제 11 강 자기상관 Auocorrelaion 111 유효성 (efficiency, accurae esimaion/predicion) 을위해서는모든체계적인정보가회귀모형에체화되어있어야함 표본의무작위성 (randomness) 은서로다른관측치들에대한오차항들이상관되어있지말아야함을의미함 자기상관 (Auocorrelaion) 은이러한표본의무작위성을위반하게만드는오차항에있는체계적패턴임 11 시계열자료 (-series daa) 를다루는경우항상연속되는오차항들이서로상관되어있을가능성이존재함 어떤특정시점에서해당시점의오차항은해당시점의충격뿐아니라과거로부터의충격으로부터이전된영향들도포함함 이러한이전된영향으로인해해당시점의충격은과거의충격들과상관될것이며, 이러한상황은오차항들이상관을낳게됨 이경우자기상관이존재한다고함 양 (posiive) 의자기상관과음 (negaive) 의자기상관이존재할수있음 113 i i Posiive auocorrelaion Cyclical: Posiive auocorrelaion Posiive auocorrelaion i 114 현재의오차항이과거의오차항과같은부호를가지려는경향이있음 u i Negaive auocorrelaion 현재의오차항이과거와다른부호를가지려는경향이있음 u i No auocorrelaion 현재의오차항이과거와는무관하게나타남 115 Posive Auo No Auo Negaive Auo 116 crosses line no enough (aracing) crosses line randomly crosses line oo much (repelling) 1
단순선형모형 117 자기상관의차수 118 + + + + zero mean: homoskedasiciy: nonauocorrelaion: auocorrelaion: ) = var( ) = cov(, s ) = s cov(, s ) s 1차 (1s Order): = 1 + 차 : = 1 1 + + 3차 : = 1 1 + + 3 3 + 1차자기상관을가정할것임 AR(1) : = 1 + 119 111 + + = 1 + where 1 < < 1 he random componen in period Carryover from he random error in he previous period Auocorrelaion coefficien A new shock o he level of he economic variable + +, = 1 + where 1 < < 1 = 1-1 = -1 = + -1 = + -1 + - + 3-3 + 1111 자기상관존재시최소제곱추정량 111 + +, = 1 + where 1 < < 1 최소제곱추정량은여전히선형불편추정량이나유효추정량은아님 E( ) = var( ) = cov(, s ) = s hese assumpions abou imply he following abou : E( ) = var( ) = = 1 cov(, k ) = k for k > corr(, k ) = k for k > 표준오차를계산하기위한통상의공식은더이상정확하지않으며, 따라서그에근거한신뢰구간이나가설검정역시잘못되게됨 Homoskedasiciy
AR(1) : = 1 + + + + + 1 + 1113 subsiue in for + + 1 + + + = y 1 1 = y 1 1 1 1114 lag he errors once Now we need o ge rid of 1 (coninued) + + y 1 1 1 + (coninued) + + y 1 1 1 + 1115 + + y 1 1 1 + y y 1 = 1 (1 ) + ( 1 )+ y = 1 1 + + =, 3, * y = y y 1 y * 1 = (1 ) = 1 1 + + 1116 * = 1 =, 3, 최소제곱으로이모형을추정함에있어의문제 : 1 변환된변수들을만들어냄에있어시차 (lagged) 변수를사용함으로써관측치하나를날려버리게되어 -1 의관측치만으로모형을추정 의값을모름 그것을추정하기위한방법을찾아야함 However, recovering he 1s observaion his way and applying leas squares is no he bes linear unbiased esimaion mehod 1117 Adding y 1 = 1 + 1 + 1 o he esimaion? (ha is, y 1* = y 1, 11* = 1, 1* = 1 ) Efficiency is los because he variance of he error associaed wih he 1s observaion is no equal o ha of he oher errors his is a special case of he heeroskedasiciy problem ecep ha here all errors are assumed o have equal variance ecep he 1s error y 1 = 1 + 1 + 1 1118 첫번째관측치는원래의모형에적합 (fi) 되어야함 wih error variance: var( 1 ) = = /(1- ) 이를변형된변수들을이용한추정에포함시킬수있으나, 다른관측치들과같은오차항의분산을가지도록변형해야만함 Noe: he oher observaions all have error variance 3
1119 11 Given any consan c : var(ce 1 ) = c var( 1 ) If c = 1-, hen var( 1-1 ) = (1- ) var( 1 ) = (1- ) = (1- ) /(1- ) = he ransformaion 1 = 1-1 has variance y 1 = 1 + 1 + 1 Muliply hrough by 1- o ge: 1- y 1 = 1-1 + 1-1 + 1-1 y 1 = 1 11 + 1 + 1 he ransformed error 1 = 1-1 has variance 이변형된첫번째관측치가다른 (-1) 관측치들에추가되어 개의관측치들을완전히복원하는것이가능 의추정 111 의추정 11 오차항 들의값을안다면다음을추정할수있음 = 1 + 우선최소제곱추정을통해다음을추정함 + + 이추정으로부터의잔차를구함 = y -b 1 -b 다음을최소제곱추정으로추정함 = 1 + 최소제곱추정량은다음과같이주어짐 : = = -1-1 = 113 H o : = vs H 1 : >, ( or < ) 대부분의경제자료에대한응용에있어서자기상관은양의자기상관의형태로나타남 더빈 - 왓슨검정통계량 (he Durbin-Wason es saisic), d 는 : d= -1 = = 1 114 d 는근사적으로 와다음과같은관련을가짐 : d (1 ) When =, he Durbin-Wason saisic is d When = 1, he Durbin-Wason saisic is d 검정통계량 d 의확률분포가설명변수들의값에의존하기때문에그임계값에대한표는일률적으로제시될수없음 ( 많은패키지들은해당 d 값에대한 p 값을제시해주시않음 ) Rejec H o if p-value <, he significance level 4
p-value 을계산해주는통계패키지가없을경우, 경계점정 (bounds es) 로알려진검정방법을통해부분적으로문제를해결할수있음 두개의다른검정통계량 d L 과 d U 은 d L <d<d U 이고, 그분포들이설명변수들에의존하지않음 ( 표로제시되어있음 ) If d d Lc, rejec H If d d Uc, accep H If d Lc <d<d Uc, he es is inconclusive 115 Lagrange Muliplier 검정 + + 1 + + + 1 + Regress y on and -1 and use a - or F-es o es he significance of he coefficien of -1 Using he firs observaion requires Se = Or, drop he firs observaion DW es is an eac valid in finie samples while LM es is an approimae large sample es his approimaion occurs because -1 is replaced by -1 DW es can only be applied o he AR(1) while LM es can be eended o he es of higher order auocorrelaion 116 자기상관존재시예측 117 오차항에자기상관에존재할경우, 이전시긴의오차항은미래의오차항을예측하는데도움을줌 다음기에대한최우수 (he bes) 예측치, y +1 는 자기상관존재시예측 h기이후에대한최우수예측치는 y +h = 1 + +h + h ~ e 118 y +1 = 1 + +1 + ~ e where 1 and are generalized leas squares ~ esimaes and e is given by: ~ e = y 1 Assuming < 1, he influence of h ~ e diminishes he furher we go ino he fuure (he larger h becomes) 5