Volume 29, Issue 3. Kazuhiko Hayakawa Hiroshima University

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1 Volume 29, Issue 3 Frst Dfference or Forward Orthogonal Devaton- Whch Transformaton Should be Used n Dynamc Panel Data Models?: A Smulaton Study Kazuhko Hayakawa Hroshma Unversty Abstract Ths paper compares the performances of the generalzed method of moments (GMM) estmator of dynamc panel data model wheren unobserved ndvdual effects are removed by the forward orthogonal devaton or the frst dfference The smulaton results show that the GMM estmator of the model transformed by the forward orthogonal devaton tends to work better than that transformed by the frst dfference Ctaton: Kazuhko Hayakawa, (2009) ''Frst Dfference or Forward Orthogonal Devaton- Whch Transformaton Should be Used n Dynamc Panel Data Models?: A Smulaton Study'', Economcs Bulletn, Vol 29 no3 pp Submtted: Jun Publshed: August 18, 2009

2 1 Introducton Snce the semnal work of Arellano and Bond (1991), there have been many papers on the GMM estmaton of dynamc panel data models One of the typcal studes n ths lterature s Arellano and Bover (1995) who show that the GMM estmator s nvarant to the choce of transformaton that removes ndvdual effects f the transformaton matrx s upper trangular and f all the avalable nstruments are used However, n emprcal studes, t s common practce not to use all nstruments snce t s well known that usng too many nstruments deterorates the fnte sample behavor, especally the bas, of the GMM estmator In ths case, the choce of transformaton s consdered to have an nfluence on the fnte sample behavor of the GMM estmator Therefore, n terms of emprcal studes, the choce of transformaton to be used s of great concern However, to the best of author s knowledge, to date, no studes have nvestgated how dfferent the performances of the GMM estmators are when dfferent transformaton methods are used Thus, ths paper compares the performances of the GMM estmators by Monte Carlo experments when dfferent transformaton methods are used Specfcally, we consder the frst dfference (DIF) and the forward orthogonal devaton (FOD) as the transformaton methods The rest of ths paper s organzed as follows Secton 2 provdes the model and the GMM estmators Secton 3 provdes Monte Carlo results, and Secton 4 concludes the paper 2 Setup We consder the followng dynamc panel data model: y t = αy,t 1 + βx t + η + v t ( =1,, N; t =1,, T ) = δ w t + η + v t where w t =(y,t 1 x t ), δ =(αβ), and δ s the parameter of nterest wth α < 1 η s the unobservable heterogenety wth E(η ) = 0 and var(η )=ση, 2 and v t s an error term wth E(v t ) = 0 and var(v t )=σv 2 For the purpose of smplcty, we consder a scalar x t and assume that x t s a weakly exogenous varable 2

3 We make standard assumptons n the sense of Ahn and Schmdt (1995), e, E(v t η )= 0, E(v t v js ) = 0, and E(v t y 0 ) = 0 for all, j, t, and s wth t s Gven a vald nstrumental varable matrx Z, the optmal one-step GMM estmator can be wrtten as ( N )( N ) 1 ( δ = W K T Z Z K T K T Z ) 1 N Z K T W =1 =1 =1 =1 =1 ( N )( N ) 1 ( ) N W K T Z Z K T K T Z Z K T y where y =(y 1,, y T ), W =(w 1,, w T ), and v =(v 1,, v T ) K T s an upper trangular matrx such that K T ι T = 0 wth ι T beng a T 1 vector of ones The typcal examples of K T are D T of the frst dfference and F T of the forward orthogonal devaton, whch are defned by D T =, T 1 1 T 1 1 T 1 1 T 1 1 T 1 [ ] T 1 1 T 2 1 T 2 1 T 2 1 T 2 F T = dag T,, =1 Now, we defne the IV matrces 1 We consder two types of nstruments, nstruments n levels commonly used n practce, and nstruments n backward orthogonal devaton recently suggested by Hayakawa (2009) Hayakawa (2009) shows that n AR(p) panel data models, nstruments n backward orthogonal devaton s asymptotcally equvalent to the nfeasble optmal nstruments when both N and T are large Therefore, nstruments n backward orthogonal devaton may work well n ths context, too Specfcally, let us 1 For the purpose of smplcty, we do not consder the addtonal moment condtons that arse from the homoskedastcty assumpton (Ahn and Schmdt, 1995) and statonary ntal condtons (Blundell and Bond, 1998) 3

4 defne the backward orthogonal devaton of w t as follows: [ w t = w t w ],t w 1 t =2,, T 1 (1) t 1 where w t =(y,t 1 x t ) Wth regard to the number of nstruments, we consder three types followng Bun and Kvet (2006) Let us defne the followng IV matrces: where Z LEV 2 LEV 2 = dag(z1,, z = dag(z BOD2 2,, z BOD2,T 1 ), Z BOD2 LEV 2 1,T 1 ), ZLEV LEV 1 = dag(z1,, z Z BOD1 LEV 1,T 1 ) = dag(z BOD1 2,, z BOD1,T 1 ) z LEV 2 t = (y 0,, y,t 1,x 1,, x t ), z LEV 1 t =(y,t 1,x t ) z BOD2 t = (y1,, y,t 1,x 2,, x t ), z BOD1 t =(y,t 1,x t ) Note that the number of nstruments of Z LEV 2 of Z LEV 1 and Z BOD2 and Z BOD1 are of order O(T ) Fnally, we defne Z LEV 0 number of nstruments are O(1) as follows: y 0 x Z LEV 0 y = 1 x 2 y 0 x 1, Z BOD0 = y,t 2 x,t 1 y,t 3 x,t 2 are of order O(T 2 ) and that y1 y2 y,t 2 and Z BOD0 whose x x 3 x,t 1 y 1 y,t 3 x 2 x,t 2 We denote, say, the GMM estmator usng IV matrx Z LEV 2 as GMM-LEV2, etc 3 Monte Carlo experments We use the same smulaton desgns as Bun and Kvet (2006) The two data generatng processes (DGPs) are gven by where y t = αy,t 1 + βx t + η + v t Scheme 1: x t = x t + φ 1 v,t 1 + πη, x t = ρ x,t 1 + ξ t Scheme 2: x t = ρx,t 1 + φ 2 y,t 1 + π 2 η + ξ t 4

5 v t, ξ t, η are generated as v t dn(0, 1), ξ t dn(0,σ 2 ξ ) and η dn(0,σ 2 η ) wth ση 2 = μ 2 (1 α)(1 + 2αβφ 1 + β 2 φ 2 1 ) (1 + α)(1 + βπ 1 ) 2 σξ 2 = 1 [ β 2 ζ (α + βφ 1) 2 ] (1 α 2 )(1 ρ 2 )(1 αρ) (1 α 2 ) (1 + αρ) for scheme 1, and ( ση 2 = μ 2 1+ρ 2 2ρ α + βφ )[ 2 + ρ 1 ρ + βπ 2 1+αρ (1 α)(1 ρ) βφ 2 [ ] 1 (αρ) 2 (1 αρ) 1 (1 + αρ) (α + βφ 2 + ρ) 2 σ 2 ξ = 1 β 2 (ζ +1) [ 1 α 2 ρ 2 1 αρ 1+αρ (α + βφ 2 + ρ) 2 φ 2 = φ 1(1 α)(1 ρ) 1+βφ 1 π 2 = π 1 (1 ρ φ 2 ) φ 2 1 α ] ] 2 1 [ β 2 1+ρ 2 2ρ α + βφ ] 2 + ρ 1+αρ for scheme 2 We consder α = {025, 075}, β =1 α, ρ = {05, 095}, φ 1 = { 1, 0, 1}, π 1 = { 1, 0, 1}, μ = {0, 1, 5}, ζ = {3, 9} Thus, we have 216 desgns n total However, for scheme 2, 6 desngs have negatve varances for σξ 2 and σ2 η Hence, we deleted these cases n the smulaton For T and N, we set T =6,N = 200 and T =15,N = 200 The number of replcatons s 1000 Snce reportng all the results requres large space, we report the summary of the smulaton results 2 The summary of smulaton results are gven n Table 1 3 In these tables, we provde the bases(bias), standard devatons (STD DEV), and root mean squared errors (RMSE) Table 1 shows the number of tmes that FOD beats DIF and DIF beats FOD over 216 desgns For nstance, n terms of the bas of α wth scheme 1 and T = 6, GMM- LEV1 from the FOD model has smaller bas n absolute value than the GMM-LEV1 from the DIF model n 130 desgns, and GMM-LEV1 from the DIF model has smaller bas n absolute value than the GMM-LEV1 from the FOD model n 86 desgns (see the total part) We decompose the total result nto two cases, e, the cases α = 025, β = 075 and α = 075, β = 025 From Table 1, the followngs are observed: 2 Complete smulaton results are avalable from the author upon request 5

6 1 In terms of bas of α, wth some exceptons n scheme 2 wth T = 6, the GMM estomators from FOD have smaller bas than that from DIF model 2 As T gets larger, the GMM estmator of α from FOD model tends to perform better than that from DIF model However, for β, ths tendency s not always true 3 In terms of standard devaton, the GMM estmator from FOD model outperforms that from the DIF model n all cases 4 In terms of RMSE, the GMM estmator from FOD model outperforms that from the DIF model n all cases 5 There s not a sgnfcant result between two cases of α = 025, β = 075 and α =075, β=025 6 If nstruments n backward orthogonal devaton s used, the GMM estmator from FOD model works better than that from DIF model In Table 2 and 3, we provde an average of the bas, standard devaton and RMSE over 216 desgns for scheme 1 and 210 desgns for scheme 2 Some remarks are n order as follows: 1 The GMM estmator from the FOD outperforms that from DIF n many cases In some cases, the dfference s sgnfcant 2 In terms of RMSE, GMM-L2 performs best n many cases 3 The GMM estmators usng nstruments n backward orthogonal devaton do not outperform that usng nstruments n levels Ths results may be explaned from the fact that the GMM estmator usng nstruments n backward orthogonal devaton uses T 1 perods whle that usng nstruments n levels uses T perods Also, the nce property of the GMM estmator usng nstruments n backward orthogonal devaton s obtaned from large N and T asymptotcs Hence, f we consder large T,sayT = 50, the result may change These results suggest that the GMM estmator from FOD model tend to outperform that from the DIF model Wth regard to the choce of nstruments, when T s as large as T = 15, usng all nstruments n levels s the best choce n terms of RMSE However, t should be noted that f T s large, ths result may change 6

7 4 Concluson In ths paper, we compared the performances of the GMM estmators of the DIF and FOD models usng sx types of IV matrces, by Monte Carlo experments The smulaton results showed that overall the GMM estmator of the FOD model performs better than that of the DIF model n many cases In terms of RMSE, we found that the GMM estmator usng all nstruments n levels tends to perform well References [1] Ahn, S C and P Schmdt (1995) Effcent Estmaton of Models for Dynamc Panel Data, Journal of Econometrcs, 68, 1, 5 27 [2] Arellano, M and S Bond (1991) Some Tests of Specfcaton for Panel Data: Monte Carlo Evdence and an Applcaton to Employment Equatons, Revew of Economc Studes, 58, 2, [3] Arellano, M and O Bover (1995) Another Look at the Instrumental Varable Estmaton of Error-Components Models, Journal of Econometrcs, 68, 1, [4] Blundell, R and S Bond (1998) Intal Condtons and Moment Restrctons n Dynamc Panel Data Models, Journal of Econometrcs, 87, 1, [5] Bun, M J G and J F Kvet (2006) The Effects of Dynamc Feedbacks on LS and MM Estmator Accuracy n Panel Data Models, Journal of Econometrcs, 127, 2, [6] Hayakawa, K (2009) A Smple Effcent Instrumental Varable Estmator n Panel AR(p) Models, Econometrc Theory, 25, 3,

8 Table 1: Number of tmes FOD(DIF) beats DIF(FOD) Scheme 1: T =6 BIAS α STD DEV α RMSE α BIAS β STD DEV β RMSE β LEV1 α =025, β = LEV1 α =075, β = LEV1 total LEV0 α =025, β = LEV0 α =075, β = LEV0 total BOD1 α =025, β = BOD1 α =075, β = BOD1 total BOD0 α =025, β = BOD0 α =075, β = BOD0 total Scheme 1: T =15 BIAS α STD DEV α RMSE α BIAS β STD DEV β RMSE β LEV1 α =025, β = LEV1 α =075, β = LEV1 total LEV0 α =025, β = LEV0 α =075, β = LEV0 total BOD1 α =025, β = BOD1 α =075, β = BOD1 total BOD0 α =025, β = BOD0 α =075, β = BOD0 total Scheme 2: T =6 BIAS α STD DEV α RMSE α BIAS β STD DEV β RMSE β LEV1 α =025, β = LEV1 α =075, β = LEV1 total LEV0 α =025, β = LEV0 α =075, β = LEV0 total BOD1 α =025, β = BOD1 α =075, β = BOD1 total BOD0 α =025, β = BOD0 α =075, β = BOD0 total Scheme 2: T =15 BIAS α STD DEV α RMSE α BIAS β STD DEV β RMSE β LEV1 α =025, β = LEV1 α =075, β = LEV1 total LEV0 α =025, β = LEV0 α =075, β = LEV0 total BOD1 α =025, β = BOD1 α =075, β = BOD1 total BOD0 α =025, β = BOD0 α =075, β = BOD0 total

9 Table 2: Average over 216 desgns Scheme 1: T =6 BIAS α STD DEV α RMSE α BIAS β STD DEV β RMSE β LEV2 α =025, β = LEV2 α =075, β = LEV2 total LEV1 α =025, β = LEV1 α =075, β = LEV1 total LEV0 α =025, β = LEV0 α =075, β = LEV0 total BOD2 α =025, β = BOD2 α =075, β = BOD2 total BOD1 α =025, β = BOD1 α =075, β = BOD1 total BOD0 α =025, β = BOD0 α =075, β = BOD0 total Scheme 1: T =15 BIAS α STD DEV α RMSE α BIAS β STD DEV β RMSE β LEV2 α =025, β = LEV2 α =075, β = LEV2 total LEV1 α =025, β = LEV1 α =075, β = LEV1 total LEV0 α =025, β = LEV0 α =075, β = LEV0 total BOD2 α =025, β = BOD2 α =075, β = BOD2 total BOD1 α =025, β = BOD1 α =075, β = BOD1 total BOD0 α =025, β = BOD0 α =075, β = BOD0 total

10 Table 3: Average over 210 desgns Scheme 2: T =6 BIAS α STD DEV α RMSE α BIAS β STD DEV β RMSE β LEV2 α =025, β = LEV2 α =075, β = LEV2 total LEV1 α =025, β = LEV1 α =075, β = LEV1 total LEV0 α =025, β = LEV0 α =075, β = LEV0 total BOD2 α =025, β = BOD2 α =075, β = BOD2 total BOD1 α =025, β = BOD1 α =075, β = BOD1 total BOD0 α =025, β = BOD0 α =075, β = BOD0 total Scheme 2: T =15 BIAS α STD DEV α RMSE α BIAS β STD DEV β RMSE β LEV2 α =025, β = LEV2 α =075, β = LEV2 total LEV1 α =025, β = LEV1 α =075, β = LEV1 total LEV0 α =025, β = LEV0 α =075, β = LEV0 total BOD2 α =025, β = BOD2 α =075, β = BOD2 total BOD1 α =025, β = BOD1 α =075, β = BOD1 total BOD0 α =025, β = BOD0 α =075, β = BOD0 total