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Часть I. Прогнозирование временных рядов традиционными эконометрическими методами и их модификациями Глава 1. Постановка проблемы и инструментарий исследования 1.1. Общие соображения.,.,,, (1), (2) (3)..
,,,,.,,.
..,,.,.
9,,,...
,,, : (a), (),.
,., (,..).
.,. (. [Zarnowitz (1978)]), 1947. 6%-, 11%. 1948.,. :
, 30-.,,., 1953.,.,, Ц,., л ,.., ( ).
.,,,,, 10..,. л - . Ц,, Ц.
.,,,,., -,.
,., [Zarnowitz (1978)],,.
,,,. -,,.
, ( )., [Klein (1984)], л/,, л,, .
,,, (), (..),,. ( [Batchelor, Dua (1990)].) 11, [McNees (1979)] [McNees, Ries (1983)],, (a) ( ); (),.
-, л () ARIMA( p, k, q),,, AR-, MA- ARMA. (.,, [, (1998)] 2 [ (2001)].
, ARIMA,,,.
-,.
1.2. Постановка задачи.,.
[ (2001)], ( ) - TS (trend stationary),, (, ) ( ) k-1 - DS (difference stationary).
, TS-, DS- -.
TS- DS-. TS- л ,,,, Ц. DS- ( ), - DS- ( ),, л,.
TS-, DS-.,, - л, -.
DS- TS- (.,, [Diebold Senhadji (1996)]).,,.,.
,.
, (.,, [Dickey, Bell, Miller (1986)]).
,, (.,, [Christiano, Eichenbaum (1990)], [Rudebusch (1993)])..,,,,,, [Box, Jenkins (1976)].,, л,,,.
, л , л л, , (, persistence),,., TS- DS-.
.
, [Clements, Hendry (1998b)] [Clements, Hendry (2000)],,.,.
1.3. Результаты некоторых предыдущих исследований 1.3.1..
[Stock, Watson (1996)],.
1., 2., 3., -,,, 4.,,,., 215.
,.. (,,..),.,,,.
, 49.,,,.,, (), 121., ( ) (AIC),, 0 12; 13. : (AR), (EX), (ANN) (LSTAR). л л .
49 ( ). [Bates, Granger (1969)], [Granger, Newbold (1977)], [Granger, Ramanathan (1984)],,.,,,,, (MSE),, MSE; (PLS).
[Markidakis et al. (1982)],,. [Meese, Geveke (1984)], 150,, AR, AIC.,. [Weigland, Gershenfeld (1994)].,, ( ).
[Swanson, White (1995, 1997)] ANN, MSE, ANN., [Stock J.H., Watson M.W. (1996)],.
:
yt+h=fi( Zt; ih)+uit+h, yt Ц,, h Ц, i - (i=1,Е,121), ih Ц, uit Ц, Zt Ц.
Zt=(yt,Е,yt-p, yt,Е,,yt-p,1,t) p Ц.
Zt.
,..
yt+h 1,2,Е,t.
yt+h ih y1, y2,Е, yt. h.
,,.. h h.
( h,, yt+h/t,ih=fi( Zt; *iht), et+h/t,ih=yt+h- yt+h/t,ih.),.,, h.
,, h.,, [Clements, Hendry (1996)].,, h,,.
[Stock, Watson (1996)] л,, л .,., ( ).
,,,,, -. л., л,.
, л.
. :, ;, 121 49,,,,. T0,. T0 T1, T1=T0+120 (). - T1 T2-1, T2=T1+24 (). - T2 T3, T3 - (12.1996) h.
,,, ( T2 T3 ). 01.1959; T0=01.1959, T1=01.1970, T2=01.1972, T3=12.1996 - h.
(AR). 18. (3 ),, (2 ), (3 ).
:, 4;, BIC (0 p12);, AIC (0 p12).
.,.. yt+h yt,Е,yt-p+1,., yt+h-yt, yt,Е,yt-p+1.
,.. Stock (1996) -. Stock, DF-GLS [Elliot, Rothenberg, Stock (1996)]., Stock, Watson DF-GLS,, DF-GLS Ц.
52 (2, 13, ). 18,, /.
:
1.,,,,,,.
2..
3.,,.
4. -,.
1.3.2. [Stock, Watson (1996)], (),,.
.
.
AR( p) Ц, BIC Ц,.
RRA1 - 120, BIC,.
RRA2 - 240, BIC,.
RLSA Ц, BIC,.
ADL( p,p), t = t-1+t, t,.
ATVP1 - TVP (Time varied parameters) = 0.0025,.
ATVP2 - TVP =0.0075,.
ATVP3 - TVP =0.015,.
ATVP4 - TVP p,.
().
VAR Ц, BIC Ц,.
RRA1 - 120, BIC,.
RRA2 - 240, BIC,.
RLSA Ц, BIC,.
ADL( p,p).
VTVP1 - VAR.
VTVP2 - RRA1.
VTVP3 - RRA2.
VTVP4 - RLSV.
TVP.
(in-sample) - [01.1959Ц12.1978].
, - [01.1979Ц12.1993].
, ( ),.
5700, 16,..
BIC,.,, VAR ADL.
ATVP1, - VTVP1. ATVP,,,,.
1.3.3. TS- DS [Franses, Kleibergen (1996)] TS- DS 14, [Nelson, Plosser (1982)], 1988. TS- DS-. : 1976., 1970.
1952.,, 1977Ц1988, 1971Ц1988 1953Ц1988.,,, 12, 18 36.
MSEP ( ), MAE ( ) TS- DS-. :
.,,.
:,, DS-.
[Franses, Kleibergen (1996)]., 1952., 1953., 1953., 1954...,. 5 14 DS-, 7 14, (Consumer Prices) TS-.
,, TS- (out-of-sample), л TS-, л DS-.
DS-,.
TS-,, Franses, Kleibergen, TS- DS-,.
[Campbell, Perron (1991)],, TS DS-, 20..
,,,,,...
[Clements, Hendry (2000)] DS- TS-, (SM) (DGP - data generating process).
, DS- TS-.
DS yt=yt-1++t, t NID( 0,2) y0=0, TS- yt=+t+ut, ut ~ NID( 0,u2).
, T h, :
DGP - DS-; SM - DS-;
DGP - DS-; SM - TS-;
DGP - TS-; SM - TS-;
DGP - TS-; SM - DS-.
VDS /DS, VTS /DS, VTS /TS VDS /TS.
,,,,,.,, DS- yT,,,. yT ( ),, DGP,.. EDS[ VTS/DS/VDS/DS] ETS[ VDS/TS / VTS/TS].
h, T h. h T.
T h, ( -).
, T ( 200) ( ) ETS[ VDS/TS / VTS/TS] 2. EDS[ VTS/DS / VDS/DS], T,,, h = 1 T = EDS[ VTS/DS / VDS/DS] 28., DGP, л DS-.
,, TS- DS,., TS- DS-.
TS DGP yt=+t+ut, ut=ut-1+at, at ~ NID( 0,2), 01.
,, DS- TS DGP, 1.
DS DGP yt=yt-1++t, t=bt-bt-1, btNID( 0,b ), 0<1.
TS- DS DGP, 1 (, = yt, ).
( ) TS- DS- DGP.
= 0.9 DS- TS DGP, T h., T< DS-, TS- ( TS DGP). [Campbell, Perron (1991)].
=0.9 T<100 TS-, DS- ( DS DGP), h.
, DGP л (DS TS) T,, DGP (.. л ),., TS- DS-.
[Diebold, Kilian (2000)],,.,.,, -.
,,.
, (DGP),., DGP, -.
DGP,,.,,,.
AR(1), (yt-a-bt)=(yt-1-a-b(t-1))+t, t ~ iid N(0,2), t =1, Е,T. :
yt=k1+k2t+yt-1+t, k1=a(1-)+b, k2=b(1-).
AR(1):
yt=Tt+xt, Tt=a+bt, xt=xt-1+t.
=1 b; <1 AR( 1) b.
, =7, b = 0. = 0.0099. {0.5, 0.9, 0.97, 0.99, 1} T{25, 30:10:80; 100:20:180; 200:40:1000},,,,.
: AR( 1) (L - л), (D - л), Ц, 5%- (P - л ). OLS. h 1 100. (PMSE), 20000 -. PMSE(D)/PMSE(L), PMSE(D)/PMSE(P) PMSE(P)/PMSE(L) h T.
Результаты D&L (разности или уровни),,,.
D&P PMSE( D)/PMSE( P),,,., Ц.
P&L PMSE( P)/PMSE( L),..
Некоторые практические советы,, DGP,,.,,. DGP,.
Частота a b Базовый ряд -6.0674 0.0173 0.0500 7.3707 0.0065 0.0099 3.3653 0.0024 0. 5.1126 0.0004 0.0095 Ц,, ( P) ( L).
( T=40-160 h=1-100), DGP 0.97. = 0.9, 70, L. = 0.., 70, Ц.
(T=80-200 h=1-16) P, 0.97. = 0.9 L. = 0.5. 0.97 L.
(T=240-480 h=1-48) P, 1 0.99. = 0.97 = 0.9 L, P. = 0.5.,, 0.99. L.
(=360-720 h=1-90) =1. =0.99, ( ). L P. = 0.97 L. = 0.9 L T<500,.. = 0.5.,,,, 0.99.
L.
, DGP. OLS 0., 0.99 0.., OLS-,.
,,.
1.3.4. Stock Watson.,, [Stock, Watson, (1994)] (76, ),,.,,, (TVP-). (., [Nyblom (1989)]), CUSUM- (.
[Quandt (1960)]).
[Stock, Watson (1994)],,., ( OLS-), (,, TVP- ) (TVP- ).
,,,.,.,, (38%).,, TVP-,.
.
[Stock, Watson, (1998a)] - ( ),,,,,,,,,. ( -).
,.,,,,, -,. -, National Bureau of Economic Research (NBER);
.
,, (), -., -,,.,,,,, -, (,.).,, -;,.
, -;
,.
- ( ),, ( ) ( ).,,.
[Stock, Watson, (1998b)],.,,,.
,,, Ц.
[Stock, Watson (1999)].. ;,.
,,,. 1959Ц1997.
,, ( ), ( ),. -,,. -,,,.
,..
,,.,,, - I(1).
[Stock, Watson (2001)].,,,. ;. OECD 1959Ц1999.
, ;
.,., (1Ц2 ),, -,,.
[Staiger, Stock, Watson (2001)], 1990-.,,,.
,.,, (non-accelerating inflation rate of unemployment, NAIRU).
, ( ), -., NAIRU.
NAIRU,,,,.