Microeconometrics (Markus Frölich) Chapter 10. Linear Models for Panel Data

Transcription

1 Mcroeconometrcs (Markus Frölch) Chapter 10 Lnear Models for Panel Data

2 Panel data: "multple observatons" for the same un e.g. Y,t, X,t for ndvdual for several tme perods t Household panel surveys SOEP: every year the same ndvduals are ntervewed t= 1,..., 25 (unbalanced panel) We examne only balanced panel data (n practce almost always unbalanced panel) Alternatve uses for panel data methods: - multple sblngs of same famly: s famly, t s chld number (e.g. twns) - several pupls n same class (n same school) herarchcal lnear models (HLM) - several workers n the same company We wll examne only N nfny and T fx dentfcaton and nference

3 7.1 Motvaton: The omted varable problem Notaton: ( x1, x2,..., x K ) observable random varables c unobservable random varable ( y, x1, x2,..., xk, c ) populaton of nterest E( y x1, x2,..., xk, c ) populaton regresson functon Assume a lnear model: x s row vector E( y xc, ) = β + xβ + c 0 f cov( xj, c ) = 0 no effect on estmated parameters f cov( xj, c) 0 serous problems for consstency Example: frm producton functon, e.g. farm producton Y s output and X nputs (land, labour, materals, fertlzer) c s sol qualy or manageral qualy

4 Potental solutons n case of cov( xj, c) 0 whout panel data - proxy varable - IV - multple ndcator IV (e.g. two ndcators wh ndependent measurement errors) Panel for T=2 E( yt xt, c) = β0 + xtβ + c t = 1,2 x β = β x + β x + + β x t 1 t1 2 t2... K tk c tme constant unobserved effect Note relaton to SEM: There are (mplc) excluson restrctons over tme!

5 Error form: yt = β0 + xtβ + c+ ut (7.1) Eu ( x, c) = 0 t= 1,2 t t Ex ( ' u) = 0 (7.2) t= 1,2 t t f Ex ( t ' c= ) 0 pooled OLS (gnorng panel structure) Alternatve n case ths assumpton does not hold: panel estmators Idea: take the dfference of (7.1) Δ y =Δ xβ +Δ u Ths equaton s just a standard cross-secton Δ y = y y 2 1 Δ x= x x 2 1 Δ u = u u 2 1 β can be estmated here, but s nterpreted n (7.1)

6 When s OLS usng the dfferenced data consstent? OLS.1: E( Δx' Δ u) = 0 OLS.2: rank E( Δx' Δ x) = K But: E( Δx' Δ u) = Ex ( ' u) Ex ( ' u) Ex ( ' u) + Ex ( ' u) = = 0 (7.2) = 0 (7.2)! Assume addonally (strct exogeney) Ex ( ' u) = 0 and Ex ( ' u) = (there s always a prce to pay...)

7 7.2 Assumpton about the Unobserved Effects and Explanatory Varables Random or fxed effects unobserved effects model: = β + β + + = y 0 x c u t 1,..., T (7.3) x s a k row vector and may contan varables: that are tme-nvarant or ndvdual-nvarant or vary wh tme and ndvdual c unobserved component, latent varable, unobserved heterogeney, ndvdual effect, ndvdual heterogeney u dosyncratc errors, dosyncratc dsturbances random effect: fxed effect: c s random varable uncorrelated wh X c s random varable correlated wh X In modern econometrcs: random effect mples cov( x, c ) = 0 t = 1,2,..., T

8 7.2.2 Strct exogeney assumpton (SEA) on the explanatory varables E y x,, ) E( y x, c ) x c (7.4) ( 1 x 2,..., xt c = = β + { x : t 1,2,..., T} = are strctly exogenous condonal on the unobserved effect c E( y x,..., x ) = E[ E( y x,.., x, c )] = x β + E( c x,..., x ) (7.5) 1 T 1 T 1 T cx E( y x, x,... x ) may be E( y x ) 1 2 T e.g. f frm nputs depend on sol qualy c, past values and/or future values of may reveal nformaton about c. Random effects model f: Ec ( x1, x2,..., xt) = Ec [ ] = 0 Strct exogeney condonal on c mples the followng moment condons E( u x, x,..., x, c ) = 0 t = 1,2,..., T 1 2 T E( x ' u ) = 0 s,t = 1,2,..., T s

9 Assumpton: no effect of x on future u no effect of u on future x Example: Patents as a functon of past R&D nvestments A large value of u n one year may ncrease (or decrease) R&D n the future

10 Example: wh lagged dependent varable ln wage = β ln wage + c + u 1, t 1 How persstent (or flexble) are wages after controllng for unobsrved heterogeney? Frst: assumpton of strct exogeney cannot be vald u affects y whch s a regressor n future perods Second: smlarly Ec [ x1,... x T] = 0 cannot be vald. models wh lagged dependent varables requre other approaches (Chapter 11) Eu [ y 1, y 2,... y0, c ] = 0 mght be a reasonable assumpton (.e. assumes that all wage dynamcs are captured by the frst lag) (regressors are pre-determned)

11 E( c x1, x2,..., x T) = 0 0 SEA yes RE (GLS) HLM FE methods no pooled OLS Chapter 11

12 7.3 Estmatng Unobserved Effects Models by Pooled OLS y = x β + v t = 1,..., T v = c + u Ex ( ' v ) = 0 Ex ( ' u ) = 0 Ex ( ' c ) = 0 OLS s consstent Autocorrelaton robust covarance matrx must be used v are serally correlated e.g. cluster command n Stata Effcency gans possble (e.g. GLS or RE below)

13 7.4 Random Effects Method Estmaton and nference wh the random effects assumpton Assumpton RE.1 (regressors not nformatve about mean of RE): (a) E( u x, c ) = 0 t = 1,..., T (b) E( c x) = E( c) = 0 t = 1,..., T x x, x,..., x ( ) 1 2 T Under assumpton RE.1 wre y = xβ + v (7.6) Ev ( x ) = 0 v = c + u apply GLS to account for the error structure more restrctve than pooled OLS (.e. less robust) 7.4 Random Effects Method (GLS) (7.6) for all t stacked (notaton lke SUR, T equatons)

14 y = X β + v v = cι + u T T ι (TX1) vector of ones. uncondonal varance: Ω E( vv ') -1 Assumpton RE.2: rank E( X'Ω X ) = K FGLS wh unrestrcted varance estmator s consstent and root-n asy normal Alternatvely: use restrcted varance estmator RE REAR(1) Proceed wh Random effect method 2 2 Eu ( ) = σ u t= 1,2,..., T (7.7) Euu ( s ) = 0 all t s (7.8) (Alternatve AR(1) )

15 7.4 Random Effects Method = restrcted GLS methods Explo partcular structure assumed: no effcency gan but better small sample prop Ev ( ) = Ec ( ) + 2 Ecu ( ) + Eu ( ) = σ + σ c u = 0 RE.1a E( v v ) = E[( c + u )( c + u )] = E( c ) = σ 2 2 s s c Ω σ + σ σ L σ c u c c σc σc + σu M 2 2 σ 2 uit c T T M O σ c σc L σc σc + σ u = Evv ( ') = = + σιι ' Instead of T (T+1) / 2 (co)varance elements, only 2 parameters have to estmated

16 For effcency of GLS assume further Evv ( ' x) = Evv ( ') (7.9) or more succnctly Assumpton RE.3 ( a) E[ u u x, c ] = σ I ( b) E[ c x ] = σ u T c

17 Implement FGLS: Defne σ v = σc + σu and assume we have a consstent estmator for both components. ˆ 2 2 = ˆ σ ˆ uit + σιι c T T ' N 1 N ˆ ˆ ˆ RE X' Ω -1 Ω -1 X X' y = 1 = 1 Ω β = ˆRE β s consstent whether or not Assumpton RE.3 holds. 1) Start wh pooled OLS v ˆ resduals from pooled OLS ˆ σ 1 ˆ N T 2 2 ˆ v = v NT K = 1 t= 1

18 2)Now estmate 2 σ c T 1 T T 1 T T 1 T T 1 = = = 2 2 s ( s ) σc σc E v v E v v ( T t) t= 1 s= t+ 1 t= 1 s= t+ 1 t= 1 s= t+ 1 t= 1 (( ) ( ) ) = σ T 1 + T = σ T( T 1) / c c σ 2 ˆc 3) N T 1 T NT ( T 1) / 2 K v ˆˆ v (7.10) 1 = ( ) ˆ σ = ˆ σ ˆ σ u v c = 1 t= 1 s=+ t 1 s Problem f negatve ˆ 2 2 = ˆ σ ˆ u T + σιι c T T ' 4) Ω I Random effect s the only source of correlaton over tme!

19 7.4.2 Robust Varance Matrx Estmator If assumpton RE.3 s wrong, ˆRE β s stll consstent, but robust varance matrx of GLS must be used N 1 N N 1 ˆ -1-1 / N = Ω -1 ) Ω -1 vv ˆˆ ' Ω -1 ) Ω -1 ) = 1 = 1 = 1 Aˆ BA X' ˆ X X' ˆ ˆ X X' ˆ X v = y -X ˆ βr ˆ E Wald statstc: 1 ( ˆ β )'( ˆ ) ( ˆ RE r βre r) R RVR' R where Vˆ s robust varance matrx Alternatve: RE wh AR(1) or Herarchcal lnear models (HLM)

20 7.4.3 A General FGLS Analyss Assume the dosyncratc errors { u : t = 1,2,..., T} to be heteroskedastc and serally correlated. Then Ω ˆ N 1 = N = 1 vv ˆˆ'. v ˆ pooled OLS resduals

21 7.4.4 Testng for the presence of an unobservable effect Null hypothess: v are serally uncorrelated. 2 Test based on (7.10) H : 0 0 σ c = the null asymptotc dstrbuton N 1/2 N T 1 T = 1 t= 1 s= t+ 1 vˆˆ v s for any dstrbuton of v N 1/2 N T 1 T = 1 t= 1 s= t+ 1 v v s has lmng normal dstrbuton

22 Under the Null the varance s E T 1 T 2 vv s and t= 1 s= t+ 1 1 N 1 N N N T 1 T = 1 t= 1 s= t+ 1 vˆˆ v s 1/2 N T 1 T 2 vv ˆˆ s s asymptotcally standard normally dstrbuted = 1 t= 1 s= t+ 1

23 Summary for pooled OLS R1: Assumptons for Pooled OLS y = x β + u t = 1,2,..., T t t t Assumpton POLS.1 E( xt ' ut) = 0 t = 1,2,..., T T Assumpton POLS.2 rank E( x ' ) t 1 t x = t = K Assumpton POLS.3 (no seral correlaton) (a) Eux x Ex x t T where Eu (b) Euux ( t s t ' xt) = 0 t s s, t = 1,2,..., T ( t t' t) = σu ( t' t) = 1,2,..., σu = ( t )

24 Revson and Summary Theorem R1: Large Sample Propertes of Pooled OLS Under Assumpton POLS.1 and POLS.2, the pooled OLS estmator s consstent and asymptotcally normal. Avar( ˆ) E( ) / N 2 If POLS.3 holds n addon, then β = σ [ X'X ] 1 approprate estmator of Avar( ˆ β ) s 2 ˆ N T ˆ σ ( X'X ) = ˆ σ x ' x = 1 t= 1 where σ s the usual OLS varance estmator from the pooled y on x t = 1,2,..., T, = 1,2,..., N. regresson The usual t- and F- statstcs are vald asymptotcally SSRr SSRur ( NT K) F = SSR Q ur 1, so that the

25 Summary for pooled OLS R2: Dynamc Completeness Dynamc Completeness of the condonal mean E( yt xt, yt 1, xt 1,..., y1, x1) = E( yt x t) (7.11) E( y z, z,..., z ) = E( y z, z,..., z ) t t t 1 1 t t t 1 t L choose x= ( zt, zt 1,..., zt L) z contemporaneous varables then E( y x, x,..., x ) E( y x ) t t t 1 1 = t t

26 Equaton (7.11) s equvalent to E( ut xt, ut 1, xt 1,..., u1, x 1) = 0 => E( utus xt, x s) = 0 t s If (7.11) holds together wh the homoskedastcy assumpton then POLS.1 and POLS.2 hold standard OLS nference 2 Var( y x ) = σ t t

27 R.3: A note on tme seres persstence Theorem poses no restrcton on the tme seres persstence n the data {( y x ) : t = 1,2,..., T}. Consder yt = β0 + β1yt 1 + ut E( ut yt 1,..., y0) = 0 N, T fxed => pooled OLS produced consstent estmates T, N fxed β 1 > 1 causes consderable problems

28 7.5 Fxed Effects methods Consstency of the Fxed Effects Estmator y = x β + c + u t = 1,2,..., T (7.12) y = X β + cι + u T Assumpton FE.1 Eu ( x, c ) = 0 t = 1,2,..., T (Strct exogeney of x condonal on c) The vectors y, X, cι T, u are ndependent draws from a cross-secton. I.e. by usng ths system of equatons, conventonal d analyss for systems of equatons can be used.

29 Whn transformaton to get moment condons ndependent of c y = xβ + c + u (7.13) T T T =, =, = t= 1 t= 1 t= 1 y T y x T x u T u (7.12)-(7.13) y y = ( x x ) β + u u && y = && x β + u&& (7.14) POLS.1 holds n (7.14) f E( && x ' u&& ) = 0 t = 1,2,..., T ths assumpton hold under FE.1 (strct exogeney requred) Note that tme constant varables dsappear due to dfferencng

30 Fxed effect estmator ˆFE β s the pooled estmator from the regresson && y on && x. && y { = X&& { β + u&& { T 1 T K T 1 tme demeanng matrx Q 1 T = IT T ι ( T ι ' T ι ) T ι ' ( T T) Q ι = 0 Q Q Q T T y T x = && x T u = && y = u&& T Q s dempotent and symmetrc

31 Assumpton FE.2: T rank E( && x '&& x ) = rank E( X'X && && ) = K t= 1 N 1 N N T 1 N T ˆ βfe = X'X && && && ' && y = && x' && x && x ' && X t y = 1 = 1 = 1 t= 1 = 1 t= 1 Whn estmator s consstent under FE.1 and FE.2 Note: Between estmator s defned as OLS to equaton (7.13). It s consstent under RE.1.

32 7.5.2 Asymptotc nference wh fxed effects Assumpton FE.3: Covarance matrx: Euu ( ' x, c) = σ I 2 u T Var( u x, c ) = σ I 2 u T ( ) 2 E u&& = E u u = Eu + Eu Euu ( ) ( ) ( ) 2 ( ) = σ + σ / T 2 σ / T = σ (1 1/ T) u u u u Euu && && = Euu Euu Euu + Eu 2 ( s ) ( s ) ( ) ( s ) ( ) corr( u&&, u&& ) = 1/( T 1) s = σ σ + σ = σ < u / T u / T u / T u / T 0 hence, negatve seral correlaton But: FE s nevertheless effcent (see below)

33 The reason why FE s effcent N 1 N 1 1/2 FE X'X && && && '&& X = 1 = 1 ( ˆ β β) N = N N u N 1 N 1 1/2 X'X && && = 1 = 1 = N N X && ' u && ' u = u = && ' u && T X X'Q X because Q s symmetrc and dempotent Euu ( ' X&& ) 2 = σ I Under FE.3 u T hence homoskedastc whout correlaton ( ˆ 2 βfe β) 0, σ u ( X'X && && ) N N E ( ) ( X'X && && ) 1 ˆ 2 var( βfe ) = σ u / A E N N 1 N T FE = u X'X && && = u && x && x = 1 = 1 t= 1 Avar( ˆ ˆ β ) ˆ σ ˆ σ ' 1

34 2 How do we obtan ˆu σ? Use Thus Eu [&& ] = u 1 σ T N T 1 Eu [&& ] NT ( 1) = 1 t= = σ u Use FE resduals uˆ = && y && x ˆ β FE [ ] 2 ˆ u / ( 1) σ SSR = SSR N T K N T = = 1 t= 1 uˆ 2

35 7.5.3 The dummy varable regresson (an equvalent way for FE) c ' s are parameters to be estmated defne dummes dn = 1 f n = dn = 0 f n Then run pooled OLS y on d1, d2,..., dn, x t = 1,2,..., T, = 1,2,..., N (7.15) But β obtaned from (7.15) equals ˆFE β (numercally equal, also resduals) cˆ = y x ˆ β = 1, 2,... N FE Note that the estmated fxed effects are unbased but not consstent!!! Here, ncluson of dummes does stll perm consstent and unbased estmaton of β Ths s usually not the case n nonlnear models!!!

36 7.5.4 Seral correlaton and the robust varance matrx estmator Problem: If FE.3 s not vald => seral correlaton T=2: corr( u&&, u&& s ) = 1/( T 1) = 1 T>2 test for seral correlaton use any two tme perods, e.g. T and T-1 regress uˆ ˆ T on ut 1 = 1,..., N δ estmated parameter H : δ = (1/ T 1) 0 Under FE.1-FE.3 t-statstc has asymptotc normal dstrbuton Alternatve: nstead of only two tme perods, use all tme perods regress wh pooled OLS uˆ on uˆ t 3,..., T; 1,..., N 1 = = but use t-statstc robust to arbrary seral correlaton (because seral correlaton under H 0 )

37 Alternatve: robust covarance matrx for FE (nstead of testng or n case test rejects H 0 ) ˆ Avar( ˆ βfe ) = ' ' = 1 uˆ && y X&& ˆ β FE N ( X'X && && ) X && uu ˆˆ && ( && && X X'X ) 1 1

38 7.5.5 Fxed effects GLS Now allow for an unrestrcted but constant condonal covarance matrx Assumpton FEGLS.3: E( uu ' x, c) = Λ ( T T) => Euu (&& && ' && x) = Euu (&& && ') u&& = Q u T Euu (&& && ') = Q Euu ( ') Q = Q ΛQ rank T T Q ΛQ T T = T 1 T T Use a generalzed nverse or as a smpler alternatve: drop one tme perod ( does not matter whch one) The results are the same n both cases

39 suppose we drop perod T && y = && x β + u&& M && yt 1 = && xt 1β + u&& T 1 (7.16) && y ( T -1) 1 X&& ( T -1) K u ( T -1) 1 Fxed effect GLS estmator N 1 N && -1 Ω && && -1 X' Ω && X X' y = 1 = 1 ˆ β ˆ ˆ FEGLS = Ω= E( uu && && ') whout tme perod T Ωˆ = N 1 1 N = 1 u ˆ = && y X&& ˆ β FE uu ˆˆ' Frst stage: use all tme perods. Second stage: drop tme perod T

40 Assumpton FEGLS.2: E ( && Ω ˆ -1 rank X' && X ) = K Under FE.1 and FEGLS.2 the FEGLS estmator s consstent. Add assumpton FEGLS.3 => effcent varance matrx can be estmated consstently as N ˆ var( ˆ ) ˆ A βfegls = && -1 X' Ω && X = 1 1

41 7.5.6 Usng fxed effects estmaton for polcy analyss y = x β + v = z γ + δw + v w polcy varable (or treatment varable) v may or may not contan an unobserved effect z controls that mght be correlated wh Suffcent for consstency of fxed effects estmator s [ ] E x '( v v ) = 0 t = 1,2,..., T So x (and z ) are permted to be correlated wh v. To obtan the above condon, we make use of the dempotent matrx Q such that X&& ' v&& = X ' v&& = X && ' v

42 7.6 Frst Dfferencng Method Often less effcent than FE but partcularly useful later to relax strct exogeney assumpton Assumpton FD.1 equals FE.1: Eu ( x, c ) = 0 t = 1,2,...,T y = x β + c + u t = 1,2,..., T Δ y =Δ x β +Δ u t = 2,3,..., T dfference the followng equaton: y = θ + θ d dt + zγ + d2 zγ dtzγ + w δ + c + u 1 2 t t 1 t 2 t T Eu ( z, w, w,..., w, c) = 0 t= 1,2,..., T 1 2 T ( 2 )... ( ) ( 2 )... ( ) Δ y = θ Δ d + + θ Δ dt + Δ d zγ + + Δ dt zγ +Δ w δ +Δ u 2 t T t t 2 t T θ1, γ1are not dentfed

43 FD estmator s pooled OLS of Δy on Δ x t = 2,3,..., T; = 1,2,..., N Under the assumpton FD.1 the FD estmator s consstent! E( Δx ' Δ u ) = 0 t = 2,3,..., T E( Δu Δx, Δx,..., Δ x ) = 0 estmator s unbased (condonal on x) 2 3 T ( t= 2 ) T Assumpton FD.2: rank ( ' ) E Δx Δ x = K FD estmator s effcent when Assumpton FD.3: 2 ( ',...,, ) E ee x x c e 1 T e T 1 ( T -1) 1 e Δ u t = 2,..., T = σ I

44 Example u = ρu 1 + ξ ; ξ d, whe nose Assumpton FD.3 mples ρ = 1 Δ u = ξ u = u 1 + ξ whch s a random walk! FD.3 s just the oppose of FE.3 (whch mples ρ = 0 ) Under assumpton FD.1-FD.3 the FD estmator s effcent n the class of estmators usng strct exogeney assumpton FE.1 A ˆ σ ˆ ˆ ( X' X ) 2 var( βfd ) = σe Δ Δ ˆ 1 N T [ ( 1) ] ˆ e = NT K e = 1 t= 2 eˆ =Δy Δx ˆ β FD because perod 1 was lost due to dfferencng

45 7.6.2 Robust varance matrx ( ) ( N X X X ee ˆˆ X )( X X ) Avar( ˆ ˆ β ) = Δ ' Δ Δ ' ' Δ Δ ' Δ 1 1 FD = Testng for seral correlaton eˆ = ρ eˆ + error t = 3,4,..., T; = 1,2,..., N ˆ1 1 In case of sgnfcant seral correlaton use the formula of for nference!

46 7.6.4 Polcy analyss usng frst dfferencng T = 2 Δ y = θ +Δ z γ + δ programme +Δu prog 2 =Δprog 2 suppose 1 0 programme = for all ndvduals When Δz 2 s omted dfference n dfferences estmator 1 ˆ δ =Δy Δ y treated control In general: e.g. f programme1 0 for some ndvduals Δ y = ξ +Δ z γ + δ Δ prog +Δ u 1 1

47 7.7 Comparson of Estmators Fxed effects versus frst dfferencng T=2 - both estmator produce the same estmates T>2 u - are serally uncorrelated FE s more effcent u - follow a random walk FD s more effcent Bas of FE estmator s of order T -1 f strct exogeney assumpton s wrong Thus, when T s moderately large, the FE estmator s preferred Both estmators requre strct exogeney Example: sblngs n a famly x ndexes famly, t ndexes chld n brth order seral correlaton n u probably small or zero

48 Testng for strct exogeney n FD H : γ = 0 o f model s correct, only Δ xt should be sgnfcant but not x t or x s for s t Δ yt =Δ xtβ + wtγ +Δ ut w t subset of x t excludng tme dummes Wald Test to account for seral correlaton or heteroskedastcy. Under FD.1-FD.3 usual F-statstc s asymptotcally vald! Testng for strct exogeney n FE H o : δ = 0 y = xβ + w+ 1δ + c + u t = 1,2,..., T 1 w. + 1 subset of 1 x + excludng tme dummes

49 7.7.2 The relatonshp between the random and the fxed effects estmator If x t does not vary much over tme, FE and FD lose much nformaton T Ω = ι ' ι T T ( ) ( ) uit c T T uit T c T T T T = σ + σιι ' = σ + σι ι ' ι ι ' = σ I + Tσ P = σ + Tσ ( P + ηq ) u T c T u c T T P u u c ( ' ) 1 I Q = ι ι ι ι ' T T T T T T T η = σ /( σ + Tσ ) 1/2 ( σu Tσc ) (1 λ) [ IT λpt] (1/ σu) [ IT λpt ] 1/ Ω = + = λ = σ σ + σ u /( u T c) 1/2

50 RE estmator s obtaned by estmatng the followng equaton by system OLS C y = C X β + C T T T [ λ ] C I P T T T v transformed equaton ( ( ( y = X β + v (7.16) (( E vv C C I 2 ( ') = Ω = σ T T u T (7.16) can be wrten as ( ) ( ) y λy = x λx β + v λv => quas demeanng due to

51 RE estmator can be wrten as N T 1 N T ˆ ( ( ( ( βre = x ' x x ' y = 1 t= 1 = 1 t= 1 determne ˆ 2 2 λ = 1 1/ 1 + T ( ˆ σ / ˆ c σu) { } 1/2 ˆ 1 λ estmates of RE and FE are close

52 7.7.3 The Hausman test comparng the RE and FE estmators Suppose Assumptons RE1-RE3 are vald => RE estmator s effcent, FE s consstent Suppose we have only tme varyng regressors 1 ( ( 1 ˆ 2 2 Avar( ˆ β ) ( ) / var( ˆ ˆ FE = σ u E N A βre ) = σ u E( ) X'X && && and X'X / N ( ( E( X'X) E( && && X'X ) = E X ' ( IT λpt ) X E X ' ( IT PT ) X ( 1 λ) E[ X ' P X ] ( 1 λ) E[ x ' x ] = = T from whch follows 1 1 Avar( ˆ ˆ β ) var( ˆ ˆ RE A βfe ) posve defne Avar( ˆ ˆ β ) var( ˆ ˆ FE A βre) posve defne

53 Hausman statstc Suppose all x are tme varyng ( ˆ ˆ 1 ) ˆ ˆ δ ' var( ˆ ) var( ˆ ) ( ˆ ˆ FE δ RE A δfe A δ RE δfe δre) ˆRE δ estmated coeffcents whout the coeffcents on tme constant varables under RE.1-RE.3 the statstc s asymptotcally 2 χ M dstrbuted (where M s # of tme varyng regressors) Often nterest n one sngle coeffcent 2 2 } 1/2 / FE RE se( FE ) se( RE ) ( ) { ˆ δ ˆ δ ˆ δ ˆ δ Dsadvantage: strct exogeney requred for FE and RE (If strct exogeney doubtful, use methods of Ch. 11)

54 F-statstc verson: ( ( y = x β + w&& ξ + error t = 1,..., T; = 1,..., N (7.17) w subset of M tme varyng elements of w&& tme demeaned verson of w ξ (MX1) x H : ξ = 0 0 F SSR SSR r ur = SSRur ( NT K M ) M a F M, NT K M SSR r SSR from (7.16) SSR ur SSR from (7.17)

55 Combnaton between FE and RE approach FE estmator has dsadvantage that may lose at lot of nformaton f x vares ltle over tme n partcular when the x of most nterest s tme constant Instead of ncludng a dummy varable for every ndvdual, perhaps a dummy for larger groups works as well E.g. we mght have panel data for students and ndcators of classes or schools Instead of ncludng a dummy for every student, nclude dummes for every school (Asymptotcs requres that number of students n school s large,.e. tendng to nfny) and then conduct a random effect analyss at the student level. (Note that many researchers wre that they ncluded school fxed effects even when the fnal estmaton s random effect at the student level)