Econ 39 - Auocorrelaion Sanjaya DeSilva Ocober 3, 008
1 Definiion Auocorrelaion (or serial correlaion) occurs when he error erm of one observaion is correlaed wih he error erm of any oher observaion. This violaes he classical assumpion ha E(u i u j ) = 0 i j 1.1 Causes of Auocorrelaion 1. Spaial dependence of observaions (cross-secional daa).. Random effecs: unobserved variables ha are independen beween bu correlaed wihin cross-secional unis (e.g. panel daa, muliple individuals in he same household) 3. Funcional form: incorrec funcional forms and smoohing. 4. Omied variables: omied variables ha are serially correlaed (e.g. ineria) Y = β 0 +β 1 X +nu where nu = β Z +u, and E(Z Z 1 ) = 0 (e.g. preference shif, oil shock, war, ineres rae) 5. Lagged dependen variable: error includes he lagged dependen variable Y = β 0 + β 1 X + nu where nu = β Y 1 + u (e.g. consumpion, invesmen, ineres rae). 6. Firs difference equaions: Y = β 0 + β 1 X + u 7. Nonsaionariy: If he mean and/or variance of he variables change over ime. 1
1. Firs Order Auoregressive Scheme: The AR1 Model Y = β 0 + β 1 X + u (1) u = ρu 1 + ɛ () ɛ (0, σɛ ) (3) cov(ɛ i ɛ j ) = 0 i j (4) where ρ is he firs-order coefficien of auocorrelaion. Wih his informaion, we can consruc he properies of u. V ar(u ) = ρ V ar(u 1 ) + σ ɛ = ρ V ar(u ) + σ ɛ = E(u ) = 0 (5) σ ɛ 1 ρ (6) Cov(u, u 1 ) = E(u u 1 ) = E((ρu 1 + u 1 ɛ )) = ρe(u ) = ρ 1 ρ (7) Cor(u, u 1 ) = ρ (8) Cov(u, u s ) = E(u u 1 ) = E((ρu 1 + u 1 ɛ )) = ρ s E(u ) = ρ (9) 1 ρ Cor(u, u s ) = ρ s (10) σ ɛ σ ɛ OLS esimaion of an AR1 Model Since E(u ) = 0, he OLS esimaor of β 1 is unbiased. However, he variance of b 1 is, E( k i u i ) = σ x i +E(k 1 k u 1 u +...+k n 1 k n u n 1 u n ) = σ x i +k 1 k E(u 1 u )+...+k n 1 k n E (11)
If X = rx 1, hen his expression reduces o V ar(b 1 ) = σ 1 + rρ x i 1 rρ (1) If boh X and u are posiively auocorrelaed, he rue variance of b 1 is greaer han wha he OLS formula repors. Therefore, we may misakenly rejec null hypoheses and consruc arificially narrow confidence inervals. 3 Properies of OLS esimaor of AR1 model 1. Coefficien esimae is unbiased: E(b 1 ) = β 1. The OLS variance formula is incorrec. Hard o ell wheher i s an overesimae or an underesimae wihou addiional assumpions. Generally, OLS formula will underesimae he variance. 3. OLS esimaor is no longer efficien. 4. The esimae of ˆσ is biased: E( ˆσ ) σ. I can be shown ha E( ˆσ ) < σ if boh X and u are boh posiively serially correlaed. This will conribue o inflaed R and o arificially low sandard errors. 4 Deecion of Auocorrelaion Noe ha, in order o es for serial correlaion, i is imporan o order he daa as a series (e.g. by year for ime series) 1. Graphical Mehod I: Plo residuals (or sandardized residuals) agains ime. (Noe: residuals are no he same as errors). 3
. Graphical Mehod II: To es for AR1, plo e agains e 1 3. The Runs Tes: Under he assumpion of no serial correlaion (i.e. independen observaions), find he mean and variance of he asympoically normally disribuion of he number of runs. These only depend on he number of observaions, posiive residuals and negaive residuals. Consruc a 95 per cen confidence inerval for he number of runs. Find wheher he observed number of runs is inside he confidence inerval. If he number of runs is smaller, here is posiive serial correlaion. 4. Durbin Wason Tes: d = (e e 1 ) (13) Assuming ha u is normally disribued and follows an AR1 process, regressors are nonsochasic, here are no lagged dependen variables, and he mode includes an inercep, Durbin Wason derived lower and upper bounds of values ha help us o es wheher here is serial correlaion. ha To see how his saisic reflec serial correlaion, noe d = (e e 1 ) (14) d = + 1 e 1 (15) Since 1, d (1 e 1 ) = (1 ˆρ) (16) 4
Noe ha 0 d 4, and 0 d < if posiive auocorrelaion, < d 4 if negaive auocorrelaion, and d = if no auocorrelaion. To carry ou he es, we need o find he lower d L and upper d U criical values for he D-W saisic for he relevan sample size and number of variables. Then compue he D-W saisic d for he regression using residuals. Now, (a) 0 < d < d L : posiive serial correlaion (b) d L d d U : inconclusive (c) d U < d < 4 d U : no serial correlaion (d) 4 d U d 4 d L : inconclusive (e) 4 d L < d < 4:negaive serial correlaion 5. Breusch-Godfrey Tes: This is no limied o AR1 and works even wih lagged dependen variables, and sochasic regressors. (a) Run regression and save residuals (b) Pick he number of lags (p) for he AR(p) process (c) Run a AR(p) regression (d) Do a hypohesis es for all slopes equal zero using (n p)r χ (p) 5