Advanced Econometrics - Chapter 9: Autocorrelation

Advanced Econometrics Chapter 9: Autocorrelation Nam T. Hoang University of New England - Australia 1 University of Economics - HCMC - Vietnam Chapter 9 AUTOCORRELATION Non-zero correlation between errors at different observations: stE tt ≠≠ 0)( εε → violated assumption (4): E(εε') = σε2I because the off-diagonals ≠ 0. Example: tttt LKQ εβββ +++= logloglog 321 t = 1,2, ... T In recession Q↓ more than inputs εt < 0 In boom Q↑ more than inputs εt > 0 Autocorrelation, also called serial correlation, can exist in any research study in which the order of the observations has some meaning, it occur most frequently in time-series data. • Pure serial correlation is caused by the underlying distribution of the error term of the true specification of an equation. • Impure serial correlation is caused by a specification error such as an omitted variable or incorrect functional form. • We here study about the pure serial correlation. Advanced Econometrics Chapter 9: Autocorrelation Nam T. Hoang University of New England - Australia 2 University of Economics - HCMC - Vietnam I. PROPERTIES OF OLS ESTIMATOR UNDER AUTOCORRELATION: 1. OLSβˆ is still unbiased. 2. OLSβˆ is still consistent. 3. OLSβˆ is longer best (efficient), it is less efficient than GLSβˆ variances. 4. VarCov( OLSβˆ ) ≠ 12 )'( −XXεσ : so the standard errors of sj 'βˆ are biased (downward) and inconsistent because they are based on incorrect formula. 5. t-statistic, R2, overall F-statistics upward. II. DISTURBANCE PROCESS: For testing or treatment we need to make more explicit assumption about the type of autocorrelation. The most common is first order autoregressive process [AR(1)]. ttt u+= −1ρεε ut satisfies all classical assumptions.      ≠= = = )(0)( )( 0)( 22 stuuE uE uE st u t t σ ρ: coefficient of autocorrelation. |ρ| → stationary of εt. • Covariance stationary of εt: the mean variance and all autocovariances of εt are constant. Autocovariances: ssstttststt XCovXCov −−+− ==Ω== γγσεεεε , 2],[],[ So ],[ XCov stt −εε does not depend on t, only depend on s. ttt u+= −1ρεε ttt uu ++= −− ][ 12ρερ ttt uu ++= −− 12 2 ρερ Advanced Econometrics Chapter 9: Autocorrelation Nam T. Hoang University of New England - Australia 3 University of Economics - HCMC - Vietnam tttt uuu +++= −−− 123 2 ][ ρρερ tttt uuu +++= −−− 12 2 3 3 ρρερ ... tε ∑ − = −− += 1 0 n j jt j nt n uρερ |ρ| < 1 n →∞ tε ∑ ∞ = −= 0j jt juρ → εt unrelated to future of ut, ut+j, ... this is called infinite moving average process. Moment of εt : )( tE ε       = ∑ ∞ = − 0j jt juE ρ 0)( 0 0 == ∑ ∞ = − j jt j uE  ρ 2 1 2 )()()( tttt uEEVar +== −ρεεε → )2()( 1 22 1 22 ttttt uuEE −− ++= ρεερε →  0 1 22 1 22 )(2)()()( 222 ttttt uEuEEE u −− ++= ερερε σσσ εε → 2222 uσσρσ εε += → 2 2 2 1 ρ σ σε − = u Autocovariance: )( 1−ttE εε [ ]11 )( −− += ttt uE ερε 2 0 1 2 1 ),()( ερσερερ =+= −−  ttt uEE Advanced Econometrics Chapter 9: Autocorrelation Nam T. Hoang University of New England - Australia 4 University of Economics - HCMC - Vietnam → )( sttE −εε 2 εσρ s= ),( 1−ttCorr εε ρσ ρσ εε εε ε ε === − − 2 2 1 1 )()( ),( tt tt VarVar Corr → ρεε =− ),( 1ttCorr ),( 2−ttCov εε ),( 2−= ttE εε [ ]21 −−= ttE εερ 22 0 2 ),( εσρε =+ −  ttuE → 22 ),( ρεε =−ttCorr → ssttCorr ρεε =− ),( We can use this to construct the matrix Ω in E(εε') = Ω 2 εσ                  =Σ −−− − − − − 1 1 1 1 321 32 2 12 1 2 2 2      TTT T T T u ρρρ ρρρ ρρρ ρρρ σ ρ σ ε 2 2 2 1 ρ σ σε − = u     = = − − s stt s stt Corr Cov ρεε σρεε ε ),( )( 2 s = 1, 2, ..., T-1 III. ESTIMATION UNDER AUTOCORRELATION: 1. Estimation with known ρ: We can find GLS estimator: YXXXGLS 111 ')'(ˆ −−− ΩΩ=β Find matrix H such that: H'H = Ω-1. Advanced Econometrics Chapter 9: Autocorrelation Nam T. Hoang University of New England - Australia 5 University of Economics - HCMC - Vietnam HY = HXβ + Hε meets all classical assumptions. Choose 2 2 1 1 1000 010 001 0001 ρ ρ ρ ρ −                 − − − =      H →                 −                 − − − = TY Y Y Y HY       3 2 1 2 2 1 1 1000 010 001 0001 ρ ρ ρ ρ                 − − − − = −1 23 12 1 21 TT YY YY YY Y ρ ρ ρ ρ  For HX:                 −−−− −−−− −−−− −−−− = −−−− kTTkTTTTTT kk kk k XXXXXXXX XXXXXXXX XXXXXXXX XXXX HX ,13,132,121,11 23233322322131 12132312221121 1 2 13 2 12 2 11 2 1111 ρρρρ ρρρρ ρρρρ ρρρρ                      =                 − 1 1 1 1 1,1 31 21 11   becan X X X X T For Hε                 − − − − = −1 23 12 1 21 TT H ρεε ρεε ρεε ερ ε  So transformed model is: Advanced Econometrics Chapter 9: Autocorrelation Nam T. Hoang University of New England - Australia 6 University of Economics - HCMC - Vietnam (i) ∑ = −+−=− k j jj XY 1 1 2 1 2 1 2 1)1(1 ερρβρ (i) ∑ = −−− −+−=− k j u tt X jttjj Y tt ttjt XXYY 1 1,11 ** )(      ρεερβρ t = 2, 3, ... T. This is also called "Autoregressive transformation" or "quasi-differencing" "rho- transformation" Note: YXXXGLS 111 ')'(ˆ −−− ΩΩ=β 2. Estimation with unknown ρ: Using Cochrane – Orcutt procedure: (1) Estimate Y X β ε= + by OLS, save te ’s (2) Use te ’s to estimate ρ from regression. 1t t te e uρ −= + → 1 2 2 1 2 ˆ T t t t T t t e e e ρ − = − = = ∑ ∑ (3) Transform the model as in (ii) by quasi-differencing the data and estimate (ii) by OLS. Stop here → Cochrane – Orcutt. (4) Use ˆ jβ from step 3 to compute new te ’s algebraically from Y X β ε= + again. 1 ˆ k t t j tj j e Y Xβ = = −∑ (5) Repeat step 2 → 4 until convergence ( ρˆ ’s at 2 successive step differ by less than 0.001). Exercise: For AR(2) process. 1 1 2 2t t t tuε ρ ε ρ ε− −= + + where ut meets classical assumptions.  Define the quasi-differencing that eliminate autocorrelation.  Spell out the iterative Cochrane – Orcutt procedure for this model. Advanced Econometrics Chapter 9: Autocorrelation Nam T. Hoang University of New England - Australia 7 University of Economics - HCMC - Vietnam Cochrane – Orcutt procedure for AR(2) model: AR(2) process: 1 1 2 2t t t tuε ρ ε ρ ε− −= + + ut meets classical assumptions: Quasi-differencing: ( ) ( )1 1 2 2 1 1, 2 2, 1 1 2 2 1 ( ) t k t t t j tj t j t j t t t j u Y Y Y X X Xρ ρ β ρ ρ ε ρ ε ρ ε− − − − − − = − − = − − + − −∑  (t = 3,4, T) Procedure: (1) Estimate Y X β ε= + by OLS, save te ’s (2) Estimate 1 1 2 2t t t tuε ρ ε ρ ε− −= + + by OLS, to get 1ρˆ , 2ρˆ . (3) Use 1ρˆ , 2ρˆ to quasi-differencing, estimate β by OLS. Stop here → 2 steps Cochrane – Orcutt. (4) Use ˆ jβ from step 3 to compute new te ’s algebraically from ˆY X β ε= + again. (5) Repeat step 2 → 4 until convergence.