제 11 강 111 자기상관 Autocorrelation 자기상관의본질 11 유효성 (efficiency, accurate estimation/prediction) 을위해서는모든체계적인정보가회귀모형에체화되어있어야함 표본의무작위성 (randomness) 은서로다른관측치들에대한오차항들이상관되어있지말아야함을의미함 자기상관 (Autocorrelation) 은이러한표본의무작위성을위반하게만드는오차항에있는체계적패턴임 1
자기상관의본질 113 시계열자료 (time-series data) 를다루는경우항상연속되는오차항들이서로상관되어있을가능성이존재함 어떤특정시점에서해당시점의오차항은해당시점의충격뿐아니라과거로부터의충격으로부터이전된영향들도포함함 이러한이전된영향으로인해해당시점의충격은과거의충격들과상관될것이며, 이러한상황은오차항들이상관을낳게됨 이경우자기상관이존재한다고함 양 (positive) 의자기상관과음 (negative) 의자기상관이존재할수있음 자기상관의본질 ^ i 0 Positive autocorrelation time ^ Positive autocorrelation i 0 114 time ^ i 0 Cyclical: Positive autocorrelation time 현재의오차항이과거의오차항과같은부호를가지려는경향이있음
자기상관의본질 ^u i Negative autocorrelation time 현재의오차항이과거와다른부호를가지려는경향이있음 115 ^u i No autocorrelation 0 time 현재의오차항이과거와는무관하게나타남 자기상관의본질 t Postive Auto 0 crosses line not enough (attracting) 116 t No Auto Negative Auto t 0 t 0 crosses line randomly crosses line too much (repelling) t t 3
자기상관의본질 단순선형모형 117 zero mean: y t = 1 homoskedasticity: + t + t ) = 0 var( t ) = nonautocorrelation: cov( t, s ) = t s autocorrelation: cov( t, s ) t s 자기상관의차수 118 y t = 1 1 차 (1st Order): + t + t t = t 1 + t 차 : 3 차 : t = 1 t 1 + t + t t = 1 t 1 + t + 3 t 3 + t 1 차자기상관을가정할것임 AR(1) : t = t 1 + t 4
1 차자기상관모형 119 y t = 1 + t + t t = t 1 + t where 1 < < 1 The random component in time period t Autocorrelation coefficient A new shock to the level of the economic variable Carryover from the random error in the previous period 1 차자기상관모형 1110 y t = 1 + t + t, t = t 1 + t where 1 < < 1 t = t 1 t t-1 = t t-1 t = t t + t-1 t = t + t-1 + t- + 3 t-3 + 5
1 차자기상관모형 1111 y t = 1 + t + t, t = t 1 + t where 1 < < 1 E( t ) = 0 var( t ) = cov( t, s ) = t s These assumptions about t imply the following about t : E( t ) = 0 var( t ) = = 1 Homoskedasticity cov( t, t k ) = k for k > 0 corr( t, t k ) = k for k > 0 자기상관존재시최소제곱추정량 111 최소제곱추정량은여전히선형불편추정량이나유효추정량은아님 표준오차를계산하기위한통상의공식은더이상정확하지않으며, 따라서그에근거한신뢰구간이나가설검정역시잘못되게됨 6
일반최소제곱추정 1113 AR(1) : t = t 1 + t y t = 1 + t + t substitute in for t y t = 1 + t + t 1 + t Now we need to get rid of t 1 (continued) 일반최소제곱추정 1114 y t = 1 + t + t 1 + t y t = 1 + t + t t t 1 = y t 1 t = y t 1 1 t 1 lag the errors once y t = 1 + t + y t 1 1 t 1 + t (continued) 7
일반최소제곱추정 1115 y t = 1 + t + y t 1 1 t 1 + t y t = 1 + t + y t 1 1 t 1 + t y t y t 1 = 1 (1 ) + ( t t 1 )+ t = 1 1t + t + t * * * y t t =, 3, T 일반최소제곱추정 * y t = y t y t 1 * 1t = (1 ) = 1 1t + t + t * * * y t 1116 * t = t t 1 t =, 3, T 최소제곱으로이모형을추정함에있어의문제 : 1 변환된변수들을만들어냄에있어시차 (lagged) 변수를사용함으로써관측치하나를날려버리게되어 T-1 의관측치만으로모형을추정 의값을모름 그것을추정하기위한방법을찾아야함 8
일반최소제곱추정 However, recovering the 1st observation this way and applying least squares is not the best linear unbiased estimation method 1117 첫번째관측치의복구 Adding y 1 = 1 + 1 + 1 to the estimation? (That is, y 1* = y 1, 11* = 1, 1* = 1 ) Efficiency is lost because the variance of the error associated with the 1st observation is not equal to that of the other errors This is a special case of the heteroskedasticity problem ecept that here all errors are assumed to have equal variance ecept the 1st error 일반최소제곱추정 첫번째관측치의복구 1118 첫번째관측치는원래의모형에적합 (fit) 되어야함 y 1 = 1 + 1 + 1 with error variance: var( 1 ) = = /(1- ) 이를변형된변수들을이용한추정에포함시킬수있으나, 다른관측치들과같은오차항의분산을가지도록변형해야만함 Note: The other observations all have error variance 9
일반최소제곱추정 첫번째관측치의복구 1119 Given any constant c : var(ce 1 ) = c var( 1 ) If c = 1-, then var( 1-1 ) = (1- ) var( 1 ) = (1- ) = (1- ) /(1- ) = The transformation 1 = 1-1 has variance 일반최소제곱추정 첫번째관측치의복구 110 y 1 = 1 + 1 + 1 Multiply through by 1- to get: 1- y 1 = 1-1 + 1-1 + 1-1 * * * y 1 = 1 11 + 1 + 1 The transformed error 1 = 1-1 has variance 이변형된첫번째관측치가다른 (T-1) 관측치들에추가되어 T 개의관측치들을완전히복원하는것이가능 10
일반최소제곱추정 의추정 111 오차항 t 들의값을안다면다음을추정할수있음 t = t 1 + t 우선최소제곱추정을통해다음을추정함 y t = 1 + t + t 이추정으로부터의잔차를구함 ^ t = y t - b 1 - b t 일반최소제곱추정 의추정 11 다음을최소제곱추정으로추정함 ^ t = ^ t 1 + t 최소제곱추정량은다음과같이주어짐 : ^ = T ^^ t t-1 t = T t-1 t = ^ 11
자기상관의탐지 DW 검정 113 H o : = 0 vs H 1 : > 0, ( 0 or < 0) 대부분의경제자료에대한응용에있어서자기상관은양의자기상관의형태로나타남 더빈 - 왓슨검정통계량 (The Durbin-Watson Test statistic), d 는 : d= T ^ t ^ t-1 T ^ t t = t = 1 자기상관의탐지 DW 검정 114 d 는근사적으로 ^ 와다음과같은관련을가짐 : d (1 ) ^ When ^ = 0, the Durbin-Watson statistic is d When ^ = 1, the Durbin-Watson statistic is d 0 검정통계량 d 의확률분포가설명변수들의값에의존하기때문에그임계값에대한표는일률적으로제시될수없음 ( 많은패키지들은해당 d 값에대한 p 값을제시해주시않음 ) Reject H o if p-value <, the significance level 1
자기상관의탐지 DW 검정 115 p-value 을계산해주는통계패키지가없을경우, 경계점정 (bounds test) 로알려진검정방법을통해부분적으로문제를해결할수있음 두개의다른검정통계량 d L 과 d U 은 d L <d<d U 이고, 그분포들이설명변수들에의존하지않음 ( 표로제시되어있음 ) If d d Lc, reject H 0 If d d Uc, accept H 0 If d Lc <d<d Uc, the test is inconclusive 자기상관의탐지 Lagrange Multiplier 검정 116 y t = 1 + t + t 1 + t y t = 1 + t + ^ t 1 + t ^ Regress y t on t and t-1 and use a t- or F-test to test the significance of the coefficient of t-1 Using the first observation requires 0 Set ^ 0 = 0 Or, drop the first observation DW test is an eact valid in finite samples while LM test is an approimate large sample test This approimation occurs because t-1 is replaced by ^ t-1 DW test can only be applied to the AR(1) while LM test can be etended to the test of higher order autocorrelation ^ ^ 13
자기상관존재시예측 117 오차항에자기상관에존재할경우, 이전시긴의오차항은미래의오차항을예측하는데도움을줌 다음기에대한최우수 (the best) 예측치, y T+1 는 ^ ^ ^ y T+1 = 1 + T+1 + ^ ~ e T ^ ^ where 1 and are generalized least squares ~ estimates and e T is given by: ~ ^ ^ e T = y T 1 T 자기상관존재시예측 h 기이후에대한최우수예측치는 118 ^ ^ ^ y T+h = 1 + T+h + ^ h ~ e T Assuming ^ < 1, the influence of ^ h ~ e T diminishes the further we go into the future (the larger h becomes) 14