Cointegration Vector Estimation by DOLS for a Three-Dimensional Panel

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1 Revista Colombiana e Estaística January 05, Volume 38, Issue,. 45 to 73 DOI: htt://x.oi.org/0.5446/rce.v38n.4880 Cointegration Vector Estimation by DOLS for a hree-dimensional Panel Estimación e un moelo e cointegración utilizano DOLS ara un anel e tres imensiones Luis Fernano elo-velania,a, John Jairo León,b, Dagoberto Saboyá 3,c Econometric Unit, Banco e la Reública, Bogotá, Colombia Deartment of Economics, University of arylan, College Park, USA 3 Deartment of athematics, Universia el Rosario, Bogotá, Colombia Abstract his aer extens the results of the ynamic orinary least squares cointegration vector estimator available in the literature to a three-imensional anel. We use a balance anel of N an lengths observe over erios. he cointegration vector is homogeneous across iniviuals but we allow for iniviual heterogeneity using ifferent short-run ynamics, iniviual-secific fixe effects an iniviual-secific time trens. We also moel cross-sectional eenence using time-secific effects. he estimator has a Gaussian sequential limit istribution that is obtaine by first letting an then letting N,. he onte Carlo simulations show evience that the finite samle roerties of the estimator are closely relate to the asymtotic ones. Key wors: Cointegration, ultiimensional, Panel Data. Resumen Este ocumento extiene los resultaos e los estimaores mínimos cuaraos inámicos ara series cointegraas isonible en la literatura a un anel e tres imensiones. Se utiliza un anel balanceao e longitues N y ara un erioo e tiemo e longitu. El vector e cointegración es homogéneo a través e los iniviuos; sin embargo, el moelo ermite cierto grao e heterogeneia al usar iferentes inámicas e corto lazo, efectos fijos y tenencias a niveles iniviuales. ambién se utilizan efectos en el a Senior Econometrician. lmelovel@banre.gov.co b PhD stuent. leon@econ.um.eu c Professor. saboyac@unal.eu.co 45

2 46 Luis Fernano elo-velania, John Jairo León & Dagoberto Saboyá tiemo ara incluir eenencias cruzaas entre los iniviuos. El estimaor tiene una istribución secuencial límite gausiana en la cual rimero y osteriormente N,. Simulaciones onte Carlo muestran eviencia e que las roieaes e muestra finita el estimaor son cercanas a las asintóticas. Palabras clave: cointegración, moelos anel, multiimensional.. Introuction his aer rooses an extension of the ynamic orinary least squares DOLS cointegration anel estimators of ark & Sul 003 to a three-imensional anel. he single equation DOLS for estimating an testing the cointegration hyothesis was roose by Phillis & Loretan 99, Saikkonen 99, an generalize by Stock & Watson 993. DOLS is a single-equation cointegration technique that overcomes the common roblems of the static an moifie OLS. he static OLS finite samle estimates of long-run relationshis are otentially biase an inferences cannot be rawn using t-statistics Banerjee, Henry & Smith 986, Kremers, Ericsson & Dolao 99. DOLS methoology is base on an equation that inclues lags an leas of right-han sie variables, which eliminates the effect of the enogeneity of these variables. herefore, it is ossible to construct asymtotically-vali test statistics an also to estimate the long-run relationshis. Panel DOLS PDOLS has been analyze by Kao & Chiang 000 an ark & Sul 003. Kao & Chiang 000 stuy the roerties of anel DOLS when there are fixe effects in the cointegration regressions. ark & Sul 003 allow for iniviual heterogeneity through ifferent short-run ynamics, iniviual-secific fixe effects an iniviual-secific time trens. hey also ermit a limite egree of cross-sectional eenence through the resence of time-secific effects. Panel analysis usually emloys two imensions, being time one of them. However, given the great availability of ata nowaays two imensions are not always enough, in these cases, a anel in three imensions is a relevant otion. hese methoologies are very useful as they moel the heterogeneity of the ata in a more rigorous way. Some emirical alication of anels in three imensions can be foun in Eilat & Einav 004, Davies 006 an Davies, Lahiri & Sheng 0, among others. For extening the results of ark & Sul 003 to a three imensions setu, we use a balance anel of three imensions with lengths N, an. he cointegration vector is homogeneous across iniviuals but we allow for iniviual heterogeneity using ifferent short-run ynamics, iniviual-secific fixe effects an iniviual-secific time trens. Both iniviual effects are consiere in the first two imensions. As in ark & Sul 003, we also moel some egree of cross-sectional eenence using time-secific effects. After obtaining the Panel he three imensions in Eilat & Einav 004 are time, country of origin an country of estination, in Davies 006 an Davies et al. 0 are iniviual, forecast horizon an forecast target erio of time. Revista Colombiana e Estaística

3 3D-Panel Cointegration Vector Estimation 47 DOLS estimator in three imensions, PDOLS-3D, we resent the sequential limit istribution of the estimator by first letting, then letting N,. Finally, to evaluate the small samle roerties of the PDOLS-3D t-tests, onte Carlo exeriments are imlemente. he remainer of the aer is organize as follows. Section escribes the cointegration reresentation for a three-imensional anel. Section 3 escribes the PDOLS-3D estimator. he asymtotic istribution of this estimator is resente in Section 4. A onte Carlo exeriment is imlemente in Section 5. Section 6 shows an emirical alication of this methoology. Finally, some concluing remarks are resente in Section 7.. Reresentation of a Cointegrate oel in Panel Data in hree Dimensions Consier the following triangular reresentation of a cointegrate system for a anel with iniviuals inexe by i =,..., N an j =,..., over time erios t =,..., y ijt = α N i α j λ N i t λ j t θ t γ x ijt u ijt x ijt = x ijt v ijt where {y ijt } is the eenent variable integrate of orer one, {x ijt } is a k- imensional vector of integrate series of orer one an {u ijt, v ijt } is a covariance stationary error rocess ineenent across i an j but ossibly eenent across t. In this case, the variables are sai to be cointegrate for each member of the anel, with cointegrate vector γ. Iniviual heterogeneity is consiere through ifferent short-run ynamics, iniviual-secific fixe effects of the first two imensions, α N i λ N i an λ j an α j, an iniviual-secific time trens in those imensions,. A limite egree of cross-sectional eenence is also ermitte by the resence of time-secific effects, θ t. In this notation, N an inicate the first an secon imension, resectively. On the other han, N an inicate the number of iniviuals in the the first an secon imension, resectively. 3. Panel DOLS Estimator in hree Dimensions PDOLS methoology is base on the estimation of the following equation y ijt = α N i α j λ N i t λ j t θ t γ x ijt δ i z ijt u ijt where z ijt = x ijt,..., x ijt,..., x ijt is a k-imensional vector of leas an lags of the first ifferences of the variables x ijt. he inclusion of lags an leas eliminates the effect of the enogeneity of these variables. o avoi erfect collinearity, α = λ = 0. Equation can be exresse as follows, y ijt = αn i α j λ N i t λ j t θ t γ x ijt u ijt 3 Revista Colombiana e Estaística

4 48 Luis Fernano elo-velania, John Jairo León & Dagoberto Saboyá where y ijt an x ijt reresent the linear rojection of the eenent variable an the variables x ijt with resect to the short run comonents, z ijt. aking average of 3 in the time imension gives t= y ijt = αn i γ α j λ N i t= x ijt λ j θ t t= t= u ijt 4 Subtracting 4 from 3 eliminates α N i an α j, an gives y ijt s= y ijs = λ N i λ j t θ t γ x ijt θ s u ijt s= s= x ijs 5 u ijs s= aking ouble average of equation 5 in the first two imensions gives the following result i= j= y ijt = t γ i= j= s= y ijs x ijt i= j= u ijt i= j= i= j= s= λ N i x ijs u ijs s= λ j θ t s= θ s 6 Subtracting 6 from 5 eliminates the common time effects, then the moel can be rewritten as y λn ijt = i λ j t γ x ijt u ijt 7 Revista Colombiana e Estaística

5 3D-Panel Cointegration Vector Estimation 49 Where the suerscrits an enote the following eviations y ijt = y ijt y ijs s= y nmt n= m= y nms, n= m= s= x ijt x ijs s= x ijt = x nmt n= m= u ijt = u ijt u ijs s= u nmt λ N i λ j = λ N i N = λ j t = t n= m= n=. where λ λ λ N =,,..., an the following matrices m= λ N N λ N n, λ m, x nms, n= m= s= n= m= s= u nms, Let us efine the gran coefficient vector of the moel as β = γ, λ λ λ N, λ λ λ λn N λ λ, λ λ λ =, λ,...,, =, q t = x t, t, 0,..., 0, t, 0,..., 0 q t = x t, t, 0,..., 0, 0, t,..., 0 λ.. q t = x t, t, 0,..., 0, 0, 0,..., t q t = x t, 0, t,..., 0, t, 0,..., 0 8. q t = x t,. q t = x t,. 0, t,...,, 0, 0, 0,..., t. 0, 0,..., t, 0, 0,..., t Revista Colombiana e Estaística

6 50 Luis Fernano elo-velania, John Jairo León & Dagoberto Saboyá hen, the moel can finally be exresse as An the PDOLS-3D estimator of β is ˆβˆβˆβ = y ijt = β q ijt u ijt 9 i= j= t= q ijt q ijt i= j= t= q ijt y ijt 4. Asymtotic Distribution of the PDOLS-3D Estimator 0 aking into account that elements in ˆβˆβˆβ have ifferent rates of convergence, we can rewrite 0 as Where G ˆβˆβˆβ β = m Ik G = 0 I N = G i= j= t= N 3 I q ijt q ijt G,,, 3, =,, 3,, 3, 3, 33,, = i= j= t= x ijt x ijt,, = 5 j= t= tx jt 5 j= t= tx Njt, 3, = N 5 i= t= tx it N 5 i= t= tx it, Revista Colombiana e Estaística

7 3D-Panel Cointegration Vector Estimation 5, = t 3 I N, t= where N is a matrix of ones, 3, = t 3 N ; t= 33, = t 3 I, t= m = G i= j= t= m, = m, = m 3, q ijt u ijt i= j= 3 3 N 3 N 3 t= x ijt u ijt tu jt j= t=. tu Njt j= t= tu it i= t=. tu it i= t= he asymtotic istribution of the PDOLS-3D estimator is resente in roosition art ii. he following lemmas are require to rove this roosition. he roofs of the lemmas follow from simle extensions of the results of ark & Sul 00. Nevertheless, they are resente in Aenix A, B an C Following the results of ark & Sul 003, a linear hyothesis of the form Rγ = r can be teste using regular Wal statistics. Let, R a r k known matrix an r a r known vector. Following the results of Phillis & oon 999 an White 00 an uner the assumtions N 0, 0 an N, we obtain similar asymtotic results when consiering joint convergence in the three imensions N, an. instea of sequential convergence. Revista Colombiana e Estaística

8 5 Luis Fernano elo-velania, John Jairo León & Dagoberto Saboyá Lemma. For each i an j as, i ii 5/ iii t= t= x ijt x ijt t= tx ijt 5/ x ijt u ijt t= x ijt x ijt tx ijt t= x ijt u ijt t= his lemma emonstrates the equivalence in robability of the rojecte series in the z ijt sace an the series which are not rojecte. his gives an asymtotic justification for ignoring the fact that we are using rojection errors instea of the original observations. Lemma. As an then N,, i, 6 i= j= Ω vv,ij 0. Where Ωvv,ij is associate with the covariance matrix of v ijt. ii, iii 3, 0 0 iv, I N v 3, 0 vi 33, I his lemma shows the convergence of each element in the matrix. Lemma 3. i For N an fixe, with, m, m, where m, = N i= j= Ωuu,ij B B B vij W uij O ii As then N,, V m D, N0, I where V = 6 i= j= Ω uu,ijω vv,ij an Ω uu,ij is associate with the variance of u ijt. iii As an N,, m, is ineenent of m, an m 3, his lemma shows the convergence in istribution of m. Proosition. For the PDOLS-3D estimator in, as an then N,, Revista Colombiana e Estaística

9 3D-Panel Cointegration Vector Estimation 53 i ˆγˆγˆγ γ is ineenent of 3 ˆλˆλˆλN λ N an N 3 ˆλˆλˆλ λ. ii C ˆγˆγˆγ γ A N0, I K. where C =, = 6 V, = 6 C C =, V,, i= j= i= j= Ω vv,ij Ω uu,ij Ω vv,ij iii D C 0. where D = V,,, = V, = i= j= i= j= t= Ω uu,ij an Ω uu,ij is a consistent estimator of Ω uu,ij., x ijt x ijt t= x ijt x ijt his roosition resents the sequential limit istribution of the PDOLS-3D estimator. he roof of art i follows from Lemma an Lemma 3.iii, the roof of art ii follows from Lemma an Lemma 3.ii. he roof of art iii is straightforwar. 5. onte Carlo Exeriments his section summarizes the onte Carlo exeriments use for evaluating some small samle roerties of the PDOLS-3D estimator erive in Section 3. he esign of the exeriment follows the structure resente in ark & Sul 003. he Data Generating Process DGP inclues two regressors in the cointegration relation, an is efine as follows: y ijt = α i α j γ x,ijt γ x,ijt µ ijt x,ijt = ν,ijt x,ijt = ν,ijt Revista Colombiana e Estaística

10 54 Luis Fernano elo-velania, John Jairo León & Dagoberto Saboyá he short run ynamics are given by: ω ijt = A ij ω ijt ɛ ijt ɛ ijt = φθ t φe ijt with ω ijt = µ ijt, ν,ijt, ν,ijt, ɛ ijt = ɛ,ijt, ɛ,ijt, ɛ 3,ijt, e ijt = e,ijt, e,ijt, e 3,ijt, e ijt N 3 0, iag σ,ij, σ,ij, σ 3,ij, θt = θ t, θ t, θ 3t, θ t N 3 0, iag σ θ, σ θ, σ θ 3, for i =,..., N, j =,...,, where iaga,..., a n is an n n iagonal matrix which iagonal elements are a,..., a n ; an A ij is a 3 3 matrix whose hr-th element is A hr,ij for h, r =,, 3. he esign of the DGP is also relate to the emirical work on the Colombian factor rouctivity by Iregui, elo & Ramírez 007, where the regressors x,ijt an x,ijt reresent the level of caital stock an labor force, resectively, for the i-th inustrial sector an j-th metroolitan area in year t. As can be seen in the secification of the DGP, both regressors are riftless I rocesses. he equilibrium error is moele to allow for a general form of Cross-Sectional Deenence CSD, where the CSD is inuce by θ t, while φ controls the egree of Cross-Sectional Deenence. he arameters θ t an θ 3t cause cross-sectional enogeneity between the regressors an the equilibrium error. he cointegration vector is simulate as γ, γ = 0.5, he values A,ij an σ hij, for each i an j, are extracte from a uniform istribution, an are ket constant throughout the exeriment. Different levels of ersistence in the short-run ynamics are obtaine by varying the limits of the uniform istribution from which the elements of A ij are rawn. hree levels of ersistence are consiere: low, A,ij U 0.3,0.5, meium, A,ij U 0.5,0.7, an high, A,ij U 0.7,0.9. Aitionally, ifferent values of φ are use in the simulations: φ = 0 for no CSD, φ = 0.3 for low CSD, an φ = 0.7 for high CSD. Other arameters use in the simulation are as follows: A,ij U -0.05,0.5, A,ij U -0.05,0.5, A 3,ij U -0.05,0.5, A 3,ij U -0.05,0.5, A 3,ij U -0.05,0.5, A 3,ij U -0.05,0.5, A,ij U 0.0,0.4, A 33,ij U 0.0,0.4, σ,ij U 0 3,53 0 3, σ,ij U , , σ 3,ij U.3 0 3, ; an σ θ =.8, σ θ = 0.645, σ θ 3 =.0 for i =,..., N, j =,...,. he rewhitene quaratic sectral methoology roose by Sul, Phillis & Choi 005 was use for estimating the long-run variances Ω uu,ij. he values of the iniviual-secific fixe effects of the first two imensions, α i an α j, are taken from the PDOLS-3D estimation with the ata in Iregui et al he simulation structure inclues,000 samles of size N = 9, = 8, an = 50 or = 00 or = 50. he number of leas an lags of x ijt inclue in the PDOLS-3D estimation is taken as two. hen, three cases are evaluate: Case : No CSD φ = 0, with low, meium an high ersistence levels. Case : Homogeneous CSD A,ij = A,lk, for i, l =,..., N, j, k =,...,, with low an high CSD egrees φ = 0.3, φ = 0.7; an with low, meium an high ersistence levels. Revista Colombiana e Estaística

11 3D-Panel Cointegration Vector Estimation 55 Case 3: Heterogeneous CSD A,ij A,lk, for i, l =,..., N, j, k =,...,, with low an high CSD egrees φ = 0.3, φ = 0.7; an with low, meium an high ersistence levels. ables to 9 reort the results of the simulation exeriments escribe in cases to 3 for N = 8 an = 9. ables, 4 an 7 resent the simulations with = 50, ables, 5 an 8 for = 00, an ables 3, 6 an 9 for = 50. able : Effective size of PDOLS-3D tests for Case, with = 50. Persistence eium est Nominal size 5% 0% H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = he effective size results for Case with 5% an 0% nominal-size tests an = 50 are resente in able. Uner low levels of ersistence, the tests effective sizes are fairly accurate; nevertheless, the results for H 0 : γ = 0.85 are slightly better than those for H 0 : γ = 0.5. For meium levels of ersistence, there is a loss of size accuracy of the tests relative to low ersistence levels, in nominal sizes of both 0% an 5%. For high levels of ersistence, the test sizes for both γ an γ are notably smaller than their nominal sizes. able shows the results obtaine for Case with = 00. Uner meium an high levels of ersistence, the effective sizes are closer to the nominal sizes than they were when = 50. When = 00 the tests for low levels of ersistence are not as accurate as those for = 50. However, the results are not very ifferent. he results for Case with = 50, resente in able 3, are very similar to the ones obtaine for = 00. able : Effective size of PDOLS-3D tests for Case, with = 00. Persistence eium est Nominal size 5% 0% H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = he size results of PDOLS-3D tests for Case, with = 50 are shown in able 4. When the CSD egree is low, the test for γ is accurate at low levels of ersistence, an effective size becomes smaller when ersistence increases to meium an high levels. For γ, the test is mis-size when the ersistence is low, an its effective size ecreases when the ersistence reaches meium an high levels. Revista Colombiana e Estaística

12 56 Luis Fernano elo-velania, John Jairo León & Dagoberto Saboyá able 3: Effective size of PDOLS-3D tests for Case, with = 50. Persistence eium est Nominal size 5% 0% H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = able 4: Effective size of PDOLS-3D tests for Case, with = 50. Homogeneous Nominal size Persistence est CSD 5% 0% H 0 : γ = H 0 : γ = eium H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = eium H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = In Case uner high CSD for the simulations resente in able 4, the tests are, in general, not as well size as they were in low CSD. For examle, for γ the effective sizes in low an meium levels of ersistence are not as accurate as uner low CSD. Even though the tests are highly mis-size for some cases, the test accuracy imroves when the level of ersistence increases. able 5 resents the test sizes for case with = 00. Uner low CSD, the test results generally imrove with resect to those resente in able 4, secially for γ. Uner high CSD, the increase in the time imension imroves the accuracy of the tests uner low, meium, an some cases of high ersistence levels. he simulations for = 50 rouce even better results, as is showe in able 6. he results for Case 3 with = 50 are reorte in able 7. Size results are closer to the nominal levels in the resence of low CSD than in the resence of high CSD, for low an meium ersistence levels. Aitionally, as in the revious cases escribe in ables an 4, size ecreases when ersistence aroaches higher levels. his leas to small an mis-size tests for γ in high ersistence levels. It is also imortant to note that, in most of the cases, the results of Case 3 are not very ifferent from those obtaine in Case. Revista Colombiana e Estaística

13 3D-Panel Cointegration Vector Estimation 57 able 5: Effective size of PDOLS-3D tests for Case, with = 00. Homogeneous Nominal size Persistence est CSD 5% 0% H 0 : γ = H 0 : γ = eium H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = eium H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = able 6: Effective size of PDOLS-3D tests for Case, with = 50. Homogeneous Nominal size Persistence est CSD 5% 0% H 0 : γ = H 0 : γ = eium H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = eium H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = Size results for Case 3 an = 00 are shown in able 8. When the CSD egree is low, there are gains in accuracy for the γ tests in all ersistence levels comare with simulations of able 7. However, there are some cases with no imrovement for γ. hese gains are also obtaine uner high CSD for most of the cases. For = 50 able 9, size istortions are, in general, smaller. In conclusion, there are four relevant observations relate to the simulation exeriments. First, nominal an effective sizes are, in general, close enough. Secon, ersistence levels an size are negatively relate; as the level of ersistence is increase, effective size systematically ecreases. hir, the results of the effective size in Cases an 3 are relatively similar, which inicates that subtracting the cross-sectional average is an effective way to control CSD, even in the resence of heterogeneous CSD. Finally, as execte, increasing the time imension, in general, imroves the accuracy of the tests. Revista Colombiana e Estaística

14 58 Luis Fernano elo-velania, John Jairo León & Dagoberto Saboyá able 7: Effective size of PDOLS-3D tests for Case 3, with = 50. Heterogeneous Nominal size Persistence est CSD 5% 0% H 0 : γ = H 0 : γ = eium H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = eium H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = able 8: Effective size of PDOLS-3D tests for Case 3, with = 00. Heterogeneous Nominal size Persistence est CSD 5% 0% H 0 : γ = H 0 : γ = eium H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = eium H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = able 9: Effective size of PDOLS-3D tests for Case 3, with = 50. Heterogeneous Nominal size Persistence est CSD 5% 0% H 0 : γ = H 0 : γ = eium H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = eium H 0 : γ = H 0 : γ = H 0 : γ = H 0 : γ = Revista Colombiana e Estaística

15 3D-Panel Cointegration Vector Estimation Emirical Alication As an emirical exercise, we estimate caital an labor elasticities associate with the total factor rouctivity for the Colombian inustry. his exercise is useful, since rouctivity is a variable that reflects how efficiently an economy uses its resources to rouce goos an services an hels to etermine the istribution of value ae between caital an labor. Assuming a Cobb-Douglas rouction function, we obtain the following equation for value ae: Y ijt = A ij K α ijtl β ijt where Y is value ae, K is caital stock, L is labor, A corresons to total factor rouctivity, α an β are the elasticities of caital an labor, resectively, an α β =. he subscrits i, j an t reresent metroolitan areas, inustry sectors an time, resectively. aking logarithms on both sies of equation we get: ln Y ijt = ln A ij α ln K ijt β ln L ijt o estimate equation we emloy the ataset use in Iregui et al hey use ata from the annual manufacturing inustry survey EA of the national aministrative eartment of statistics DANE. he value ae is calculate as the ifference between gross outut an intermeiate inuts, where the latter corresons to the value of omestic an foreign consume raw materials an the value of urchase electricity kw/h. Labor is efine as the number of emloyees. An finally, caital stock was calculate using the eretual inventory metho for gross investment 3. his ata inclues annual information for 9 metroolitan areas an 8 inustrial sectors three-igits CIIU from 975 to 000. he metroolitan areas consiere are: Bogotá, Cali, eellín, anizales, Barranquilla, Bucaramanga, Pereira an Cartagena an the rest of the country. he estimate arameters of are resente in able 0. As inicate in section 3, these estimations are obtaine controlling for iniviual-secific fixe effects in the metroolitan areas an inustrial sector imensions, iniviual-secific time trens in those imensions an cross-sectional eenence. hese results show that the elasticities of caital an labor are 0.4 an 0.76, resectively. hese elasticities are similar to those foun in Colombian literature. For examle, a technical ocument from Secretaría e Haciena Distrital 003 estimate coefficients of 0.7 an 0.7 for caital an labor, resectively; an Eslava, Haltiwanger, Kugler & Kugler 004 foun elasticities of 0.3 for caital an 0.74 for labor for the erio his ata is in 994 Colombian esos. Revista Colombiana e Estaística

16 60 Luis Fernano elo-velania, John Jairo León & Dagoberto Saboyá 7. Final Remarks able 0: PDOLS-3D estimation results. Parameter Estimation Stanar Error α β his aer extens the asymtotic results of the ynamic orinary least squares cointegration vector estimator of ark & Sul 003 to a three imensional anel PDOLS-3D. his metho allows for iniviual heterogeneity using ifferent short-run ynamics, iniviual-secific fixe effects an iniviual secific time trens. Also some egree of cross-sectional eenence is consiere by the use of time-secific effects. A convenient feature of this metho is that it ermits the construction of asymtotically-vali test statistics for hyothesis testing. he roose estimators have also accetable finite samle roerties. hroughout the onte Carlo exeriments it was foun that the effective sizes of the PDOLS-3D t-tests are relatively close to the nominal sizes, for ifferent ersistence levels of the series, an ifferent forms an egrees of cross-sectional eenence. Acknowlegement We thank two anonymous referees for their valuable comments an suggestions. Receive: Setember 03 Accete: ay 04 References Banerjee, A., Henry, D. & Smith, G. 986, Exloring equilibrium relationshis in econometrics through static moels: Some onte Carlo evience, Oxfor Bulletin of Economics an Statistics 5, Davies, A. 006, A framework for ecomosing shocks an measuring volatilities erive from multi-imensional anel ata of survey forecasts, International Journal of Forecasting, Davies, A., Lahiri, K. & Sheng, X. 0, Analyzing three-imensional anel ata of forecasts, in. Clements & D. Henry, es, he Oxfor Hanbook of Economics Forecasting, Oxfor university Press, Eilat, Y. & Einav, L. 004, Determinants of international tourism: A threeimensional anel ata analysis, Alie Economics 36, Eslava,., Haltiwanger, J., Kugler, A. & Kugler,. 004, he effects of structural reforms on rouctivity an rofitability enhancing reallocation: evience from Colombia, Journal of Develoment Economics 75, Revista Colombiana e Estaística

17 3D-Panel Cointegration Vector Estimation 6 Hamilton, J. 994, ime Series Analysis, Princeton University Press. Iregui, A., elo, L. & Ramírez,. 007, Prouctivia regional y sectorial en Colombia: Análisis utilizano atos e anel, Ensayos sobre Política Económica 553, Kao, C. & Chiang,.-H. 000, On the estimation an inference of a cointegrate regression in anel ata, in B. Baltagi, e., Avances in Econometrics: Nonstationary Panels, Panel Cointegration an Dynamic Panels, JAI Press. Kremers, J., Ericsson, N. & Dolao, J. 99, he ower of cointegration tests, Oxfor Bulletin of Economics an Statistics 54, ark, N. & Sul, D. 00, Aenix to cointegration vector estimation by anel ols an long-run money eman. Unublishe manuscrit, available at htt:// ark, N. & Sul, D. 003, Cointegration vector estimation by anel DOLS an long-run money eman, Oxfor Bulletin of Economics an Statistics 655, Phillis, P. & Loretan,. 99, Estimating long-run economic equilibria, Review of Economic Stuies 58, Phillis, P. & oon, H. 999, Linear regression limit theory for nonstationary anel ata, Econometrica 675, 057. Saikkonen, P. 99, Asymtotically efficient estimation of cointegration regressions, Econometric heory 7,. Secretaría e Haciena Distrital 003, Cambio tecnológico, rouctivia y crecimiento e la inustria en Bogotá, Cuaernos e la Ciua, Serie Prouctivia y Cometivia. Stock, J. & Watson, J. 993, A simle estimator of cointegrating vectors in higher orer integrate systems, Econometrica 6, Sul, D., Phillis, P. & Choi, C. 005, Prewhitening bias in HAC estimation, Oxfor Bulletin of Economics an Statistics 674, White, H. 00, Asymtotic heory for Econometricians, Acaemic Press. Revise Eition. Revista Colombiana e Estaística

18 6 Luis Fernano elo-velania, John Jairo León & Dagoberto Saboyá Aenix A. Proof of Lemma Proof. Following the efinitions resente in Section 3, x ijt = x ijt x ijs s= i= j= x ijt x ijs i= j= s= where x ijt = x ijt Φ ij z ijt, is the linear rojection of each x ijt into z ijt, with Φ ij a matrix of rojections coefficients, then x ijt = x ijt Φ ijz ijt x ijs Φ ijz ijt s= x nmt Φ nmz nmt x nms Φ nmz nms n= m= n= m= s= = x ijt x ijs x nmt x nms s= n= m= n= m= s= Φ ij z ijt z ijt Φ nm z nmt z nms s= n= m= s= = x ijt Φ ij z ijt Φ nm z nmt n= m= 3 i Using 3, we obtain the following exression then x ijt x ijt = t= = x ijt t= Φ ij z ijt x ijt z ijt Φ ij x ijt x ijt t= x ijt t= t= z ijt Φ ij = t= x ijt x ijt t= t= Φ nm z nmt n= m= z nmt Φ nm n= m= Φ ij z ijt Φ nm z nmt z ijt Φ ij Φ ij z ijt n= m= z nmt Φ nm n= m= n= m= z nmt Φ nm n= m= Φ nm z nmt x ijt x ijt O O O t= x ijt x ijt = O O 0 x ijt Revista Colombiana e Estaística

19 3D-Panel Cointegration Vector Estimation 63 ii Base on 3, we also fin where then tx 5/ ijt = t= 5/ = 5/ = 5/ = 5/ t x ijt t= t= t= t= x ijt = x ijt t= t= s= = x ijt t= s= =0 tx ijt 5/ t n= m= s= Φ ij z ijt O 3 tx ijt 3/ x ijs x ijs x nms tx 5/ ijt t= 5/ t= t= x ijt 5/ n= m= t= x ijt t= Φ nm z nmt O 3 5/ x nmt n= m= x nmt n= m= t= tx ijt = 5/ O 3 t= O 3 = t= 5/ 0 = 0 t= O 3 t= n= m= s= x nms iii Again, from equation 3 we get x ijt u ijt = x ijt u ijt Φ ij z ijt Φ nm z nmt u ijt t= t= t= n= m= = x ijt u ijt Φ ij z ijtu ijt Φ nm z nmtu ijt t= t= n= m= t= = x ijt u ijt O t= then x ijt u ijt t= x ijt u ijt = O 0 t= Revista Colombiana e Estaística

20 64 Luis Fernano elo-velania, John Jairo León & Dagoberto Saboyá Aenix B. Proof of Lemma Proof. i First, we nee to analyze the term x ijt x ijt = x t= ijt x ijs x nmt x nms t= s= n= m= n= m= s= x ijt x ijs x nmt x nms s= n= m= n= m= s= = x ijtx ijt x ijs x ijs t= s= s= a b x nmt x nmt t= n= m= n= m= c x nms x nms n= m= s= n= m= s= x ijt x nms x nms x ijt t= n= m= s= n= m= s= e x ijs x nmt x nmt x ijs t= s= n= m= n= m= s= f x ijt x ijs x ijs x ijt t= s= s= g x ijt x nmt x nmt x ijt t= n= m= n= m= h x ijs x nms x nms x ijs t= s= n= m= s= n= m= s= s= i x nmt x nms x nms x nmt t= n= m= n= m= s= n= m= s= n= m= as, we have the following results for the ten factors of the revious exression a t= x ijt x ijt s= x ijs s= ijs x Bvij B vij Hamilton 994, roosi- b tions 8.g an 7.3f B vij B vij j Revista Colombiana e Estaística

21 3D-Panel Cointegration Vector Estimation 65 c t= n= m= x nmt N n= m= Bvnm B vnm N n= m= s= x nms N = n= m= 3/ s= x nms n= m= Bvnm N n= m= x nmt n= m= s= x nms n= m= B vnm n= m= Hamilton 994, roosition 8.g N e t= x ijt n= m= s= x nms N n= m= s= x nms x ijt = 3/ t= x N ijt n= m= N n= m= 3/ s= x nms N B vij n= m= B vnm Hamilton 994, roosition 6.g f t= s= x N ijs n= m= x nmt N n= m= x nmt s= x ijs = 3/ s= x N ijs n= m= N n= m= 3/ t= x nmt N B vij n= m= B vnm x ijt s= x ijs g t= = 3/ t= x ijt B vij B vij x ijt h t= = N t= N t= = 3/ s= x ijs 3/ s= x nms 3/ s= x ijt 3/ s= x nms n= m= Bvnm B vij 3/ t= x nmt s= x ijs n= m= x nmt n= m= x ijt x nmt n= m= x nmt x ijt 3/ s= x ijs n= m= Bvnm B vij x ijt 3/ s= x ijs n= t= x ijt x ijt t= x ijt x ijt Bvij B vij Bvij B vij i t= s= x N ijs n= m= s= x nms N n= m= s= x nms s= x ijs = 3/ s= x N ijs n= m= 3/ s= x nms 3/ t= x ijt m= x nmt x ijt Revista Colombiana e Estaística

22 66 Luis Fernano elo-velania, John Jairo León & Dagoberto Saboyá N n= m= 3/ s= x nms 3/ s= x ijs N B vij n= m= B N vnm n= m= Bvnm B vij Hamilton 994, roosition 7.3f N j t= n= m= x N nmt n= m= s= x nms N n= m= s= x N nms n= m= x nmt N = n= m= 3/ t= x N nmt n= m= 3/ s= x nms N n= m= 3/ s= x N nms n= m= 3/ t= x nmt N N n= m= Bvnm n= m= B vnm Using the efinition of,, Lemma art i, an the revious results of the terms of x ijt x ijt, for fixe N an as, then t=,, where, = = i= j= i= j= i= j= t= x ijt x ijt a b c e f g h i j {a c h b g j e f i}, = B vij B vij B vij i= j= i= j= B vij B vij i= j= As N an we have the following result i= j= N i= j= Bvij B vij i= j= Ω vv,ij 0 N i= j= Bvij B vij 3 i= j= Ω vv,ij 0 N N i= j= Bvij i= j= B vij 0 B vij Revista Colombiana e Estaística

23 3D-Panel Cointegration Vector Estimation 67 then, 6 i= j= Ω vv,ij 0 ii Analyzing the i th column of,, efine in section 4, an using the result of lemma art ii N 5/ Examining k for fixe j N 5/ then j= t= tx ijt = j= N 5/ t= tx ijt k tx ijt = t x ijt x ijs x nmt x nms t= N 5/ t= s= n= m= n= m= s= = tx ijt t x ijs t x N 5/ t= N 7/ N t= s= 3/ 5/ nmt t= n= m= t x N 3/ 7/ nms t= n= m= s= = tx N 5/ ijt t x t= N t= 3/ ijs s= tx N 3/ n= m= 5/ nmt t= t x N t= 3/ n= m= 3/ nms s= rb vij N B vij rb N N 3/ vnm n= m= B N 3/ vnm n= m= = rb vij B vij rb N N 3/ vnm B vnm n= m=, = tx 5 jt,, tx 5 Njt, j= t= j= t= where, is the i-th column of the matrix i, an, = i j= rb vij N As N,, 0 for all i an fixe, then i B vij rb vnm B vnm n= m=, 0 Revista Colombiana e Estaística

24 68 Luis Fernano elo-velania, John Jairo León & Dagoberto Saboyá iii Similar to the roof of lemma art ii. iv From revious efinitions,, = t 3 t= I N, with then 3 v From section 4. t= t = 3 = 3 = 3 t= t= t t t t t= 3 = = t t=, I N 3, = t 3 t= N 4 4 N 0 as N an vi Using lemma art iv, the following result is obtaine 33, I 4 Revista Colombiana e Estaística

25 3D-Panel Cointegration Vector Estimation 69 Aenix C. Proof of Lemma 3 Proof. i First, we analyze the following term x ijt u ijt = x ijt x ijs x nmt x nms t= t= s= n= m= n= m= s= u ijt u ijs u nmt u nms s= n= m= n= m= s= = x ijtu ijt x nmt u nmt t= t= n= m= n= m= a x ijs u ijs t= s= s= b c x nms u nms x ijt u nmt t= n= m= s= n= m= s= t= n= m= x ijt u ijs x ijt u nms t= s= t= n= m= s= f g x nmt u ijt x nmt u ijs t= n= m= t= n= m= s= h i x nmt u nms x ijs u ijt t= n= m= n= m= s= t= s= j x ijs u nmt t= s= n= m= l x ijs u nms t= s= n= m= s= m x nms u ijt t= n= m= s= n x nms u nmt t= n= m= s= n= m= o x nms u ijs t= n= m= s= s= k e Revista Colombiana e Estaística

26 70 Luis Fernano elo-velania, John Jairo León & Dagoberto Saboyá with a t= x ijt u ijt Ωuu,ij Bvij W uij b t= m= x nmt n= n= m= u nmt n= m= Ωuu,nm Bvnm W unm c t= t= s= x ijs s= u ijs Bvij Ω uu,ij W uij N n= m= s= x N nms n= m= s= u nms N Bvnm n= m= Ωuu,nm W unm n= m= e t= x N ijt n= m= u nmt Ωuu,ij Bvij W uij f t= x ijt s= u ijs Bvij Ω uu,ij W uij g t= x N ijt n= m= s= u nms N B vij n= m= Ωuu,nm W unm h t= i t= j t= k t= l t= n= n= m= x nmt u ijt Ωuu,ij Bvij W uij m= x nmt s= u ijs n= m= Bvnm Ωuu,ij W uij n= n= m= m= x nmt N n= m= s= u nms N Bvnm n= m= Ωuu,nm W unm s= x ijs u ijt Bvij Ω uu,ij W uij B vij m t= B vij n t= s= x ijs N n= m= u nmt N n= m= Ωuu,nm W unm s= x ijs n= m= s= u nms N n= m= Ωuu,nm W unm n= m= s= x nms u ijt n= m= Bvnm Ωuu,ij W uij Revista Colombiana e Estaística

27 3D-Panel Cointegration Vector Estimation 7 o N t= n= m= s= x N nms n= m= u nmt N N n= m= Bvnm n= m= Ωuu,nm W unm t= n= m= s= x nms s= u ijs n= m= Bvnm Ωuu,ij W uij Using the revious results an lemma art iii, we have t= x ijt u ijt Ωuu,ij B vijw uij Ωuu,nm B vnmw N unm n= m= Ωuu,ijW B vij uij B vnm Ωuu,nmW unm n= m= n= m= B vij Ωuu,nmW unm B vnm Ωuu,ijW uij n= m= n= m= Summing over i an j an iviing the result by, we obtain Ωuu,ij i= j= B vij W uij Ωuu,nm B vnmw 3 unm B vij Ωuu,ij W uij n= m= i= j= N B 3 vnm Ωuu,nmWunm n= m= n= m= B vij Ωuu,nmWunm i= j= n= m= B vnm Ωuu,ij W uij n= m= i= j= = Ωuu,ij B vij W uij i= j= B vij W uij 3/ i= j= B vij Ωuu,ij W uij 3/ B vij Ωuu,nmWunm i= j= n= m= = Ωuu,ij B vij W uij i= j= B vij W uij Ωuu,ij 3/ B vij W uij Ωuu,nmWunm i= j= n= m= Revista Colombiana e Estaística

28 7 Luis Fernano elo-velania, John Jairo León & Dagoberto Saboyá herefore m, = N i= j= Ωuu,ij B B B vij W uij O where B vij = B vij B vij ii First, we nee to establish the asymtotic normality of N i= i= j= U ij j= e ij, where U ij = B vij B vij an e ij = Ω uu,ij Bvij W uij. Working with {e Lij } i,j=0 an L =, we have a e Lij = i= j= e Lij b µ Lij = Ee Lij = E{Ee Lij U Lij} = E{0} = 0 c µ Lij = i= j= µ Lij = 0 V Lij = V are Lij = Ee Lije Lij = E{Ee Lije Lij U Lij} = E{Ω uu,liju Lij} = Ωuu,LijΩ 6 vv,lij e E e Lij δ = E Ω uu,lij Bv,LijW u,lij δ = E Ω uu,lijω / vv,lij W v,lijw u,lij δ Ω uu,lijω / vv,lij δ E W v,lijw u,lij δ < < f V L = N i= j= VLij = N i= j= V are Lij = N 6 i= j= Ωuu,LijΩ vv,lij Since Ω vv,lij is ositive efinite for all i, j, V L is O an uniformly ositive efinite. Using theorem 5. of White 00, it follows that as, N an V N i= j= e ij N0, I where V = V ar N i= j= e ij = 6 i= j= Ω uu,ijω vv,ij. iii Analyzing art of the i th element of m,, for fixe N, as Revista Colombiana e Estaística

29 3D-Panel Cointegration Vector Estimation 73 tu 3/ ijt = tu t= 3/ ijt t= = tu 3/ ijt t u t= t= / ijs tu s= n= m= 3/ nmt t= t u t= n= m= / nms s= = tu 3/ ijt u t= / ijs tu s= n= m= 3/ nmt t= u n= m= / nms s= Ω uu,ij W uij W uijrr Ωuu,ijW uij 0 Ωuu,nm W unm W unmrr Ωuu,nmW unm n= m= 0 n= m= = Ω uu,ij rw uij Ωuu,ijW uij n= m= = Ω uu,ij hen Ωuu,nm rw unm rw uij Wuij n= m= Ωuu,nm Wunm n= m= Ωuu,nm rw unm Wunm tu 3/ ijt Ω uu,ij t= rw uij Wuij n= m= Ωuu,nm rw unm Wunm from Lemma 3 art i, m, m, = m,, an N i= j= Ωuu,ij B B B vij W uij O Using these results an an extension of roosition 4 art a from ark & Sul 00, we obtain that as, N an, m, is ineenent of m,. he roof of the asymtotic ineenence of m, an m 3, follows in a similar way. Revista Colombiana e Estaística