Chapter 5. The linear fixed-effects estimators: matrix creation

Transcription

1 haper 5 he lnear fed-effecs esmaors: mar creaon In hs chaper hree basc models and he daa marces needed o creae esmaors for hem are defned. he frs s ermed he cross-secon model: alhough ncorporaes some panel aspecs, s lle dfferen from a dumm varable cross-secon. he second model allows for ndvdual heerogene, reang hs as a fed effec o be removed b he covarance ransformaon; hs s called he fed-effecs model. An alernave approach o he heerogene problem s me-dfferencng of he daa. Secon 5.3 and 5.4 consder dfferencng n balanced and unbalanced panels. hese are he dfferencng models. For each model, hree cases are consdered: unresrcedslopes and nerceps var over me pooledslopes and nerceps are consan over me resrcedslopes are consan, bu nerceps ma var over me. 5.ross-secons: he smple panel model hs model has no ndvdual heerogene bu allows for slopes and nerceps o var over me. hs chaper nvolves a large amoun of mar algebra whch s sraghforward bu eensve. Alhough he dfferen models are defned b smlar equaons, he basc equaons are descrbed n some deal here as he are drecl mplemened n he sofware and so form par of he valdaon of he programs. A shorer verson of hs chaper s n preparaon as a dscusson paper. he ssue of panel balance does no maerall affec he cross-secon or fed effecs, alhough smplfes he daa requremens for he laer. As hs can be done pos eracon, we gnore he ssue here and oulne he adjusmens n he ne chaper. However, he balance of he daase wll deermne wheher he marces for a dfferencng model have o be formed durng eracon or wheher he can be creaed poseracon. 55

2 5..he unresrced case he "unresrced" regresson s E( u β + λ +u E( u u s σ s (5. where s a row vecor, β s a column vecor, and he oher erms are all scalars. Sacked over all ndvduals for me, β + J λ +u (5. where J s an vecor of ones and s he mar of he sacked. Defne - I J J (5.3 where I s he den mar. oe ha J (5.4 hen premulplng (5. b wll remove he me effecs: β + J λ + u β + u (5.5 For he ssem of equaons over all and n, he equvalen of (5.5 s PY ζ + PU (5.6 where Y Z u β u β U ζ u β (5.7 and P s he ssem equvalen of, namel 56

3 P (5.8 57

4 58 P s also smmerc and dempoen. he OLS soluon o hs wll be o mnmse whch gves he normal equaons he consuens of (5. are block-dagonal: where and herefore (5. s equvalen o separae esmaons of oe ha where ha s, he mean of over all ndvduals for perod. Smlarl, and Z PY - Z PU Y PY + U ζ ζ ζ (5.9 Z PY (Z - ζˆ (5. R R R Z PY S S S Z (5. R S (5. ( R S - - βˆ (5.3 - J J - (5.4 J (5.5 - (5.6

5 59 reae a mar v'v b summng over cross-producs for ndvduals for a perod : learl ' s he op-lef corner of v'v and he op rgh, bu he mar also conans all he oher nformaon necessar o calculae he erms n (5.4 and (5.6. he OLS soluon requres he summaon of v'v over for each separae. hs s he forma creaed b he eracon sofware. Fnall, noe ha he value of he me effecs can be calculaed from he mean for each perod: hs nformaon s readl obaned from he cross-produc mar. 5..he pooled case he "pooled" model consrans boh slopes and nerceps o be consan over all perods: J (5.7 v v ] J [ v (5.8 β λ β λ λ β λ β ˆ ˆ - u - - +u + u + + (5.9

6 β + λ +u (5. Sackng over and : Y β + J λ +U (5. where [ ' '... ' ]' and Σ. Premulplng b he mar (as defned above gves Y β + J β + λ + U U (5. he me effec has been removed. he ransformaon mar akes means over he whole regresson, as all observaons are reaed alke. he OLS soluon s - βˆ ( Y (5.3 hs dffers from he unresrced verson n ha he regressor marces are no longer block dagonal and he summaon s aken over he whole regresson. hs me he regressors spl no - J - - J (5.4 In oher words, he mean s hs me aken from he whole se of observaons. Summng he raw cross-produc mar gves v v (5.5 and so he cross-produc mar once more provdes all he nformaon necessar o calculae 6

7 he esmaor. In hs case he nercep s found from β + λ + u ˆ λ - ˆ β (5.6 wh he means aken over all varables. 5..3he resrced case In he me-specfc nercep case wh consan slopes (he "whn" esmaor he model s β + λ +u (5.7 o remove he me effec, sack over all ndvduals and premulpl b as for he unresrced model β + J λ + u β + u (5.8 For he ssem of equaons, he approprae ransformaon mar s P, above: PY Pβ + PU (5.9 bu noe ha β s as defned for he pooled model, nsead of he Zζ n he unresrced model. Agan, he normal equaons are βˆ ( P - PY (5.3 Unlke he unresrced model, hese erms are no longer block-dagonal; however, P PY (5.3 and from (5.4 and (5.6 ma be observed ha 6

8 - - (5.3 herefore, he elemens of he regresson n hs case are he sum of hose n he unresrced case afer he laer has been adjused o ake devaons from me means (n he pooled case, he relevan marces were summed before akng devaons. hus he whn esmaor s also achevable from he v'v cross-produc mar. For he whn case, he esmaes of λ are gven b β + λ + u ˆ λ - ˆ β (5.33 ha s, b akng means for each perod. hs mehod of akng devaons from me means s a one-wa analss-of-covarance approach. learl he coeffcens could also be esmaed b usng me dummes, so wh boher akng devaons o remove hese dummes? he man reason s ha smplfes esng he dfferen specfcaons, as he ess are carred ou on he same number of coeffcens n all hree models. A second reason s ha he analss-of-covarance mehod merel ess for consanc over me; usng he sandard F-ess n he me-dumm specfcaon ess for boh consanc and a common level of he nercep n all hree models Varances and esng n he smple model 3 A levels cross-secon wh me dummes ncluded s avalable n he sofware, alhough hs opon does no calculae he specfcaon ess. he srucure of he npu mar s block-dagonal as before, and he basc daa requremen s sll he mar v 'v. 6

9 onsder varances for he unresrced model frs. Defnng e as he resdual error and E s ssem equvalen, hen e ˆ β ˆ - - λ (5.34 or PE PY - ζˆ (5.35 usng he same noaon of (5.6. he resdual sum of squares s gven b E PE Y PY - Y ˆ ζ + ˆ ζ Z ˆ ζ - Y PY - (Y - Y (Z Z ˆ ζ Y PY - (Y - Y ˆ ζ Y PY - Y ˆ ζ (5.36 or RSSSS-ESS. Subsung he esmaed coeffcens agan, E PE Y PY - Y (Z Y P( I (U P + Z P( I - (Z - (Z Z PY Z PY Z ( + PU (5.37 where s Σ. As he mddle erms drop ou, E PE U P( I - (Z U PU -U (Z - - Z PU Z PU (5.38 E'PE s a scalar, and so he soluon o (5.38 s he race of E'PE. akng epeced values, _(E PE _(r[u PU -U (Z _(r[p UU - (Z r[p _(U U - (Z Z P UU Z PU ] Z P _(U U ] ] (5.39 On he assumpon ha _(UU'σ u I, 63

10 _ (E PE σ u r[p - (Z σ u(r P - r((z σ u( σ u(r σ u ( P - r( I Z ( Z P ] ( - - (5.4 herefore E PE σˆ u ( A smlar resul holds for he pooled and whn esmaors. he man dfferences are he value of "" and he race of he frs mar. If s he number of varables n, hen ˆ σ u ˆ σ p ˆ σ r E PE - u - E PE p - - E PE r - - (5.4 where he u, p, and r subscrps refer o he unresrced, pooled and resrced models. On hese error assumpons, F-sascs for esng hpoheses of he laer wo specfcaons are 64

11 F u vs. up p (E PE p - E PE u /(( - ( + E PE u/( - - F ur u vs. r (E PE E PE r u - E PE /( u /( - - ( - (5.43 F rp r vs. p (E PE p - E PE E PE r /( r - /( - - Large values mpl a rejecon of he more resrced hpohess. One refnemen s o noe ha, as ζ s beng esmaed for he unresrced model over separae regressons, s relavel eas o calculae separae esmaes of he varance for each perod. In hs case, he me-heeroscedasc errors for perod are (from equaon (5.4: e e σˆ u ( hese are he errors repored b he regresson program. However, he F-ess n (5.43 make he assumpon ha he varance s homoscedasc; he poenal benef of me-heeroscedasc errors appears small relave o he addonal comple of he correced F-sasc. 5.Fed-effecs: allowng for ndvdual heerogene he models n hs secon are he more usual "panel" models n ha he allow for ndvdual heerogene. hs s reaed as a fed effec and removed b akng devaons from ndvdual means. me dummes are lef n he regresson. hs s because he ransformaon mar whch removes ndvdual dummes canno remove me dummes, and vce versa. I s 65

12 possble o consruc a mar whch removes boh effecs, bu he srucure of he resulng marces are oo complcaed for our purposes. In an case, wll be demonsraed ha s no necessar o remove he me dummes o consruc esmaors for he sabl of coeffcens over me he unresrced case Le he fundamenal equaon be β + α + λ +u E( u E( u u s σ s (5.45 or, sacked for ndvdual, w β + J α + L λ +u (5.46 where J s a vecor of ones, λ s a vecor of he me effecs, he mar L s a den mar wh he rows removed for whch an observaon s mssng, and w β β β β (5.47 wh rows of w removed where observaons are mssng. Defne - I J J (5.48 where I s he den mar. oe ha J ( Perfec collnear beween he me dummes and he ndvdual dummes means ha one me dumm should be dropped o remove he lnear dependenc. However, hs can easl be done pos eracon, and does no change he qualave resuls of hs chaper a all. hus, alhough he ' of hs secon s acuall sngular, hs s gnored solel o smplf he eposon. he ne chaper dscusses approprae correcons. 66

13 Assume he ndvdual effecs α are fed (whch enables sngle-sage regresson. o remove hem from he equaon, premulpl b : w β + J α + L λ + u w β + L λ + u zζ + u (5.5 wh z[w L] and ζ[β' λ']'. he OLS soluon o hs wll be o mnmse (5.5 he normal equaons for hs are ζˆ z z z - (5.5 he breakdown of he frs elemen here s: z z z ( I - J J z z z - z J J z z z - z z (5.53 Smlarl, z z - z (5.54 where z J z (5.55 are he mean values of he elemens of z and. Because of he block naure of w and L, hs s equvalen o creang a vecor of / mes all varables: 67

14 z [ ] (5.56 I can be seen ha (5.57 herefore, f defnng v [ ] (5.58 o gve he cross-produc mar 68

15 69 hen s clear ha z'z - v'v and z'z s he block dagonal of v'v. A smlar sor holds for z'. As s a column vecor, hen z' s he horzonal summaon of - v'' and z' are he dagonal erms '..' and.. of v''. Unlke he smple case, he me effecs λ are now drecl esmaed, raher han beng eraced from he me means as n Secon 5.. hs makes no real dfference o he oucome. he ssue of esng for levels and consanc of he nerceps does no arse as all hree models are based around devaons from he mean of he whole regresson; herefore he ess for a consan nercep n all perods amoun o esng for zero nerceps n all perods. v v (5.59 (5.6

16 5..he pooled case For he pooled model, he hpohess s ha β and λ are consan over me: β + α +u (5.6 or, sacked for ndvdual, β + J α +u (5.6 where [ ] (5.63 and conans he consan erm 5. Usng as above and sll assumng he ndvdual effecs α are fed, he laer are removed b premulplng b : β + J α + u β + u (5.64 he OLS mnmsaon problem s (5.65 gvng βˆ - (5.66 Agan, hs breaks down no: - J J - - ( Includng he consan erm whn he a hs sage s merel a smplfcaon and has no bearng on he resuls. 7

17 wh J (5.68 he mean values of he elemens of and. oe ha he mean of s dfferen from he mean of z because of he block naure of he laer. In hs case (5.69 and, summng over, (5.7 In addon (5.7 Summng over gves s s (5.7 s s All he summaons over me are from o. Alhough can var from ndvdual o ndvdual, mssng values are se o zero and so he mean calculaons are correc wheher he summaon s over or. Usng he defnon of v from (5.58, s apparen ha ' s he sum of he dagonal blocks of vv and ' s he sum of he all of he blocks of he means mar. ' has a smlar srucure: 7

18 s s s s ( he resrced case ow consder β consan and λ varng over me: β + α + λ +u (5.74 or, sacked, β + J α + L λ +u (5.75 where all he erms are as defned above. As for Secon 5.., he consan erm s separaed ou from. hs s he commones form of lnear panel esmaor found n appled work, even hough nvolves sgnfcan (and ofen no esed resrcons on he basc model (5.45. Premulplng b o remove he ndvdual effecs: β + J α + L λ + u β + L λ + u χ + u (5.76 wh [ L] and χ[β' λ']'. he OLS soluon s - χˆ (5.77 Breakng hs down, - J J - - (5.78 wh 7

19 73 for he mean values. he mean of s he same n all hese alernave hpoheses, bu agan, he mean of has some slghl dfferen elemens: herefore and As for he pooled model, Σ'Σ can be calculaed pos eracon. For he smple means of over, f column c of conans he consan erm hen J (5.79 (5.8 (5.8 (5.8

20 c (5.83 From he defnon of v s apparen ha ' s he block dagonal of vv and ' s - mes one of he consan nersecons of v'v. However, n boh cases he non-consan secons (he bs mus be summed. Agan, ' has a smlar srucure, wh (5.84 learl all ha s needed o esmae all models are he wo marces Σv'v and Σ - v'v. As hese can be creaed pecewse (ha s, for each ndvdual, v'v s found and hen summed no a oallng cross-produc mar, and as hs mar can be creaed whou reference o he parcular varables used n a parcular regresson (ha s, Σv'v conans all varables of neres of whch a subse are used n an parcular regresson, hs presens no real dffcules o he eracon or analss sofware. 6 I s clear ha he daa marces for he pooled and resrced versons are consruced merel b summng over me he relevan elemens from he unresrced mar. hs s wha mgh be epeced. In he smple cross-secon models, akng devaons for each perod led o dfferen marces beng needed for dfferen hpoheses abou varaon over me, as he mean used depended upon he model n queson. In he fed effecs models, he ransformaon s 6 he fac ha he consan erms are somemes ncluded n and somemes represened as separae elemens s merel for noaonal convenence and makes lle dfference o he analss. However, for analcal purposes s easer f he consan erms are alwas ncluded n raher han beng grouped separael, and so hs s he srucure he eracon sofware uses. 74

21 beng made wh respec o devaons from ndvdual means. In ha cone, dfferen assumpons abou me-varng slopes and nerceps merel amouns o a rearrangemen of he varables; he same mean (over all perods for one ndvdual s used. Had assumpons been made abou coeffcens varng over ndvduals, hen esng he dfferen models would have nvolved calculang dfferen means - as he cross-secon models necessaed. oe ha here are wo sgnfcan dsadvanages o he fed-effecs formulaon used here. Frsl, each cross-produc means mar Σ - v'v s dependen upon a parcular value of : as he dvsor n he means mar s, and as, a dfferen value of alers he range of accepable s. Onl n he case of he balanced mar can he means for several ears be calculaed from one Σv'v mar, because n hs case he consan dvsor can be aken ousde all he summaons over. In he general case of an unbalanced panel, s no longer possble o run regressons on mulple combnaons of ears usng he same cross-produc means mar. Separae means marces need o be creaed for each selecon of ears. Secondl, creaon of hese marces nvolves much compuer me and memor - sgnfcanl more han for he smple panel model. However, noe ha Σv'v for a gven conans Σv'v for an smaller. As he wo marces can be creaed ndependenl, s sensble o erac Σv'v n one run and hen he means mar Σ - v'v separael - possbl for several dfferen values of. hs s a relavel effcen soluon o he eracon problem, and s mplemened n he eracon sofware. In fac, he daa requremen s less onerous han saed above. onsder he wo marces requesed for hs secon and he prevous one. I s clear ha each of he dagonal blocks n Σv'v (5.59 corresponds o one of he v'v marces n (5.8. here s no necess o calculae he whole mar (5.59; he me-effecs cross-producs marces wll do equall well. he analss sofware akes accoun of hs. However, as he praccal dfference beween creang Σv'v and Σ - v'v s neglgble, he eracon sofware wrer allows for he 75

22 creaon of full Σv'v marces. Such a mar can conan useful nformaon on nerperod correlaons; n addon, allows for me-dfferencng models o be consruced, as wll be shown n Secon Varances and esng n he heerogeneous model onsder varances for he general model frs. he sum of squared resduals s e e [ - z ˆ ζ + ˆ ζ z z ˆ ζ ] ( - z z z RSS SS + ESS - z (5.85 where e s he resdual error. Defne he smmerc, dempoen mar P: P (5.86 hen e e E PE Y PY z z Z z Z PY (5.87 where E[e' e'... e']', Y[' '... ']', and Z[z' z'... z']'. Defne U[u' u'... u']' and I as he den mar wh Σ rows. Subsung (5.87 n (5.85 gves E PE Y PY - Y (Z (U P + Z P( I - (Z U PU -U (Z Z PY Z ( + PU Z PU (5.88 As for secon 5., E'PE s a scalar wh s soluon equal o s race. akng epeced 7 he full Σv'v mar should also allow for a "mnmum-dsance" model o be generaed, whch allows for a more general error srucure (Hsao (986 ch.3; hamberlan (984. hs s currenl beng nvesgaed. 76

23 values, _(E PE _(r[u PU -U (Z r[p _(U U - (Z - - Z PU Z P _(U U ] ] (5.89 ananng he earler assumpon ha_(uu'σ u I leads o _ (E PE σ u r[p - (Z σ u(r σ u( P - r( I Z P ] (5.9 herefore E PE σˆ u ( hs resul holds for all hree models, as he srucure of P s dencal n all hree. he onl dfference beween hem s he value of. If s he number of varables n (ecludng an consan erm, hen ˆ σ u ˆ σ p ˆ σ r E PE - - ( u E PE - - ( E - - ( PE p r (5.9 where he u, p, and r subscrps refer o he unresrced, pooled and resrced models. On he error assumpons, F-sascs for esng hpoheses of unresrced, pooled and medumm are 77

24 F u vs. up p (E PE p - E PE E PE u /( - u /(( - ( - ( + + F ur u vs. r (E PE E PE u /( r - E PE - u /( - ( ( - + (5.93 F p r vs. p (E PE p - E PE E PE r /( r - - ( /( - + One dffcul s he calculaon of and Σ. However, from he boom-rgh hand corner of - v'v we have he block (5.94 hs wll acuall be sored as a block wh zeroes n he approprae places, bu hs s qualavel he same resul. Summng he dagonal gves (5.95 herefore, summng he consan dagonal for he whole mar gves (5.96 eanwhle, for he oal number of observaons, le d be a marker, f ndvdual was observed n perod and oherwse. hen d d (5.97 learl Σ s hen he sum of he dagonal of he uns secon of Σv'v, as hs gves he number of observaons n each perod. 78

25 Observe ha comes from he means-mar and hus s me dependen. hs s correc: f an ndvdual onl has observaons ousde he range of ears used for a parcular regresson, hen he ma no be ncluded n he "" for ha regresson. he oal number of observaons s jus a sraghforward all of he number of observaons n each perod, and so he oal number of mes an ndvdual has been observed s rrelevan. 5.3Dfferencng models (complee observaons ses An alernave approach o ndvdual heerogene s o ake me dfferences. hs also removes he ndvdual effec; however, for unbalanced panels he arhmec s more complcaed. hs secon consders he case where all ndvduals have he same number of observaons. 5.3.he unresrced case he unresrced case s he same as for he heerogenous model of Secon 5.: w β + J α + L λ +u (5.98 where he marces are as defned for equaon (5.46. o remove heerogene b medfferencng, he ransformaon mar s (5.99 We sll have J; however, s no longer dempoen. Insead, 79

26 - - - I + S ( where S ( he normal equaons for he coeffcens are (from (5.5: ζˆ z z z - (5. oe ha one me dumm wll have o be deleed because he mar s no of full rank. Secon 5.4 dscusses he ssue of denfcaon of me dummes n more deal. Breakng down he componens of (5., z z z z + z S z z z + z S (5.3 Defne (-; ha s, he negave of. hen, from he defnon of S and z, 8

27 (5.4 I can be seen ha z'sz s jus he Hadamard produc of z'z and he (+(+ equvalen of S: z S z z z _ H (5.5 where J s a -vecor of ones and H - J J - J J - J J - J J - J J J - J J - J J - J (5.6 J J - J J - J J - J - J J - J - J - J herefore, all he nformaon o calculae hs specfcaon of he me-dfferencng model s n he mar v'v. As he mar H s a consan mar dependng onl on he ears of he analss and no on an ndvdual varables, hen Σ v'v can be bul up and used for regressons on mulple ears. Fnall, calculang z'' requres 8

28 z S (5.7 whch has a smlar srucure o z'sz and can also be calculaed from v'v. Resrcons can be placed on he specfcaon n a manner analogous o he earler secons. 5.3.he pooled case Le β and λ be consan over me: β + α +u (5.8 conans he consan erm. Sackng over and usng as above and sll assumng he ndvdual effecs α are fed, he laer s removed b premulplng b : β + J α + u β + u (5.9 he OLS normal equaons are βˆ - (5. Agan, hs breaks down no: 8

29 + S + S (5. Bu n hs case S +( ( (5. - ( where (-, as before. 'S has a smlar srucure: S +( ( (5.3 - ( Once more he resul s a consan mar mulple of he consruced Σv'v; n fac, he resul s merel he summaon over of he ' secon of he unresrced case. hus he pooled case can be feasbl consruced from a correcl formed cross-produc he resrced case Fnall, consder β consan and λ varng over me: β + α + λ +u (5.4 or, sacked, β + J α + L λ +u (5.5 where all he erms are as defned above. Agan, he consan erm s made eplc. Premulplng b removes he ndvdual effecs: β + L λ + u (5.6 wh [ L] and χ[β' λ']', as n Par II. he OLS soluon s 83

30 84 Breakng hs down, where, as would be epeced from prevous cases wh 'S as defned above and Agan, hs presens no especal dffcules n he consrucon of he OLS esmaor from a cross-produc mar. he nformaon requremens are less han for he esmaors n secon 5., as all ha s needed o esmae all models s he mar Σv'v, whch can be creaed pecewse 8. hus, alhough he me-dfferencng model requres more compuer power han he smple panel models of Secon 5., s less of a burden han he devaons models. In addon, he cross-produc mar used for he me-dfferencng model (Σv'v, s no dependen on he acual number of ears used n regresson. 8 Agan, he represenaon of he consan erms somemes as beng ncluded n and somemes as separae elemens s merel for noaonal convenence. - χˆ (5.7 S + S + (5.8 - S - S S S (5.9 [ ] ( ( S (5.

31 On he oher hand, he me-dfferencng model s a less effcen soluon han he devaons model for wo reasons. Frsl, does no ake full accoun of all observaons, as he end observaons onl conrbue once o he model; secondl, he devaons esmaors use he cross-correlaon beween all he eplanaor varables o deermne he coeffcens, whereas he me dfferencng approach merel uses correlaons beween wo perods. Fnall, he dfferencng esmaor descrbed n hs secon suffers from he problem of mssng daa. he above soluon s onl approprae where all ndvduals have a full se of observaons for he perod of neres. A balanced sample s ver much he ecepon n ES eracons, and so secon 5.4 consders he ssue of me dfferencng from an unbalanced panel Varances and esng n he me-dfferencng model Agan, consder varances for he general model frs. As he ssem s sll block-dagonal, he ssem equvalen of ' s P: P (5. Usng he logc and noaon of he earler models E PE Y PY - Y (Z (U P + Z P( I - (Z U PU -U (Z Z PY Z ( + PU Z PU (5. he soluon o hs scalar s he race of E'PE. akng epeced values, _(E PE _(r[u PU -U (Z r[p _(U U - (Z - - Z PU Z P _(U U ] ] (5.3 85

32 and assumng ha_(uu'σ u I 86

33 _ (E PE σ u r[p - (Z σ u(r ( I σ u(r + σ u( + P - r( I r ( S ( σ u( ( r( I Z P ] (5.4 hus E PE σˆ u (5.5 ( - - Agan, he resuls s common o all hree models, wh onl he value of changng. akng as he number of varables n (ecludng he consan, he relevan adjusmens are E PEu ˆ σ u ( - - ( + E PE p ˆ σ p ( - - ( + E PE r ˆ σ r ( - - ( + (5.6 where he u, p, and r subscrps now refer o he unresrced, pooled and resrced models. oe ha he neffcenc of he dfferencng approach s o some een refleced n he hgher denomnaors n he esmaon of he sandard error. he approprae F-ess for hs specfcaon are F u vs. up p (E PE p - E PE u E PEu /(( /(( - ( ( + F ur u vs. r (E PE r - E PEu /( ( - E PE u/(( - - ( + (5.7 F r vs. rp p (E PE p - E PE r /( - E PE r /(( - - ( + 87

34 he value of can be easl eraced from he cross-produc mar. and are known. 5.4me dfferencng wh mssng observaons If he panels are unbalanced, hen he approach of he prevous secon wll no work. hs s because he mar H s dfferen for each ndvdual, and so he pos faco creaon of he necessar daa marces from Σv'v s no possble. Observaons ma onl be used where boh and - are observed, and hese canno be denfed from an of he group marces. In hs case, marces mus be consruced nall n frs dfferences. Defne ˆ - ˆ -, - observed oherwse (5.8 and smlar erms for andu. learl, as he ndvdual-specfc erm α s consan (α α for all, drops ou of he fnal equaon. Wh no ndvdual heerogene n he equaon for, he approach of Secon 5. can be used, and so all he resuls of ha secon appl here. he fac ha he "means ransformaon" (he mar s beng appled o dfferenced daa does no have an mplcaons for he esmaon of he slope coeffcens. However, here ma be some confuson over he equaon o be esmaed. onsder a sragh dfferencng of he sacked equaon (5.: - - β - - β -+ J λ - J -λ-+ u - u- β - - β -+ J ( λ - λ- + u - u- ( - - β + -( β - β + J ( λ - λ- +u - u - - (5.9 where he requremen ha an ndvdual mus be observed n and - o be ncluded ensures ha JJ-. o esmae hs equaon requres ha boh and - be ncluded eplcl as regressors. hs s feasble for eracon and analss goes, bu requres a degree of carefulness n ensurng ha he correc varables are ncluded. 88

35 o smplf he analss, he ES eracon sofware makes he assumpon ha β β - n ( Alhough here s no concepual necess for hs, s jusfed on he grounds of praccal. he form of (5.9 wh hs assumpon s - - ( - - β + J ( λ - λ- +u ˆ ˆ β + J ˆ λ +uˆ - u - (5.3 learl he form of (5.3 s dencal o (5., wh dependen and eplanaor varables and a un vecor. Premulplcaon b he mar defned n (5.3 wll sll remove he me dummes and so, once he marces have been consruced correcl, all he mahemacs of he frs secon can be appled. he presence of he un vecor n (5.3 means ha can be recovered from he cross-produc mar. hs me-dfferencng approach mples a resrcon on he slope coeffcens ha s no presen n he earler models. A necessar assumpon for he dfferencng approach of (5.3 s ha he slopes do no change sgnfcanl from one ear o he ne, alhough over he whole perod he slopes ma shf. he calculaed βs now represen he bes-fng slope for an wo consecuve perods; ha s, nsead of he ζ of equaon (5.7 beng esmaed slopes for 975, 976, and so on, now represens a separae slope coeffcen for wo ears a a me (975/6, 976/7, 977/8... hs has mplcaons for he nerpreaon of hese coeffcens. If he slopes do change over me, hen he dfferencng model wll show less varaon han he basc me-effecs model, smpl because he dfferencng approach esmaes average slopes. A larger degree of auocorrelaon ma also be epeced, wh he slopes evolvng over me. odels of evolvng coeffcens have been developed b some auhors, bu he are ouwh he scope of hs work. he F-ess descrbed wll onl check for parameer consanc, no ssemac change. 9 hs means ha he auomac sofware procedures produce daa approprae for (5.3. Obvousl, here s no reason wh a cross-produc mar should no conan he daa o esmae (

36 We should remark ha, alhough he pracce descrbed above nvolves a lmaon on he values of he coeffcens, remans a more general specfcaon han s ofen found n dfferencng models. In mos appled analss, all coeffcens save he nercep are kep consan over me; here onl consanc over an wo consecuve perods s assumed. For he me dummes, akng me dfferences means ha he change n λ over an wo perods s now beng esmaed. hs holds for boh (5.9 and (5.3, alhough few auhors seem o recognse hs pon. As hs s a reparameersaon and no a resrcon (such as ha mpled b movng from (5.9 o (5.3, hs s unaffeced b an evoluon of he nercep, n he sense ha he esmaon of an change s unbased. However, n conras o he slope coeffcens, we no longer have an absolue measure of he level of he nercep, and so esmaes of he nercep are no longer drecl comparable wh he resuls from models n levels Varances and esng n he dfferenced unbalanced panels akng dfferences before calculang he mar means ha he epeced error erm s no longer he same as n secon In he dfferenced model, he error erm s (u-u-. Reanng he assumpon ha _(UU'Iσ, he correc varance for hs erm s herefore [ - - ] σ _ ( u u- ( u u- _ ( u u- + _ ( u u - (5.3 I In hs case, hen he epeced value of he error erm from (5.4 becomes _ (E PE σ u( ( - - (5.3 and so he esmaed error erms for he hree models are now he levels of he nerceps can be recovered from he means of he regresson. 9

37 ˆ σ u ( ˆ σ p ˆ σ r E PE - - E PE p ( - - E PE r ( - - u (5.33 he F-sascs are unchanged as he era s cancel ou. As n he prevous secons, noe ha esmang ζ for he unresrced model over separae regressons enables he calculaon of me heeroscedasc errors. Allowng for heeroscedasc n (5.33 gves _ [ ] ( σ + σ - σ u ( u u ( u u I I (5.34 and herefore e e σˆ u ( oe ha hs esmaed varance s a compound erm, and so here s no need o dvde he RSS b wo. hese are he repored errors, bu agan, he F-ess for model resrcons are based on homoscedasc errors. 9