Econometrics 2. Panel Data Methods. Advanced Panel Data Methods I
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1 Panel Data Methods Econometrcs 2 Advanced Panel Data Methods I Last tme: Panel data concepts and the two-perod case (13.3-4) Unobserved effects model: Tme-nvarant and dosyncratc effects Omted varables bas (heterogeney bas) Frst-dfference estmaton Polcy analyss wh two-perod panel data Today: Panel data wh more perods and fxed effects estmaton (13.5, 14.1) Frst-dfferencng wh more than two perods Whn transformaton Dummy varable regresson A smulated llustraton Example: Job tranng grants. Advanced Panel Data Methods 1 Advanced Panel Data Methods 2 1
2 Data structure and model Three approaches to fxed effect estmaton { y, X }, = 1,2,..., n, t = 1, 2,..., T * Consder the case that n s a large number whereas T s small or moderate. * Balanced panel: Same T for all. If the number of tme perods s not the same across ndvduals: Unbalanced panel. Indvduals leave due to attron, new ndvduals mght then be ncluded n the panel. Fxed effect model: y = δ + δ d2 + δ d δ dt + β x β x + a + u 1 2 t 3 t T t 1 1 k k * One dummy varable for each perod (except the frst perod) * Assume: Cov( x, u ) = 0 for all, j, t, s (x's are strctly exogenous). j s Compare three approaches to fxed effect estmaton: 1. Frst-dfferencng (FD): y = δ d2 + δ d δ dt + β x β x + u 2 t 3 t T t 1 1 k k Convenent to reparameterze tme dummes: y = α + α d α dt + β 1 x β x + u 0 3 t T t k k nt ( 1) observatons: Pool and estmate by OLS. Could be dfferences over more than one perod: "Long dfferences". Advanced Panel Data Methods 3 Advanced Panel Data Methods 4 2
3 Three approaches to fxed effect estmaton (2) Three approaches to fxed effect estmaton (3) 2. Whn transformaton: Form ndvdual tme averages of each varable: y 1 T 1 T = y, T x = T x t= 1 j t= 1 j Average equatons over tme and subtract from orgnal equatons (tme dummes can be ncluded, wll be gnored for now): y = δ + β x β x + a + u (the "between" equaton) k k y y = β ( x x ) β ( x x ) + ( u u ) k k k 3. Dummy varable regresson (LSDV): Dummy varable for each ndvdual: d = 1 f ndvdual s observed, d = 0 otherwse. Include full set of n dummes n regresson and nterpret a as the coeffcent of the dummy for ndvdual : n y = β1x β k xk + ad 1 + u = n * Leave out constant term snce d (1,1,...,1)'. 1 = = * Regresson has k + n rght-hand sde varables. Impractcal f n s large. Fxed effect a cancels. OLS regresson on tme-demeaned data s consstent. Advanced Panel Data Methods 5 Advanced Panel Data Methods 6 3
4 Comparson of fxed effect approaches Comparson of fxed effect approaches (2) * OLS estmates of δ 2, δ3,..., δtβ1,..., βk n LSDV are dentcal to those obtaned by the Whn transformaton: Frsch-Waugh theorem (Problem Set #2). Shows that Whn regresson has only Tn k n degrees of freedom. Indvdual-specfc ntercepts can be obtaned from aˆ = y ˆ β x... ˆ β x (and allowance for tme dummes, f present) 1 1 k k * The fxed effects approaches cannot dentfy the coeffcent of tme-nvarant varables. Effects of nteractons between tme dummes (or other tme-varyng varables) and tme-nvarant varables can be dentfed. * Whn estmator and LSDV estmator are exactly equvalent. Whn/LSDV and FD are exactly equvalent only for T = 2. For general case of T 3, the Whn/LSDV and FD estmators dffer but: * Both are consstent (as n ) and become very close for large n. * If Whn and FD estmates are very dfferent whle n s large, then some assumpton for consstency s lkely to be volated. * In fne samples the relatve effcency of each estmator depends on the presence of any correlaton over tme n. - If no correlaton then Whn estmator s the most effcent - If there s substantal correlaton then FD s best. Tme-seres ssues! Next topc n ths course. u Advanced Panel Data Methods 7 Advanced Panel Data Methods 8 4
5 A smulated llustraton: n=5, T=3 Case 1: a and x are posvely correlated: Crossplots of levels and devatons from tme-averages Consder smulated data from the followng model: y = β x + a + u 1 The true value of β s 0.5. The u are ndependent random drawngs from 1 some dstrbuton (unform) and ndependent of x for all j and s. u and x are repeated for each of the three cases. js y x y ȳ ydevm xdevm Compare pooled OLS and whn estmaton for 3 cases: y Case 1: a and x are posvely correlated Case 2: a and x are negatvely correlated Case 3: a and x are uncorrelated x x x Advanced Panel Data Methods 9 Advanced Panel Data Methods 10 5
6 Case 1: a and x are posvely correlated: Analytcal graph Case 2: a and x are negatvely correlated 10 y x Levels 0.6 ydevm xdevm y1 x1 0.6 ydevm1 xdevm Devatons from tme-averages y ȳ y ^β 1 =2.16 (0.15) ^β 1 = 0.53 (0.13) x x x Advanced Panel Data Methods 11 Advanced Panel Data Methods 12 6
7 Case 3: a and x are uncorrelated Job tranng example wh three perods y2 x ydevm2 xdevm2 Example: The effect of a grant to frms for job tranng. Am of program: Enhance the productvy of workers n the frm. Data for 1987, 1988 and Effect of grant mght extend over tme. Model: log( scrap ) = β + δ d88 + δ d89 + β grant + β grant + a + u 0 0 t 1 t Estmate by Whn estmaton and FD estmaton. Use panel data module of PcGve. Advanced Panel Data Methods 13 Advanced Panel Data Methods 14 7
8 Next tme Monday! Random effects models Applyng panel data methods to other data structures Wooldrdge sec Exercses ths week! Advanced Panel Data Methods 15 8
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