Lecture 15 Panel Data Models

Transcription

1 Lecture 15 Panel Data Models

2 Panel Data Sets A panel data set, or longtudnal data set, s one where there are repeated observatons on the same unts. The unts may be ndvduals, households, enterprses, countres, or any set of enttes that reman stable through tme. The Natonal Longtudnal Survey (NLS) of Youth s an example. The same respondents were ntervewed every year from 1979 to Snce 1994 they have been ntervewed every two years. Panel data sets are often very large. If there are N unts and T tme perods => Number of observatons: NT. Great for estmaton! Tme seres y y y1 y t T y y y y Cross secton t 2T y y y y 1 2 t T y y y y N1 N 2 Nt NT

3 A standard panel data set model stacks the y s and the x s: y = X + c + X s a Σ T xk matrx s a kx1 matrx c s Σ T x1 matrx, assocated wth unobservable varables. y and are Σ T x1 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 y y y y y y y y y y ;...; ;...; Notaton:

4 Panel Data Sets Fnancal data COMPUSTAT provdes fnancal data by frm (N =99,000) and by quarter (T = 1962:I, 1962:II,..., ) CRSP daly and monthly stock and ndex returns from 1962 on. Datastream provdes economc and fnancal data for countres. It also covers bonds and stock markets around the world. Intra-daly data: Olsen (exchange rates) and TAQ (stock market transacton prces). Essentally nfnte T, large N. OptonMetrcs, also known as Ivy DB, s a database of hstorcal prces, mpled volatlty for lsted stocks and opton markets. Internatonal Fnancal Statstcs (IFS from the IMF) covers economc and fnancal data for almost all countres post-wwii.

5 Balanced and Unbalanced Panels Notaton: y,t, = 1,,N; t = 1,,T Mathematcal and notatonal convenence: - Balanced: NT (that s, every unt s surveyed n every tme perod.) - Unbalanced: N T =1 Q: Is the fxed T assumpton ever necessary? SUR models. The NLS of Youth s unbalanced because some ndvduals have not been ntervewed n some years. Some could not be located, some refused, and a few have ded. CRSP s also unbalanced, some frms are lsted from 1962, others started to be lsted later.

6 Panel Data Models Wth panel data we can study dfferent ssues: - Cross sectonal varaton (unobservable n tme seres data) vs. tme seres varaton (unobservable n cross sectonal data) - Heterogenety (observable and unobservable ndvdual heterogenety) - Herarchcal structures - Dynamcs n economc behavor - Group effects (ndvdual effects) - Tme effects

7 Panel Data Models: Smplest Model - SUR In Zellner s SUR formulaton we have: (A1) y t = x t + t (A2) E[ X] = 0, (A3 ) Var[ X] = 2 I T --groupwse heteroscedastcty. E[ t jt X] = j --contemporaneous correlaton E[ t jt X] = 0 when t t (A4) Rank(X) = full rank In (A1)- (A4), we have the a GR model wth heteroscedastcty. 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.

8 Panel Data Models: Smple Model - Poolng Assumptons (A1) y t = x t + z γ + t - the DGP = 1, 2,..., N - we thnk of as ndvduals or groups. t = 1, 2,..., T - usually, N >> T. (A2) E[ X,z] = 0, - X and z: exogenous (A3) Var[ X,z] = 2 I. - Heteroscedastcty 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, qualty of management, growth opportuntes, etc.

9 Panel Data Models: Basc Model The DGP (A1): y t 1 k X j jt j 2 p 1 - The ndex refers to the unt of observaton, t refers to the tme perod, and j and p are used to dfferentate between dfferent observed and unobserved explanatory 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 by a set of dummy varables, one for each tme perod except the reference perod. s p Z p t t -TheX j varables are usually the varables of nterest, whle the Z p varables are responsble for unobserved heterogenety and as such consttute a nusance component of the model. 25

10 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. Usually, t s convenent to defne a term c known as the unobserved effect, representng the jont mpact of the Z p varables on y. s c Z p 1 p p We can rewrte the regresson model as: y t 1 k j 2 j X The characterzaton of the c component s crucal. Dfferent ways of thnkng about z p create dfferent panel data models. jt c t t 30

11 Panel Data Models: Basc Model y t 1 k j 2 j X Note that f the X j s are so comprehensve that they capture all the relevant characterstcs of the ndvdual, there wll be no relevant unobserved characterstcs. jt => Now, c can be dropped. Pooled OLS may be used, treatng all the observatons for all of the tme perods as a sngle sample. c t t In general, droppng c leads to mssng varables problem: bas! We usually thnk of c as contemporaneously exogenous to the condtonal error. That s, E[ t c ] = 0, t =1,..., T 30

12 Panel Data Models: Basc Model A stronger assumpton can solve the estmaton problem. Strct exogenety can also be mposed. In ths case, Then, E[ t x 1, x 2,...,x T,c ] = 0, E[ y x t t, c ] 1 k j 2 j t =1,..., T X jt c t => The β j s are partal effects holdng c constant. Volatons of strct exogenety are not rare. If x t contans lagged dependent varables or f changes n t affect x t+1 (a feedback effect). But to estmate β we stll need to say somethng about the relaton between x t and c. Dfferent assumptons wll gve rse to dfferent models. 30

13 Panel Data Models: Types The basc DGP: y t = x t + z γ + t & (A2)-(A4) apply. Cases: (1) Pooled (Constant Effect) Model z γ s a constant. z = α. In ths case, y t = x t + α + t => OLS estmates α and consstently. We estmate k+1 parameters. (2) Fxed Effects Model (FEM) The z s are correlated wth X Fxed Effects: E[z X ] = g(x ); effects are correlated wth ncluded varables.e., OLS wll be nconsstent. Assume z γ = α (constant; t does not vary wth t). Then, y t = x t + α + t =>We have a lot of parameters: k+n. We have N ndvdual effects! OLS can be used to estmates α and consstently.

14 Panel Data Models: Types (3) Random Effects Model (REM) The z s are uncorrelated wth the X. That s: E[z X ] = μ. If X contans a constant term, μ=0 WLOG. Add and subtract E[z γ] from (*): y 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. We wll have contemporaneous cross correlatons across the group. =>OLS estmates μ and consstently, but GLS wll be effcent. (4) Random Parameters Model y t = x t ( + h ) + α + t h s a random vector that nduces parameter varaton. Let h ~D(0, σ 2 h), then, we ntroduce heteroscedastcty.

15 Compact Notaton Compact Notaton: y = X + c + X s a T xk matrx s a kx1 matrx c s a T x1 matrx y and are T x1 matrces Recall we stack the y s and X s: X s a Σ T xk matrx s a kx1 matrx c, y and are Σ T x1 matrces y = X + c + Or y = X* * +, wth X * = [X ι] - Σ T x(k+1) matrx. * = [ c] - (k+1)x1 matrx

16 Assumptons for Asymptotcs (Greene) Convergence of moments nvolvng cross secton X. Usually, we assume N ncreasng, T or T assumed fxed. Fxed T asymptotcs (see Greene, p. 196) Tme seres characterstcs are not relevant (may be nonstatonary) If T s also growng, need to treat as multvarate tme seres. Ranks of matrces. X must have full column rank. (X may not, f T < K.) Strct exogenety and dynamcs. If x t contans y,t-1 then x t cannot be strctly exogenous. X t wll be correlated wth the unobservables n perod t-1. Inconsstent OLS estmates! (To be revsted later.)

17 Panel Data Models: No Homoscedastcty We can relax assumpton (A3). The new DGP model: y = X* * +, wth X * = [X ι] - Σ T x(k+1) matrx. * = [ c] - (k+1)x1 matrx Now, we allow E[' X] = Σ σ 2 I ΣT Potentally, a lot of dfferent elements n E[' X] n a panel: - Indvdual heteroscedastcty. Usual groupwse heteroscedastcty. - Autocorrelaton (Indvdual/group/frm) effects. Errors have arbtrary correlaton across tme for a partcular ndvdual : - Temporal correlaton (Tme) effects. Errors have arbtrary correlaton across ndvduals at a moment n tme (SUR-type 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 may be gnored after L perods).

18 Panel Data Models: No Homoscedastcty 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 f2 ) and η t = ϕ η t-1 + ς t, ς t ~D(0, σ ς2 ) 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-varyng crosscorrelatons between and j. (Autorrelatons de out after L perods.) -If f t s uncorrelated across t, =>only contemporaneous (SUR) effects. -If f t s persstent n t, =>both SUR and persstent common effects.

19 Panel Data Models: No Homoscedastcty Dfferent forms for E[' X]: - Indvdual heteroscedastcty. E[ 2 X] = σ 2 => standard groupwse heteroscedastcty. - Autocorrelaton (Indvdual) effects: E[ t s X] 0 (t s) => auto-/tme-correlaton for errors, t. - Temporal correlaton effects: E[ t jt X] 0 ( j) => contemporary cross-correlaton for errors. - Persstent common shocks: E[ t js X] 0 ( j) and t s < L => tme-varyng cross-correlaton for errors. Remark: Heteroscedastcty ponts to GLS effcent estmaton, but, as before, for consstent nferences we can use OLS wth (adjusted for panels) Whte or NW SE s.

20 Panel Data Models: No Homoscedastcty For consstent nferences, we can use OLS wth Whte or NW SE s: - The Whte s adjust only for heteroscedastcty: S 0 = (1/T) e 2 x x. - The NW SE s adjust for heteroscedastcty and autocorrelaton: S T = S 0 + (1/T) l w L (l) t=l+1,...,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. We may have dfferent covarance structures wthn the data, that vary by some characterstc (a cluster), say ndustry. We wll need to group or cluster the errors for these dependences. These new robust SE s are generally called Panel corrected SE s (PCSE).

21 Pooled Model General DGP y t = x t + c + t & (A2)-(A4) apply. The pooled model assumes that unobservable characterstcs are constant, ndependent of --no heterogenety. That s, we have: y t = x t + α + t Now, we have a CLM, wth k+1 parameters. Stackng the varables n matrces, we have: y = X + α ι + Dmensons: -y, ι and are Σ T x1 -X s Σ T xk - s kx1

22 Pooled Model We can re-wrte the pooled equaton model as: y = X* * +, X * = [X ι] - Σ T x(k+1) matrx: * = [ α] - (k+1)x1 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 nconsstency of pooled OLS. The magntude of these problems depends on how the true model behaves: fxed or random.

23 Pooled Regresson: Heterogenety Bas In the pooled model, there s no model for group/ndvdual heterogenety. Thus, pooled regresson may result n heterogenety bas: y Pooled regresson: y t = β 0 +β 1 x t +ε t j True model: Frm 1 True model: Frm 4 True model: Frm 3 True model: Frm 2 x

24 Pooled Regresson: Wthn Transformaton We can estmate by centerng the observatons around ther group/ndvdual means. That s, y y t t y 1 k j2 k j2 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-subject nformaton reflected n the changes wthn subjects across tme. We are estmatng usng the tme-seres nformaton n the data. j ) t t

25 Pooled Regresson: Wthn Transformaton There s a cost n the smplcty of the wthn-groups estmaton. Frst, the ntercept and any X varable that remans constant for each ndvdual wll drop out of the model. The elmnaton of the ntercept may not matter, but the loss of the unchangng explanatory varables may be frustratng. Under the usual assumptons, pooled OLS s consstent and unbased.

26 Pooled Regresson: Between Transformaton There s an addtonal alternatve to estmate, by expressng the model n terms of group/ndvdual means. That s, y y t 1 1 k j k 2 j 2 j j x x j It s called the between estmator because t reles on varatons between groups/ndvduals. We are estmatng usng the cross-sectonal nformaton n the data (the tme-seres varaton s gone!). We lose observatons (and power!): we have only N data ponts. Under the usual assumptons, pooled OLS usng the between transformaton s consstent and unbased. jt t

27 Useful Analyss of Varance Notaton (Greene) Decomposton of Total varaton: N T 2 N T Σ Σ (z z) Σ Σ (z z.) 2 Σ N T z. z =1 t=1 t =1 t=1 t =1 2 Total varaton = Wthn groups varaton + Between groups varaton

28 WHO Data (Greene)

29 Pooled Model: Lvng wth (A3 ) We start wth the pooled model: y = X* * +, wth X * = [X ι] - Σ T x(k+1) matrx. * = [ α] - (k+1)x1 matrx Now, we allow E[ j ' X ] = σ j Ω j Potentally a lot of dfferent forms for E[ j ' X ] n a panel: - Indvdual heteroscedastcty. E[ 2 X ] = σ 2 - 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

30 Pooled Model: Lvng wth (A3 ) Heteroscedastcty ponts to GLS effcent estmaton, but, as before, for consstent nferences we can use OLS wth Whte or NW SE: But, cross-sectonal (contemporaneous or tme-varyng) dependences are gnored. If present, the Whte s or NW s HAC need to be adjusted. We may have dfferent covarance structures wthn the data, that vary by some characterstc (a cluster), say ndustry. We need to group or cluster the errors for these dependences. Drscoll and Kraay (1998) provde an easy 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.

31 Pooled Model: Lvng wth (A3 ) Drscoll and Kraay (1998) provde an easy extenson to estmate robust NW SE s n panels wth cross-sectonal dependences. The NW SE s adjust for heteroscedastcty and autocorrelaton: S T = S 0 + (1/T) l w L (l) t=l+1,...,t (x t-l e t-l e t x t + x t e t e t-l x t-l ) The w(l ) are the usual Bartlett or QS weghts other weghts are OK. The KxK sandwch matrx s defned as ^ X ' X T h t j 1 t ( b) h t j ( b)' wth h t ( b) N ( t ) 1 h t ( b) The NW method s appled to the tme seres of cross-sectonal averages of h t (b). By usng cross-sectonal averages, estmated SE are consstent ndependently of the panel s cross-sectonal dmenson N.

32 Pooled Model: Lvng wth (A3 ) - PCSE If we do not suspect autocorrelaton problems not a strange stuaton, gven that many panel data sets have heavy temporally spaced observatons-, we can rely on Whte SE (S 0 ). We can cluster them by one varable (say, ndustry) or by several varables (say, year and ndustry) -- mult-level clusterng. We assume that the correlatons wthn a cluster (a group of frms, a regon, dfferent years for the same frm, dfferent years for the same regon) are the same for dfferent observatons. These clusetered standard errors are called panel corrected SE (PCSE s) or clustered SE. The clustered Whte-style SE are, sometmes, called Rogers SE.

33 Pooled Model: PCSE - Remarks We assume that the correlatons wthn a cluster are the same for dfferent observatons. Dfferent clusters can produce very dfferent SE. Usually, we cluster usng economc theory (clusterng by ndustry, year, ndustry and year). It s not a bad dea to try dfferent ways of defnng clusters and see how the estmated SE are affected. Be conservatve, report largest SE. These PCSE s are robust to very general forms of cross-sectonal (and temporal) dependence.

34 Pooled Model: PCSE - Remarks In general, we fnd that PCSE tend to be bgger than the usual OLS SE s. The bgger the cross-sectonal correlaton, the bgger the SE. That s, NW SE s tend to be smaller than Drscoll and Kraay 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 may be a good dea, especally when results are not robust to dfferent SE. LM tests can be easly mplemented. Pesaran (2004) proposes an easy test. More on PCSE later.

35 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, 1 f blue collar, IND = 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 UNION = 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 122 for further analyss. The data were downloaded from the webste for Baltag's text.

36 Applcaton 1: Cornell and Rupert (Greene)

37 Applcaton 2: Bd-Ask Spread (Hoechle)

38 Pooled Model wth (A3 ) - GLS We start wth the pooled model: y = X* * + where X * = [X ι] - Σ T x(k+1) matrx: * = [ α] - (k+1)x1 matrx Now, we allow E[ j ' X ] = σ j Ω j We can use OLS wth PCSE s or we can do GLS. Note: Why GLS? Effcency. Suppose Ω j = I T Then, we only have cross-equaton correlaton, not tme correlaton. We are back n the (aggregaton) SUR framework

39 Pooled Model wth SUR - GLS Suppose Ω j = I T. We are n the (aggregaton) SUR framework: ˆ GLS ( X' V ( X'[ 1 1 X) 1 X' V I] X) 1 1 y ( X'[ I] X'[ 1 I] y 1 X) 1 X'[ I] For FGLS, use the pooled OLS resduals e and e j to estmate the covarance σ j. Note that ˆ 1 T T t1 e e t t ' 1 T E' E where E s a TxN matrx and e t =[e 1t e 2t... e Nt ]' s Nx1 vector. We need to nvert ˆ (NxN matrx). Note: In general, the rank(e) T. Then, rank( ˆ ) T < N => sngularty, FGLS cannot be computed. Ths s a problem of the data, not the model. 1 y

40 Pooled Model wth Heteroscedastcty - GLS Now, suppose we have groupwse heteroscedastcty. That s, E[ j ' X ] = 0 for j E[ 2 X ] = σ 2 We do FGLS, as usual, usng the pooled OLS resduals e to estmate the varance σ 2 and, thus, to estmate Σ: N We can test ths model wth H 0 : σ 12 =σ 22 =...=σ N2. We can use: W = Σ (s 2 s 2 pooled)/var(s 2 ) χ 2 N where s 2 s computed usng the pooled OLS e resduals.

41 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, Var[ t X ] = σ 2 t t 1 u t We do FGLS, as usual, usng the pooled OLS resduals e to estmate the ρ and, thus, to estmate Σ : 2 u 1 ( ) T T 2 We can test ths model wth H 0 : ρ 1 = ρ 2 =...= ρ N =0. We can use an LM test to test H T

42 Pooled OLS wth Frst Dfferences From the general DGP: y t = x t + c + t & (A2)-(A4) apply. It may stll be possble to use OLS to estmate, when we have ndvdual heterogenety. We can use OLS f we elmnate the cause of heterogenety: c We can do ths by takng frst dfferences of the DGP. That s, Δy t = y t y t-1 = Δx t + Δ c + Δ t = (x t -x t-1 ) + 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 usually called dffs n dffs.

43 Pooled OLS wth Frst Dfferences Wth strct exogenety of (X,c ), the OLS regresson of Δy t on Δx t s unbased and consstent, but neffcent. Why? The error s not longer t, but u t. The Var[u] s gven by: 2 2,2, ,3, ,T,T 1 Var (Toepltz form) That s, frst dfferencng produces heteroscedastcty. Effcent estmaton method: GLS. It turns out that GLS s complcated. Use OLS n frst dfferences and use Newey-West SE/PCSE wth one lag.

44 OLS wth Frst Dffs: Treatment Applcaton Suppose there s random assgnment to treatment and control groups, lke n a typcal medcal experment. We compare the change n outcomes across the treatment and control groups to estmate the treatment effect Wth two perods.e., before and after- and strct exogenety: y t = y 2 y 1 = δ 0 + δ 1 Treatment + (x 2 ' x 1 ' ) u t (Ths s a CLM. OLS s consstent and BLUE). In medcal experments, they use the dffs n dffs estmaton: ˆ y treatment - y control 1 = "dfference n dfferences" estmator. ˆ Average change n y for the "treated" 0

45 OLS wth Frst Dffs: Natural Experment Suppose you are studyng the effect of X t (say, transparency) on y t (say, analysts forecastng sklls). But, you suspect X t s endogenous. Recall that t s very dffcult to fnd a good IV Z t.e., Z t s exogenous to t and vald (Z t has a good correlaton wth X t ). Fndng a good exogenous event may help n the study of the effect of X t on y t. There are a lot of natural, exogenous events: asterods httng the Pacfc Ocean, earthquakes n New Zealand, etc. But, few of them have an effect on y t, the fnancal varables we are studyng (say, analysts forecastng sklls).

46 OLS wth Frst Dffs: Natural Experment A change n a law, a polcy or a regulaton can have the same exogenous characterstcs as a natural event., but have a drect effect on y t. In the context of studyng the effect of X t on y t, these exogenous events are usually referred as natural experments. Now, we have two perods: Before and after the natural experment. 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 the dffs n dffs estmaton.

47 OLS wth Frst Dffs: Natural Experment - 1 Example 1: We are nterested n the effect of labor shocks on wages and employments. Natural experment: The 1980 Marel boatlfs, a temporary lftng of emgraton restrctons n Cuba. Most of the mareltos (the 1980 Cuban mmgrants) settled n Mam. Two perods: Before and after the 1980 Marel boatlfs. Control group: Low sklled workers n Houston, LA and Atlanta. Calculate unemployment and wages of low sklled workers n both perods. Then, regress y t aganst a set of control varables (ndustry, educaton, age, etc.) and a treatment dummy: y t = y 2 y 1 = δ 0 + δ 1 Treatment + (x 2 ' x 1 ' ) u t H 0 : δ 1 =0. Card (1990) found no effect of massve mmgraton.

48 OLS wth Frst Dffs: Natural Experment - 2 Example 2: Suppose we are nterested n the effect of accountng transparency on analysts forecast errors. We can use the July 30, 2002 Sarbanes-Oxley (SOX) law as a natural experment. Two perods: Before and after SOX. Control group (frms under no SOX law): It may be dffcult to fnd a good control group. Maybe, we can use analysts n Canada or n Germany as a control group. Calculate the dsperson of the analysts forecasts, y, n both perods & regress y t aganst a set of control varables (experence of analyst, educaton, age, etc.) and a treatment dummy: y t = y 2 y 1 = δ 0 + δ 1 Treatment + (x 2 ' x 1 ' ) u t H 0 : δ 1 =0.

49 Dealng wth Attrton Attrton problem: If an unbalanced panel s a result of some selecton process related to ε t, then endogenety s present and need to be dealt wth usng some correcton methods. Otherwse, we have attrton bas. Example: In the "Qualty of Lfe for cancer patents" study 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 study) 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. (!)

50 Pooled Model: ML Estmaton In the pooled model, y = X +, we assume t ~N(0, Σ), where t = [ 1t, 2t,..., Nt ]' and Σ s an NxN matrx. We can wrte the log lkelhood functon as: L = log L(, Σ X) = -NT/2 ln(2π) T/2 ln Σ - ½ Σ t t 'Σ -1 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 2 - ln ˆ ), whch follows a χ 2 wth N(N-1)/2 d.f.

51 Estmaton wth Fxed Effects The two man approaches to the fttng of models usng panel data are known as (1) Fxed effects regressons. (2) Random effects regressons. The key dfference between these two approaches s how the unobservable characterstcs the ndvdual effects- are modeled.

52 Estmaton wth Fxed Effects The fxed effects (FE) model y t = x t + c + t -observaton for ndvdual at tme t. The unobserved component, c, s arbtrarly correlated wth x t : E[c X ] = g(x ) => Cov[x t,c ] 0 => Under the FE assumpton, pooled OLS s based and nconsstent. We summarze these ndvdual effects on a constant α. => All tme nvarant characterstc of ndvdual (locaton, gender, natonalty, etc.) wll be swept away under ths formulaton. But, E[ε t X s,c s ]=0, for all t and s. - X and c strct exogenous.

53 Estmaton wth Fxed Effects Matrx notaton - In matrx notaton for ndvdual : y = x + c + - c s a T x1 vector. (Each ndvdual has T observatons.) - In matrx notaton for all ndvduals.e., stackng: y = X + c+ - we have Σ T observatons. Now, c, y, and are Σ T x1 vectors. Dummy varable representaton y t x t ' N j 1 c j d jt t ; d jt 1 ( j)

54 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 vary wth t): y = X + d α + ε, for each ndvdual. Stackng 1 1 y X d y2 X2 0 d2 0 0 β α y X d N N N = = β [X,D] ε α Zδ ε ε

55 FEM: Estmaton The FEM s the CLM, but wth many ndependent varables: k+n. OLS s unbased, consstent, effcent, but mpractcal f N s large. The OLS estmates of β and α are gven by: b X X X D X y a D X D D D y Usng the Frsch-W aug h theorem 1 1 b =[ X M DX ] X M Dy Note (Greene): LS s an estmator, not a model. Gven the formulaton wth a lot of dummy varables, ths partcular LS estmator s called Least Squares Dummy Varable (LSDV) estmator.

56 FEM: Estmaton 1 M D MD 0 M D (The dummy varables are orthogonal) N 0 0 MD M I d dd d = I dd 1 D T ( ) T (1/T) N T D =1 D D k,l t=1 t,k.,k t,l.,l XM X= XM X XM X N D =1 D D, (x -x )(x -x ) XM y= XM y, XM y k T t=1 t,k.,k t. (x -x )(y -y ) That s, we subtract the group mean from each ndvdual observaton. Then, the ndvdual effects dsappear. Now, OLS can easly be used to estmate the k β parameters, usng the demeaned data. We know ths method to estmate the FEM: The wthn-groups estmaton.

57 t k j j jt j t t t X X Y Y ) ( ) ( 2 t k j jt j t t X Y 2 1 k j j j t X Y 2 1 The wthn-groups method estmates the parameters usng demeaned data. That s, Recall: It s called wthn-groups 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 explanatory varables for the group of observatons relatng to a gven ndvdual. FEM: Wthn Transformaton Removes Effects

58 FEM: Wthn Transformaton Removes Effects y t y k j 2 j ( x jt x j ) ( t t ) There are costs n the smplcty of the wthn-groups estmaton. t Frst, all X varables (ncludng constant) that reman constant for each ndvdual wll drop out of the model. The elmnaton of the ntercept may not matter, but the loss of the unchangng explanatory varables may matter. For example, f we are studyng CEO compensaton, the wthn group transformaton wll lose ndvdual schoolng, years of experence, prevous job network, etc. That s, schoolng effects on CEO compensaton cannot be study n ths context.

59 FEM: Wthn Transformaton Removes Effects y t y k j 2 j ( x jt x j ) ( t t ) t Second, the dependent varables are lkely to have much smaller varances than n the orgnal specfcaton. Now, they are measured as devatons from the ndvdual mean, rather than as absolute amounts. Ths s lkely to adversely affect the precson of the estmates of the coeffcents. It s also lkely to aggravate measurement error bas f the explanatory varables are subject to measurement error. Thrd, the manpulaton nvolves the loss of N degrees of freedom (we are estmatng N means!).

60 FEM: LS Dummy Varable (LSDV) Estmator b s obtaned by wthn-groups least squares (group mean devatons). Then, we use the normal equatons to estmate a: D Xb + D Da=D y a = (D D) -1 D (y Xb) Note: T a=(1/t)σ (y - x b)= e t= 1 t t - Ths s smple algebra the estmator s just OLS - Agan, LS s an estmator, not a model. Ths partcular least squares estmator s called LSDV estmator. - Note what a s when T =1. Follow ths wth y t -a -x t b=0 f T =1.

61 FEM: LSDV Estmator Recall the dummy varable trap: If a constant s present n the model, the number of dummy varable should be N-1. The omtted ndvdual or group becomes the reference category. However, the choce of reference category s often arbtrary and, thus, the nterpretaton of the wll not be partcularly nterestng. Alternatvely, we can drop the 1 ntercept and defne dummy 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.

62 FEM: Frst-Dfference (FD) Method We can also elmnate the ndvdual FE usng the frst-dfference method. The unobserved effect s elmnated by subtractng the observaton for the prevous tme perod from the observaton for the current tme perod, for all tme perods. Y t k Y 1 )) t t k j2 j 2 j ( X jt X jt1) ( t ( t 1 t 1 Y t j X jt t t The error term s now ( t t 1 ). As before, dfferencng nduces a movng average autocorrelaton f t satsfes the CLM assumptons. Note: If t s subject to AR(1) autocorrelaton and s close to 1, takng frst dfferences may approxmately solve the problem. 1

63 FEM: Estmaton FE or FD? Fxed-effects (or Wthn) Estmator Each varable s demeaned -.e., subtracted by ts average. Dummy Varable Regresson -.e., put n a dummy varable for each cross-sectonal unt, along wth other explanatory varables. Ths may cause estmaton dffculty when N s large. FD Estmator Each varable s dfferenced once over tme, so we are effectvely estmatng the relatonshp between changes of varables.

64 FEM: Estmaton FE or FD? Theoretcally, when N s large and T s small but greater than 2 (for T=2, FE=FD), FE s more effcent when ε t are serally uncorrelated whle FD s more effcent when ε t follows a random walk (ρ=1). When T s large and N s small FD has advantage for processes wth large postve autocorrelaton. (If s near 1, FD solves the nonstatonary problem!) FE s more senstve to nonnormalty, heteroskedastcty, and seral correlaton n ε t. On the other hand, FE s less senstve to volaton of the strct exogenety assumpton. Then, FE s preferred when the processes are weakly dependent over tme

65 FEM: Calculaton of Var[b X] Assume strct exogenety: Cov[ε t,(x js,c j )]=0. Every dsturbance n every perod for each person s uncorrelated wth varables and effects for every person and across perods. Now, we have OLS n a CLM. Asy.Var[b X] = 2 N 2 N N ( / T )plm[( / T ) XM X] 1 =1 =1 =1 D whch s the usual estmator for OLS T 2 =1t=1 t tb ˆ N 2 N =1 (y -a -x ) T - N - K (Note the degrees of freedom correcton)

66 FEM: PCSE If heteroscedastcty s suspected, we can use PSCE (clustered SE) for robust nferences. All prevous comments and remarks apply to the FEM. If we do not suspect autocorrelaton problems not a strange stuaton, gven that many panel data sets have heavy temporally spaced observatons-, we can rely on Whte SE s (S 0 ). We can clustered them by one varable (say, ndustry) or by several varables (say, year and ndustry) -- mult-level clusterng. The clustered Whte-style SE s are sometmes called Rogers SE. If we suspect autocorrelaton problems, then the Drscoll and Kraay SE should be used.

67 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 dummy varable model: constant or zero coeffcents for D. Test follows an F( N-1,NT-N-K ) dstrbuton. F-test based on FEM (the unrestrcted model) vs. pooled model (the restrcted model). Test follows an F( N-1,NT-N-K ) dstrbuton. A LR can also be done usually, assumng normalty. Test follows a χ 2 N-1 dstrbuton.

68 FEM: Hypothess Testng Based on estmated resduals of the fxed effects model. (1) Estmate FEM: y t = x t β + α + t => Keep resduals e FE,t (2) Tests as usual: Heteroscedastcty Breusch and Pagan (1980) Autocorrelaton: AR(1) Breusch and Godfrey (1981) LM 2 NT e ' 1 FE efe d T 1 e FE ' e FE 2 1

69 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 2 WKS = weeks worked OCC = occupaton, 1 f blue collar, (IND = 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 UNION = 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 (Y) 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

70 Applcaton: Cornwell and Rupert (Greene) (1) Returns to Schoolng - Pooled OLS Results K RSS & R 2 X only

71 Applcaton: Cornwell and Rupert (Greene) (2) Returns to Schoolng - LSDV Results N+K RSS & R 2 X and group effects

72 FEM: Testng for FE (and other formulatons) Pooled FEM Calculatons: F-test 594,3566 = [( )/594]/[83.89/3566] = 40.64

73 The Random Effects Model (REM) Recall the general DGP: y 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: (1)It s possble to treat each of the unobserved Z p varables as beng drawn randomly from a gven dstrbuton. (2) The Z p varables are dstrbuted ndependently of all of the X j varables.

74 The Random Effects Model (REM) Condtons: (1) The unobserved Z p varables are drawn randomly from a gven dstrbuton. Thus, the c may be treated as RV (thus, the name of ths approach) drawn from a gven dstrbuton. Let s call t u. Then, Y t 1 k j2 X j jt u t t 1 k j2 X j jt t w t w t u t We deal wth the unobserved effect by 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( ) t t t 0

75 The Random Effects Model (REM) The zero mean assumpton E[u ] = 0-- s not crucal, any nonzero component s beng absorbed by the ntercept, 1. (2) The Z p varables are dstrbuted ndependently 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 (1) and (2) are satsfed, we can use the REM, but there s a complcaton. Now, w t wll be heterosccedastc.

76 REM: Error Components Model REM Assumptons: y t = x t + z γ + t = x t + u + t = x t + w t E[ t X ]=0 E[ t2 X ]= σ 2 ε E[u X ]=0 E[u 2 X ]= σ 2 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. 2 w w t t1 w t2 2 u t ( u t1 2 u )( u 2 t2 t ) 2 2 2u, u 2 u t

77 REM: Notaton (Greene) y1 X1 ε1 u11 T 1 observatons 2 u y2 X2 ε 22 T 2 observatons β yn XN εn unn T N observatons N = Xβ+ ε+ u 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

78 REM: Notaton (Greene) u u u u u u Var[ +u ] ε u u u = I T T Var[ w X] = = 2 2 T u I Ω 2 2 T u Ω Ω Ω N (Note these dffer only n the dmenson T ) Note: If E[ t jt X ] 0 ( j) or E[ t js X ] 0 (t s), we no longer have ths nce dagonal-type structure for Var[w X].

79 REM: Assumptons - Convergence of Moments XX T N 1 XΩX T N 1 f N 1 f N 1 XX T XΩ X T a weghted sum of ndvdual moment matrces XX = f f xx 2 N 2 N 1 u 1 T a weghted sum of ndvdual moment matrces XX Note asymptotcs are wth respect to N. Each matrx s the T 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).

80 REM: Pooled OLS Estmaton (Greene) Standard results for the pooled OLS estmator b n the GR model - Consstent and asymptotc 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 1T 1T 1T 1T Q Q * Q 0 as N wth our convergence assumptons

81 REM: Sandwch Estmator for OLS (Greene) Var[ b X] X Ω X T N 1 N 1 N 1 N XX XΩ X XX N N N N 1T 1T 1T 1T X Ω X f, where = Ω =E[ w w X ] T In the sprt of the W hte estm ator, use X Ω X T XwwX ˆ ˆ f, T w ˆ = Hypothess tests are then based on Wald statstcs. y - 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 possbly correlated, but outsde of whch (across groups) t s not.

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

83 Clustered Standard Errors (PCSE) Key Assumpton Correlatons wthn a cluster (a group of frms, a regon, dfferent years for the same frm, dfferent years for the same regon) are the same for dfferent observatons. Procedure (1) Identfy clusters usng economc theory (clustered by ndustry, year, ndustry and year) (2) Calculate clustered standard errors (3) Try dfferent ways of defnng clusters and see how estmated standard errors are affected. Be conservatve, report largest SE. Performance Not a lot of studes some smulatons done for smple DGPs. PCSE s coverage rates are not very good (typcally below ther nomnal sze).

84 Pooled OLS Results (Greene) Ordnary least squares regresson... LHS=LWAGE Mean = Resduals Sum of squares = Standard error of e = Ft R-squared = Model test F[ 8, 4156] (prob) = 362.8(.0000) Panel Data Analyss of LWAGE [ONE way] 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 *** D OCC *** SMSA.14856*** MS.06798*** FEM *** UNION.09410*** ED.05812*** Constant ***

85 Alternatve Varance Estmators (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.

86 REM: GLS Standard results for GLS n a GR model - Consstent - Unbased - Effcent (f functonal form for Ω correct) ˆ =[ -1 ] 1 [ -1 β X Ω X X Ω y ] N -1 1 N -1 1 X Ω X 1 X Ω y = [ ] [ ] Ω I 2 T 2 2 T u (note, depends on only throug h T ) As usual, the matrx Ω -1/2 =P wll be used to transform the data.

87 REM: GLS The matrx Ω -1/2 =P s used to transform the data. That s, y t y where Asy. Var [ ˆ x GLS t x 1 2 ] ( X ' 2 1 t T X ) 1 2 u 2 ( X *' X *) 1 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 = 1 (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 heterogenety s bgger-,, the closer t s to FE (pooled OLS). Also, when T s large, t becomes more lke FE

88 REM: FGLS - Estmators for the Varances To transform the data, we need to estmate σ ε2 and σ u2, consstently. Usual steps (assume a balanced panel): (1) Start wth a consstent estmator of β. For example, pooled OLS, b. (2) Compute Σ Σ t (y t - x t b) 2 -It estmates Σ Σ t (σ ε2 + σ u2 ) (3) Dvde by a functon of NT. For example: NT K 1 => We have an estmator of σ 2, s 2 pooled = e pooled e pooled /(NT K 1) We wll use s 2 pooled to estmate the sum: σ ε2 + σ 2 u (4) Use LSDV estmaton to get a and b LSDV. Keep resduals, e FE,t. (5) Compute Σ Σ t (y t -a - x t b LSDV ) 2 -It estmates Σ Σ t (σ ε2 ) (6) Dvde by NT-K-N. => We have an estmator of σ ε2, s 2 ε=σ Σ t (e FE,t ) 2 /(NT-K-N) (7) Estmate σ u2 as s 2 u= s 2 pooled -s 2 ε

89 REM: FGLS - Estmators for the Varances Feasble GLS requres (only) consstent estmators of and. Canddates: From the robust LSDV estmator: ˆ From the pooled OLS estmator: From the group means regresson: T 2 2 u N t1 t t LSDV N 1TK N N u (y a xb ) (y a xb ) T 2 t1 t OLS t OLS N 1TK 1 (y a xb ) N t MEANS / T u NK 1 N ww ˆ ˆ T 1 T t1 st1 t s t s X u u N 1TK N (Wooldrdge) Based on E[w w ] f t s, ˆ There are many others. Note: A slght chance n notaton, x does not contan the constant term.

90 REM: Practcal Problems wth FGLS All of the precedng regularly produce negatve estmates of. Estmaton s made very complcated n unbalanced panels. A bulletproof soluton (orgnally used n TSP, now LIMDEP and others). From the robust LSDV estmator: ˆ From the pooled OLS estmator: ˆ N T 2 N T (y a xb ) (y T N t1 t t LSDV N 1T N t1 t OLS t OLS 2 u N ˆ 1T 2 1 t1 t OLS t OLS 1 t1 t t LSDV u N 1T 2 u (y a xb ) T (y a xb ) a xb ) 2 0 Bullet proof soluton: Do not correct by degrees of freedom. Then, gven that the unrestrcted RSS (LSDV) wll be lower than the restrcted (pooled OLS) RSS, σ u 2 wll be postve!

91 Applcaton: Fxed Effects Estmates (Greene) Least Squares wth Group Dummy Varables... LHS=LWAGE Mean = Resduals Sum of squares = Standard error of e = These 2 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.11346*** EXPSQ *** D OCC SMSA ** MS FEM (Fxed Parameter)... UNION.03413** ED (Fxed Parameter)

92 REM: Computng Varance Estmators (Greene) Usng full lst of varables (FEM and ED are tme nvarant) OLS sum of squares = = / (4165-9) = 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 were negatve, we would use estmators wthout DF correctons. 2 u

93 REM: Applcaton (Greene) Random Effects Model: v(,t) = e(,t) + u() Estmates: Var[e] = Var[u] = Corr[v(,t),v(,s)] = Lagrange Multpler Test vs. Model (3) = ( 1 degrees of freedom, prob. value = ) (Hgh values of LM favor FEM/REM over CR model) Fxed vs. Random Effects (Hausman) =.00 (Cannot be computed) ( 8 degrees of freedom, prob. value = ) (Hgh (low) values of H favor F.E.(R.E.) model) Sum of Squares R-squared Varable Coeffcent Standard Error b/st.er. P[ Z >z] Mean of X EXP EXPSQ D OCC SMSA MS FEM UNION ED Constant

94 Testng for Random Effects: LM Test We want to test for RE. That s, H 0 : σ u2 =0. We can use the Breusch-Pagan (1980) Test for RE effects. Smlar to the LM-BP test for autocorrelaton, t s based on the pooled OLS resduals, e. It s easy to compute dstrbuted as 12 : Breusch and Pagan Lagrange Multpler statstc Assumng normalty (and for convenence now, a balanced panel) 2 2 N 2 N 2 1(Te ) NT 1[(Te ) ee ] N N 2 N 1 1 t 1ee NT LM= 1 2(T-1) e 2(T-1) Converges to ch-squared[1] under the null hypothess of no common effects. (For unbalanced panels, the scale n front becom es ( T ) /[2 T (T 1)].) N 2 N 1 1

95 REM: LM Test Applcaton - Cornwell-Rupert Note: Check the dfferent standard errors from both models.

96 FE vs. RE Q: RE estmaton or FE estmaton? Case for RE: - In prncple, the REM s more attractve: Why should we assume one set of unobservables fxed and the other random? - RE can deal wth observed characterstcs that reman constant for each ndvdual. In FE, they have to be dropped from model. - In contrast wth FE, RE estmates a small number of parameters - It s effcent. - We do not lose N degrees of freedom.

97 FE vs. RE Case aganst RE: - If ether of the condtons for usng RE s volated, we should use FE. Condton (1): The unobserved effects are drawn randomly from a gven populaton. Ths s a reasonable assumpton n many cases: Many of the panels are desgned to be a random sample (for example, NLSY). But, t would not be a reasonable assumpton f the unts of observaton n the panel data set were data from the S&P 500 frms. Condton (2): The unobserved effect be dstrbuted ndependently of the X j varables. A volaton of condton (2) causes nconsstency n the RE estmaton

98 FE vs. RE FE estmaton s always consstent. On the other hand, a volaton of condton (2) causes nconsstency n the RE estmaton. Q: How can we tell f condton (2) s volated? A: A DHW test can help.

99 DHW (Hausman) Specfcaton Test: 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 Under an H 0 (RE s true), we have one estmator that s effcent (RE) and one neffcent (LSDV). We can use a Durbn-Hausman-Wu test. As n ts other applcatons, the DHW test determnes whether the estmates of the coeffcents, taken as a group, are sgnfcantly dfferent n the two regressons.

100 DHW (Hausman) Specfcaton Test: FE vs. RE 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] N[ 0V, RE] (effcent) nt[ˆ β FE FE RE - β 0V d ] N[, FE] (neffcent) Note: 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 Note: β does not contan the constant term.

101 Computng the DHW Statstc ˆ 2 N 1 Est.Var[ β ] ˆ 1 I FE X X T N ˆ Tˆ u 1 ˆ 2 2 T ˆ Tˆu Est.Var[ βˆ RE] ˆ X I X, 0 = 1 As long as ˆ and are consstent, as N, Est.Var[ˆ β ] Est.Var[ βˆ ] 2 2 ˆu wll be nonnegatve defnte. In a fnte sample, to ensure ths, both must be computed usng the same estmate of ˆ generally be the better choce. 2 FE RE. The one based on LSDV wll Note that columns of zeros wll appear n Est.Var[ ˆ nvarant varables n X. β FE ] f there are tme Note: Pooled OLS s consstent, but neffcent under H 0. Then, the RE estmaton s GLS.

102 DHW Specfcaton Test: Applcaton (Hoechle) Bd-Ask Spread Panel estmaton. Rejecton at the 5% level, lke n ths case, ndcates that β FE β RE - Usually, ths result s taken as an ndcaton of a FEM.

103 DHW Specfcaton Test: Applcaton (Greene) Random Effects Model: v(,t) = e(,t) + u() Estmates: Var[e] = D-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).) The DHW statstc s used to tests the dfference n coeffcents between an RE and FE models - A rejecton, lke n ths case, ndcates that β FE β RE - But, rejectng H 0 does not mply necessarly H 1 s accepted. - Ether the model s msspecfed or u and x t are correlated - Q: Is the model msspecfed (any varable mssng)?