Lecture 15 Panel Data Models

Transcription

1 Lecture 5 Panel Data Models Panel Data Sets A panel data set, or longtudnal data set, s one where there are repeated observatons on the same unts. The unts ma be ndvduals, households, enterprses, countres, or an set of enttes that reman stable through tme. Repeated observatons create a potentall ver large panel data sets. Wth N unts and T tme perods => Number of observatons: NT. Repeated observatons can be great for estmaton: Large sample! We use the repeated observatons to our advantage. But, repeated observatons from the same ndvdual ntroduce a problem: Dependence, snce the observatons are, ver lkel, not ndependent. Modelng the potental dependence creates dfferent models.

2 The Natonal Longtudnal Surve (NLS) of Youth s an example. The same respondents were ntervewed ever ear from 979 to 994. Snce 994 the have been ntervewed ever two ears. Panel data allows us a researcher to stud cross secton effects.e., along N, varaton across the frms- & tme seres effects.e., along T, varaton across tme. Panel Data Sets NT T T T Nt t t t N N Tme seres Cross secton A standard panel data set model stacks the s and the x s: = X + c + X s a Σ T xk matrx s a kx matrx c s Σ T x matrx, assocated wth unobservable varables. and are Σ T x matrces Panel Data Sets j j j kt T T k t t k k kt T T k t t k k T t T t w w w w w w x w w w w w X x x x x x x x x x x x x X ;...; ;...; Notaton:

3 Panel Data Sets Fnancal data COMPUSTAT provdes fnancal data b frm (N =99,000) and b quarter (T = 96:I, 96:II,..., ) CRSP dal and monthl stock and ndex returns from 96 on. Datastream provdes economc and fnancal data for countres. It also covers bonds and stock markets around the world. Intra-dal data: Olsen (exchange rates) and TAQ (stock market transacton prces). Essentall nfnte T, large N. OptonMetrcs, also known as Iv DB, s a database of hstorcal prces, mpled volatlt for lsted stocks and opton markets. Internatonal Fnancal Statstcs (IFS from the IMF) covers economc and fnancal data for almost all countres post-wwii. Balanced and Unbalanced Panels Notaton:,t, =,,N; t =,,T Mathematcal and notatonal convenence: - Balanced: NT (that s, ever unt s surveed n ever tme perod.) N - Unbalanced: T = Q: Is the fxed T assumpton ever necessar? SUR models. The NLS of Youth s unbalanced because some ndvduals have not been ntervewed n some ears. Some could not be located, some refused, and a few have ded. CRSP s also unbalanced, some frms are lsted from 96, others started to be lsted later. 3

4 Panel Data Models Wth panel data we can stud dfferent ssues: - Cross sectonal varaton (unobservable n tme seres data) vs. tme seres varaton (unobservable n cross sectonal data) - Heterogenet (observable and unobservable ndvdual heterogenet) - Herarchcal structures (sa, zp code, ct and state effects) - Dnamcs n economc behavor - Indvdual/Group effects (ndvdual effects) - Tme effects Panel Data Models: Example - SUR In Zellner s SUR formulaton (no lnear dependence on t ) we have: (A) t = x t + t (A) E[ X] = 0, (A3 ) Var[ X] = I T --groupwse heteroscedastct. E[ t jt X] = j --contemporaneous correlaton E[ t jt X] = 0 when t t (A4) Rank(X) = full rank In (A)- (A4), we have the a GR model wth heteroscedastct. OLS n each equaton s OK, but not effcent. GLS s effcent. We are not takng advantage of poolng.e., usng NT observatons! Use LR or F tests to check f poolng (aggregaton) can be done. 4

5 Panel Data Models: Example - Poolng Assumptons (A) t = x t + z γ + t - the DGP =,,..., N - we thnk of as ndvduals or groups. t =,,..., T - usuall, N >> T. (A) E[ X,z] = 0, - X and z: exogenous (A3) Var[ X,z] = I. - Heteroscedastct can be allowed. (A4) Rank(X) = full rank We thnk of X as a vector of observed characterstcs. For example, frm sze, Market-to-book, Z-score, R&D expendtures, etc. We thnk of z as a vector of unobserved characterstcs (ndvdual effects). For example, qualt of management, growth opportuntes, etc. Panel Data Models: Basc Model The DGP (A): t k X j jt j p -The ndex refers to ndvduals.e., the unt of observaton, t refers to the tme perod, and j and p are used to dfferentate between dfferent observed and unobserved explanator varables. - A tme trend t allows for a shft of the ntercept over tme, capturng tme effects technologcal change, regulatons, taxes, etc. - If the mplct assumpton of a constant rate of change s strong, the trend can be replaced b a set of dumm varables, one for each tme perod except the reference perod. -TheX j varables are usuall the varables of nterest, whle the Z p varables are responsble for unobserved heterogenet (& dependence on the s) and as such consttute a nusance component of the model. 5 s p Z p t t 5

6 Panel Data Models: Basc Model Snce the Z p varables are unobserved, there s no means of obtanng nformaton about the p Z p component of the model. Usuall, t s convenent to defne a term c known as the unobserved effect, representng the jont mpact of the Z p varables on. s c Z p p p We can rewrte the regresson model as: t k j j X jt c t t The characterzaton of the c component s crucal. Dfferent was of thnkng about z p create dfferent panel data models. 30 Panel Data Models: Basc Model t k j j X jt c t Note that f the X j s are so comprehensve that the capture all the relevant characterstcs of the ndvdual, there wll be no relevant unobserved characterstcs. => Now, c can be dropped. Pooled OLS ma be used, treatng all the observatons for all of the tme perods as a sngle sample. t In general, droppng c leads to mssng varables problem: bas! We usuall thnk of c as contemporaneousl exogenous to the condtonal error. That s, E[ t c ] = 0, t =,..., T 30 6

7 Panel Data Models: Basc Model A stronger assumpton can solve the estmaton problem. Strct exogenet can also be mposed. In ths case, Then, E[ t x, x,...,x T,c ] = 0, E[ x t t, c ] t =,..., T c t => The β j s are partal effects holdng c constant. k j Volatons of strct exogenet are not rare. If x t contans lagged dependent varables or f changes n t affect x t+ (a feedback effect). j X jt But to estmate β we stll need to sa somethng about the relaton between x t and c. Dfferent assumptons wll gve rse to dfferent models. 30 Panel Data Models: Tpes The basc DGP: t = x t + z γ + t & (A)-(A4) appl. Dependng on how we model the heterogenet n the panel, we have dfferent models. Four Popular Models: () Pooled (Constant Effect) Model z γ s a constant. z = α. Dependence on the t ma enter through the varance. That s, repeated observatons on ndvdual are lnearl ndependent. In ths case, t = x t + α + t OLS estmates α and consstentl. We estmate k+ parameters. 7

8 Panel Data Models: Tpes () Fxed Effects Model (FEM) The z s are correlated wth X Fxed Effects: E[z X ] = g(x ) = α* ; effects are correlated wth ncluded varables.e., pooled OLS wll be nconsstent. Assume z γ = α (constant; t does not var wth t). Then, t = x t + α + t the regresson lne s rased/lowered b a fxed amount for each ndvdual (the dependence created b the repeated observatons!). In econometrcs terms, ths s the source of the fxed-effects. We have a lot of parameters: k+n. We have N ndvdual effects! OLS can be used to estmates α and consstentl. Panel Data Models: Tpes (3) Random Effects Model (REM) The dfferences between ndvduals are random, drawn from a gven dstrbuton wth constant parameters. We assume the z s are uncorrelated wth the X. That s, E[z X ] = μ (f X contans a constant term, μ=0 WLOG). Add and subtract E[z γ] = μ* from (*): t = x t + E[z γ] +( z γ) - E[z γ] + t = x t + μ* + u + t We have a compound ( composed ) error.e., u + t = w t. Ths error w t ntroduces contemporaneous cross-correlatons across the group. OLS estmates μ and consstentl, but GLS wll be effcent. 8

9 Panel Data Models: Tpes (4) Random Parameters /Coeffcents Model We ntroduce heterogenet through. But, ths ma ntroduce addtonal N parameters. A soluton s to model. For example, t = x t ( + h ) + α + t h s a random vector that nduces parameter varaton, where h ~D(0, σ h). That s, we ntroduce heteroscedastct. Now, the coeffcents are dfferent for each ndvdual. It s possble to complcate the model b makng them dfferent through tme: t = ( + h ) + θ t where θ t ~D(0, σ t). Estmaton: GLS, MLE. Long hstor: Rao (965) and Chow (975) worked on these models. Compact Notaton Compact Notaton: = X + c + X s a T xk matrx s a kx matrx c s a T x matrx and are T x matrces Recall we stack the s and X s: X s a Σ T xk matrx s a kx matrx c, and are Σ T x matrces = X + c + Or = X* * +, wth X * = [X ι] - Σ T x(k+) matrx. * = [ c] - (k+)x matrx 9

10 Assumptons for Asmptotcs (Greene) Convergence of moments nvolvng cross secton X. Usuall, we assume N ncreasng, T or T assumed fxed. Fxed T asmptotcs (see Greene, p. 96) Tme seres characterstcs are not relevant (ma be nonstatonar) If T s also growng, need to treat as multvarate tme seres. Rank of matrces. X must have full column rank. (X ma not, f T < K.) Strct exogenet and dnamcs. If x t contans,t- then x t cannot be strctl exogenous. X t wll be correlated wth the unobservables n perod t-. Inconsstent OLS estmates! (To be revsted later.) Panel Data Models: (A3 ) - No Homoscedastct We can relax assumpton (A3). The new DGP model: = X* * +, wth X * = [X ι] - Σ T x(k+) matrx. * = [ c] - (k+)x matrx Now, we assume (A3 ) E[ ' X] = Σ σ I ΣT Potentall, a lot of dfferent elements n E[' X] n a panel: - Indvdual heteroscedastct. Usual groupwse heteroscedastct. - Autocorrelaton (Indvdual/group/frm) effects. Errors have arbtrar correlaton across tme for a partcular ndvdual : - Temporal correlaton (Tme) effects. Errors have arbtrar correlaton across ndvduals at a moment n tme (SUR-tpe correlaton). - Persstent common shocks: Errors have some correlaton between dfferent frms n dfferent tme perods (but, these shocks are assumed to de out over tme, and ma be gnored after L perods). 0

11 Panel Data Models: (A3 ) - Error Structures To understand the dfferent elements n Σ, consder the followng DGP for the errors, ε t s: ε t = θ f t + η t, f t ~D(0, σ f ) & η t = ϕ η t- + ς t, ς t ~D(0, σ ς ) f t : vector of random factors common to all ndvduals/groups/frms. θ : vector of factor loadngs, specfc to ndvdual. ς t : random shocks to ndvdual, uncorrelated across both and t. η t : random shocks to. Ths term generates autocorrelaton effects n. θ f t generates both contemporaneous (SUR) and tme-varng crosscorrelatons between and j. (Autorrelatons de out after L perods.) -If f t s uncorrelated across t, =>onl contemporaneous (SUR) effects. -If f t s persstent n t, =>both SUR and persstent common effects. Panel Data Models: (A3 ) - Error Structures Dfferent forms for E[ ' X]: - Indvdual heteroscedastct. E[ X] = σ => standard groupwse heteroscedastct drven b ς t. - Autocorrelaton (Indvdual) effects: E[ t s X] 0 (t s) => auto-/tme-correlaton for errors, t drven b η t. - Temporal correlaton effects: E[ t jt X] 0 ( j) => contemporar cross-correlaton for errors drven b f t. - Persstent common shocks: E[ t js X] 0 ( j) and t s < L => tme-varng cross-correlaton for errors drven b f t. Remark: Heteroscedastct ponts to GLS effcent estmaton, but, as before, for consstent nferences we can use OLS wth (adjusted for panels) Whte or NW SE s.

12 Panel Data Models: (A3 ) Clustered SE For consstent nferences, we can use OLS wth Whte or NW SE s: - Whte SE s adjust onl for heteroscedastct: S 0 = (/T) e x x. - NW SE s adjust for heteroscedastct and autocorrelaton: S T = S 0 + (/T) l w L (l) t=l+,...,t (x t-l e t-l e t x t + x t e t e t-l x t-l ) But, cross-sectonal (SUR) or spatal dependences are gnored. If present, the Whte s or NW s HAC need to be adjusted. Smple ntuton: Repeatng a dataset 0 tmes should not ncrease the precson of parameter estmates. However, the..d. assumpton wll do ths: Now, we dvde b NT, not T or N. We cannot gnore the dependence n the data. An obvous soluton s to aggregate the repeated data. That s, aggregate n groups! Panel Data Models: (A3 ) Clustered SE In general, the observatons are not dentcal, but correlated wthn a cluster.e., a group that share certan characterstc. We assume correlaton wthn a cluster, but ndependence across clusters. Smple dea: Aggregate over the clusters. The ke s how to cluster. Canoncal example: We want to stud the effect of class sze on st graders' grades, the unobservables of st graders belongng to the same classroom wll be correlated (sa, teachers qualt, recess routnes) whle wll not be correlated wth st graders n far awa classrooms. Then, we can cluster b school/teacher. In fnance, t s reasonable to expect that shocks to frms n the same ndustr are not ndependent. Then, we can cluster b ndustr.

13 Panel Data Models: (A3 ) Clustered SE We remove the dependence b assumng correlaton wthn a cluster, but ndependence across clusters. That s, we thnk of the data as g E[ x g jg '] ' g c 0 jg g t f g g ' f g g g ' g,..., G Or stackng the data b cluster: g = X g + c g + g g =,..., G. Eas extenson of the Whte SE: S G N N g g 0 g j x ' gx jg w g, jg where w g,jg = E[ g jg X g ] s the error covarance for the g th and jg th observatons. We use S 0 to calculate clustered Whte SE. Panel Data Models: (A3 ) - PCSE Drscoll and Kraa (998) provde an eas extenson to estmate robust NW SE s n panels wth cross-sectonal dependences: Average the x t e t over the clusters we suspect cause dependence. Wthn a cluster, we assume that the correlatons wthn a cluster are the same for dfferent observatons. The KxK sandwch matrx s defned as ^ X ' X T t t j h ( b) h t j ( b)' wth h ( b) N ( t ) The NW method s appled to the tme seres of cross-sectonal averages of h t (b). We average over the G clusters to get G h t (b). We use the sandwch matrx to estmate clustered NW SE, usng w(l) as usual Bartlett or QS weghts other weghts are OK. t h ( b) t 3

14 Panel Data Models: (A3 ) - PCSE The NW method s appled to the tme seres of cross-sectonal averages of h t (b). B usng cross-sectonal averages, estmated SE are consstent ndependentl of the panel s cross-sectonal dmenson N. Clearl, these clustered NW SE reduce to the usual NW SE f each cluster onl has one observaton. If we do not suspect autocorrelaton problems not rare, gven that man panel data sets have heav temporall spaced observatons-, we can rel on Whte SE (S 0 ). These clustered standard errors are called panel corrected SE (PCSE s) or clustered SE. The clustered Whte-stle SE are, sometmes, called Rogers SE. Panel Data Models: (A3 ) - PCSE These PCSE s are robust to ver general forms of cross-sectonal (and temporal) dependence. These PCSE s usng HAR estmators (sa, based on KVB SE) are also possble. (See Hansen (007). Consstenc of the PCSE s dscussed b Whte (984), Lang and Zegger (986) for panels wth fnte number of observatons per cluster, as G. Hansen (007) shows that PCSE can be used wth N g --.e., long panels--, n addton to G. 4

15 Pooled Model General DGP t = x t + c + t & (A)-(A4) appl. The pooled model assumes that unobservable characterstcs are constant, ndependent of --no heterogenet. That s, we have: t = x t + α + t Now, we have a CLM, wth k+ parameters. Stackng the varables n matrces, we have: = X + α ι + Dmensons: -, ι and are Σ T x -X s Σ T xk - s kx Pooled Model We can re-wrte the pooled equaton model as: = X* * +, X* = [X ι] - Σ T x(k+) matrx: *=[ α] - (k+)x matrx In ths context, OLS produces BLUE and consstent estmator. In ths model, we refer to pooled OLS estmaton Of course, f our assumpton regardng the unobservable varables s wrong, we are n the presence of an omtted varable, c. Then, we have potental bas and nconsstenc of pooled OLS. The magntude of these problems depends on how the true model behaves: fxed or random. 5

16 Pooled Regresson: Heterogenet Bas In the pooled model, there s no model for group/ndvdual heterogenet. Thus, pooled regresson ma result n heterogenet bas: Pooled regresson: t = β 0 +β x t +ε t j True model: Frm True model: Frm 4 j True model: Frm 3 True model: Frm x Pooled Regresson: Wthn Transformaton We can estmate b centerng the observatons around ther group/ndvdual means. That s, t t k j k j j j x ( x jt jt x Ths method s called the wthn-groups estmaton because t reles on varatons wthn ndvduals rather than between ndvduals. That s, ths estmator reflects the tme-seres or wthn-ndvdual nformaton reflected n the changes wthn ndvduals across tme. We are estmatng usng the tme-seres nformaton n the data. j t ) t 6

17 Pooled Regresson: Wthn Transformaton There s a cost n the smplct of the wthn-groups estmaton. Frst, the ntercept and an X varable that remans constant for each ndvdual (sa, gender or College degree) wll drop out of the model. The elmnaton of the ntercept ma not matter, but the loss of the unchangng explanator varables ma be frustratng. Obvousl, f we are nterested on the effect of gender on CEO compensaton, wthn transformaton wll not work. But t wll work well f we are nterested on the effect of an ndependent Board of Drectors, b lookng at the compensaton pre-/post-bod. Under the usual assumptons, pooled OLS s consstent and unbased. Pooled Regresson: Between Transformaton There s an addtonal alternatve to estmate, b expressng the model n terms of group/ndvdual means. That s, t k jk j j j x x jt j It s called the between estmator because t reles on varatons between ndvduals (sa, & j). We are estmatng usng the cross-sectonal nformaton n the data (the tme-seres ndvdual varaton s gone!). Obvousl, f we are nterested on the effect of a new ndependent BOD durng the tenure of a CEO on that CEO s compensaton, between transformaton wll not work. But t wll work well f we stud the effect of gender on CEO s compensaton. t 7

18 Pooled Regresson: Between Transformaton We lose observatons (and power!): we have onl N data ponts. Under the usual assumptons, pooled OLS usng the between transformaton s consstent and unbased. Useful Analss of Varance Notaton (Greene) The varance (total varanton) quantfes the dea that each ndvdual sa, each frm-- dffers from the overall average. We can decompose the varance nto two parts: a wthn-group/ndvdual part and a between group/ndvdual part. Decomposton of Total varaton: N T N T Σ N =Σ t=(zt z) Σ = Σ t=(zt z.) Σ =T z. z Total varaton = Wthn groups varaton + Between groups varaton Interpretaton: - Wthn group varaton: Measures varaton of ndvduals over tme. - Between group varaton: Measures varaton of the means across ndvduals. 8

19 WHO Data (Greene) Note: The varablt s drven b between groups varaton Pooled Model: Lvng wth (A3 ) We start wth the pooled model: = X* * +, wth X * = [X ι] - Σ T x(k+) matrx. * = [ α] - (k+)x matrx Now, we allow E[ j ' X ] = σ j Ω j Potentall a lot of dfferent forms for E[ j ' X ] n a panel: - Indvdual heteroscedastct. E[ X ] = σ - Indvdual/group effects: E[ t s X ] 0 (t s) - Tme (SUR or spatal) effects: E[ t jt X ] 0 ( j) - Persstent common shocks: E[ t js X ] 0 ( j) and t s < L 9

20 Pooled Model: Lvng wth (A3 ) Heteroscedastct ponts to GLS effcent estmaton, but, for consstent nferences we can use OLS wth Whte/NW SE. Before calculatng the NW SE, we cluster the data to remove the dependence caused b the wthn group correlaton of the data. Drscoll and Kraa (998): Average the x t e t over the clusters. Then, the KxK sandwch matrx s defned as ^ X ' X T t t j h ( b) h t j ( b)' wth h ( b) t N ( t) h ( b) t The NW method s appled to the tme seres of cross-sectonal averages of h t (b). If we do not suspect autocorrelaton problems, we can rel on Whte SE (S 0 ), also referred as Rogers SE. Pooled Model: PCSE - Clusterng We can cluster the SE b one varable (sa, ndustr) or b several varables (sa, ear and ndustr) -- mult-level clusterng. If these several varables are nested (sa, ndustr and state), cluster at hghest level. We assume that the correlatons wthn a cluster (a group of frms, a regon, dfferent ears for the same frm, dfferent ears for the same regon) are the same for dfferent observatons. Dfferent clusters can produce ver dfferent SE. We want to cluster n groups that produce correlated errors. Usuall, we cluster usng economc theor (clusterng b ndustr, ear, ndustr and ear). 0

21 Pooled OLS: Clustered SE Results (Greene) Ordnar least squares regresson... LHS=LWAGE Mean = Resduals Sum of squares = Standard error of e = Ft R-squared =.4 Model test F[ 8, 456] (prob) = 36.8(.0000) Panel Data Analss of LWAGE [ONE wa] Uncondtonal ANOVA (No regressors) Source Varaton Deg. Free. Mean Square Between Resdual Total Varable Coeffcent Standard Error b/st.er. P[ Z >z] EXP.04085*** EXPSQ ***.48048D OCC *** SMSA.4856*** MS.06798*** FEM *** UNION.0940*** ED.058*** Constant *** Pooled OLS: Clustered SE Results (Greene) Varable Coeffcent Standard Error b/st.er. P[ Z >z] Constant EXP EXPSQ D OCC SMSA MS FEM UNION ED Clustered SE Constant EXP EXPSQ D OCC SMSA MS FEM UNION ED Note: Clustered SE s tend to be bgger. The more correlaton allowed, the hgher the SE.

22 Pooled Model: PCSE Clusterng: Remarks Snce we allow for correlaton between observatons, clustered SE wll ncrease CIs. The hgher the clusterng level, the larger the resultng SE. Thus, dfferent clusters can produce dfferent SE. Rel when possble on economc theor/ntuton to cluster. It s not a bad dea to tr dfferent was of defnng clusters and see how the estmated SE are affected. Be conservatve, report largest SE. Practcal rules - If aggregate varables (sa, b ndustr, or zp code) are used n the model, clusterng should be done at that level. - When the data correlates n more than one wa, we have two cases: - If nested (sa, ct and state), cluster at hghest level of aggregaton - If not nested (e.g., tme and ndustr), use mult-level clusterng. Pooled Model: PCSE - Remarks The bgger the cross-sectonal correlaton, the bgger the SE. That s, NW SE s tend to be smaller than Drscoll and Kraa SE. In smulatons, t s found (as expected) that the PCSE perform better when there s cross-sectonal dependence n the data. But, when there s no dependence n the cross-secton, the standard Whte or NW SE do better. In some cases, these dfferences can be sgnfcant Testng for cross-sectonal dependence ma be a good dea, especall when results are not robust to dfferent SE. LM tests can be easl mplemented. Pesaran (004) proposes an eas test.

23 Pooled Model: PCSE - Remarks The computatonal ssues are straghtforward for balanced data. We need onl the vector of resduals, the model matrx (X), and ndcators for group (and, usuall, tme) to form the clusters. But for unbalanced there are two approaches - Create a balanced subset of the panel to estmate Ω. Advantage: Computatonall smple. - Loop over e t e jt pars to estmate covarances over avalable overlappng tme frames (loop over all pars can take a long tme). Advantage: Informaton s not thrown out. More on PCSE later. Cornwell and Rupert Data (Greene) Cornwell and Rupert Returns to Schoolng Data, 595 Indvduals, 7 Years Varables n the fle are EXP = work experence WKS = weeks worked OCC = occupaton, f blue collar, IND = f manufacturng ndustr SOUTH = f resdes n south SMSA = f resdes n a ct (SMSA) MS = f marred FEM = f female UNION = f wage set b unon contract ED = ears of educaton BLK = f ndvdual s black LWAGE = log of wage = dependent varable n regressons These data were analzed n Cornwell, C. and Rupert, P., "Effcent Estmaton wth Panel Data: An Emprcal Comparson of Instrumental Varable Estmators," Journal of Appled Econometrcs, 3, 988, pp See Baltag, page for further analss. The data were downloaded from the webste for Baltag's text. 3

24 Applcaton : Cornell and Rupert (Greene) Applcaton : Bd-Ask Spread (Hoechle) 4

25 Pooled Model wth (A3 ) - GLS We start wth the pooled model: = X* * + where X * = [X ι] - Σ T x(k+) matrx: * = [ α] - (k+)x matrx Now, we allow E[ j ' X ] = σ j Ω j We can use OLS wth PCSE s or we can do GLS. Note: Wh GLS? Effcenc. Suppose Ω j = I T Then, we onl have cross-equaton correlaton, not tme correlaton. We are back n the (aggregaton) SUR framework Pooled Model wth SUR - GLS Suppose Ω j = I T. We are n the (aggregaton) SUR framework: ˆ GLS ( X' V ( X'[ X) X' V I] X) ( X'[ I] X'[ I] X) X'[ I] For FGLS, use the pooled OLS resduals e and e j to estmate the covarance σ j. Note that ˆ T T t et et ' E' E T where E s a TxN matrx and e t =[e t e t... e Nt ]' s Nx vector. We need to nvert ˆ (NxN matrx). Note: In general, the rank(e) T. Then, rank( ˆ ) T < N => sngulart, FGLS cannot be computed. Ths s a problem of the data, not the model. 5

26 Pooled Model wth Heteroscedastct - GLS Now, suppose we have groupwse heteroscedastct. That s, E[ j ' X ] = 0 for j E[ X ] = σ We do FGLS, as usual, usng the pooled OLS resduals e to estmate the varance σ and, thus, to estmate Σ: We can test ths model wth H 0 : σ =σ =...=σ N. We can use: W = Σ (s s pooled)/var(s ) χ N where s s computed usng the pooled OLS e resduals. N Pooled Model wth Autocorrelaton - GLS Now, suppose we have ndvdual autocorrelaton. That s, E[ t js ' X ] = 0 for j E[ t t-p ' X ] 0 -for example, t t u Var[ t X ] = σ t We do FGLS, as usual, usng the pooled OLS resduals e to estmate the ρ and, thus, to estmate Σ : u ( ).. T.. T T We can test ths model wth H 0 : ρ = ρ =...= ρ N =0. We can use an LM test to test H

27 Pooled OLS wth Frst Dfferences From the general DGP: t = x t + c + t & (A)-(A4) appl. It ma stll be possble to use OLS to estmate, when we have ndvdual heterogenet. We can use OLS f we elmnate the cause of heterogenet: c We can do ths b takng frst dfferences of the DGP. That s, Δ t = t t- = (x t -x t- ) + Δ c + Δ t = Δx t + u t Note: All tme nvarant varables, ncludng c dsappear from the model (one dff ). If the model has a tme trend economc fluctuatons-, t also dsappear, t become the constant term (the other dff ). Thus, ths method s usuall called dffs n dffs (DD or DD). Pooled OLS wth Frst Dfferences Wth strct exogenet of (X,c ), the OLS regresson of Δ t on Δx t s unbased and consstent, but neffcent. Wh? The error s not longer t, but u t. The Var[u] s gven b:,, 0 0,3, 0 0,T,T Var (Toepltz form) That s, frst dfferencng produces heteroscedastct. Effcent estmaton method: GLS. It turns out that GLS s complcated. Use OLS n frst dfferences and use Newe-West SE/PCSE wth one lag. 7

28 OLS wth Frst Dffs: Treatment Applcaton Suppose there s random assgnment to treatment and control groups, lke n a tpcal medcal experment. We compare the change n outcomes across the treatment and control groups to estmate the treatment effect. (We used ths method natural experment n Lecture 8 to deal wth endogenet.) Wth two perods.e., before and after and strct exogenet: Then, t = = δ 0 + δ Treatment + (x ' x ' ) u t (Ths s a CLM. OLS s consstent and BLUE). E[ t Treatment = ] = δ 0 + δ + E[(x ' x ' ) Treatment = ] E[ t Treatment = 0] = δ 0 + E[(x ' x ' ) Treatment = 0] OLS wth Frst Dffs: Treatment Applcaton Assumng that controls are orthogonal to Treatment: δ = E[ t Treatment = ] E[ t Treatment = 0] δ s the dfference n average change n the two perods.e., before and after between the treated and control groups. Ths s the dffs n dffs (DD, DD) estmator. Tpcal problem: Exogenet (randomness) of treatment. That s, (A) E[u t Treatment ] = 0. In medcal experments, dffs n dffs estmaton s routnel used to evaluate the effectveness of a new treatment and/or medcaton. Usual H 0 : δ = 0. It can be tested wth a t-test (usng HAR/PCSEs). 8

29 OLS wth Frst Dffs: Treatment Applcaton Same result can be derved b lookng at levels DGP ( t, x t ) ncludng two dummes: One for Treatment t and one for after Treatment t (Post ): t = x t + c + γ 0 + γ Treatment + γ Post + δ Treatment x Post u t Now, t s eas to separate cross-sectonal dfferences from tme-seres dfferences. - Cross-sectonal dfference E[ t Treatment =, Post =] E[ t Treatment = 0, Post =] = γ + δ Note: Unbased f γ =0 No permanent dfference between the treatment and control groups. OLS wth Frst Dffs: Treatment Applcaton - Tme-sectonal dfference E[ t Treatment =, Post =] E[ t Treatment =, Post =0] = γ + δ Note: Unbased f γ =0 => No common trend over the pre- and posttreatment tmes. In Lecture 8 we emphaszed that we need to make sure that the treatment s the onl dfference between the two groups. That s, n the absence of treatment, the average change n t would have been the same for both groups. Ths s a ke assumpton behnd the DD estmator, usuall tested b lookng at a graph of the behavor of both groups before treatment recall example n Lecture 8 from Reddng & Sturm (008). 9

30 OLS wth Frst Dffs: Natural Experment In Fnance & Economcs, especall n Corporate Fnance, we appl the DD method when we use natural experments (a change n a law, a polc or a regulaton) to stud the effect of X t on t. (Recall Lecture 8.) We have two perods: Before and after the natural experment (the treatment). If we also have a well-defned control group, where the treatment was not admnstered.e., the natural experment never occurred, then, we can use DD estmaton. The number of groups, S, (treated & not treated) under consderaton s usuall small tpcall. N s usuall ver large. Dffs n Dffs: Natural Experment - Example : We are nterested n the effect of labor shocks on wages and emploments. Natural experment: The 980 Marel boatlfs, a temporar lftng of emgraton restrctons n Cuba. Most of the mareltos (the 980 Cuban mmgrants) settled n Mam. Two perods: Before and after the 980 Marel boatlfs. Control group: Low sklled workers n Houston, LA and Atlanta. Calculate unemploment and wages of low sklled workers n both perods. Then, regress t aganst a set of control varables (ndustr, educaton, age, etc.) and a treatment dumm: t = = δ 0 + δ Treatment + (x ' x ' ) u t H 0 : δ =0. Card (990) found no effect of massve mmgraton. 30

31 Dffs n Dffs: Natural Experment - Example : Suppose we are nterested n the effect of accountng transparenc on analsts forecast errors. We can use the Jul 30, 00 Sarbanes-Oxle (SOX) law as a natural experment. Two perods: Before and after SOX. Control group (frms under no SOX law): It ma be dffcult to fnd a good control group. Mabe, we can use analsts n Canada or n German as a control group. Calculate the dsperson of the analsts forecasts,, n both perods & regress t aganst a set of control varables (experence of analst, educaton, age, etc.) and a treatment dumm: t = = δ 0 + δ Treatment + (x ' x ' ) u t H 0 : δ =0. Dffs n Dffs: Remarks We express the DGP n terms of (ndvuals), s (groups), and t (tme): st = δ s + δ t + δ Treatment st + x st ' st Usuall, we have small S and T; but large N. Snce, n general, we have wthn group correlaton (treated ndvduals show smlar errors), the asmptotcs of the t-test are drven b S*T. Donald and Lang (004): Under the usual (generous) assumptons, t converges to a normal dstrbuton (a t ST-K ma work better). Intuton: Suppose that wthn s,t groups the errors are perfectl correlated. Then, we onl have S*T ndependent observatons! Gven the potental (tme-varng) correlatons n the errors, OLS SE can be terrble. PCSE tend to do better. 3

32 Dealng wth Attrton Attrton problem: If an unbalanced panel s a result of some selecton process related to ε t, then endogenet s present and need to be dealt wth usng some correcton methods. Otherwse, we have attrton bas. Example: In the "Qualt of Lfe for cancer patents" stud dscussed n Greene, appearance for the second ntervew was low for people wth ntal low QOL (death or depresson) or wth ntal hgh QOL (don t need the treatment). Solutons to the attrton problem Heckman selecton model (used n the stud) Prob[Present at ext covarates] = Φ(z θ) (Probt model) Addtonal varable added to dfference model = ϕ(z θ)/φ(z θ) The FDA soluton: fll wth zeros. (!) Pooled Model: ML Estmaton In the pooled model, = X +, we assume t ~N(0, Σ), where t = [ t, t,..., Nt ]' and Σ s an NxN matrx. We can wrte the log lkelhood functon as: L = log L(, Σ X) = -NT/ ln(π) T/ ln Σ - ½ Σ t t 'Σ - t The ML estmator s equal to the terated FGLS estmator. Testng s straghtforward wth lkelhood rato test. Example: H 0 : No cross correlaton across equatons: The off-dagonal elements of Σ are zero. LR = T (ln ˆ R - ln ˆ U ) = T (Σ ln s - ln ˆ ), whch follows a χ wth N(N-)/ d.f. 3

33 Man Models: FEM and REM Two man approaches to fttng models usng panel data: () Fxed effects regressons. () Random effects regressons. The ke dfference between these two approaches s how the unobservable characterstcs the ndvdual effects- are modeled. Termnolog from expermental desgn (sa, pscholog or medcne), where the emphass was on the knd of sample at hand and nferences: - FE: The ndvduals are fxed. The dfferences between them are not of nterest, onl s nterestng. No ntent on generalzng the results. - RE: The ndvduals come from a random sample drawn from a larger populaton, and the varance between them s nterestng and can be nformatve about the larger populaton. The Fxed Effects Model (FEM) The fxed effects (FE) model (A) t = x t + c + t -observaton for ndvdual at tme t. (A) E[ε t X s,c s ]=0, for all t and s. - X and c strct exogenous. The unobserved component, c, s arbtrarl correlated wth x t : E[c X ] = g(x ) = constant Cov[x t,c ] 0 Under the FE assumpton, snce we omt c, pooled OLS s based and nconsstent. We summarze ( control for ) these unobservable effects wth α, a constant. All tme nvarant characterstc of ndvdual (locaton, gender, natonalt, etc.) are swept awa under ths formulaton. Note: In a FEM, ndvduals serve as ther own controls. 33

34 Estmaton wth Fxed Effects Whatever effects the omtted varables have on the ndvdual at one tme, the wll also have the same effect at a later tme, thus, ther effects wll be constant, or fxed. For ths, we need the omtted varables to have tme-nvarant values wth tme-nvarant effects. Tpcal example, a CEO s IQ/manageral sklls. We expect ths varable to have the same effect at t= or t=0. As we wll see, FEM are estmated usng the wthn transformaton. Thus, f ndvduals do not change much (or at all) across tme, a FEM ma not work ver well. We need wthn-ndvduals varablt n the varables f we are to use ndvduals as ther own controls. Estmaton wth Fxed Effects Matrx notaton - In matrx notaton for ndvdual : = x + c + - c s a T x vector. (Each ndvdual has T observatons.) - In matrx notaton for all ndvduals.e., stackng: = X + c + - we have Σ T observatons. Now, c,, and are Σ T x vectors. Dumm varable representaton t x t ' N j c j d jt t ; d jt ( j) 34

35 Assumptons for the FE Model (FEM) The ndvdual unobservable characterstcs (effects) are correlated wth the ncluded varables: E[c X ] = g(x ) Cov[x t,c ] 0 The FE model assumes c = α (constant; t does not var wth t): = X + d α + ε, for each ndvdual. Stackng X d X 0 d 0 0 β ε α N XN dn β = [X,D] ε α = Zδ ε FEM: Estmaton The FEM s the CLM, but wth man ndependent varables: k+n. OLS s unbased, consstent, effcent, but mpractcal f N s large. The OLS estmates of β and α are gven b: b X X X D X a D X D D D Usng the Frsch-W aug h theorem b =[ X M DX ] X M D Note (Greene): LS s an estmator, not a model. Gven the formulaton wth a lot of dumm varables, ths partcular LS estmator s called Least Squares Dumm Varable (LSDV) estmator. 35

36 FEM: Estmaton M D MD 0 M D (The dumm varables are orthogonal) N 0 0 MD M I d dd d = I d d D T ( ) (/T) T N T D = D D k,l t= t,k.,k t,l.,l N T D = D, D k t=(xt,k -x.,k )(t-. ) XM X= XM X, XM X (x -x )(x -x ) XM = XM XM That s, we subtract the group mean from each ndvdual observaton. Then, the ndvdual effects dsappear. Now, OLS can easl be used to estmate the k β parameters, usng the demeaned data. We know ths method: The wthn-groups estmaton. FEM: Wthn Transformaton Removes Effects The wthn-groups method estmates the parameters usng demeaned data. That s, Y t Y Y t Y k j ( X j k j k j X jt j X j jt j X j t t ) ( t t ) t t Recall: It s called wthn-groups/ndvduals method because the model s explanng the varatons about the mean of the dependent varable n terms of the varatons about the means of the explanator varables for the group of observatons relatng to a gven ndvdual. 36

37 FEM: Wthn Transformaton Removes Effects t k j j ( x jt x j ) ( t t ) t There are costs n the smplct of the wthn-groups estmaton. Frst, all tme-nvarant varables (ncludng constant) for each ndvdual drop out of the model. Ths elmnates all betweenndvduals varablt (whch ma be contamnated b omtted varable bas) and leaves onl the wthn-subject varablt to analze. The loss of the unchangng explanator varables ma matter. For example, f we are studng CEO compensaton, the wthn group transformaton wll lose ndvdual schoolng, ears of experence, prevous job network, etc. That s, schoolng effects on CEO compensaton cannot be stud n ths context. FEM: Wthn Transformaton Removes Effects t k j j ( x jt x j ) ( t t ) t Second, the dependent varables are lkel to have much smaller varances than n the orgnal specfcaton. Now, the are measured as devatons from the ndvdual mean, rather than as absolute amounts. Ths s lkel to adversel affect the precson of the estmates of the coeffcents. It s also lkel to aggravate measurement error bas f the explanator varables are subject to measurement error. Thrd, the manpulaton nvolves the loss of N degrees of freedom (we are estmatng N means!). 37

38 FEM: LS Dumm Varable (LSDV) Estmator b s obtaned b wthn-groups least squares (group mean devatons). Then, we use the normal equatons to estmate a: D Xb + D Da=D a = (D D) - D ( Xb) Note: a=(/t)σ ( - x b)= e T t= t t - Ths s smple algebra the estmator s just OLS - Agan, LS s an estmator, not a model. Ths partcular OLS estmator s called LSDV estmator. - Note what a s when T =. Follow ths wth t -a -x t b=0 f T =. FEM: LSDV Estmator Recall the dumm varable trap: If a constant s present n the model, the number of dumm varable should be N-. The omtted ndvdual or group becomes the reference categor. However, the choce of reference categor s often arbtrar and, thus, the nterpretaton of the wll not be partcularl nterestng. Alternatvel, we can drop the ntercept and defne dumm varables for all of the ndvduals. Ths s the more common approach. The now become the ntercepts for each of the s. If E[ε t X s,c s ] 0, then LSDV cannot be used. It s nconsstent. In ths case, we need to use IVs. Or a good natural experment. 38

39 FEM: Frst-Dfference (FD) Method We can also elmnate the ndvdual FE usng the frst-dfference method. The unobserved effect s elmnated b subtractng the observaton for the prevous tme perod from the observaton for the current tme perod, for all tme perods. Y t k Yt )) t j k j j ( X jt X jt) ( t ( t t Y t j X jt t t The error term s now ( t t ). As before, dfferencng nduces a movng average autocorrelaton f t satsfes the CLM assumptons. Note: If t s subject to AR() autocorrelaton and s close to, takng frst dfferences ma approxmatel solve the problem. FEM: Estmaton FE or FD? Fxed-effects (or Wthn) Estmator Each varable s demeaned -.e., subtracted b ts average. Dumm Varable Regresson -.e., put n a dumm varable for each cross-sectonal unt, along wth other explanator varables. Ths ma cause estmaton dffcult when N s large. FD Estmator Each varable s dfferenced once over tme, so we are effectvel estmatng the relatonshp between changes of varables. 39

40 FEM: Estmaton FE or FD? Theoretcall, when N s large and T s small but greater than (for T=, FE=FD), FE s more effcent when ε t are serall uncorrelated whle FD s more effcent when ε t follows a random walk (ρ=). When T s large and N s small FD has advantage for processes wth large postve autocorrelaton. (If s near, FD solves the nonstatonar problem!) FE s more senstve to nonnormalt, heteroskedastct, and seral correlaton n ε t. On the other hand, FE s less senstve to volaton of the strct exogenet assumpton. Then, FE s preferred when the processes are weakl dependent over tme FEM: Calculaton of Var[b X] Assume strct exogenet: Cov[ε t,(x js,c j )]=0. Ever dsturbance n ever perod for each person s uncorrelated wth varables and effects for ever person and across perods. Now, we have OLS n a CLM. As.Var[b X] = N N N ( / T )plm[( / T ) XM X] = = = D whch s the usual estmator for OLS ( -a -x b) ˆ N T = t= t t N =T - N - K (Note the degrees of freedom correcton) 40

41 FEM: PCSE All prevous comments and remarks appl to the FEM. If dependence structure s suspected, we cluster the SE. We can clustered them b one varable (sa, ndustr, regon) or b several varables (sa, ear and ndustr) -- mult-level clusterng. We buld the SE accordng to the tpe of data we have: - If we do not suspect autocorrelated errors not a strange stuaton-, we can rel on clustered Whte SE s (S 0 ). - If we suspect autocorrelated errors, then the Drscoll and Kraa SE should be used. FEM: Testng for Fxed Effects Under H 0 (No FE): α = α for all. => That s, we test whether to pool or not to pool the data. Dfferent tests: F-test based on the LSDV dumm varable model: constant or zero coeffcents for D. Test follows an F( N-,NT-N-K ) dstrbuton. F-test based on FEM (the unrestrcted model) vs. pooled model (the restrcted model). Test follows an F( N-,NT-N-K ) dstrbuton. A LR can also be done usuall, assumng normalt. Test follows a χ N- dstrbuton. 4

42 FEM: Hpothess Testng Based on estmated resduals of the fxed effects model. () Estmate FEM: t = x t β + α + t => Keep resduals e FE,t () Tests as usual: Heteroscedastct Breusch and Pagan (980) Autocorrelaton: AR() Breusch and Godfre (98) LM NT e ' FE efe d ' T efe efe Applcaton: Cornwell and Rupert Data (Greene) Cornwell and Rupert Returns to Schoolng Data, 595 Indvduals, 7 Years Varables n the fle are: (Not used n regressons) EXP = work experence, EXPSQ = EXP WKS = weeks worked OCC = occupaton, f blue collar, (IND = f manufacturng ndustr) (SOUTH = f resdes n south) SMSA = f resdes n a ct (SMSA) MS = f marred FEM = f female UNION = f wage set b unon contract ED = ears of educaton (BLK = f ndvdual s black) LWAGE = log of wage = dependent varable n regressons (Y) These data were analzed n Cornwell, C. and Rupert, P., "Effcent Estmaton wth Panel Data: An Emprcal Comparson of Instrumental Varable Estmators," Journal of Appled Econometrcs, 3, 988, pp

43 Applcaton: Cornwell and Rupert (Greene) () Returns to Schoolng - Pooled OLS Results K RSS & R X onl Applcaton: Cornwell and Rupert (Greene) () Returns to Schoolng - LSDV Results N+K RSS & R X and group effects 43

44 FEM: Testng for FE (and other formulatons) Pooled FEM Calculatons: F-test 594,3566 = [( )/594]/[83.89/3566] = The Random Effects Model (REM) Recall the general DGP: t = x t + z γ + t -observaton for ndvdual at tme t. When the observed characterstcs are constant for each ndvdual, a FEM s not an effectve tool because such varables cannot be ncluded. An alternatve approach, known as a random effects (REM) model that, subject to two condtons, provdes a soluton to ths problem. Condtons: ()It s possble to treat each of the unobserved Z p varables as beng drawn randoml from a gven dstrbuton. () The Z p varables are dstrbuted ndependentl of all of the X j varables. 44

45 The Random Effects Model (REM) Condtons: () The unobserved Z p varables are drawn randoml from a gven dstrbuton. Thus, the c ma be treated as RV (thus, the name of ths approach) drawn from a gven dstrbuton. Let s call t u. Then, Y t k j k j X j X j jt jt u t t w t t w t u t We deal wth the unobserved effect b subsumng t nto a compound dsturbance term, w t. We wll assume that the Z p s drawn from a dstrbuton wth zero mean and constant varance. Then, E( w ) E( u ) E( u ) E( ) 0 t t t The Random Effects Model (REM) The zero mean assumpton E[u ] = 0-- s not crucal, an nonzero component s beng absorbed b the ntercept,. () The Z p varables are dstrbuted ndependentl of all of the X j varables. Otherwse, u and the compounded error, w t, wll not be uncorrelated wth X j. The RE estmaton wll be based and nconsstent. Note: We would have to use the FEM, even f the frst condton seems to be satsfed. If () and () are satsfed, we can use the REM, but there s a complcaton. Now, w t wll be heterosccedastc. 45

46 REM: Error Components Model REM Assumptons: t = x t + z γ + t = x t + u + t = x t + w t E[ t X ]=0 E[ t X ]= σ ε E[u X ]=0 E[u X ]= σ u E[u t X ] = E[u jt X ] =0 - u and are ndependent. E[u u j X ] = 0 ( j) -no cross-correlaton of RE. E[ t jt X ]= 0 ( j) -no cross-correlaton for the errors, t. E[ t js X ]= 0 (t s) -there s no autocorrelaton for t. w w t w t t u t ( u t u )( u t t ) u, u u t REM: Notaton (Greene) X ε u T observatons u X ε T observatons β N XN εn unn T N observatons N = Xβ+ ε+ u T observatons = Xβ+ w = In all that follows, except where explctl noted, X, X and x contan a constant term as the frst element. t To avod notatonal clutter, n those cases, x etc. wll smpl denote the counterpart wthout the constant term. Use of the smbol K for the number of varables wll thus be context specfc but wll usuall nclude the constant term. t 46

47 REM: Notaton (Greene) u u u u u u Var[ +u ] ε u u u = IT u T T = I T u = Ω Ω 0 0 Var[ ] 0 Ω 0 w X (Note these dffer onl n the dmenson T ) 0 0 ΩN Note: If E[ t jt X ] 0 ( j) or E[ t js X ] 0 (t s), we no longer have ths nce dagonal-tpe structure for Var[w X]. REM: Assumptons - Convergence of Moments XX T N XΩX T N f N f N XX T XΩ X T a weghted sum of ndvdual moment matrces N N u T a weghted sum of ndvdual moment matrces XX = f f xx Note asmptotcs are wth respect to N. Each matrx XX T s the moments for the T observatons. Should be 'well behaved' n mcro level data. The average of N such matrces should be lkewse. T or T s assumed to be fxed (and small). 47

48 REM: Pooled OLS Estmaton (Greene) Standard results for the pooled OLS estmator b n the GR model - Consstent and asmptotc normal - Unbased - Ineffcent We can use pooled OLS, but for nferences we need the true varance.e., the sandwch estmator: XX XΩX XX Var[ b X] N N N N T T T T Q Q * Q 0 as N wth our convergence assumptons REM: Sandwch Estmator for OLS (Greene) N N N XX XΩ X XX Var[ b X] N N N N T T T T X Ω X N X Ω X f, where = Ω =E[ w w X ] T T In the sprt of the Whte estmator, use X Ω X T X ˆ ˆ w w X f, T w ˆ = Hpothess tests are then based on W ald statstcs. - X b THIS IS THE 'CLUSTER' ESTIMATOR Recall: Clustered standard errors or PCSE There s a groupng, or cluster, wthn whch the error term s possbl correlated, but outsde of whch (across groups) t s not. 48

49 REM: Sandwch Estmator Mechancs (Greene) N ˆ ˆ Est.Var[ b X] XX XwwX XX wˆ = set of T OLS resduals for ndvdual. X = TxK data on exogenous varable for ndvdual. Xw ˆ = K x vector of products ( Xw ˆ )( wx ˆ ) N Xw wx KxK matrx (rank, outer product) ˆ ˆ = sum of N rank matrces. Rank K. N N ˆ ˆ ˆ We could compute ths as X w w X = X Ω X. Wh not do t that wa? REM: GLS Standard results for GLS n a GR model - Consstent - Unbased - Effcent (f functonal form for Ω correct) ˆ - - β =[ X Ω X ] [ X Ω ] N - N - X Ω X X Ω =[ ] [ ] - Ω I T T u (note, depends on onl through T ) As usual, the matrx Ω -/ =P wll be used to transform the data. 49

50 REM: GLS The matrx Ω -/ = P s used to transform the data. That s, t where As. Var [ ˆ x GLS t x ] ( X ' t T X ) u ( X *' X *) We call the transformed data: quas tme-demeaned data. As expected, GLS s just pooled OLS wth the transformed data. Note: The RE can be seen as mxture of two estmators: - when θ = 0 (σ u =0) => pooled OLS estmator - when θ = (σ ε =0 or σ u ) => LSDV estmator (u s become the FE) Then, the bgger (smaller) the varance of the unobserved effect.e., ndvdual heterogenet s bgger-, the closer t s to FE (pooled OLS). Also, when T s large, t becomes more lke FE REM: FGLS - Estmators for the Varances To transform the data, we need to estmate σ ε and σ u, consstentl. Usual steps (assume a balanced panel): () Start wth a consstent estmator of β. For example, pooled OLS, b. () Compute Σ Σ t ( t - x t b) -It estmates Σ Σ t (σ ε + σ u ) (3) Dvde b a functon of NT. For example: NT K => We have an estmator of σ, s pooled = e pooled e pooled /(NT K ) We wll use s pooled to estmate the sum: σ ε + σ u (4) Use LSDV estmaton to get a and b LSDV. Keep resduals, e FE,t. (5) Compute Σ Σ t ( t -a - x t b LSDV ) -It estmates Σ Σ t (σ ε ) (6) Dvde b NT-K-N. => We have an estmator of σ ε, s ε=σ Σ t (e FE,t ) /(NT-K-N) (7) Estmate σ u as s u= s pooled -s ε 50

51 REM: FGLS - Estmators for the Varances Feasble GLS requres (onl) consstent estmators of and. Canddates: u ( a xb ) From the robust LSDV estmator: ˆ From the pooled OLS estmator: N T t t t LSDV N TK N N u ( a xb ) T t t OLS t OLS N TK N (t a xb MEANS) From the group means regresson: / T u NK ww ˆ ˆ (Wooldrdge) Based on E[w w ] f t s, ˆ There are man others. N T T t s t st t s X u u N TK N Note: A slght chance n notaton, x does not contan the constant term. REM: Practcal Problems wth FGLS All of the precedng regularl produce negatve estmates of. Estmaton s made ver complcated n unbalanced panels. A bulletproof soluton (orgnall used n TSP, now LIMDEP and others). From the robust LSDV estmator: ˆ u ( a xb ) N T t t t LSDV N T ( a xb ) From the pooled OLS estmator: ˆ ( a xb ) ( N T t t OLS t OLS u N ˆ T a xb ) N T N T t t OLS t OLS t t t LSDV u N T 0 Bullet proof soluton: Do not correct b degrees of freedom. Then, gven that the unrestrcted RSS (LSDV) wll be lower than the restrcted (pooled OLS) RSS, σ u wll be postve! 5

52 Applcaton: Fxed Effects Estmates (Greene) Least Squares wth Group Dumm Varables... LHS=LWAGE Mean = Resduals Sum of squares = Standard error of e =.505 These varables have no wthn group varaton. FEM ED F.E. estmates are based on a generalzed nverse Varable Coeffcent Standard Error b/st.er. P[ Z >z] Mean of X EXP.346*** EXPSQ *** D OCC SMSA ** MS FEM (Fxed Parameter)... UNION.0343** ED (Fxed Parameter) REM: Computng Varance Estmators (Greene) Usng full lst of varables (FEM and ED are tme nvarant) OLS sum of squares = = / (465-9) = u Usng full lst of varables and a generalzed nverse (same as droppng FEM and ED), LSDV sum of squares = u = / ( ) = = Both estmators are postve. We stop here. If u were negatve, we would use estmators wthout DF correctons. 5