DEPARTMENT OF ECONOMICS WORKING PAPERS

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1 DEPARTMENT OF ECONOMICS WORKING PAPERS economics.ceu.hu The Esimaion of Muli-dimensional Fixed Effecs Panel Daa Models by Laszlo Mayas and Laszlo Balazsi 0/ Deparmen of Economics, Cenral European Universiy, Deparmen of Economics, Cenral European Universiy

2 Absrac The paper inroduces for he mos frequenly used hree-dimensional fixed effecs panel daa models he appropriae wihin esimaors. I analyzes he behaviour of hese esimaors in he case of no-self-flow daa, unbalanced daa and dynamic auoregressive models. Then he main resuls are generalised for higher dimensional panel daa ses as well. Key words: panel daa, unbalanced panel, dynamic panel daa model, mulidimensional panel daa, fixed effecs, rade models, graviy models, FDI. JEL classificaion: C, C, C4, F7, F47.

3 . Inroducion Mulidimensional panel daa ses are becoming more readily available, and used o sudy phenomena like inernaional rade and/or capial flow beween counries or regions, he rading volume across several producs and sores over ime hree panel dimensions, he air passenger numbers beween muliple hubs deserved by differen airlines four panel dimensions and so on. Over he years several, mosly fixed effecs, specificaions have been worked ou o ake ino accoun he specific hree or higher dimensional naure and heerogeneiy of hese kinds of daa ses. In his paper in Secion we presen he differen fixed effecs formulaions inroduced in he lieraure o deal wih hree-dimensional panels and derive he proper Wihin ransformaions for each model. In Secion 3 we firs have a closer look a a problem ypical for such daa ses, ha is he lack of self-flow observaions. Then we also analyze he properies of he Wihin esimaors in an unbalanced daa seing. In Secion 4 we invesigae how he differen Wihin esimaors behave in he case of a dynamic specificaion, generalizing he seminal resuls of Nickell [98], in Secion 5 we exend our resuls for higher dimensional daa ses and finally, we draw some conclusions in Secion 6.. Models wih Differen Types of Heerogeneiy and he Wihin Transformaion In hree-dimensional panel daa ses he dependen variable of a model is observed along hree indices such as y ij, i =,..., N, j =,..., N, and =,..., T. As in economic flows such as rade, capial FDI, ec., here is some kind of reciprociy, we assume o sar wih, ha N = N = N. Implicily we also assume ha he se of individuals in he observaion ses i and j are he same, hen we relax his assumpion laer on. The main quesion is how o formalize he individual and ime heerogeneiy, in our case he fixed effecs. Differen forms of heerogeneiy yield naurally differen models. In heory any fixed effecs hree-dimensional panel daa model can direcly be esimaed, say for example, by leas squares LS. This involves he explici incorporaion in he model of he fixed effecs hrough dummy variables see for example formulaion 3 laer on. The resuling esimaor is usually called Leas Squares Dummy Variable LSDV esimaor. However, i is well known ha he We mus noice here, for hose familiar wih he usual panel daa erminology, ha in a higher dimensional seup he wihin and beween groups variaion of he daa is somewha arbirary, and so he disincion beween Wihin and Beween esimaors would make our narraive unnecessarily complex. Therefore in his paper all esimaors using a kind of projecion are called Wihin esimaors.

4 firs momen of he LS esimaors is invarian o linear ransformaions, as long as he ransformed explanaory variables and disurbance erms remain uncorrelaed. So if we could ransform he model, ha is all variables of he model, in such a way ha he ransformaion wipes ou he fixed effecs, and hen esimae his ransformed model by leas squares, we would ge parameer esimaes wih similar firs momen properies unbiasedness as hose from he esimaion of he original unransformed model. This would be simpler as he fixed effecs hen need no o be esimaed or explicily incorporaed ino he model. We mus emphasize, however, ha hese ransformaions are usually no unique in our conex. The resuling differen Wihin esimaors for he same model, alhough have he same bias/unbiasedness, may no give numerically he same parameer esimaes. This comes from he fac ha he differen Wihin ransformaions represen differen projecion in he i, j, space, so he corresponding Wihin esimaors may in fac use differen subses of he hreedimensional daa space. Due o he Gauss-Markov heorem, here is always an opimal Wihin esimaor, excaly he one which is based on he ransformaions generaed by he appropriae LSDV esimaor. Why o boher hen, and no always use he LSDV esimaor direcly? Firs, because when he daa becomes larger, he esimaion of a model wih he fixed effecs explicily incorporaed ino i is quie difficul, or even pracically impossible, so he use of Wihin esimaors can be quie useful. Then, we may also exploi he differen projecions and he resuling various Wihin esimaors o deal wih some daa generaed problems. The firs aemp he properly exend he sandard fixed effecs panel daa model see for example Balagi [995] or Balesra and Krishnakumar [008] o a mulidimensional seup was proposed by Mayas [997]. The specificaion of his model is y ij = β x ij α i γ j λ ε ij i =,..., N j =,..., N, =,..., T, where he α, γ and λ parameers are ime and counry specific fixed effecs, he x variables are he usual covariaes, β K he focus srucural parameers and ε is he idiosyncraic disurbance erm. The simples Wihin ransformaion for his model is y ij ȳ ij ȳ ȳ An early parial overview of hese ransformaions can be found in Mayas, Harris and Konya [0].

5 where ȳ ij = /T ȳ = /N i ȳ = /N T i y ij j y ij j y ij However, he opimal Wihin ransformaion which acually gives numerically he same parameer esimaes as he direc LS esimaion of model, ha is he LSDV esimaor is in fac y ij ȳ i ȳ j ȳ ȳ 3 where ȳ i = /NT j ȳ j = /NT i y ij y ij Anoher model has been proposed by Egger and Pfanffermayr [003] which akes ino accoun bilaeral ineracion effecs. The model specificaion is y ij = β x ij γ ij ε ij 4 where he γ ij are he bilaeral specific fixed effecs his approach can easily be exended o accoun for mulilaeral effecs as well. The simples and opimal Wihin ransformaion which clears he fixed effecs now is y ij ȳ ij where ȳ ij = /T y ij 5 I can be seen ha he use of he Wihin esimaor here, and even more so for he models discussed laer, is highly recommended as direc esimaion of he model by LS would involve he esimaion of N N parameers which is no very pracical for larger N. For model his would even be pracically impossible. A varian of model 4 ofen used in empirical sudies is y ij = β x ij γ ij λ ε ij 6 As model is in fac a special case of his model 6, ransformaion can be used o clear he fixed effecs. While ransformaion leads o he opimal Wihin 3

6 esimaor for model 6, is is clear why i is no opimal for model : i overclears he fixed effecs, as i does no ake ino accoun he parameer resricions γ ij = α i γ i. I is worh noicing ha models 4 and 6 are in fac sraigh panel daa models where he individuals are now he ij pairs. Balagi e al. [003], Baldwin and Taglioni [006] and Baier and Bergsrand [007] suggesed several oher forms of fixed effecs. A simpler model is y ij = β x ij α j ε ij 7 The Wihin ransformaion which clears he fixed effecs is y ij ȳ j where ȳ j = /N i y ij Anoher varian of his model is y ij = β x ij α i ε ij 8 Here he Wihin ransformaion which clears he fixed effecs is y ij ȳ i where ȳ i = /N j y ij The mos frequenly used variaion of his model is y ij = β x ij α i α j ε ij 9 The required Wihin ransformaion here is y ij /N i y ij /N j y ij /N i y ij j or in shor y ij ȳ j ȳ i ȳ 0 Le us noice here ha ransformaion 0 clears he fixed effecs for model as well, bu of course he resuling Wihin esimaor is no opimal. The model which encompasses all above effecs is y ij = β x ij γ ij α i α j ε ij 4

7 By applying suiable resricions o model we can obain he models discussed above. The Wihin ransformaion for his model is y ij /T y ij /N y ij /N y ij /N y ij i j i j /NT y ij /NT y ij /N T y ij i j i j or in a shorer form y ij ȳ ij ȳ j ȳ i ȳ ȳ j ȳ i ȳ We can wrie up hese Wihin ransformaions in a more compac marix form using Davis [00] and Hornok s [0] approach. Model in marix form is y = Xβ D γ D α D 3 α ε 3 where y, N is he vecor of he dependen variable, X, N T K is he marix of explanaory variables, γ, α and α are he vecors of fixed effecs wih size N T N, N T NT and N T NT respecively, D = I N l, D = I N l N I T and D 3 = l N I NT l is he vecor of ones and I is he ideniy marix wih he appropriae size in he index. Le D = D, D, D 3, Q D = DD D D and P D = I Q D. Using Davis [00] mehod i can be shown ha P D = P Q Q 3 where P = I N J N I NT Q = I N J N J N I T Q 3 = I N J N I N J N J T J N = N J, JT = T J and J is he marix of ones wih is size in he index. Collecing all hese erms we ge P D = [ I N J N I N J N I T J T ] = I N T J N I N T I N J N I T I N J T I N J NT J N I N J T J N I T J N T The ypical elemen of P D gives he ransformaion. By appropriae resricions on he parameers of 3 we ge back he previously analysed Wihin ransformaions. Now ransforming model 3 wih ransformaion leads o P D y = P }{{} D X β P }{{} D D }{{ } y p X p =0 γ P D D }{{ α P } D D }{{ 3 } =0 =0 and he corresponding opimal Wihin esimaor is β W = X p X p X p y p α P D ε }{{} ε p 5

8 3. Some Daa Problems As hese mulidimensional panel daa models are frequenly used o deal wih flow ypes of daa like rade, capial movemens FDI, ec., i is imporan o have a closer look a he case when, by naure, we do no observe self flow. This means ha from he ij indexes we do no have observaions for he dependen variable of he model when i = j for any. This is he firs sep o relax our iniial assumpion ha N = N = N and ha he observaion ses i and j are equivalen. For mos of he previously inroduced models his is no a problem, he Wihin ransformaions work as hey are mean o and eliminae he fixed effecs. However, his is no he case unforunaely for models ransformaion 3, 9 and. Le us have a closer look a he difficuly. For model and ransformaion 3, insead of canceled ou fixed effecs, we end up wih he following remaining fixed effecs α i = α i N T N T α i NN N N α i i= = α i α i N γ j = γ j N T NN = γ j N and for he ime effecs λ = λ N i=; i =j N j=; j i α i N T γ j N N γ j j= N j=; j i N T γ j γ j N N T NN T N i=; i =j T α i N N T α i i= N α i = N α j i= NN N T N T γ j NN T N N T γ j j= N γ j = N γ i j= T N λ = NN NN λ = λ T T λ T = T λ λ T = 6 N T NN T NN N i=; i =j N j=; j i T N λ = T NN λ = = T λ = 0 = α i γ j

9 So clearly his Wihin esimaor now is biased. The bias is of course eliminaed if we add he ii observaions back o he above bias formulae, and also, quie inuiively, when N. On he oher hand, luckily, ransformaion as seen earlier, alhough no opimal, leads o an unbiased Wihin esimaor for model and remains so even in he lack of self flow daa. Now le us coninue wih model 9. Afer he Wihin ransformaion 0, insead of canceled ou fixed effecs we end up wih he following remaining fixed effecs and α i = α i N = NN N i=;i =j N k=;k j α i N N α i α k N α j γ j = γ j N N γ j N = NN N l=;l i γ l N γ i N j=;j i γ j NN NN N N α i i= N N γ j As long as he α and γ parameers are no zero, he Wihin esimaors will be biased. Similarly for model, he remaining fixed effecs are now γ ij = γ ij T T γ ij N NN N T N i= j=;j i N j=;j i N N i=;i =j γ ij Tγ ij γ ij N N T NN T N i=;i =j N N j=;j i Tγ ij N i= j=;j i j= γ ij Tγ ij = 0 7

10 bu α i = α i T = and, finally T = NN N T α i N N N α i i= T N α i = NN T α j = α j T = T = NN N T NN T N T i=;i =j = N i=;i =j α i N N α i N T NN T α i NT N T i=;i =j = N T α j = i= = α j N N α j N N N α j j= N j=;j i = N T α j T j=;j i = N T NN T α j NT α i T N α i NN N j=;j i T N α j = N j= = T α i = N i=;i =j α j T N α j NN N j=;j i α i N α j α j N α i where in order o avoid confusion wih he wo similar α fixed effecs α j is now denoed by α j. I can be seen, as expeced, hese remaining fixed effecs are indeed wiped ou when ii ype observaions are presen in he daa. When N he remaining effecs go o zero, which implies ha he bias of he Wihin esimaors go o zero as well. We can go furher along he above lines and see wha going o happen if he observaion ses i and j are differen. If he wo se are compleely disjunc, say for example if we are modeling expor aciviy beween he EU and APEC counries, inuiively enough, for all he models considered he Wihin esimaors are unbiased, even in finie samples, as he no-self-rade problem do no arise. If he wo ses are no compleely disjunc, on he oher hand, say for example in he case of rade beween he EU and OECD counries, as he no-self-rade do arise, we are face wih he same biases oulined above. 8

11 Like in he case of he usual panel daa ses, jus more frequenly, one may be faced wih he siuaion when he daa a hand is unbalanced. In our framework of analysis his means ha for all he previously sudied models, in general =,...,, i j = T and is ofen no equal o T i j. For models 4, 7, 8 and 9 he unbalanced naure of he daa does no cause any problems, he Wihin ransformaions can be used, and have exacly he same properies, as in he balanced case. However, for models and we are facing rouble. In he case of model and ransformaion we ge for he fixed effecs he following erms le us remember: his in fac is he opimal ransformaion for model 6 α i = α i = N = NT = N α i T i= N α i N i= γ j = γ j = N = NT = N γ j T j= N γ j N j= α i N N Nα i i= N α i i= N j= N T j= γ j N N Nγ j j= N γ j j= N i= N i= N T i= N i= N j= N j= N T N ij i= j= = N T N ij i= j= = α i γ j and λ = λ = λ = = λ N N λ T λ λ T = λ T = N N T N ij i= j= = N i= j= = λ T N ij i= j= = T N ij λ λ 9

12 These erms clearly do no add up o zero in general, so he Wihin ransformaion does no clear he fixed effecs, as a resul his Wihin esimaor will be biased. I can easily checked ha he above α i, γ j and λ erms add up o zero when i, j = T. As is he opimal Wihin esimaor for model 6, his is bad news for he esimaion of ha model as well. We, unforunaely, ge very similar resuls for ransformaion 3 oo. The good news is, on he oher hand, as seen earlier, ha for model ransformaion 0 clears he fixed effecs, and alhough no opimal in his case, i does no depend on ime, so in fac he corresponding Wihin esimaor is sill unbiased in his case. Unforunaely, no such luck in he case of model and ransformaion. The remaining fixed effecs are now γ ij = γ ij γ ij N = N i= = γ ij γ ij N = N N j= N γ ij N i= T N ij γ ij i= = N γ ij N i= N j= N j= N j= N γ ij T j= N γ ij N i= N j= N j= γ ij N N γ ij T j= γ ij N N i= j= N γ ij N N i= j= T N ij γ ij T j= = N N γ ij i= j= N i= j= N γ ij N γ ij i= j= N γ ij N i= N γ ij N T N ij i= j= = N i= γ ij N γ ij i= N γ ij i= 0

13 α i = α i and α i N = N i= = α i = N α i N i= T N ij α i i= = α i N = N i= α i = α j = α j N j= N j= N α i α i N i= T N ij α i i= = N i= α j N = N i= = α j = N j= T N ij α i i= = N α j N i= T N ij α j i= = α j α j N = N j= α j = T N ij α j T j= = N i= α i N N i= j= T N ij α i T j= = N α i i= T N ij α i T j= = N j= N j= N α j N i= N N j= α j N N α i N T N ij i= j= = N T N ij i= j= = T N ij α i T j= = N i= j= T N ij α j T j= = T N ij i= j= = T N ij α j i= = N α j i= α j N j= N α j N T N ij i= j= = N i= T N ij α j T j= = α i α i N T N ij i= j= = α j T N ij α j i= = N T N ij i= j= = These erms clearly do no cancel ou in general, as a resul he corresponding Wihin esimaor is biased. Unforunaely, he increase of N does no deal wih he problem, so he bias remains even when N. I can easily be checked, however, ha in he balanced case, i.e., when each = T/N he fixed effecs drop ou indeed from he above formulaions. Therefore, from a pracical poin of view, he esimaion of model is quie problemaic. The direc esimaion of he model by LSDV is no feasible, even for moderae N, as he number of dummies fixed effecs is becoming very large quie quickly. On he oher hand, he esimaion of he model wih he Wihin esimaor is unbiased only when here are no daa problems such α i α j

14 as no-self-rade or unbalanced observaions. A pragmaic way ou of his rap is o follow a wo-sep procedure. Firs, ransform model, 3 including all dummy variables wih he Wihin ransformaion 5 D will drop ou, hen esimae his ransformed model wih OLS including he ransformed D and D 3. Using his procedure we need o use subsanially less dummy variables, and i can be shown afer some algebra ha he esimaor remains unbiased even in he case of he above daa problems. 4. Dynamic Models In he case of dynamic auoregressive models, he use of which is unavoidable if he daa generaing process has parial adjusmen or some kind of memory, he Wihin esimaors in a usual panel daa framework are biased. In his secion we generalize hese well known resuls o his higher dimensional seup. We derive he finie sample bias for each of he models inroduced in Secion. In order o show he problem, le us sar wih he simple linear dynamic model wih bilaeral ineracion effecs, ha is model 4 Wih backward subsiuion we ge and y ij = ρy ij γ ij ε ij 4 y ij = ρ y ij0 ρ ρ γ ij ρ k ε ij k 5 k=0 y ij = ρ y ij0 ρ ρ γ ij ρ k ε ij k Wha needs o be checked is he correlaion beween he righ hand side variables of model 4 afer applying he appropriae Wihin ransformaion, ha is he correlaion beween y ij ȳ ij where ȳ ij = /T y ij and ε ij ε ij where ε ij = /T ε ij. This amouns o check he correlaions y ij ε ij, ȳ ij ε ij and ȳ ij ε ij because y ij ε ij are uncorrelaed. These correlaions are obviously no zero, no even in he semi-asympoic case when N, as we are facing he so called Nickell-ype bias Nickell [98]. This may be he case for all oher Wihin ransformaions as well. Model 4 can of course be expanded o have exogenous explanaory variables as well y ij = ρy ij x ijβ γ ij ε ij 6 k=0

15 Le us urn now o he derivaion of he finie sample bias and denoe in general any of he above Wihin ransformaions by ȳ rans. Using his noaion we can derive he general form of he bias using Nickell ype calculaions. Saring from he simple firs order auoregressive model 4 inroduced above we ge y ij ȳ rans = ρy ij ȳ rans ε ij ε rans 7 Using OLS o esimae ρ, we ge ρ = N i= N j= y ij ȳ rans y ij ȳ rans N i= N j= y ij ȳ rans 8 So he bias is E [ˆρ ] = E [ N i= N j= y ] ij ȳ rans ρy ij ȳ rans ε ij ε rans N j= y = ij ȳ rans N i= [ N N ρ i= j= = E y ij ȳ rans N N i= j= y ij ȳ rans = ρ E [ N i= N i= N j= y ] ij ȳ rans ε ij ε rans N j= y ij ȳ rans N i= N j= y ] ij ȳ rans ε ij ε rans N j= y ij ȳ rans N i= = ρ A B 9 Coninuing wih model 4 and using now he appropriae 5 Wihin ransformaion we ge y ij ȳ ij = ρy ij ȳ ij ε ij ε ij For he numeraor A from above we ge E[y ij ε ij ] = E [ k=0 [ E[ȳ ij ε ij ] = E T [ E[ȳ ij ε ij ] = E T E[y ij ε ij ] = 0 ] ρ k T ε ij k T ε ij = ] T ρ k ε ij k ε ij = k=0 T ρ k ε ij k = k=0 T ] T ε ij = = σ ε T ρ ρ = σ ε T ρt ρ = σ ε T ρ T ρ T ρ 3

16 And for he denominaor B E[yij ] = E E[y ij ȳ ij ] = E = E[ȳij ] = E = [ k=0 ρ k ε ij k = σε ρ ρ ρ k ε ij k k=0 σ ε T ρ T σ ε T ρ T ] T ρ k ε ij k = = k=0 ρ ρ ρ ρt ρ T ρ k ε ij k = = k=0 So he finie sample bias for his model is E [ˆρ ρ] = σ ε T ρ ρ ρ ρt ρ ρ ρt T ρ ρt ρ T ρ ρ σ ε T ρ T ρ ρ σ ε ρ σ ε T A B ρ T ρ T ρ where and A = B = σε ρ T ρ ρ ρ ρt ρ ρ ρt ρ σε T ρ ρ ρt T ρ ρt ρ T ρ ρ I can be seen ha hese resuls are very similar o he original Nickell resuls, and he bias is persisen even in he semi-asympoic case when N. Le us urn now our aenion o model. In his case he Wihin ransformaion leads o y ij ȳ ij ȳ ȳ = ρ y ij ȳ ij ȳ ȳ ε ij ε ij ε ε Afer lenghy derivaions see he Appendix we ge for he finie sample bias E [ˆρ ρ] = N N ρ T ρ N N ρ T N T ρ N N N ρ ρ B C 4 T A

17 where A = T ρ ρ ρ ρ ρ ρ N B = N σε ρ T ρ ρ ρ ρt ρ ρ ρt ρ and N C σ ε = N T ρ ρ ρt T ρ ρt ρ T ρ ρ I is worh noicing ha in he semi-asympoic case as N we ge back he bias derived above for model 4. As seen earlier, he opimal Wihin ransformaion for model is in fac 3 y ij ȳ i ȳ j ȳ ȳ For his Wihin esimaor he bias is see he derivaion in he Appendix E [ˆρ ρ] = N N σ ɛ T ρ ρ N σ N ɛ T ρt ρ N N ρ ρ B C A where and N A = 4 4N B = N N σ ɛ T ρ σ ɛ T ρ T ρ ρ ρ ρt ρ ρ T ρ ρ ρt ρ N 4 C σ ɛ = N T ρ ρ ρt T ρ ρt ρ T ρ ρ I can be seen as N he bias goes o zero, so his esimaor is semi-asympoically unbiased unlike he previous one. Le us now coninue wih models 7 and 8 which can be considered as he same models from his poin of view y ij = ρy ij α j ε ij 5

18 Wih he Wihin ransformaion we ge where y ij ȳ j = ρ y ij ȳ j α j N ȳ j = N N y ij ȳ j = N i= N i= α j } {{ } N Nα j N y ij ε j = N i= ε ij ε j, N ε ij. Following he derivaion presened in deails in he Appendix he bias for Model 7 is in fac zero, so his Wihin esimaor is unbiased. Le us carry on wih model 9. Using he Wihin ransformaion we ge y ij ȳ j ȳ i ȳ = ρy ij ȳ j ȳ i ȳ ε ij ε j ε i ε The finie sample bias now is see he Appendix for deails, as above, zero, so again, his Wihin esimaor is unbiased. And finally, le us urn o model i= The Wihin ransformaion gives so we ge y ij = ρy ij γ ij α i α j ε ij y ij ȳ ij ȳ j ȳ i ȳ ȳ j ȳ i ȳ y ij ȳ ij ȳ j ȳ i ȳ ȳ j ȳ i ȳ = = ρ y ij ȳ ij ȳ j ȳ i ȳ ȳ j ȳ i ȳ ε ij ε ij ε j ε i ε ε j ε i ε And for he finie sample bias of his model we ge E[ˆρ ρ] = N N T ρ ρ N N N N ρ ρ B C T ρt ρ A where N A = N T T ρ ρ ρ ρ ρ ρ 6

19 N B σ ε ρ = N T ρ ρ ρ ρt ρ ρ ρt ρ and N C σ ε = N T ρ ρ ρt T ρ ρt ρ T ρ ρ I is clear ha if N goes o infiniy and T is finie, hen we ge back he bias of model Exensions o Higher Dimensions Le us assume ha we would like o sudy he volume of expors y from a given counry o counries i, for some producs j by firms s a ime. This would resul in four dimensional observaions for our variable of ineres y ijs, i =,..., N i, j =,..., N j, s =,..., N s and, in he balanced case =,..., T. If we had daa no only for a given counry, bu for several, hen we would end up wih a five dimensional panel daa, and so on. In order o analyse he higher dimensional seup, le us use he all encompassing model, 3 wih pair-wise ineracion effecs: y ijs = x ijsβ γ 0 ijs γ ij γ js γ 3 is ε ijs 0 The fixed effecs of his model in a more compac and general form are γ 0 IS M γi M k, k= where i k is any pair-wise, combinaion of he individual index-se IS, in he above case IS = i, j, s, and M is he number of such pair-wise combinaions in 0 M = 3. In he case of unbalanced panel daa =,..., T IS. The Wihin ransformaion for model 0 is y ijs ȳ js ȳ is ȳ ij ȳ ijs ȳ s ȳ j ȳ js ȳ i ȳ is ȳ ij ȳ ȳ s ȳ j ȳ i ȳ 7

20 or in marix form P D = I Ni J Ni INj J Nj INs J Ns IT J T = I Ni N j N s T JNi I Nj N s T INi J Nj I Ns T INi N j J Ns I T I Ni N j N s J T JNi N j I Ns T JNi I Nj J Ns I T JNi I Nj N s J T INi J Nj N s I T INi J Nj I Ns J T I Ni N j J Ns T JNi N j N s I T JNi N j I Ns J T JNi I Nj J Ns T I Nj J Nj N s T JNi N j N s T I can be shown easily, ha he properies of he Wihin esimaor based on ransformaion in he case of no-self-rade, unbalanced daa and dynamic models are exacly he same as seen earlier for he hree dimensional model. The generalizaion of his Wihin esimaor for any higher dimensions can be done using he general form. There are basically wo ypes of fixed effec, γis 0, depending on all indices excep, and he res, which are symmeric in a sense, since all consis wo indices from IS and. Le us see he mehod for γis 0, and hen for a represenaive fixed effec, from he oher group, le i be γ ij. Le denoe IS {} by IS, and is elemens by s,...s M in he hree-dimensional case s = i, s = j and s 3 =. The Wihin ransformaion hen is M M M M y IS ỹ si ỹ si s j i= i= j= M M i= j=;i =j k=;k i,j ỹ si s j s k ± ỹ IS where ỹ si s i...s i m = N si...n si m N si,...,n si m s i =,...s i m = y IS The mehod in fac is he following. Firs, we subrac he firs order sums wih respec o each variables from he original unransformed variable y IS. Then we add up he second order sums in every possible pair-wise combinaion, hen subrac he hird order sums, and so on. The sum wih respec o equals o γis 0, clearing i ou. All oher firs order sums sill remain. In he nex sep we add he second order sums. All he previously remaining erms appear addiionally summed wih respec o, bu wih an opposie sign, canceling ou all he remaining erms from period. Coninuing he process, all he remaining erms in period i appear in he nex one, also summed wih respec o, and wih an opposie sign, again clearing ou all he erms from period i. The inducion should now be clear. In he las bu one period, 8

21 he only remaining erm is going o be he sum wih respec o all indices bu, wih a sign deermined by he pariy of he indices. In he las period, we are summing up γis 0 wih respec o all indices including, bu wih an opposie sign, which herefore cancels ou he only previously remaining erm. 6. Conclusion In he case of hree and higher dimensional fixed effecs panel daa models, due o he many ineracion effecs, he number of dummy variables in he model increases dramaically. As a consequence, even when he number of individuals is no oo large, he LSDV esimaor becomes, unforunaely, pracically unfeasible. The obvious answer o his challenge is o use appropriae Wihin esimaors, which do no require he explici incorporaion of he fixed effecs ino he model. Alhough hese Wihin esimaors are more complex han for he usual wo dimensional panel daa models, hey are quie useful in hese higher dimensional seups. However, unlike in he wo dimensional case, hey are biased and inconsisen in he case of some very relevan daa problems like he lack of self-rade, or unbalanced observaions. These properies mus be aken ino accoun by all researchers relying on hese mehods. 9

22 Appendix Finie sample bias derivaions for he dynamic model. Model In his case he Wihin ransformaion leads o y ij ȳ ij ȳ ȳ = ρ y ij ȳ ij ȳ ȳ ε ij ε ij ε ε Componens of he numeraor of he bias are E[y ij ε ij ] = 0 E[y ij ε ij ] = σ ε T E[y ij ε ] = 0 E[y ij ε] = σ ε N T ρ ρ ρ ρ ρ T ρ E[ȳ ij ε ij ] = σ ε T E[ȳ ij ε ij ] = σ ε T ρ T ρ T ρ E[ȳ ij ε ] = σ ε N T E[ȳ ij ε] = σ ε N T ρ T E[ȳ ε ij ] = 0 E[ȳ ε ij ] = σ ε N T E[ȳ ε ] = 0 E[ȳ ε] = σ ε N T ρ ρ ρ ρ ρ T ρ E[ȳ ε ij ] = σ ε N T E[ȳ ε ij ] = σ ε N T ρ T 0 ρ T ρ ρ T ρ ρ T ρ

23 ρ T ρ E[ȳ ε ] = σ ε N T E[ȳ ε] = σ ε N T ρ T ρ T ρ Considering he signs of he componens, we ge he following expeced value for he numeraor N σ ε N T ρ N ρ N N σ ε N T ρ T Componens of he denominaor are σ ε ρ T ρ T ρt ρ E[y ij ȳ ij ] = E[y ij ȳ ] = E[ȳ ij ] = E[ȳ ij ȳ ] = E[ȳ ij ȳ ] = E[ȳ ȳ ] = E[ȳ ] = E[y ij ] = σ ε ρ ρ σε ρ T ρ ρ ρ ρt ρ E[y ij ȳ ] = σ ε N ρ ρ σε ρ N T ρ ρ ρ ρt ρ σ ε T ρ σ ε N T ρ σ ε N T ρ ρ ρt ρ ρ ρt ρ ρ ρt T ρ ρt ρ T ρ ρ ρ ρ ρ ρt ρ ρ ρt ρ ρ ρt T ρ ρt ρ T ρ ρ E[ȳ ] = σ ε N ρ ρ σε ρ N T ρ ρ ρ ρt ρ σ ε N T ρ ρ ρt ρ ρ ρt T ρ ρt ρ T ρ ρ

24 Thus he expeced value of he denominaor is N N ε ρ σ ρ N N σ ε T ρ N N σε ρ T ρ ρ ρ ρt ρ ρ ρt T ρ ρt ρ T ρ ρ The bias of his Wihin esimaor for is herefore he following: E [ˆρ ρ] = N N ρ T ρ N N ρ T N T ρ N N N ρ ρ B C T A ρ ρt ρ where A = T ρ ρ ρ ρ ρ ρ N B = N σε ρ T ρ ρ ρ ρt ρ ρ ρt ρ and N C σ ε = ρ ρt N T ρ T ρ ρt ρ T ρ ρ Now for he same model ransformaion 3 leads o he following erms. For he numeraor: E[y ij ε ij ] = 0 E[y ij ε i ] = E[y ij ε j ] = σ ε ρ NT ρ E[y ij ε ] = 0 E[y ij ε] = σ ε N T ρ ρ E[ȳ i ε ij ] = E[ȳ j ε ij ] = σ ε NT ρt ρ E[ȳ i ε i ] = E[ȳ j ε j ] = σ ε NT ρ T E[ȳ i ε j ] = E[ȳ j ε i ] = σ ε N T ρ T ρ T ρ ρ T ρ

25 And for he denominaor E[ȳ i ε ] = E[ȳ j ε ] = σ ε N T ρt ρ E[ȳ i ε] = E[ȳ j ε] = σ ε N T ρ T E[ȳ ε ij ] = 0 E[ȳ ε i ] = E[ȳ ε j ] = σ ε N T ρ ρ E[ȳ ε ] = 0 E[ȳ ε] = σ ε N T ρ ρ E[ȳ ε ij ] = σ ε N T ρt ρ E[ȳ ε i ] = E[ȳ ε j ] = σ ε N T ρ T E[ȳ ε ] = σ ε N T ρt ρ E[ȳ ε] = 4σ ε N T ρ T ρ T ρ ρ T ρ ρ T ρ E[yij ] = σε ρ ρ σ ε ρ E[y ij ȳ i ] = E[y ij ȳ j ] = NT ρ ρ ρ ρt ρ E[y ij ȳ ] = E[ȳ i ] = E[ȳ j ] = E[ȳ i ȳ ] = E[ȳ j ȳ ] = E[ȳ i ȳ ] = E[ȳ j ȳ ] = E[y ij ȳ ] = σ ε N ρ ρ σε ρ N T ρ ρ ρ ρt ρ σ ε NT ρ ρ ρt ρ ρ ρt ρ ρ ρt T ρ ρt ρ T ρ ρ ρ ρ ρ ρt ρ σ ε N T ρ σ ε N T ρ ρ ρt ρ ρ ρt T ρ ρt ρ T ρ ρ 3

26 E[ȳ ȳ ] = E[4ȳ ] = E[ȳ ] = σ ε N ρ ρ σε ρ N T ρ ρ ρ ρt ρ 4σ ε N T ρ ρ ρt ρ ρ ρt T ρ ρt ρ T ρ ρ Taking ino accoun he sign and he frequency of he above elemens he bias of his Wihin esimaor is E [ˆρ ρ] = N N σ ε T ρ ρ N σ N ε T ρt ρ N N ρ ρ B C A where N A = 4 4N B = N N σ ε T ρ σ ε T ρ T ρ ρ ρ ρt ρ ρ T ρ ρ ρt ρ and N 4 σ ε ρ ρt N T ρ T ρ ρt ρ T ρ ρ Models 7 and 8 Le us coninue wih models 7 and 8 which can be considered as he same models from his poin of view y ij = ρy ij α j ε ij Wih he Wihin ransformaion we ge where y ij ȳ j = ρ y ij ȳ j α j N ȳ j = N N y ij ȳ j = N i= The componens of he bias are he following N i= α j } {{ } N Nα j N y ij ε j = N i= E[y ij ε ij ] = 0 since hey are uncorrelaed 4 ε ij ε j, N ε ij. i=

27 [ ] N E[ȳ j ε ij ] = E N ρ k ε ij k ε ij = 0 E[y ij ε j ] = E [ i= k=0 ρ k ε ij k k=0 i= k=0 N ] N ε ij = 0 [ ] N E[ȳ j ε j ] = E N ρ k N ε ij k N ε ij = 0 The elemens in he denominaor are E[yij ] = E E[y ij ȳ j ] = E [ E[ȳj ] = E k=0 ρ k ε ij k N k=0 N i= i= ρ k ε ij k = σε ρ ρ ] N ρ k ε ij k i= k=0 N ρ k ε ij k i= k=0 = N σ ε ρ ρ = N N σ ε ρ ρ So he bias for Model 7 is nil as he nominaor of he bias is zero, and he denominaor finie. Model 9 Using he Wihin ransformaion we ge y ij ȳ j ȳ i ȳ = ρy ij ȳ j ȳ i ȳ ε ij ε j ε i ε As in he numeraor of he bias all elemens are zero, while he denominaor is finie, his Wihin esimaor is obviously unbiased. Model And finally, le us urn o model The Wihin ransformaion gives y ij = ρy ij γ ij α i α j ε ij y ij ȳ ij ȳ j ȳ i ȳ ȳ j ȳ i ȳ, 5

28 so we ge y ij ȳ ij ȳ j ȳ i ȳ ȳ j ȳ i ȳ = = ρ y ij ȳ ij ȳ j ȳ i ȳ ȳ j ȳ i ȳ ε ij ε ij ε j ε i ε ε j ε i ε The expeced value of he componens are he following. For he numeraor: E[y ij ε ij ] = 0 E[y ij ε ij ] = σ ε T ρ ρ E[y ij ε i ] = 0 E[y ij ε j ] = 0 E[y ij ε ] = 0 E[y ij ε i ] = σ ε NT ρ ρ E[y ij ε j ] = σ ε NT ρ ρ E[y ij ε] = σ ε N T ρ ρ E[ȳ ij ε ij ] = σ ε T ρt ρ E[ȳ ij ε ij ] = σ ε T ρ T ρ T ρ E[ȳ ij ε j ] = σ ε NT ρt ρ E[ȳ ij ε i ] = σ ε NT ρt ρ E[ȳ ij ε ] = σ ε N T ρt ρ E[ȳ ij ε j ] = σ ε NT ρ T ρ T ρ 6

29 E[ȳ ij ε i ] = σ ε NT ρ T E[ȳ ij ε] = σ ε N T ρ T E[ȳ i ε ij ] = E[ȳ j ε ij ] = 0 ρ T ρ ρ T ρ E[ȳ i ε ij ] = E[ȳ j ε ij ] = σ ε NT ρ ρ E[ȳ i ε i ] = E[ȳ j ε j ] = 0 E[ȳ i ε j ] = E[ȳ j ε i ] = 0 E[ȳ i ε ] = E[ȳ j ε ] = 0 E[ȳ i ε i ] = E[ȳ j ε j ] = σ ε NT ρ ρ E[ȳ i ε j ] = E[ȳ j ε i ] = σ ε N T ρ ρ E[ȳ i ε] = E[ȳ j ε] = σ ε N T ρ ρ E[ȳ ε ij ] = 0 E[ȳ ε ij ] = σ ε N T ρ ρ E[ȳ ε j ] = 0 E[ȳ ε i ] = 0 E[ȳ ε ] = 0 E[ȳ ε i ] = σ ε N T ρ ρ E[ȳ ε j ] = σ ε N T ρ ρ E[ȳ ε] = σ ε N T ρ ρ E[ȳ i ε ij ] = E[ȳ j ε ij ] = σ ε NT ρt ρ E[ȳ i ε ij ] = E[ȳ j ε ij ] = σ ε NT ρ T 7 ρ T ρ

30 E[ȳ i ε i ] = E[ȳ j ε j ] = σ ε NT ρt ρ E[ȳ i ε j ] = E[ȳ j ε i ] = σ ε N T ρt ρ E[ȳ i ε ] = E[ȳ j ε ] = σ ε N T ρt ρ E[ȳ i ε i ] = E[ȳ j ε j ] = σ ε NT ρ T E[ȳ i ε j ] = E[ȳ j ε i ] = σ ε N T ρ T E[ȳ i ε] = E[ȳ j ε] = σ ε N T ρ T E[ȳ ε ij ] = σ ε N T ρt ρ E[ȳ ε ij ] = σ ε N T ρ T E[ȳ ε j ] = σ ε N T ρt ρ E[ȳ ε i ] = σ ε N T ρt ρ E[ȳ ε ] = σ ε N T ρt ρ E[ȳ ε i ] = σ ε N T ρ T E[ȳ ε j ] = σ ε N T ρ T E[ȳ ε] = σ ε N T ρ T ρ T ρ ρ T ρ ρ T ρ ρ T ρ ρ T ρ ρ T ρ ρ T ρ So he expeced value of he numeraor, considering he signs of he componens is N σ ε N T ρ N ρ σ ε N T ρt ρ N σ ε N T ρ T ρ T ρ 8

31 The componens of he denominaor are E[y ij ȳ ij ] = E[y ij ȳ i ] = E[y ij ȳ j ] = E[y ij ȳ ] = E[ȳ ij ] = E[ȳ ij ȳ i ] = E[ȳ ij ȳ j ] = E[ȳ ij ȳ ] = E[ȳ ij ȳ i ] = E[ȳ ij ȳ j ] = E[ȳ ij ȳ ] = E[y ij ] = σ ε ρ ρ σε ρ T ρ ρ ρ ρt ρ E[y ij ȳ i ] = σ ε N ρ ρ E[y ij ȳ j ] = σ ε N ρ ρ E[y ij ȳ ] = σ ε N ρ ρ σε ρ NT ρ ρ ρ ρt ρ σε ρ NT ρ ρ ρ ρt ρ σ ε N T ρ σ ε T ρ σ ε NT ρ σ ε NT ρ σ ε N T ρ σ ε NT ρ σ ε NT ρ σ ε N T ρ ρ ρt ρ ρ ρt ρ ρ ρt ρ ρ ρ ρ ρt ρ ρt ρ ρ ρ ρt T ρ ρt ρ T ρ ρ ρ ρ ρ ρt ρ ρt ρ ρ ρ ρ ρ ρt ρ ρ ρ ρ ρt ρ ρ ρt ρ ρ ρt ρ ρ ρt T ρ ρt ρ T ρ ρ ρ ρt T ρ ρt ρ T ρ ρ ρ ρt T ρ ρt ρ T ρ ρ 9

32 E[ȳi ] = E[ȳ j ] = σ ε N ρ ρ E[ȳ i ȳ j ] = σ ε N ρ ρ E[ȳ i ȳ ] = E[ȳ j ȳ ] = σ ε N ρ ρ σ ε ρ E[ȳ i ȳ i ] = E[ȳ j ȳ j ] = NT ρ ρ ρ ρt ρ ρ E[ȳ i ȳ j ] = E[ȳ j ȳ i ] = E[ȳ i ȳ ] = E[ȳ j ȳ ] = E[ȳ ȳ i ] = E[ȳ ȳ j ] = E[ȳ ȳ ] = E[ȳ i ] = E[ȳ j ] = E[ȳ i ȳ j ] = E[ȳ i ȳ ] = E[ȳ j ȳ ] = E[ȳ ] = σ ε N T ρ σ ε N T ρ ρ ρ ρt ρ ρ ρ ρ ρt ρ E[ȳ ] = σ ε N ρ ρ σε ρ N T ρ ρ ρ ρt ρ σε ρ N T ρ ρ ρ ρt ρ σ ε N T ρ σ ε NT ρ σ ε N T ρ σ ε N T ρ σ ε ρ ρt ρ ρ ρt ρ ρ ρt ρ ρ ρt ρ ρ ρt ρ ρ ρ ρ ρt ρ ρt ρ ρ ρ ρt T ρ ρt ρ T ρ ρ ρ ρt T ρ ρt ρ T ρ ρ N T ρ ρ ρt T ρ ρt ρ T ρ ρ ρ ρt T ρ ρt ρ T ρ ρ 30

33 Thus he expeced value of he denominaor afer aking ino accoun he signs of he componens is N σ N ε ρ ρ N σ ε ρ N T ρ N σε N T ρ ρ ρ ρt ρ ρ ρt ρ ρ ρt T ρ ρt ρ T ρ ρ To sum up he bias we ge for his model is E[ ρ ρ] = N N T ρ ρ N N N N ρ ρ B C T ρt ρ A where A = N N σ ε T ρ T ρ T ρ B = N N σε ρ T ρ ρ ρ ρt ρ ρ ρt ρ and N C σ ε = ρ ρt N T ρ T ρ ρt ρ T ρ ρ 3

34 References Sco L. Baier and Jeffrey H. Bergsrand [007]: Do Free Trade Agreemens Acually Increase Members Inernaional Trade? Journal of Inernaional Economics, 7, Piero Balesra and Jayalakshmi Krishnakumar [008]: Fixed Effecs and Fixed Coefficiens Models, in Mayas and Sevesre, The Economerics of Panel Daa, 3rd ediion, Sringer Verlag, pp Badi Balagi [995]: Economeric Analysis of Panel Daa, John Wiley & Sons. Badi H. Balagi, Peer Egger and Michael Pfaffermayr [003]: A generalized Design for Bilaeral Trade Flow Models, Economic Leers, 80, pp Richard Baldwin and Daria Taglioni [006]: Graviy for Dummies and Dummies for he Graviy Equaions, NBER Working Paper 56. Peer Davis [00]: Esimaing muli-way error componens models wih unbalanced daa srucures, Journal of Economerics, 06, Peer Egger and Michael Pfaffermayr [003]: The Proper Economeric Specificaion of he Graviy Equaion: 3-Way Model wih Bilaeral Ineracion Effecs, Empirical Economics, 8, Cecilia Hornok [0]: Graviy or Dummies? The Limis of Idenificaion in Graviy Esimaions, CeFig Working Papers, 5, Sepember 0. Laszlo Mayas [997]: Proper Economeric Specificaion of he Graviy Model, The World Economy, 0, Laszlo Mayas, Mark N. Harris and Laszlo Konya [0]: Wihin Transformaions for Three-Way Fixed Effecs Models of Trade, unpublished manuscrip, 0/05. Sephen Nickell [98]: Biases in Dynamic Models wih Fixed Effecs, Economerica, 49,