Random Effects Linear Model - Econometric Analysis of Panel Data - Lecture Slides, Slides of Econometrics and Mathematical Economics

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Random Effects Linear Model Docsity.com The Random Effects Model  The random effects model  ci is uncorrelated with xit for all t;  E[ci |Xi] = 0  E[εit|Xi,ci]=0 it it i it i i i i i i i i i i i i N i=1 i 1 2 N y = +c +ε , observation for person i at time t = +c + , T observations in group i = + + , note (c ,c ,...,c ) = + + , T observations in the sample c=( , ,... ) , ′ ′= Σ ′ ′ ′ ′ x β y X β i ε X β c ε c y Xβ c ε c c c Ni=1 iT by 1 vectorΣ Docsity.com Notation 2 2 2 2 u u u 2 2 2 2 u u u i i 2 2 2 2 u u u 2 2 u i i 2 2 u i 1 2 N Var[ +u ] = T T = = Var[ | ]  +  + =     +   ′+ × ′+   =    i i T T ε i I ii I ii Ω Ω 0 0 0 Ω 0 w X 0 0 Ω               ε ε ε ε ε σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ i (Note these differ only in the dimension T )        Docsity.com Regression Model-Orthogonality iN N i 1 i i 1 i i i iN i 1 i i i i i i i i N i i i 1 i 1 plim #observations 1 1 plim plim ( +u ) T T 1 plim T + Tu T T T T plim f + f u , 0 < f < 1 T T T pli = = = = = ′ ′Σ = Σ = Σ Σ  ′ ′ Σ Σ Σ    ′ ′ Σ Σ =  Σ  N N i=1 i i i=1 i i N Ni i i i i=1 i=1 N Ni i i i i=1 i=1 X'w 0 X w X ε i 0 X ε X i X ε X i i i i i i m f + f u T  ′ Σ Σ =    N Ni i i=1 i=1 X ε x 0 Docsity.com Convergence of Moments N i 1 iN i 1 i N i 1 iN i 1 i N N i 1 i u i 1 i i f a weighted sum of individual moment matrices T T f a weighted sum of individual moment matrices T T = f f T Note asymptoti = = = = = = ′′ = Σ = Σ ′′ = Σ = Σ ′ ′Σ + Σ i i i i i 2 2i i i i X XX X X Ω XX ΩX X X x xεσ σ i i i cs are with respect to N. Each matrix is the T moments for the T observations. Should be 'well behaved' in micro level data. The average of N such matrices should be likewise. T or T is assum ′i iX X ed to be fixed (and small). Docsity.com Estimating the Variance for OLS  1 1 N N N N i 1 i i 1 i i 1 i i 1 i N i 1 iN i 1 i N i 1 iN i 1 i Var[ | ] T T T T f , where = =E[ | ] T T In the spirit of the White estimator, use ˆ ˆ ˆf , T T − − = = = = = = = =      ′ ′ ′ =     Σ Σ Σ Σ      ′′ ′= Σ Σ ′ ′′ = Σ Σ i i i i i i i i i i i i 1 X X X ΩX X Xb X X Ω XX ΩX Ω w w X X w w XX ΩX w = Hypothesis tests are then based on Wald statistics. i iy - X b THIS IS THE 'CLUSTER' ESTIMATOR Docsity.com Mechanics ( )1 1Ni 1 i i ˆ ˆEst.Var[ | ] ˆ = set of T OLS residuals for individual i. = TxK data on exogenous variable for individual i. ˆ = K x 1 vector of products ˆ ˆ( )( ) − − =′ ′ ′ ′= Σ       ′ ′ ′ ′ = i i i i i i i i i i i i b X X X X w w X X X w X X w X w w X ( ) ( )( ) ( )( ) ( )( ) N i 1 N N i 1 i 1 KxK matrix (rank 1, outer product) ˆ ˆ = sum of N rank 1 matrices. Rank K. ˆˆ ˆWe could compute this as = . Why not do it that way? = = = ′ ′Σ ≤ ′ ′ ′Σ Σ i i i i i i i i i i i X w w X X w w X X Ω X Docsity.com Cornwell and Rupert Data Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 Years Variables in the file are EXP = work experience, EXPSQ = EXP2 WKS = weeks worked OCC = occupation, 1 if blue collar, IND = 1 if manufacturing industry SOUTH = 1 if resides in south SMSA = 1 if resides in a city (SMSA) MS = 1 if married FEM = 1 if female UNION = 1 if wage set by unioin contract ED = years of education BLK = 1 if individual is black LWAGE = log of wage = dependent variable in regressions These data were analyzed in Cornwell, C. and Rupert, P., "Efficient Estimation with Panel Data: An Empirical Comparison of Instrumental Variable Estimators," Journal of Applied Econometrics, 3, 1988, pp. 149-155. See Baltagi, page 122 for further analysis. The data were downloaded from the website for Baltagi's text. Docsity.com Generalized Least Squares i 1 N 1 N i 1 i 1 2 T2 2 2 i u i ˆ=[ ] [ ] =[ ] [ ] 1 I T (note, depends on i only through T ) − − = = ε ε ε ′ ′ ′ ′Σ Σ  σ ′= − σ σ + σ  -1 -1 -1 -1 i i i i i i -1 i β X Ω X X Ω y X Ω X X Ω y Ω ii Docsity.com Panel Data Algebra (1) 2 2 ε u i i 2 2 2 2 ε u ε 2 2 ε ε 2 ε = σ +σ , depends on 'i' because it is T T = σ [ ], = σ / σ = σ [ ] = σ [ ], = , = . Using (A-66) in Greene (p. 822) 1 1 σ 1+ = ′ × ′ρ ρ ′ ′ρ + ρ  ′=  ′  i 2 i 2 i -1 -1 -1 -1 i -1 Ω I ii Ω I + ii Ω I + ii A bb A I b i Ω A - A bb A b A b 2 2 u 2 2 2 2 2 ε i ε ε i u σ1 1 1 = σ 1+T σ σ +Tσ    ′ ′ρ   ρ    I - ii I - ii Docsity.com Panel Data Algebra (2) 2 2 2 2 2 2 ε u ε i u ε i u 2 2 ε i u 2 2 2 2 2 ε i u i ε ε i u 2 2 ε i u 2 ε (Based on Wooldridge p. 286) σ +σ σ +Tσ σ +Tσ σ +Tσ ( ) (σ +Tσ )[ ( )], = σ /(σ +Tσ ) (σ +Tσ )[ ] (σ ′ ′ ′= = = = − = + η − η = + η = -1 i i D i D i i D D i i D D Ω I ii I i(i i) i I P I I M P I P P M 2 i u 1 i 1 / 2 i i i i i a a i +Tσ ) (1 / ) (Prove by multiplying. .) 1 (1 / ) , =1- 1 (Note ) − − = + η =  = + η = − θ θ η − θ = + η i i i i i i D D D D i i i i D D D i i i D D S S P M P M 0 S P M I P S P M Docsity.com Estimators for the Variances i i it it i OLS T TN 2 N 2 2 i 1 t 1 it i 1 t 1 U 2 2 2 U i LSDV i y u With a consistent estimator of , say , (y ) estimates ( ) Divide by something to estimate = With the LSDV estimates, a and , = = = = ε ε = ′= + ε + ′Σ Σ Σ Σ σ + σ σ σ + σ Σ it it x β β b - x b b  i iT TN 2 N 2 1 t 1 it i 1 t 1 2 2 2 2 2 U U (y a ) estimates Divide by something to estimate Estimate with ( ) - .ˆ = = = ε ε ε ε ′Σ Σ Σ σ σ σ σ + σ σ i it- - x b Docsity.com Feasible GLS  i 2 2 u TN 2 2 i 1 t 1 it i it LSDV N i 1 i N 2 2 i 1 u Feasible GLS requires (only) consistent estimators of and . Candidates: (y a ) From the robust LSDV estimator: ˆ T K N From the pooled OLS estimator: ε = = ε = = ε σ σ ′Σ Σ − − σ = Σ − − Σ Σ σ + σ = x b  i i i T 2 t 1 it OLS it OLS N i 1 i N 2 2 2 i 1 it i MEANS u T 1 TN 2 2 i 1 t 1 s t 1 it is it is i u u N i 1 i (y a ) T K 1 (y a ) From the group means regression: / T N K 1 ˆ ˆw w (Wooldridge) Based on E[w w | ] if t s, ˆ T K = = = ε − = = = + = ′− − Σ − − ′Σ − − σ + σ = − − Σ Σ Σ = σ ≠ σ = Σ − − x b x b X  N There are many others. x´ does not contain a constant term in the preceding. Docsity.com Practical Problems with FGLS 2 uAll of the preceding regularly produce negative estimates of . Estimation is made very complicated in unbalanced panels. A bulletproof solution (originally used in TSP, now LIMDEP and others). From th σ  i i i i TN 2 2 i 1 t 1 it i it LSDV N i 1 i TN 2 2 2 2i 1 t 1 it OLS it OLS u N i 1 i T TN 2 N 2 i 1 t 1 it OLS it OLS i 1 t 1 i u (y a ) e robust LSDV estimator: ˆ T (y a ) From the pooled OLS estimator: ˆ T (y a ) (y ˆ = = ε = = = ε ε = = = = = ′Σ Σ − − σ = Σ ′Σ Σ − − σ + σ = ≥ σ Σ ′Σ Σ − − − Σ Σ σ = x b x b x b 2t i it LSDV N i 1 i a ) 0 T= ′− − ≥ Σ x b x´ does not contain a constant term in the preceding. Docsity.com Testing for Effects: LM Test 2 2N 2 N 2 i 1 i i 1 i i N N 2 N i 1 i 1 it i 1 i Breusch and Pagan Lagrange Multiplier statistic Assuming normality (and for convenience now, a balanced panel) (Te ) [(Te ) ]NT NT LM= 1 2(T-1) e 2(T-1) Co = = = = =    ′Σ Σ − − =   ′Σ Σ Σ   i i e e e e i N 2 N i 1 i i 1 i nverges to chi-squared[1] under the null hypothesis of no common effects. (For unbalanced panels, the scale in front becomes ( T ) /[2 T (T 1)].)= =Σ Σ − Docsity.com Testing for Effects: Moments ( ) N T-1 T i=1 t=1 s=t+1 it is 2N T-1 T i=1 t=1 s=t+1 it is Wooldridge (page 265) suggests based on the off diagonal elements e e Z= e e which converges to standard normal. ("We are not assuming any particular distribut Σ Σ Σ Σ Σ Σ it 2 ion for the . Instead, we derive a similar test that has the advantage of being valid for any distribution...") It's convenient to examine Z which, by the Slutsky theorem converges (also) to chi- sq ε uared with one degree of freedom. Docsity.com Testing (2) T-1 T t=1 s=t+1 it is T-1 T t=1 s=t+1 it is i e e = 1/2 of the sum of all off diagonal elements of = 1/2 the sum of all the elements minus the diagonal elements. e e =1/2[ ( ) ]. But, = Te . S Σ Σ ′ ′ ′ ′ ′Σ Σ − i i i i i i i e e i e e i e e i e { } T-1 T 2 t=1 s=t+1 it is i 2N 2 N 2 i 1 i2 i 1 N 2 2 N 2 2 i 1 i i 1 i N 2i 1 i i iN 2 ri 1 i o, e e = (1/2)[(Te ) ] [(Te ) ] [ ]2(T 1) Z LM [(Te ) ] NT [(Te ) ] Note, also r N r Z= ,where r (Te ) . sr The claim th = = = = = = ′Σ Σ − ′Σ − ′Σ− = = × ′ ′Σ − Σ − Σ ′= = − Σ i i i i i i i i i i i i e e e e e e e e e e e e i i 1,...,Nat one function of [e , ] is more valid than the other seems a little dubious. =′i ie e Docsity.com Hausman Test for FE vs. RE Estimator Random Effects E[ci|Xi] = 0 Fixed Effects E[ci|Xi] ≠ 0 FGLS (Random Effects) Consistent and Efficient Inconsistent LSDV (Fixed Effects) Consistent Inefficient Consistent Possibly Efficient Docsity.com Hausman Test for Effects -1 d ˆ ˆBasis for the test, ˆ ˆˆ ˆ ˆ ˆWald Criterion: = ; W = [Var( )] A lemma (Hausman (1978)): Under the null hypothesis (RE) ˆ nT[ ] N[ , ] (efficient) ˆ nT[ ′ → FE RE FE RE RE RE FE β - β q β - β q q q β - β 0 V β d] N[ , ] (inefficient) ˆ ˆˆNote: = ( )-( ). The lemma states that in the ˆ ˆjoint limiting distribution of nT[ ] and nT , the limiting covariance, is . But, = → − FE FE RE RE Q,RE Q,RE FE,R - β 0 V q β - β β β β - β q C 0 C C - . Then, Var[ ] = + - - . Using the lemma, = . It follows that Var[ ]= - . Based on the preceding ˆ ˆ ˆ ˆ ˆ ˆH=( ) [Est.Var( ) - Est.Var( )] ( ′ ′ E RE FE RE FE,RE FE,RE FE,RE RE FE RE -1 FE RE FE RE FE RE V q V V C C C V q V V β - β β β β - β ) β does not contain the constant term in the preceding. Docsity.com Computing the Hausman Statistic 1 2 N i 1 i i -1 2 2 N i ui i 1 i i 2 2 i i u 2 2 u 1ˆEst.Var[ ] Iˆ T Tˆ ˆˆEst.Var[ ] I , 0 = 1ˆˆ T Tˆ ˆ ˆAs long as and are consistent, as N , Est.Var[ˆ ˆ − ε = ε = ε ε    ′ ′= σ Σ −          σγ′ ′= σ Σ − ≤ γ ≤   σ + σ    σ σ → ∞ FE i RE i F β X ii X β X ii X β 2 ˆ] Est.Var[ ] will be nonnegative definite. In a finite sample, to ensure this, both must be computed using the same estimate of . The one based on LSDV willˆ generally be the better choice. Note ε − σ E REβ ˆthat columns of zeros will appear in Est.Var[ ] if there are time invariant variables in . FEβ X β does not contain the constant term in the preceding. Docsity.com