Dnamc Panel Data L Gan Septembe 00 The classc panel data settng assumes the stct eogenet: ( t,, T ( t t t + c Once t and c ae contolled fo, s has no patal effect on t fo s t. In ths case, t s stctl eogenous, condtonal on the unobseved effect c. Anothe wa to state the stct eogenet s: (u t,, T 0 ( t u s 0. Ths assumpton s much stonge than ( t u t 0. Howeve, ths stct eogenet assumpton s athe stong. Consde the followng model: ample.: (lagged dependent vaable Statc model wth feedback: We ae nteested n the how the spead of HIV affect condoms sales. t t γ + ρ t- + w t + c + u t t s pe capta condom sales, w t measues the HIV spead (HIV nfecton ate, fo eample; ρ 0, then state dependence; t s stctl eogenous and fnall w t s sequentall eogenous. (u t t, w t, w, c 0. Howeve, w t (HIV spead s nfluenced b past t (condom sales. w t t δ + ρ t- + c + t ample: patents and R&D spendng. Consde a model, k t θ t + tγ + δ τ τ 0 patents RD + c + u tτ Is RD t-j coelated wth toda s u t? A shock n patents ma affect the eanng ablt of the fm, and hence affect futue spendng n R&D. Cov(u t, t+ 0, So stct eogenet fals. t
These models allow u t to be coelated wth futue values of t,.e., t+, t+,, T, but not wth past t,,.e., t-, t-,,. In othe wods, the eo tem u t eman to be andom, but ts effect ma sta n the sstem to affect futue τ fo τ > t. stmate such a model wthout outsde nstument vaables eques the so-called sequental eogenet,.e, Cov( t,u s 0 fo all t s. All pevous models can be consdeed as one veson of the sequental eogenet. It s necessa to ecogne hee that u t cannot be affect b an pevous t o RD t. It has to be andom n the sense that t s the act of God. If we allow u t to be affected b pevous t o RD t, the basc causalt that the model mples s wong. What happens f the stct eogenet fals? Both F and FD estmatos ae nconsstent and based. Fo the F estmato: ( + ( ( u p lm F t t t t T t T t Unde sequental eogenet, t s possble to bound the bas: Theefoe, T (& u [ ( u ] ( u (snce ( 0 T t t t t t t (& u ( u ( u t t T T t To bound the nconsstenc, va ( and ( u v B Cauch-Schwat nequalt [ ] / O( T ( u va( va( u t t t va ae of ode T. t u t So when T s lage, the bas could be small. The ke condton fo ths to hold s that ρ <. Fo the FD estmato: When T > : st dffeencng: Δ t Δ t + Δu t,. t, 3,,T
p lm FD T ( Δ Δ ( Δ Δu T + t t T t T t t t n whch (Δ t Δu t - ( t u t- 0. Theefoe, the FD estmate s based and nconsstent. Note ths bas does not depend on T. How to estmate ths tpe of models? GMM. Agan, st dffeencng to emove c : Δ t Δ t + Δu t,. t, 3,,T Now unde the sequental eogenet assumpton (s t: ( s u t 0, s,,, t ( s Δu t 0, s,,,t- So at tme t we can use t- o as potental nstuments fo Δ t, whee t o (,.,, t Obvousl, fo Δ t, t s lkel that t- would be coelated wth Δ t. Howeve, when the sequental eogenet condton mples ( s u t 0 when s < t, onl the set of vaables t- o can be nstumental vaables fo Δ t. Vaous tpes of models wth the sequental eogenet would mpl dffeent sets of nstuments. These sets of nstuments dffe on the numbe of lags necessa to ensue that the nstuments and eos ae uncoelated. When the numbe lags nceases, t s less lkel that the nstument vaables ae coelated wth Δ t. To estmate such a model, we have to appl GMM. Geneal Method of Moments (GMM Assumpton : ( u 0 Assumpton : ( 0 t +u, wth ( u 0 So the moment condtons ae: ( u ( ( - t 0 Fnd, such that: 3
( ( N N W / mn The soluton to ths poblem:, ( ( u W W Y W W MM + and the covaance of the estmato: ( ( ( W W W W Va MM whee ( u u. Ths s a ve long but ntutve covaance mat. The net step s to fnd the optmal weghtng mat W, whch s the GMM. When W -, then the optmal covaance mat s eached. In ths case,, and ( Y GMM ( ( ( ( ( ( Va GMM How to do t? SLS ( A egesson of Δ t on t- o ( Use n pevous equaton. Δt The potental poblem of ths s that Δ t on t- o could be pool dentfed. The esdual of SLS could be used to constuct optmal weghtng mat GMM. Suppose usng all t- o as IVs. The fst-dffeenced model s: Δ Δ + Δu. Defne o T o 0 0 O ffcent GMM estmato: 4
( Consstent estmato (SLS: Δ Δ Δ u SLS ( Optmal weghtng mat: N W N o Δu Δu Fnall, the GMM estmate: N N (3 o o o o GMM W W N N o Ava N GMM ( W If ( u u ( Ω, the GMM s equvalent to 3SLS. Dffeent Tpes of Models wth the Sequental ogenet Model : t ρ t- + c + u t. (u t t-,,, c 0 Dffeencng: Δ t ρδ t- + Δu t, t. Obvousl the stct eogenet fals: Cov(Δ t-, Δu t Cov( t-,-u t- 0. Note that t-, t-3,, ae uncoelated wth Δu t. Theefoe, n ths case, the set of nstuments fo Δ t s: { t-, t-3,, }. ample: Testng fo pesstenc n count cme ates log(cme ate t + ρlog(cme ate t- + c + u t. Fst dffeencng: Δlog(cme ate t ρδlog(cme ate t- + Δu t. IVs ae: log(cme ate t-, log(cme ate t-3. The fst stage egesson s: Δlog(cme ate t- on log(cme ate t-, log(cme ate t-3. Ths egesson elds a p-value of.03 (whch means F s lage. 5
Howeve, the estmated ρ tuns out to be not sgnfcant fom eo. Model : t ρ t- + c + u t, and u t au t- + ε t. o tem u t s AR(. Plug the eo equaton nto the ognal equaton: t ρ t- + c + au t- + ε t. Note ths mples that Cov( t, u t- 0. Take the fst dffeence of the ognal equaton (we need to wok wth the ognal equaton fst: Δ t ρδ t- + Δu t Note Cov( t-, Δu t Cov( t-, -u t- 0. Theefoe, the nstuments fo Δ t- have to be t-3, t-4,,. Model 3: When both eogenous and endogenous vaables ae pesent Consde a model: t t γ+ w t δ + c + u t. ( t u s 0, fo an t and s, stct eogenet. (w t u s 0, f t s, sequental eogenet. Fst dffeencng: Δ t Δ t γ+ Δw t δ + Δu t. Note that Δu t ae uncoelated wth (w t-, w t-,, w and s fo an s. IVs avalable at tme t ae ( s, w t-, w t-,, w fo an s. Model 4: contempoaneous coelaton between the eo and some eplanato vaables t t γ+ w t δ + c + u t. ( t u s 0, fo an t and s, stct eogenet. (w t u s 0, f t < s, sequental eogenet. (w t u t 0, contempoaneous coelaton. Agan, fst dffeencng: Δ t Δ t γ+ Δw t δ + Δu t. Note that Δu t ae uncoelated wth (w t-, w t-3,, w and s fo an s. 6
IVs avalable at tme t ae ( s, w t-, w t-3,, w fo an s. ample: the effect of cgaette smokng on wage log(wage t t γ+δ cgs t + c + u t. If cgs t and ncome t ae coelated, and ncome t and wage t ae coelated, then cgs t and u t ae coelated. IVs n ths case would be (cgs t-, cgs t-3,, cgs, s ample: effect of pson populaton on cme ates. Dffeencng: log(cme t θ t + log(pson t + t γ + c + u t Δlog(cme t ξ t + Δlog(pson t + Δ t γ + Δu t Smultanet between Δlog(cme t and Δlog(pson t makes estmaton nconsstent. IVs fo Δlog(pson t ea. ( whethe a fnal decson was eached on ovecowdng ltgaton n the cuent ( whethe a fnal decson was eached n the pevous two eas. Ltgatons have seveal stages pe-flng flng pelmna decson b the cout fnal decson futhe acton elease fom these estctons States have ovecowdng ltgatons n an gven ea, states whee such ltgaton occued mght be n an on of these stages. OLS estmate: -0.8(S 0.048 SLS estmate: -.03(S 0.37 ample: we ae nteested n how pe student spendng would affect aveage scoes of a student. avgscoe t θ t + δ spendng t + t γ + c + u t 7
Note n ths model, t should contan aveage faml ncome fo school at tme t. Howeve, we ae unable to collect such data. Theefoe, ncome t ae mssng but the ae coelated wth spendng. st dffeencng : Δavgscoe t θ t + δ Δspendng t + Δ t γ + Δu t We need IVs fo Δspendng t. In ths model, usng lagged spendng ma not be a good dea snce spendng ma affect avgscoe wth a lag past spendng ma affect cuent avgscoe. Theefoe, one cannot use the lagged vaables as IVs. Addtonal IVs ae necessa. Possblt fo addtonal IVs: use eogenous changes n popet ta that ase because of an unepected change n ta laws (Such changes occu n Calfona n 978 and n Mchgan 994 ample (Gan and hang, 007, NBR Wokng Pape. The esults ae not epoted n the pape: the effect of maket se on housng maket tansactons, usng Teas ct-ea data. We ae nteested n how the maket se, chaacteed b the log of populaton se, lpop ct, affect the log of the aveage housng pce lpce ct, and log of the month n the maket to sell (nvento ct. We ae also nteested n how the eogenous shocks, such as unemploment ate, uate ct, affect lpce ct and nvento ct. We t thee dffeent specfcatons. The ke paamaete s the coeffcent on lpop ct. The pedcton of the model s that coeffcent s postve fo lpcect. lpce ct lpce ct- + nteest-ate t + ponts t + uate ct + lpop ct -.035 -.57 -.006.335 F (.0055 (.07 (.00 (.046.05 -.57 -.003.94 F & AR( (.007 (.08 (.0038 (.05 (ho.705.43 -.00 -.0056 -.00.009 Dnamc panel (.006 (.0000 (.004 (.000 (.0089 (ho.009 (.0089 nvento ct nvento ct- + nteest-ate t + ponts t + uate ct + lpop ct.309 -.0377.633 -.44 F (.94 (.453 (.090 (.509.034.73.3.363 F & AR( (.46 (.43 (.086 (.69 (ho.803 8
.065.608.748.447.509 Dnamc (.0 (.09 (.0 (.03 (.7 The thee dffeent specfcatons eld ve dffeent estmates. Model 5: Measuement eo nduced endogenet Consde the classc measuement eo model: t t * + c + u t, t,, T, and,, N. whee (u t t *,, c, stct eogenet. Let Pooled OLS: t t * + t, whee t * and t ae uncoelated. POLS Cov + Cov + whee σ va( cov(, t ( t, c + ut t Va( t ( t, c σ Va( t t t. Fom pevous equaton, thee ae two souces of asmptotc bas: coelaton between c and t, and the measuement bas. If t and c ae postvel coelated, then two souce of bas tend to cancel each othe out. Now consde the fst dffeencng: Δ t Δ t * + Δu t Δ t - Δ t * + Δu t FD Cov + Cov σ ( Δt, Δut Δt Va( t ( Δt, Δt Va( Δt Cov( Δt, Δt Va( Δt [ σ Cov( Δt, Δt ] Va( Δt σ ( ρ ( + ( ρ σ ρ 9
whee ρ co( t *, t- *, ρ co( t, t-. Va [ ] ( Δ σ ( ρ + σ ( ρ t. As ρ, bas -. How to estmate such a model? Consstent a moe geneal fom: Let t w t w t *. t t + w t * + c + u t, Assume stct eogenet of w t * and t. Replacng w t * wth w t and st dffeencng: Δ t Δ t + w t - t + Δu t The standad classc eo n vaable assumpton: ( t, w *, c 0, t,,, T, whch mples t s uncoelated wth s and w s * fo all t and s. Howeve, Δ t and Δw t ae coelated, so we need IV fo Δw t. Two appoaches: ( ( Measue w * t twce. If ( t s 0 fo an t s, no seal coelaton n measuement eo. In ths case, we have nstuments eadl avalable. Theefoe, the set of vaables (w t-, w t-3,, w and (w t+ w t+,, w T as nstuments fo Δw t. 0
Models wth Indvdual Specfc Slopes t t + c + g t+ u t, (9 Fo eample, each ct, ndvdual, fm etc. s allowed to have ts own tme tend. andom tend model. If t s natual log of some vaables (such as GDP, then (9 s sometmes efeed as andom gowth model, n whch g s oughl the aveage gowth ate ove a peod (holdng the eplanato vaables fed. We want to allow (c, g to be abtal coelated wth t. Ou pma nteest s to consstentl estmate. Stct eogenet, (u t,, t, c, g 0 One appoach: ( fst dffeencng. Δ t Δ t + g + Δu t, (0 ( fed effect model of (*. Obvousl t eques T 3 ample: Fedbeg, L (998: Dd Unlateal Dvoce Rase Dvoce Rates? vdence fom Panel Data, Amecan conomc Revew 88, 608-67. Issue: do unlateal and no-fault dvoce laws encouage moe dvoces? A substantal ncease n dvoce ates:. pe thousand people n 960 to 5.0 n 985. At the same tme, states substantall lbealed and smplfed the dvoce laws: ( most states used to eque both spouses had to consent n the absence of fault; no longe so. ( Most states also adopted some fom of no-fault dvoce, elmnatng the need fo one spouse to pove a tansgesson b the othe. A ecent tend to abandon ths faml values, etc. dvoce ate st c 0 + c *unlateal st + c s *state s + c 3t *ea t + c 4s *state s *tme t + c 5s *state s *tme t + u st Data and estmates:
Fom the gaph, t looks lke a quadatc gowth (Data s lsted n the pape. Theefoe, havng quadatc tem n tme s mpotant. Note f no ea effect and state effect (column 3., the estmate s bg and sgnfcant. Howeve, wth ea and state effect (columns 3. and 3.3, the estmate ae no longe sgnfcant; fnall, allowng tend, the estmates (columns 3.4 and 3.5 become sgnfcant agan and estmates ae
easonable. Model: A geneal model wth ndvdual-specfc slopes: t t + t + u t. (0. Ths model allows some tme-nvaant paametes. Note thee s no longe a need to have unobseved heteogenet tem c. ample: ( Polachak and Km 994 etun to educaton ma be dffeent fo dffeent people. ( Lemeu (998 unobseved heteogenet s awaded dffeentl n the unon and nonunon sectos. Assumpton (stct eogenet: (u t,, a 0 Rewte equaton (0.: + + u. ( ( Defne M I T. It s nteestng to note that M 0. So pemultplng M to ( would elmnate the t tem. Note that: M γ, ( whch s the esdual fom the egesson of on. M γ whch s the s OLS estmate of the egesson of on. Theefoe, the M pemultpled ( s: M M + M u. If we let M & γ t, and M γ t &. Then ( becomes: & + u&, (3 The OLS of (3 s: 3
+ F u Gven the stct eogenet assumpton, ( 0 u & & & &. So the estmate s consstent. Futhe, f we assume that (u u,, a σ u I O we could use obust standad eo. To obtan a consstent estmato of n equatons (0. o (: Pemultpl ( b ( and eaange to get: ( ( ( ( u + + So we get: ( ( ( u Unde Assumpton, we have ( ( ( [ ] So, a consstent estmato of ( s: ( ( N F N (4 Note n (4 aveages ove all. Wth fed T, we cannot consstentl estmate each, as n the case of lnea panel data model. Howeve, fo each, the tem could be unbased. To get asmptotc covaance mat we defne: ( ( F It s eas to see that ( ( ( ( ( ( ( F +, 4
Matched Pas and Cluste Samples ample, husband and wfe, o sblngs, + + f f + u + u We just have to ecogne that ths s equvalent to a two-peod panel data model. O moe geneall, a cluste model, whee numbe of people n a cluste ma va. Ths wll cause poblems of homoscedastct theefoe, t s natual to use obust standad eo. Pee effect model: δ + v s s + w( s s Thee ae man eamples fo ths tpe of models. The ssue hee s f the cluste o goups ae endogenousl fomed. If the ae, ths s smla to Hausman-Talo model. Howeve, f the ae endogenousl fomed, then we have to fnd outsde nstuments. A ecent pape usng A Foce Academ n whch case the goup s eogenousl fomed s qute nteestng. ` 5