Advanced Econometrics - Chapter 10: Models for panel data

Advanced Econometrics Chapter 10: Models for panel data Nam T. Hoang University of New England - Australia 1 University of Economics - HCMC - Vietnam Chapter 10 MODELS FOR PANEL DATA I. GENERAL FRAMEWORK FOR PANEL DATA Panel data (longitudinal data): same entities are observed overtime. The basic framework for panel data is a regression model of the form: (1) ∑ = ×× ++= k j it ll iitjjit zXY 1 )1)(1( εαβ it ll i kk it zX εαβ ++= ×××× )1)(1()1)(1( Where: [ ]21 ...i i ilz z z= 1 2 ... l α α α α      =       [ ]1 2 ...it it it itkX X X X= There are k regressors in itX , not including a constant term. The heterogeneity or individual effect is ( )iz α where iz contains a constant term and a set of individual or group specific variable, which may be observed, such as race, sex, location, or unobserved, such as family specific characteristics, individual heterogeneity in skill or preferences, All of which are taken to be constant over time t. Therefore: [ ]21 ...i i ilz z z= = constant over time t. If iz is observed for all cross-sections (individuals), then the entire model can be treated as an ordinary linear model and fit by least squares. When iz is not observed (most of the cases), complications arise that leads to main objective of the analysis will be consistent and efficient estimation of the partial effects. Advanced Econometrics Chapter 10: Models for panel data Nam T. Hoang University of New England - Australia 2 University of Economics - HCMC - Vietnam it it it E Y X X β ∂   = ∂ Assumption of strict exogeneity: ( )1 2, ,..., 0it i i iTE X X Xε = That is, the current disturbance is uncorrelated with the independent variable in every period of t. Assumption of mean independence: ( )1 21 , , ,i i i iTlE z X X Xα×  1 1α×= Or = (fixed effect) = ( )i ih X α= II. POOLED REGRESSION If [ ]21 ...i i ilz z z= =1, which contains only a constant term. Then OLS provides consistent and efficient estimates of the intercept α and the slope vector β (common effect model). Thus equation (1) → itY it ll i kk it zX εαβ ++= ×××× )1)(1()1)(1( Where itX it obs of all k explanatory variables:  1 ( 1) 2 ( 1) ( 1) T N T NT Y Y Y × × ×                 =  1 ( 1) 2 ( 1) ( ) ( 1) T k N T k NT X X X β × × × ×                  ( 1) (1 1) ( 1) ( 1) T T NT i i i α × × × ×       +           1 ( 1) 2 ( 1) ( 1) T N T NT ε ε ε × × ×       +           ( 1) 1 1 1 T i ×      =        → 1st country: 1 1 1Y X iβ α ε= + + with classical assumptions: 2 ( ) 0 ( ) ( , ) 0 it it it js E Var Cov ε ε ε σ ε ε =  =  = with ( ) ( )i j or t s∀ ≠ ≠ Advanced Econometrics Chapter 10: Models for panel data Nam T. Hoang University of New England - Australia 3 University of Economics - HCMC - Vietnam III. FIXED EFFECTS 1. Assumption: 11 ( )i ll E z α ×× = 1 1 iα × = ( )ih X Different across units can be captured in differences in the constant term: it it i itY X zβ α ε= + + ( )it it i it i iY X zβ α ε α α= + + + − White noise ( )i i izυ α α= − = ( )i iz h Xα − Because ( )i iz h Xα − is uncorrelated with Xi, we may absorb it in the disturbance: i it iη ε υ= + it it i itY X β α ε= + + Each iα is treated as an unknown parameter to be estimated: ( 1)NT Y × 1 ( 1) (1 1) 2 ( ) ( 1)( 1) T NT k NTk N i iX i α αβ ε α × × × ××      = + +        [ ] 1 (1 1) 2 1 2 ( 1) ( ) N NT NT N D N X d d d α αβ ε α × × ×      = + +          1 1 1 1 0 0 0 0 0 0 d                  =                       0 0 0 1 1 1 0 0 0 id                  =                       Advanced Econometrics Chapter 10: Models for panel data Nam T. Hoang University of New England - Australia 4 University of Economics - HCMC - Vietnam  1 2 ( 1) N NT Y Y Y ×              =  1 2 ( ) N NT k X X X ×              ( 1)k β × ( ) 0 0 0 0 0 0 NT N i i i ×      +                1 ( 1) 2 ( 1) ( 1) T N T N α α α × × ×                  1 ( 1) 2 ( 1) ( 1) T N T NT ε ε ε × × ×       +           ( 1)NT Y × ( ) ( ) ( 1) ( 1)( 1)NT k NT N N NTk X Dβ α ε × × × ×× = + + This model is also called “Least Squares Dummy Variables” (LSDV) because it can be estimated directly with the intercept dummies. 2. The Within-Groups and Between-Groups Estimators There are 3 ways to estimate the pooled regression model: it it itY Xα β ε= + + Using OLS: → ( 1) ˆ OLS k β × 1 1 1 ( ) '( ) N T it it i t X X X X − = =   = − −    ∑∑ 1 1 ( ) '( ) N T it it i t X X Y Y = =   − −    ∑∑ Note: (1 ) ( )it k X X × − and ( 1) ( ) 'it k X X × − (1 )k X × and ( 1)kY× are overall means: (1 ) (1 ) 1 itk k X X NT× ×   =     ∑∑ ( 1) ( 1) 1 itk k Y Y NT× ×   =     ∑∑ Using the deviation from the group means ( ). . . ( 1)( 1) (1 ) (1 ) it i it i it i kk k k Y Y X Xα β ε ε ×× × ×     − = + − + −        We get the Within – Group estimator: → ˆLSDVβ ( 1) ˆ within k β × = 1 . . 1 1 ( ) '( ) N T it i it i i t X X X X − = =   = − −    ∑∑ . . 1 1 ( )( ) N T it i it i i t X X Y Y = =   − −    ∑∑ This is also the LSDV or fixed effect estimator of 𝛽 Advanced Econometrics Chapter 10: Models for panel data Nam T. Hoang University of New England - Australia 5 University of Economics - HCMC - Vietnam Where . (1 ) 1 . (1 1) 1 1 1 T i it k t T i it t X X T Y Y T × = × =  =   =  ∑ ∑ group means. (i = 1, 2, ..., N) We can write the model in terms of the group means: . . .i i iY Xα β ε= + + (i = 1, 2, ..., N) We use only N observations, (the group means). Apply OLS to N observations, we get the Between – Group estimator: → ˆBetweenβ 1 . . 1 ( ) '( ) N i i i X X X X − =   = − −    ∑ . . 1 ( )( ) N i i i X X Y Y =   − −    ∑ Back to the fixed effects: → ˆFEβ [ ] [ ] 1' 'D DX M X X M Y −= where 1( ' ) 'DM I D D D D −= − DM = 0 0 0 0 0 0 0 0 0 M M M                    0 1 'TM I iiT = − [ ] 12ˆ ˆ( ) 'FE DVarCov X M Xβ σ −= 2 1 ˆ ˆˆ D DY M X Y M XNT N K σ β β′   = − −   − − Note: Why pooled estimator, within group estimator and between group estimator are different? Because: • Pooled estimator: 2 ˆ 1 1pooled N T it i t Min e β = =       ∑∑ • Within-group estimator: 2 .ˆ 1 1 ( ) within N T it i i t Min e e β = =   −    ∑∑ • Between-group estimator: 2 .ˆ 1 1 ( ) between N T i i t Min e β = =       ∑∑ There are 3 different minimum problems, eit are the same from: Advanced Econometrics Chapter 10: Models for panel data Nam T. Hoang University of New England - Australia 6 University of Economics - HCMC - Vietnam ˆ it it itY X eα β= + + ˆ it it ite Y Xα β = − −  Note that in deviation form: ( ) ( ) ˆi i i iY Y X X e eβ− = − + − and ˆi i iY X eα β= + + , βˆ and ei are the same. 3. Fixed time and Group Effects The LSDV approach can be extended to include a time specific effect as well: it it i t itY X β α γ ε= + + + 1 2 2 31 2 2 3N T it it it it it it it it it N T Y X d d d g g g α γ α γ β ε α γ               = + + +                  (one of the time effects must be dropped to avoid perfect co linearity – the group effects and time effects both sum to one) 1 0 j any t it if i j d if not ==   1 0 s any i it if t s g if not ==   For panel data now we can use • Pooled • Fixed effects o Time effects only o Group effects only o Time and group effect 4. Unbalanced Panel: ˆ LSDVβ ˆFEβ= ( 1) ˆ within k β × = 1 . . 1 1 ( ) '( ) iTN it i it i i i X X X X − = =    = − −      ∑ ∑ 1 . . 1 1 ( ) '( ) iTN it i it i i i X X Y Y − = =    − −      ∑ ∑ 5. Allow for autocorrelation and heteroskedasticity in error term: , 1it i i t ituε ρ ε −= + 1iρ < Advanced Econometrics Chapter 10: Models for panel data Nam T. Hoang University of New England - Australia 7 University of Economics - HCMC - Vietnam ( ) 0itE u = ( ) 0it isE u u = for t s≠ (no temporal auto in uit) ( ) 0it jtE u u = for i j≠ (no spatial auto) ( ) 0it jsE u u = for i j≠ and t s≠ 2 2( )itE u σ= i = 1, 2, ..., N (spatial heteroskedasticity but no temporal auto) Estimation: • Estimate (1) by OLS → get eit's • 1 2 2 1 2 ˆ T it it t T it t e e e ρ − = − = = ∑ ∑ for i = 1, 2, ..., N (use all NT observations). or 1 1ˆ ˆ n i iN ρ ρ = = ∑ if T is small. • Use ˆiρ 's to quasi-difference (1): ( ) ( ) ( )1 1 1ˆ ˆ ˆ ˆ(1 ) it it i it it i it i it i it u Y Y X Xρ ρ β α ρ ε ρ ε− − − ≈ − = − + − + −  Estimate by OLS with N(T-1) observations → ˆitu • 2 2 2 ˆ ˆ 1i T it t u u T k σ == − − ∑ or 2 2 2 ˆ ˆ 1i T it t u u T σ == − ∑ • From WLS: * * 1 ˆ ˆ ˆ ˆ i i i i it it i it u u u u Y X uρβ α σ σ σ σ        − = + +                       OLS → estimator ⇒ BLUE 6. Testing for Group - Special Effects: The standard F test can be used to test whether the pooled or fixed-effect model is more appropriate: H0: 1 2 NC C C α= = = = * i iC z α= Advanced Econometrics Chapter 10: Models for panel data Nam T. Hoang University of New England - Australia 8 University of Economics - HCMC - Vietnam ( ) ( ) 2 2 2 ( ) ( 1) 1, 1 ( ) LSDV pooled LSDV U R R NF F N NT N k R ESS NT N k − −= − − − − − −  7. Shortcoming of the fixed effects: The coefficients on the time-invariant variables cannot be estimated by within estimators. IV. RANDOM EFFECTS MODEL: 1. Model: * it it i itY X zβ α ε= + + Denote: *i iz cα = Fixed effects assume ci are correlated with Xi ( )i iE c X = ( )ih X = 1 1i α × Random effects assume ci are uncorrelated with Xi ( )i iE c X = α = constant. → i ic uα= + with ui ∼ iid. then ( )it it i itY X uβ α ε= + + + ( )it i itX uα β ε= + + + (ci and Xi are not correlated). The following assumptions are made: ( ) ( ) 0it NT kE Xε × = ( )( ) 0it NT kE u X× = 2 2( )itE εε σ= 2 2( )it uE u σ= ( ) 0it jsE ε ε = for i j≠ or t s≠ or both. ( ) 0i jE u u = for i j≠ ( ) 0i jsE u ε = for all i,j,s Let: it i ituη ε= + 2 2 2 2( ) ( )it i it uE E u εη ε σ σ= + = + ( ) ( )( ) 0it jt i it j jtE E u uη η ε ε= + + = Advanced Econometrics Chapter 10: Models for panel data Nam T. Hoang University of New England - Australia 9 University of Economics - HCMC - Vietnam 2( ) ( )( )it is i it i is uE E u uη η ε ε σ= + + = ( 1) ( 1)NT NT Y X β η × × = + 1 2 ( 1) ( 1) ( 1) NT N T NT η η η η × × ×      =         2 2 2 2 2 2 2 2 ' 00 ( )( ) 2 2 2 2 ( ) u u u u u u i j T TT T u u u E ε ε ε σ σ σ σ σ σ σ σ ηη σ σ σ σ ××  +  + = = Σ     +         ' ( )( ) ( ) 0i j T TT T E ηη × × = So: ' ( ) ( ) ( )i jNT NT NT NT E ηη × × Σ = = 00 00 00 0 0 0 0 0 0 Σ   Σ     Σ         We can estimate ˆREβ by GLS estimation: ( 1) ˆ RE k β × 1 1 1( ' ) ( ' )X X X Y− − −= Σ Σ ' 1 ' * * * *( ) ( )X X X Y −= [ ] [ ]1( ) '( ) ( ) '( )PX PX PX PY−= 1'P P −= Σ → 1/2 00NP I −= ⊗ Σ 1/2 00 1/2 00 1/2 00 0 0 0 0 0 0 − − −  Σ  Σ =     Σ         1/2 ' 00 1 T T TI i iTε θ σ −  Σ = −    2 2 1 uT ε ε σθ σ σ = − + 1 . 2 . * . 1 i i i i i iT i Y Y Y Y Y Y Y ε θ θ σ θ −   − =    −   and the same for X*i Advanced Econometrics Chapter 10: Models for panel data Nam T. Hoang University of New England - Australia 10 University of Economics - HCMC - Vietnam 2. Feasible Estimation: We need to estimated 2εσ and 2 uσ Estimation of 2εσ : ( )it it i itY X uα β ε= + + + → . . .( )i i i iY X uα β ε= + + + → ( ) ( ). . .( )it i it i it iY Y X X β ε ε− = − + + (*) Estimate (*) using OLS and use the residuals to get the estimation of 2εσ 2 . 2 1 1 ( ) ˆ N T it i i t e e NT N kε σ = = − = − − ∑∑ Estimation of 2uσ : * . . .( )i i i i e u Y Xε α β+ = − −  .( )i iVar u ε+ = 2 2 u T εσσ + Estimation of .( )i iVar u ε+ is ' * * ˆe e N k T εσ− − Insert 2εσ , 2 uσ into Σ and calculate ˆREβ V. CHOOSING BETWEEN FIXED AND RANDOM EFFECTS MODELS: Whether *i ic z α= are correlated with (Xi) or not. If they are → RE will produce inconsistent estimates. If they are not → RE model may be preferable. 1. Think through the problem: 2. Hausman Specification Test: ( ) ( ) ( ) ( ) 1 2 ( 1) ˆ ˆ ˆ ˆ ˆ ˆ within RE within RE within RE kW Var Varβ β β β β β χ − − ′  = − − −   H0: no correlation between Ci and Xi HA: correlation between Ci and Xi Advanced Econometrics Chapter 10: Models for panel data Nam T. Hoang University of New England - Australia 11 University of Economics - HCMC - Vietnam Under H0: Both ˆFEβ and ˆREβ are consistent estimators but only ˆREβ is efficient. VI. FINDING BIG Σ: Example 1: , 1it i i t ituε ρ ε −= + 1iρ < ( ) 0itE u = 2 2( ) iit u E u σ= ( ) 0it jtE ε ε = ( ) 0it isE ε ε = for t s≠ (no temporal auto) ( ) 0it jsE ε ε = for t s≠ (no temporal auto)  1 ( 1) 2 ( 1) ( 1) T N T NT ε ε ε ε × × ×       =           ( ')E εεΣ = off-diagonal blocks t s≠ : ( 1) ( 1) ( ' ) 0i j T T E ε ε × × = diagonal blocks ( 1) ( 1) ( ' )i i T T E ε ε × × 2 1 2 2 2 ( ) 1 2 3 1 1 1 i T i i i T i i i T T T T T i i i ε ρ ρ ρ ρ ρ ρ σ ρ ρ ρ − − × − − −      = = Ω              ( ) ( ) NT NT E εε × ′ 1 2 2 1 2 2 2 0 0 0 0 0 0 N N ε ε ε σ σ σ  Ω  Ω =     Ω          with 2 2 2 ˆ ˆ ˆ1 i i u i ε σ σ ρ = − Example 2: Now assume no temporal autocorrelation. But allow spatial autocorrelation and cross-section heteroskedasticity. ( ) 0itE u = 2 2( ) iit u E u σ= Advanced Econometrics Chapter 10: Models for panel data Nam T. Hoang University of New England - Australia 12 University of Economics - HCMC - Vietnam ( ) 0it jt ijE ε ε σ= ≠ spatial autocorrelation ( ) 0it isE ε ε = for t s≠ (no temporal auto) ( ) 0it jsE ε ε = for t s≠ (no temporal auto) ( ')E εεΣ =  1 ( 1) 2 ( 1) ( 1) T N T NT ε ε ε ε × × ×       =           Same country: ( ) ( ) NT NT E εε × ′ 0 0 0 0 0 0 ii ii ii ii I σ σ σ σ      = =              Different country i j≠ : ( 1) ( 1) ( ' )i j T T E ε ε × × 0 0 0 0 0 0 ij ij ij ij I σ σ σ σ      = =              ( ) ( ) NT NT E εε × ′ 11 12 1 21 22 2 1 2 N N N N NN I I I I I I I I I σ σ σ σ σ σ σ σ σ      =             