , yt) = Cov(xt , xt + ut) = Cov(xt , xt ) = D(xt), .. = Cov(xt , yt ) / D(xt). ( DGP)Cov(xt , yt ) = 0, = 0, ( SM) , T 0 . ,Tt-tH0:=0,t- , .. ,xt yt ., t, ( ) T / 1 t , . (DW), T DW 0 , . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm8 . , . DGP: xt = xt 1 + 1t , yt= yt 1 + 2t , 1t 2t, ,N(0,1), SM:yt = + xt + t. 51100 : T = 0.598, t = 7.708,DW = 0.214. -6-4-202455 60 65 70 75 80 85 90 95 100RESIDS . , ,, ( , .)., , , , , ( ). , . , , , ., . () SM:yt = + xt + t 51 100 yt = 8.616 +0.598 xt + e t. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm9, , .. e t= e t 1 + t

t-t H0:=0, . H0 HA: < 0 ( ),t 0. (II) AR(p) (p < ), ECM ( ) ,, 1111 1 1 1 1 tpjj t j j t j t ty x z x + + + + = = ( ) ,, 2112 2 1 2 2 tpjj t j j t j t ty x z y + + + + = = : xt,yt~I(1),ECM . (xt,yt)T ~I(1)((xt,yt)T

) ECM , xtyt. (,ECM,zt1, ; zt 1.) . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm18xt,yt~I(1),VAR .(,xtyt .) zt = yt xt , = E(yt xt), ,t, ytxt=0.xtyt

zt 1 . zt , , , , . 1 xt yt, xt yt . yt xt, yt xt .( .) xt,yt~I(1), ,,. ECM, 12 + 22 > 0.xt 1 zt 1 yt(.. xt yt), 2 0.yt 1 zt 1 xt(.. yt xt),

1 0. 2 xt , yt ~ I(1) wt ~ I(0). k xt yt k + wt , 0., xt ~ I(1), xt

xtk.(,xt xtk =xt +xt1 ++xtk I(0)-, I(0)-.) , xt , yt ~ I(1) () yt = + xt ; ECM, . ,,,ECM (, ).,. , . [Engle,Granger(1987)], yt

xt yt = + xt + ut . (K-), . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm19tz = yt xt

. ,,( !) ( ) , 111 1 1 1 1 tpjj t j j t j t ty x z x + + + + = = ( ) , 112 2 1 2 2 tpjj t j j t j t tw y x z y + + + + = = (.. VAR(p) xt , yt). , ( ) (1,)T .( .),,, ECM, 1 tz , , , 1 tz. ( .)- . ,, tz , . , (, 1)T , xt yt , , , . ,(), , ., ,, . , . , (, ). DGP: xt = xt 1 + t , yt= 2 xt+ t , x1=0,t t ,, . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm20N(0,1).xtyt -40-30-20-1001010 20 30 40 50 60 70 80 90 100Y X (xt , yt) xt =xt 1 + t , yt= 2 xt 1 + t , t = t + 2t~ i.i.d. N(0, 5). ECM xt = t , yt = (yt 1 2 xt 1) + t = zt + t , zt= yt 2 xt , xt = 1 zt 1 + t , yt = 2 zt 1 + t , 1 = 0,2 = 1, 12 + 22 > 0. ,, ,VARDGP., ECM xt = 1 zt 1 + 11xt 1 + 11yt 1 + vt , yt = 2 zt 1 + 21xt 1 + 21yt 1 + wt , , (p = 2). 100 . (I ) yt = + xt + ut . : Dependent Variable: Y VariableCoefficientStd. Errort-StatisticProb. C-0.0067640.165007-0.0409920.9674 X1.9833730.02085295.116540.0000 R-squared0.989284Durbin-Watson stat2.217786 ..yt = 0.006764 + 1.983373 xt+ tu , tz

=tu =yt + 0.006764 1.983373 xt . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm21, VAR 2, ytxt : tz= 1 tz + 11 tz + t . , : Augmented Dickey-Fuller Test Equation Dependent Variable: D(Z) Sample(adjusted): 3 100 Included observations: 98 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. Z(-1)-1.1535150.151497-7.6140880.0000 D(Z(-1))0.0381560.1001900.3808370.7042 t = 7.614 5% 3.396(.[Patterson(2000),8.7]). .( ,, .t =11.423, .) ,ytxt , . ( II) xt : Dependent Variable: D(X) VariableCoefficientStd. Errort-StatisticProb. C-0.0280160.100847-0.2778100.7818 Z(-1)0.2509420.1766131.4208580.1587 D(X(-1))0.6399670.2578232.4822010.0148 D(Y(-1))-0.2587400.116654-2.2180190.0290 P- Dependent Variable: D(X) VariableCoefficientStd. Errort-StatisticProb. D(X(-1))0.1151410.1002491.1485540.2536 , , xt = t , xt . yt , Dependent Variable: D(Y) VariableCoefficientStd. Errort-StatisticProb. C-0.0601010.211899-0.2836300.7773 Z(-1)-0.6410600.371097-1.7274720.0874 D(X(-1))1.3138720.5417332.4253110.0172 D(Y(-1))-0.4829810.245111-1.9704590.0517 . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm22 , : Dependent Variable: D(Y) VariableCoefficientStd. Errort-StatisticProb. Z(-1)-0.6388880.369218-1.7303810.0868 D(X(-1))1.3177630.5389322.4451380.0163 D(Y(-1))-0.4837220.243908-1.9832170.0502 1 tz , ytxt . , ECM 12 + 22 > 0. zt1 xt, yt .yt 1 , Dependent Variable: D(Y) VariableCoefficientStd. Errort-StatisticProb. Z(-1)-1.1864110.248876-4.7670720.0000 D(X(-1))0.3314110.2107321.5726710.1191 xt 1 , yt = 2 1 tz

+ wt, , Dependent Variable: D(Y) VariableCoefficientStd. Errort-StatisticProb. Z(-1)-1.2735840.247887-5.1377600.0000 H0: 2 = 1 : Null Hypothesis:C(1)= -1 F-statistic1.218077Probability0.272441 Chi-square1.218077Probability0.269738 , ECMxt = t , yt = 1 tz + wt , 1 tz =yt 1 + 0.006764 1.983373 xt 1 . 1 tz yt yt = 0.0068 + 1.983 xt 1 + wt , yt= 2 xt 1 + t , DGP. ,,wt=yt + 1 tz 4.62(DGP5.00),t=xt

1.04( DGP 1.00). ECM xt = t , yt = 1 tz + wt , . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm23 , yt : 1 tz , .. yt 1 ( 0.0068 + 1.983 xt 1) > 0, yt 1 tz yt ., 1 tz yt . xt 1 tz yt, .. xt yt . , yt xt , yt xt . , ECMCov(vt, wt) 0, , . DGP: xt = xt 1 + t , yt= 2 xt+ t , , . ( ) yt= xt+ t , xt = xt 1 + t ,(t,t)T~i.i.d.N2(0,), , .( .) , Cov(t , t) = 0, xt , . Cov(t , t) 0, xt ,..Cov(xt,t)=Cov(xt1 +t,t)0. . , . Ny1t,,yNt , 1. = (1, ... , N)T , , 1 y1t + ... + N yN t ~ I(0) , ,(); . c = E(1 y1t + ... + N yN t), . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm241 y1t + ... + N yN t = c . t , zt = 1 y1t + ... + N yN t c . zt , ,, , , .. . y1t,,yNt N1 zt . , , , y1t= 1 + 2 y2 t + ... + N yN t + ut , tu = y1t (1 + 2 y2 t + ... + NyN t), ., [MacKinnon (1991)] (. [Patterson (2000), A8.1]). , = (1, 2 , , N) . tz = tu. , , , .=(1,...,N)T , 1 y1t + ... + N yNt . , y1t,,yNt (,), , , . , . 7.3. . . (1) . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm25 1 y1t + ... + N yN t . , ; N . . . (2) y1t , , yN t (, E(yk t) = 0).(2a) (SM) . SM:y1t = 2 y2t + ... + N yN t + ut , ( )t N N t t ty y y u2 2 1 + + = K , t K t K t t tu u u u + + + + = 1 1 1 K H0: = 0 H0: < 0.t-t N. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm26,[Hamilton(1994),TableB.9,Case1]. . (2b) (SM) . SM:y1t = + 2 y2t + ... + N yN t + ut , ( )t N N t t ty y y u2 2 1 + + + = K , t K t K t t tu u u u + + + + = 1 1 1 KH0: = 0 H0: < 0.(2a). , [Hamilton(1994),TableB.9,Case2].T , [MacKinnon (1991), 1 ( no trend)] [Patterson (2000)]. (3) y2t , , yN t , E(yk t) 0 . (3a) . SM: y1t = + 2 y2t + ... + N yN t + ut . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm27(2b),. , [Hamilton (1994), Table B.9, Case 3]. T ,[MacKinnon(1991), Table 1 ( with trend)] [Patterson (2000)]. (3b) . SM:y1t = + t + 2 y2t + ... + N yN t + ut . , , , (3a), N , N + 1. -. , , y1t, y2t , , yN t . y1t,y2t ,y3t,y4t, DGP: y1t =y2, t +y3, t + y4, t + 1t , y2t =y2, t 1 + 2t ,y3t =y3, t 1 + 3t ,y4t =y4, t 1 + 4t , 1t , 2t , 3t , 4t , 1 2t , 3t , 4t21t . T = 200 . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm28-60-40-20020406020 40 60 80 100 120 140 160 180 200Y1Y2Y3Y4 , . .t-, 2.18,1.78,0.57,1.70,.4. , y1t , y2t , y3t , y4t AR(1) , .. 1. . (1) y1t =y2, t + y3, t + y4, t , y1t y2, t y3, t y4, t . -6-4-20246820 40 60 80 100 120 140 160 180 200COINT , :t- 15.07. . , . (2a) : Dependent Variable: Y1 Method: Least Squares VariableCoefficientStd. Errort-StatisticProb. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm29Y20.9960840.00997399.881610.0000 Y30.9925500.009578103.62960.0000 Y41.0023050.01239380.879220.0000 Augmented Dickey-Fuller Test Equation Dependent Variable: D(RESID_2A) VariableCoefficientStd. Errort-StatisticProb. RESID_2A(-1)-1.0755520.070892-15.171780.0000 t-15.17,5% 3.74([Hamilton(1994),TableB.9,Case1]). . (2b) : Dependent Variable: Y1 VariableCoefficientStd. Errort-StatisticProb. C0.3321830.3735420.8892790.3749 Y21.0025830.01236981.058430.0000 Y30.9873690.01121588.040480.0000 Y40.9990220.01293777.221290.0000 Augmented Dickey-Fuller Test Equation Dependent Variable: D(RESID_2B) VariableCoefficientStd. Errort-StatisticProb. RESID_2B(-1)-1.0790490.070861-15.227640.0000 t-15.235% ,4.11([Hamilton(1994),TableB.9,Case2]). . (3) y1t, y*1t= y1t + 0.75t , y1t : -5005010015020020 40 60 80 100 120 140 160 180 200Y1 Y1_STAR 4 . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm30-5005010015020020 40 60 80 100 120 140 160 180 200Y1_STARY2Y3Y4 (3a) : Dependent Variable: Y1_STAR VariableCoefficientStd. Errort-StatisticProb. C11.490532.7048024.2481950.0000 Y2-1.3337620.089561-14.892240.0000 Y32.8569520.08120735.181150.0000 Y40.0726300.0936770.7753230.4391 : -15-10-505101520 40 60 80 100 120 140 160 180 200RESID_3A : Augmented Dickey-Fuller Test Equation VariableCoefficientStd. Errort-StatisticProb. RESID_3A(-1)-0.1198050.033630-3.5624310.0005 t- 3.565% , 4.16([Hamilton(1994),TableB.9,Case3]). . (3b) : Dependent Variable: Y1_STAR VariableCoefficientStd. Errort-StatisticProb. C0.3040680.3907390.7781870.4374 . . .. www.iet.ru www.iet.ru/mipt/2/text/[email protected] Y21.0084700.02646838.101660.0000 Y30.9826580.02183045.014530.0000 Y41.0013560.01594262.812470.0000 : -8-6-4-20246820 40 60 80 100 120 140 160 180 200RESID_3B , : Augmented Dickey-Fuller Test Equation Dependent Variable: D(RESID_3B) VariableCoefficientStd. Errort-StatisticProb. RESID_3B(-1)-1.0794920.070859-15.234480.0000 t-15.2345% ,4.49([Hamilton(1994),TableB.9,Case3]). . , . , . y1t , y2t , y3t , y4t ,(). . y*1t , y2t , y3t , y4t., ., , . . . ,, ( )., . ,, ,, . , . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm32,.. , . , . ..7.1, yt=+xt+ut ,.. T - xt yt . , T, .,[Entorf(1992)], . DGP:xt = x + xt 1 + 1t , yt= y + yt 1 + 2t , 1t2t , x , y 0. SM:yt = + xt + u t T ,T,T, T T y / x . SM:yt = + xt + t + u t, (T) T y, T , . 7.4. Ny1t,,yNt , 1. = (1, ... , N)T , , 1 y1t + ... + N yN t ~I(0) , (); . c = E(1 y1t + ... + N yN t), 1 y1t + ... + N yN t = c . t , zt = 1 y1t + ... + N yN t c . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm33zt , ,, , , .. . ,,,I(1) . y1t,,yNt r,r .,N, r = 1, , N 1 . (, ,r=0.r=N ,N.) I(1) r-, .r , , , . I(1) y1t , , yN t r VAR(p)p (VAR vector autoregression)A(L) yt = + t , yt = (y1t , , yN t )T , = (1 , , N )T , A(L) = A0 A1L ApLp,A0 , A1 , ,Ap (N N), A0 = IN ( ), ..yt = + A1 yt 1 + + Ap yt p + t . A(1)rankA(1)=r(N=2) VAR ECM( )( ) , 111, , 1 , 1 , 111 , 1 1 , 1 11 1 1tpjj t N j N j t jt r r t ty yz z y + + + ++ + + + = = KK .. ( ) , 11, , , 1 , 11 , 1 , 1 1t Npjj t N j NN j t j Nt r r N t N N t Ny yz z y + + + ++ + + + = = KK z1,t ,...,zr,t I(0),r (1) , ... (r) , (11,...,N1)T,,(1r,...,Nr)T . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm34 yt = + T yt 1 + 1 yt 1 + p 1 yt p + 1 + t , 1 , , p 1 N N ,(N r)- r.(1),,(r) , i j z1, t 1 = T(1) yt 1, , zr, t 1 = T(r) yt 1 ( t 1 r y1t , , yN t) y1t , , yN t . VAR ,(1),,(r) ., . I(1) . r ,0 < r < N , I(1) y1t , , yN t , 1 , 1, 1 1|||.|

\|+|||.|

\|+|||.|

\|=|||.|

\|+t rtt Nt rr t rtvvyyCyyM M M M, , 1, 1 , 1,|||.|

\|+|||.|

\|=|||.|

\|+ + +t Nt rNrt Nt rvvyyM M M

( ) , C = (i j) r (N r) , vt =(v1t ,,vNt)T -() E(vt) = 0, yr + 1, t , , yN , t . : y1 t c11yr + 1, t c1, N r yN, t = 1+ v1 t, . . .yr t cr 1 yr + 1, t cr, N r yN, t= r+

vr t , (1) = (1, 0, 0, , 0, c11, , c1, N r )T , (2) = (0, 1, 0, , 0, c21, , c2, N r )T , . . . (r) = (0, 0, 0, , 1, cr 1, , cr, N r )T

. r y1 t , , yN , t z1, t= T(1) yt = y1 t c11yr + 1, t c1, N r yN, t ,zr, t= T(r) yt = yr t cr 1 yr + 1, t cr, N r yN, t. v1 t , , vr t vr + 1, t , , vN t, yr +1,t ,,yN,t, y1 t , , yr t yr + 1, t , ,yN,t. j ic , . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm35( )j i j ic c T T.r=1 y1 t =1+ c11y 2, t c1, N 1 yN, t + v1 t, (

y 2, t , , yN, t ) , ,t- F- (, ), ,v1t ., .7.1,v1t . j ic r(N1)- (1) , , (r) :

) 1 ( = (1, 0, 0, , 0, 1 1 c , , r Nc, 1)T , . . .) (r = (0, 0, 0, , 1, 1rc , , r N rc,)T . ) 1 ( ,..., ) (r , , ... ,1 ) ( 1 , 1 ) 1 ( 1 , 1 = =tTr t r tTty z y z . ECM yt = + 1 tz+ 1 yt 1 + p 1 yt p + 1 + t , |||.|

\|=1 ,1 , 11t rttzzz M. , ECM. ,(c) , z1, t , ... , zr, t y1t,,yNt , (N +1) y1t , , yN t t , . A(1)r,r . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm36 11 y1t + ... + 1N y1N + 1, N + 1 t , r1 y1t + ... + r N y1N + r, N + 1 t . ((N + 1)1)- (1) = ( 11 , ... , 1N , 1, N + 1)T .. (r)= ( r1 , ... , r N , r, N + 1)T. . (ECM), ,, . I(1) r ; . , y1t = + 2 y2t + ... + N yN t + ut ( y1t = + 2 y2t + ... + N yN t + N + 1 t + ut). , r > 1, (1, 2 , , N )T ( (1, 2 , , N , 1+ N )T) . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm37r , . r-,r>1 . (1) = (1, 0, 0, , 0, c11, , c1, N r )T , (2) = (0, 1, 0, , 0, c21, , c2, N r )T , . . . (r) = (0, 0, 0, , 1, cr 1, , cr, N r )T . ( ), r.( ) y1t , , yN t , (1) , ,(r) . () , ,. 8. ,[Johansen(1988)]. . . 8.1,,r=1,..( ) . ,N=2., y1t y2t , ., ,, ( ). , . . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm38 yt= xt+ t , xt = xt 1 + t , tt . , [West (1988)]:yt= + xt+ ut , xt~I(1),E(xt)=0(xt ), ut~I(0), . (,)T.ut , t-,, . t- TT , : S( ) 111X XT T ,S( ) 122X XT T . X(T2)-(1xt)T, S2 ut , ut ~ i.i.d., , 2112 2==TttunSt t tx y u =. ,ut ~ i.i.d., t- (,N(0,1))t-(T), S2 . ,ut . h =Cov(ut,ut+h) .West, S2 ut(..6.8.1), == 2hh . ,,2 , .,, . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm39 h ,h=0,1,2,,,,, . - , , , , ut , 2 h =Cov(ut,ut+h). , h h ,() h 2 h . ,ut MA(q)q, h =0|h|>q, h h., , hqhqh 11 2 102=||.|

\|+ + =, 11 1+ ==tTh tt hu uT h . (., ,[Hamilton(1994),p.513]),q=O(T1/5) 2( NeweyWest). EVIEWS . : Newey-West.(,, , .) . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm40,ut AR(p) p , ut = a1 ut 1 + a2 ut 2 + + ap ut p + t , t D(t) =2 , 2 = 2 (1 a1 a2 ... ap)2 . 2 ( )22 122 1pa a a =K, pa a a , , , 2 1K a1 , a2 , ... , ap , + ==Tp tp T12 21 , t tu . ,S2 2 t-, , ( ) S . DGP: yt= 2 xt+ ut , xt = 1 + xt 1 + vt , ut = 0.4 ut 1 + 0.2 ut 2 + t AR(2) , t,t , : Cov(t , t) = 0.8 : . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm41-5005010015020010 20 30 40 50 60 70 80 90 100X Y SM:yt=+xt +ut : Dependent Variable: Y VariableCoefficientStd. Errort-StatisticProb. C-0.3980710.172093-2.3131110.0228 X2.0319380.007241280.63360.0000 R-squared0.998757Mean dependent var37.39809 Adjusted R-squared0.998745S.D. dependent var30.23477 S.E. of regression1.071308Akaike info criterion2.995436 Sum squared resid112.4748Schwarz criterion3.047539 Log likelihood-147.7718F-statistic78755.23 Durbin-Watson stat1.080957Prob(F-statistic)0.000000 , ,DGPyt

. -4-202410 20 30 40 50 60 70 80 90 100RESIDS AR(2). AR(2) : Dependent Variable: RESIDS VariableCoefficientStd. Errort-StatisticProb. RESIDS(-1)0.3635220.1004943.6173440.0005 RESIDS(-2)0.2050740.1004272.0420240.0439 R-squared0.240364Mean dependent var-0.008547 Adjusted R-squared0.232451S.D. dependent var1.075097 S.E. of regression0.941891Akaike info criterion2.738343 Sum squared resid85.16727Schwarz criterion2.791098 : . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm42 = 0.941891/ (1 0.363522 0.205074) = 2.183, (S ) =1.071/ 2.183 = 0.491. t-P-: t : -2.313111(P-value = 0.0228) -1.135738(P-value = 0.2588) t :280.6336(P-value = 0.0000) 137.791098(P-value = 0.0000). . N=2 ()NI(1) y1t , , yN t . y1t = +2 y2t + ... +N yN t + ut ,, , .,.7.2, . T .N=2, . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm43y1t = + 2 y2t + ... + N yN t + 1t y2t = y2, t 1 + 2t , ... yN t = y N, t 1 + N t ,t=(1t,2t,..., Nt)T N-(..1,2,...,, N- =(ij) ), 2t , ... , Nt , 1t ( 1j =0j=2,,N).y2t,,yNt ,=(1, 2 ,, N )T . ( c , 2 2, , N

N)T {y2t , , yN t , t = 1, , T} N-,,F- , 2 , ... , N F-, t- t-. , y1t = + 2 y2t + ... + N yN t + u1t y2t = y2, t 1 + u2t , ... yN t = y N, t 1 + u N t ,ut=(u1t,u2t,..., uNt)T N-( N- ), u2t , ... , u Nt ,u1t , Cov(u1t, uk s) = 0k 1 t, s. . ( ut, TNT- .) F-t-F-t-,S2 u1t 2 2 u1t . F- 2 2 S t- S. . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm44 ,- y2t , , yN t . [Stock,Watson(1993)][Saikonnen(1991)] , (lags) (leads).( leadsandlags.), y1t = + 2 y2t + ... + N yN t ++ =pp j(2j y2, t j+ ... + Nj yN, t j )+ut . p(), 2 , ... , N (, ), t- F-, ut . , y2t , , yN t . , , . , . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm45(a)y1t 2 y2t ... N yN t ; (b) y1t 2 y2t ... N yN t . (,(a)(b), , .) DGP: yt= 5 + zt+ ut , zt = zt 1 + vt , z1 = 0, ut = t+ 0.25 t 1 + 0.25 t + 1 + 0.1 t 2 + 0.1 t + 2+ 0.1 t, t , t .ut t ,t1 ,t+1 ,t2 ,t+2 , . DGP (50 ): -40481210 20 30 40 50 60 70 80 90 100Y Z ytzt 50 .. yt=+zt +t : Dependent Variable: Y Method: Least Squares Sample(adjusted): 3 98 Included observations: 96 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. C4.8512620.13315236.434100.0000 Z1.0888700.04757022.889770.0000 H0: = 1, t-, , Cov(t , zs)0 t , s. -, et , zt , et , . . .. www.iet.ru www.iet.ru/mipt/2/text/curs_econometrics.htm46yt = + zt+ t . -Cov(et , zt - i) i = 0, 1, 2, ; - lag. -Cov(et,zt+i)i=0,1,2,; - lead. Included observations: 96 Correlations are asymptotically consistent approximations e , Z(-i)e , Z(+i)ilag lead . |********* . |********* 00.90170.9017 | .. |* 1-0.02170.0830 | .*| . 2-0.0956-0.0413 | .. | . 30.00640.0341 *| .. | . 4-0.05100.0118 *| .. | . 5-0.0824-0.0228 . | .. | . 6-0.01710.0150 **| .**| . 7-0.1858-0.1579 . | .. | . 8-0.0292-0.0272 . |*. |*90.08330.0701 . | .. | . 100.01250.0216 - -DGP7-., zt 7- . Dependent Variable: Y Sample(adjusted): 9 93 Included observations: 85 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. C4.9873620.020874238.92360.0000 Z1.0006890.007818127.99550.0000 D(Z)1

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