Econometric Analysis of Panel Data. William Greene Department of Economics Stern School of Business

Transcription

1 Econometrc Analyss of Panel Data Wllam Greene Department of Economcs Stern School of Busness

2 Econometrc Analyss of Panel Data 5. Random Effects Lnear Model

3 The Random Effects Model The random effects model y= x β+c+, observaton for person at tme t t t t y= Xβ+c +, T observatons n group = X β+ c +, note c = (c,c,...,c ) y = Xβ+ c +, Σ T observatons n the sample 1 c s uncorrelated wth x t for all t; E[c X ] = 0 E[ t X,c ]=0 =1 c=( c, c,... c ), Σ =1 T by 1 vector

4 Error Components Model Generalzed Regresson Model y = xb + +u t t t E[ X ] = 0 t E[ X ] = σ t E[u X ] = 0 E[u X ] = σ u y= Xβ+ +u for T observatons Var[ +u ] σ + σu σu σ u σu σ + σu σu = σ u σu σ + σu

5 otaton y1 X1 1 u11 T 1 observatons u y X T observatons = β + + y X u T observatons = Xβ+ + u Σ T observatons = Xβ+ w =1 In all that follows, except where explctly noted, X, X and x contan a constant term as the frst element. t To avod notatonal clutter, n those cases, x etc. wll smply denote the counterpart wthout the constant term. Use of the symbol K for the number of varables wll thus be context specfc but wll usually nclude the constant term. t

6 otaton σ + σu σu σu u u u Var[ +u ] σ σ + σ σ = σu σu σ + σu = σ I + σ T T Var[ w X] = = = T u σ I Ω T u 1 + σ Ω Ω Ω (ote these dffer only n the dmenson T )

7 Regresson Model-Orthogonalty 1 plm #observatons X'w = plm Σ Xw = plm Σ X ( +u ) = 0 Σ =1 =1 = 1T Σ= 1T 1 X X plm Σ T + Σ Tu =1 =1 Σ= 1T T T X X plm Σ f + Σ f =1 =1 u, 0 < f = < 1 T T Σ= 1T X plm Σ=1f + Σ =1 fxu = 0 T T

8 Convergence of Moments XX XX =Σ = 1f = a weghted sum of ndvdual moment matrces Σ T T = 1 X ΩX Σ T = 1 X ΩX =Σ = 1f = a weghted sum of ndvdual moment matrces T XX = σ Σ f + σ Σ f xx = 1 u = 1 T XX ote asymptotcs are wth respect to. Each matrx s the T moments for the T observatons. Should be 'well behaved' n mcro level data. The average of such matrces should be lkewse. T or T s assumed to be fxed (and small).

9 Random vs. Fxed Effects Random Effects Small number of parameters Effcent estmaton Objectonable orthogonalty assumpton (c X ) Fxed Effects Robust generally consstent Large number of parameters

10 Ordnary Least Squares Standard results for OLS n a GR model Consstent Unbased Ineffcent True Varance Var[ b X] XX XΩX XX = Σ= 1T Σ= 1T Σ= 1T Σ= 1T 0 Q Q * Q as wth our convergence assumptons

11 Estmatng the Varance for OLS = XX XΩX XX Var[ b X] = Σ= 1T Σ= 1T Σ= 1T Σ= 1T X ΩX X ΩX =Σ= 1f, where = Ω =E[ ww X] Σ T T In the sprt of the Whte estmator, use X ΩX XwwX ˆ ˆ =Σ ˆ = 1f, w = y -Xb Σ T T = 1 Hypothess tests are then based on Wald statstcs. THIS IS THE 'CLUSTER' ESTIMATOR

12 Mechancs ( ˆ ˆ ) = Est.Var[ b X] = XX Σ X w w X XX wˆ = set of T OLS resduals for ndvdual. X = TxK data on exogenous varable for ndvdual. Xw ˆ = K x 1 vector of products ( Xw ˆ )( wx ˆ ) = ( Xw )( wx ) ( Σ ) = 1 KxK matrx (rank 1, outer product) ˆ ˆ = sum of rank 1 matrces. Rank K. ( ˆ ˆ ) ( ) ( ) ( ) Σ ˆ = 1 Σ = 1 We could compute ths as X w w X = X Ω X. Why not do t that way?

13 Cornwell and Rupert Data Cornwell and Rupert Returns to Schoolng Data, 595 Indvduals, 7 Years Varables n the fle are EXP = work experence, EXPSQ = EXP WKS = weeks worked OCC = occupaton, 1 f blue collar, ID = 1 f manufacturng ndustry SOUTH = 1 f resdes n south SMSA = 1 f resdes n a cty (SMSA) MS = 1 f marred FEM = 1 f female UIO = 1 f wage set by unon contract ED = years of educaton BLK = 1 f ndvdual s black LWAGE = log of wage = dependent varable n regressons These data were analyzed n Cornwell, C. and Rupert, P., "Effcent Estmaton wth Panel Data: An Emprcal Comparson of Instrumental Varable Estmators," Journal of Appled Econometrcs, 3, 1988, pp See Baltag, page 1 for further analyss. The data were downloaded from the webste for Baltag's text.

14 OLS Results Resduals Sum of squares = Standard error of e = Ft R-squared = Adjusted R-squared = Varable Coeffcent Standard Error b/st.er. P[ Z >z] Mean of X Constant EXP EXPSQ D OCC SMSA MS FEM UIO ED

15 Alternatve Varance Estmators Varable Coeffcent Standard Error b/st.er. P[ Z >z] Constant EXP EXPSQ D OCC SMSA MS FEM UIO ED Robust Constant EXP EXPSQ D OCC SMSA MS FEM UIO ED

16 Generalzed Least Squares βˆ=[ X Ω X] [ XΩ y] =[ Σ X Ω X ] [ Σ XΩ y ] = 1 = σ I Ω = T σ σ + Tσu (note, depends on only through T )

17 Panel Data Algebra (1) Ω = σ I+σ, depends on '' because t s T T u Ω = σ [ I+ ρ ], ρ = σ / σ u Ω = σ [ I+ ρ ] = σ [ A+ bb ], A= I, b= ρ. Usng (A-66) n Greene (p. 8) 1 1 Ω = A - A bba σ 1+ ba b σ u = = σ I- ρ 1+Tρ σ I- σ +Tσ u

18 Panel Data Algebra () (Based on Wooldrdge p. 86) Ω = σ I+σ = σ I+Tσ () = σ I+Tσ P -1 u u u D = σ I+Tσ ( I M ) u = (σ +Tσ )[ P +η( I P )], η = σ /(σ +Tσ ) u D D u = (σ +Tσ )[ P +ηm ] = (σ u +Tσ ) S u D D D S = P + (1 / η ) M (Prove by multplyng. P M = 0.) 1 D D D D 1 S = P + (1 / η ) M = I θp, θ=1- η 1/ D D 1 θ D S = P +η M a a (ote D D)

19 Panel Data Algebra (, cont.) 1 Ω = (P + (1 / η ) M ) 1/ D D σ + Tσu u =1-1 σ + Tσu σ + Tσu 1/ 1 1 = T 1 I θp σ + σ θ σ θ = 1/ 1 [ Ω = I θpd ] σ Var[ Ω ( + u )] = σ I 1/ Ω y = (1 / σ )( y θy. ) for the GLS transformaton. σ D,

20 GLS (cont.) GLS s equvalent to OLS regresson of y* = y θ y. on x * = x θx., t t t t where θ = 1 ˆ σ σ + Tσ u Asy.Var[ β] = [ X Ω X] = σ[ X *X*]

21 Estmators for the Varances y = x β + + u t t t Wth a consstent estmator of β, say b, OLS (y -x b) estmates ( ) T T Σ Σ Σ Σ σ +σ = 1 t= 1 t t = 1 t= 1 U Dvde by somethng to estmate σ = σ +σ Wth the LSDV estmates, a and b, Σ U LSDV Σ (y - a - x b) estmates Σ Σ σ T T = 1 t= 1 t t = 1 t= 1 Dvde by somethng to estmate Estmate σ wth ( σ +σ ) - σˆ. U U σ

22 Feasble GLS Feasble GLS requres (only) consstent estmators of σ and σ. Canddates: From the robust LSDV estmator: σ ˆ = From the pooled OLS estmator: From the group means regresson: σ +σ = T Σ Σ u = 1 t= 1 t t LSDV Σ= 1T K Σ = 1 u (y a xb ) Σ σ / T +σ = (y a xb ) T t= 1 t OLS t OLS Σ= 1T K 1 Σ (y a xb ) = 1 t MEAS u K 1 T 1 Σ Σ Σ ww ˆ ˆ T = 1 t= 1 s= t+ 1 t s t s X =σu σ u = Σ= 1T K (Wooldrdge) Based on E[w w ] f t s, ˆ There are many others. x does not contan a constant term n the precedng.

23 Practcal Problems wth FGLS All of the precedng regularly produce negatve estmates of σ. Estmaton s made very complcated n unbalanced panels. A bulletproof soluton (orgnally used n TSP, now LIMDEP and others). From the robust LSDV estmator: σ ˆ = From the pooled OLS estmator: σ ˆ = σ +σ = T T Σ Σ (y a xb ) Σ Σ (y T Σ Σ = 1 t= 1 t t LSDV Σ= 1T T Σ Σ = 1 t= 1 t OLS t OLS u σ ˆ Σ= 1T = 1 t= 1 t OLS t OLS = 1 t= 1 t t LSDV u Σ= 1T u (y a xb ) (y a xb ) a xb ) 0 x does not contan a constant term n the precedng.

24 Computng Varance Estmators Usng full lst of varables (FEM and ED are tme nvarant) OLS sum of squares = σ + σ = / (4165-9) = u Usng full lst of varables and a generalzed nverse (same as droppng FEM and ED), LSDV sum of squares = = / ( ) = σ σ = = u Both estmators are postve. We stop here. If σ were negatve, we would use estmators wthout DF correctons. u

25 Applcaton Random Effects Model: v(,t) = e(,t) + u() Estmates: Var[e] =.31188D-01 Var[u] =.10531D+00 Corr[v(,t),v(,s)] = (Hgh (low) values of H favor FEM (REM).) Sum of Squares.14114D+04 R-squared D Varable Coeffcent Standard Error b/st.er. P[ Z >z] Mean of X EXP EXPSQ D OCC SMSA MS FEM UIO ED Constant

26 Testng for Effects: LM Test Breusch and Pagan Lagrange Multpler statstc Assumng normalty (and for convenence now, a balanced panel) = 1(Te ) T = 1[(Te ) ee ] = = 1 = 1 t = 1ee T Σ Σ LM= 1 (T-1) Σ Σ e (T-1) Σ Converges to ch-squared[1] under the null hypothess of no common effects. (For unbalanced panels, the scale n front becomes ( Σ T ) /[Σ T (T 1)].) = 1 = 1

27 Testng for Effects: Moments Wooldrdge (page 65) suggests based on the off dagonal elements Z= Σ Σ Σ ee T-1 T =1 t=1 s=t+1 t s ( ee ) Σ Σ Σ T-1 T =1 t=1 s=t+1 t s whch converges to standard normal. ("We are not assumng any partcular dstrbuton for the t. Instead, we derve a smlar test that has the advantage of beng vald for any dstrbuton...") It's convenent to examne Z whch, by the Slutsky theorem converges (also) to chsquared wth one degree of freedom.

28 Σ Σ T-1 T t=1 s=t+1 t s ee Testng () e e = 1/ of the sum of all off dagonal elements of = 1/ the sum of all the elements mnus the dagonal elements. Σ Σ e e =1/[ ( ee ) ee ]. But, e = Te. So, T-1 T t=1 s=t+1 t s Σ Σ e e = (1/)[(Te ) ee ] Z T-1 T t=1 s=t+1 t s { 1 ee } Σ = [(Te ) ] (T 1) [ Σ= 1ee ] = = LM Σ= 1[(Te ) ee ] T Σ= 1[(Te ) ee ] ote, also Σ r r = 1 Z= =, where r = (Te ) ee. Σ s = 1r r The clam th ee = 1,..., seems a lttle dubous. at one functon of [e, ] s more vald than the other

29 Applcaton: Cornwell-Rupert

30 Testng for Effects Regress; lhs=lwage;rhs=fxedx,varyngx;res=e$ Matrx ; tebar=7*gxbr(e,person)$ Calc ; lst;lm=595*7/(*(7-1))* (tebar'tebar/sumsqdev - 1)^$ Create ; e=e*e$ Matrx ; e=7*gxbr(e,person)$ Matrx ; r=drp(tebar,tebar)-e$ Matrx ; sumr=r'1$ Calc ; lst;z=(sumr)^/r'r$ LM = D+04 Z = D+03

31 Hausman Test for FE vs. RE Estmator FGLS (Random Effects) LSDV (Fxed Effects) Random Effects E[c X ] = 0 Consstent and Effcent Consstent Ineffcent Fxed Effects E[c X ] 0 Inconsstent Consstent Possbly Effcent

32 Hausman Test for Effects Bass for the test, β ˆ - βˆ Wald Crteron: = ˆ ˆ ; W = [Var( )] -1 qˆ β ˆ ˆ ˆ FE - βre q q q A lemma (Hausman (1978)): Under the null hypothess (RE) ˆ d nt[ β RE - β] [ 0V, RE] (effcent) nt[ˆ β FE FE RE - β 0V d ] [, FE] (neffcent) ote: qˆ = ( β ˆ - β)-( βˆ β). The lemma states that n the FE RE jont lmtng dstrbuton of nt[ β ˆ - β] and nt qˆ, the lmtng covarance, C s 0. But, C = C - V. Then, RE Q,RE Q,RE FE,RE Var[ q] = V + V - C - C. Usng the lemma, C = V. FE RE FE,RE FE,RE FE,RE RE It follows that Var[ q]= V - V. Based on the precedng FE ˆ ˆ ˆ ˆ -1 H=( β - β ) [Est.Var( β ) - Est.Var( β )] ( β ˆ - β ˆ ) FE RE FE RE FE RE RE RE β does not contan the constant term n the precedng.

33 Computng the Hausman Statstc ˆ 1 Est.Var[ β ] ˆ = 1 I FE =σ Σ X X T 1-1 ˆγ Tσˆ u = 1 ˆ T σ ˆ + Tσˆu Est.Var[ βˆ RE] =σˆ Σ X I X, 0 γ = 1 As long as σˆ and σ are consstent, as, Est.Var[ˆ β ] Est.Var[ βˆ ] ˆu wll be nonnegatve defnte. In a fnte sample, to ensure ths, both must be computed usng the same estmate of σˆ generally be the better choce. FE RE. The one based on LSDV wll ote that columns of zeros wll appear n Est.Var[ ˆ nvarant varables n X. β does not contan the constant term n the precedng. β FE ] f there are tme

34 Hausman Test Random Effects Model: v(,t) = e(,t) + u() Estmates: Var[e] =.3536D-01 Var[u] = D+00 Corr[v(,t),v(,s)] = Lagrange Multpler Test vs. Model (3) = ( 1 df, prob value = ) (Hgh values of LM favor FEM/REM over CR model.) Fxed vs. Random Effects (Hausman) = ( 4 df, prob value = ) (Hgh (low) values of H favor FEM (REM).)

35 Varable Addton Test Asymptotc equvalent to Hausman Also equvalent to Mundlak formulaton In the random effects model, usng FGLS Only apples to tme varyng varables Add expanded group means to the regresson (.e., observaton,t gets same group means for all t. Use standard F or Wald test to test for coeffcents on means equal to 0. Large F or ch-squared weghs aganst random effects specfcaton.

36 Fxed vs. Random Effects ˆ ˆγ,Model ˆγ,Model βmodel = Σ= 1X I X = 1 I Σ X y T T ˆγ = 1 for fxed effects. Model Tˆ σ ˆγ = σ + σ u,model ˆ Tˆu,RE -1 for random effects. As T, ˆγ 1, random effects becomes fxed effects As σˆ 0, ˆγ 0, random effects becomes OLS (of course) u u,re As σˆ, ˆγ 1, random effects becomes fxed effects,re For the C&R applcaton, σˆ = , σ =.03531, ˆγ = u ˆ Looks lke a fxed effects model. ote the Hausman statstc agrees. β does not contan the constant term n the precedng.