Chapter7自相关 Autocorrelation or Serial correlation
Chapter 7 自相关 Autocorrelation or Serial correlation
1. Suppose the linear regression model is X=b+bx1+…+b2X+E 如果cov(6,E1)=0(i≠j不恒成立,则称误差项是 序列相关的(自相关的)。( serial correlation) 2.产生序列相关的原因及序列相关的影响 1)原因 (1)经济行为的惯性或冲击的惯性(SARS) (2)模型误设( Model misspecification) 2)序列相关对估计与检验的影响与异方差性 类似
1. Suppose the linear regression model is 如果 不恒成立,则称误差项是 序列相关的(自相关的)。(serial correlation) 2. 产生序列相关的原因及序列相关的影响 1)原因 (1)经济行为的惯性或冲击的惯性(SARS) (2)模型误设(Model misspecification) 2)序列相关对估计与检验的影响与异方差性 类似。 Y = b b X b X + t t k kt t 0 1 1 + + + cov( , ) 0 ( ) i j = i j
3.如误差项的序列相关具有形式 Et=pEt-1+Ut 则称为一阶序列相关,其中p<1。 0<p<1:正序列相关 1<p<0:负序列相关 p=0:不序列相关 Assumption E(U,)=0,E(U 2=0u 2 0-= const E(UU,)=0(t≠s)
3. 如误差项的序列相关具有形式 则称为一阶序列相关,其中 。 :正序列相关 :负序列相关 :不序列相关 Assumption: t t t 1 = + − 1 0 1 − 1 0 = 0 2 2 E( ) 0, E( ) const. t t = = = E( ) 0 ( ) t s = t s
4.有关推论 )a=∑ 2)E(U,E)=0(>1) S=0 3)Va()=o2/(1-p2)会a2t=1…,n n-2 4)E(t8)=a 2
4. 有关推论 1) 2) 3) 4) 0 s t t s s − = = ( ) 0 ( ) E s t s t = 2 2 2 var( ) (1 ) 1, , t t n = − = 1 2 2 1 2 1 1 ( ) 1 n n n n E − − − − = εε
5) coVe.8 coVIE. 8 2 lvar(E,)Ⅴa(E;)2
5) 1 1 2 1 1 2 2 1 cov( , ) cov( , ) [var( )] [var( )] t t t t t t − − − = =
Tests for autocorrelation 1. Durbin-Watson test 1 Suppose the linear regression model is =b+bX1n+…+bX+E(b≠0) E=nE1+U,(AR(1)) cov(8 阶自相关系数:P t2t-1 2 。将E用残差 代替并运用OLS于以上AR(1)可得估计
Tests for Autocorrelation 1. Durbin-Watson test 1) Suppose the linear regression model is 一阶自相关系数: 。 将 用残差 代替并运用OLS于以上AR(1)可得估计 0 1 1 0 ( 0) Y = b b X b X + b t t k kt t + + + 1 (AR(1)) t t t = + − 1 2 cov( , ) t t − = 2 1 1 2 2 ˆ n n t t t t t e e e − − = = = t
2)Hypothesis: Ho: p=0 VS H:p*0 Construct the d-w statistic ∑( 2|1 t-1 2(1-p) ∑ t=1 00(正相关)→0d>2
2) Hypothesis: Construct the D-W statistic 0 1 H H : 0 vs : 0 = 2 1 2 1 2 2 1 1 ( ) 2 1 2(1 )ˆ n t t t t t n t t t e e e e d e e − = − − = − = − = − 0 4 d 0( 2 0( 0 2 0( 4 2 d d d = 不相关) 正相关) 负相关)
3)检验判断: 上临界值:a,下临界值:a (1)0<d<d1→ reject H (2)4-di <d<4= reject Ho B)du<d<4-du= not reject Ho (4)Otherwise, no conclusions
3) 检验判断: U L 上临界值: , d d 下临界值: 0 0 0 (1) reject (2) 4 - 4 reject (3) 4 not reject (4) Otherwise, no conclusions L L U U o d d H d d H d d d H −
2. Test for autocorrelation when a lagged dependent variable serves as an independent variable Suppose that Y=a+ Bra+rX,+E Durbin h statistic(T is the observation No T h= N(O,1) 1-Tvar(β) d T h=(1 T var(B
2. Test for autocorrelation when a lagged dependent variable serves as an independent variable Suppose that Durbin h statistic (T is the observation No.) Y Y X t t t t = + + + −1 ˆ (0,1) ˆ 1 var( ) T h N T = − (1 ) 2 ˆ 1 var( ) d T h T = − −
自相关修正 1. Generalized Differencing(广义差分法) It is necessary that p is known 2. Cochrane-Orcutt Approach(P101)
自相关修正 1. Generalized Differencing(广义差分法) It is necessary that is known. 2. Cochrane-Orcutt Approach(P101)