Volume 29, Issue 3. Kazuhiko Hayakawa Hiroshima University

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 2008-2017 Submtted: Jun 15 2009 Publshed: August 18, 2009

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

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 1 1 0 0 0 0 1 1 0 0 D T =, 0 0 0 1 1 1 1 T 1 1 T 1 1 T 1 1 T 1 1 T 1 [ ] 0 1 1 T 1 1 T 2 1 T 2 1 T 2 1 T 2 F T = dag T,, 2 0 0 0 1 1 2 1 2 0 0 0 0 1 1 =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

defne the backward orthogonal devaton of w t as follows: [ w t = w t w ],t 1 + + 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 1 0 0 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 2 0 0 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

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

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

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, 277 97 [3] Arellano, M and O Bover (1995) Another Look at the Instrumental Varable Estmaton of Error-Components Models, Journal of Econometrcs, 68, 1, 29 51 [4] Blundell, R and S Bond (1998) Intal Condtons and Moment Restrctons n Dynamc Panel Data Models, Journal of Econometrcs, 87, 1, 115 143 [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, 409 444 [6] Hayakawa, K (2009) A Smple Effcent Instrumental Varable Estmator n Panel AR(p) Models, Econometrc Theory, 25, 3, 873 890 7

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 =075 60 48 70 38 70 38 64 44 75 33 76 32 LEV1 α =075, β =025 70 38 91 17 92 16 52 56 90 18 90 18 LEV1 total 130 86 161 55 162 54 116 100 165 51 166 50 LEV0 α =025, β =075 59 49 97 11 97 11 69 39 98 10 98 10 LEV0 α =075, β =025 70 38 105 3 105 3 60 48 100 8 100 8 LEV0 total 129 87 202 14 202 14 129 87 198 18 198 18 BOD1 α =025, β =075 99 9 90 18 99 9 72 36 90 18 90 18 BOD1 α =075, β =025 108 0 99 9 108 0 99 9 81 27 81 27 BOD1 total 207 9 189 27 207 9 171 45 171 45 171 45 BOD0 α =025, β =075 72 36 99 9 99 9 54 54 81 27 81 27 BOD0 α =075, β =025 90 18 108 0 108 0 63 45 108 0 108 0 BOD0 total 162 54 207 9 207 9 117 99 189 27 189 27 Scheme 1: T =15 BIAS α STD DEV α RMSE α BIAS β STD DEV β RMSE β LEV1 α =025, β =075 74 34 70 38 70 38 75 33 74 34 74 34 LEV1 α =075, β =025 85 23 86 22 86 22 61 47 89 19 89 19 LEV1 total 159 57 156 60 156 60 136 80 163 53 163 53 LEV0 α =025, β =075 81 27 106 2 106 2 82 26 103 5 103 5 LEV0 α =075, β =025 82 26 107 1 107 1 78 30 104 4 104 4 LEV0 total 163 53 213 3 213 3 160 53 207 9 207 9 BOD1 α =025, β =075 108 0 99 9 108 0 90 18 90 18 90 18 BOD1 α =075, β =025 108 0 99 9 99 9 99 9 72 36 72 36 BOD1 total 216 0 198 18 207 9 189 27 162 54 162 54 BOD0 α =025, β =075 99 9 108 0 108 0 36 72 108 0 108 0 BOD0 α =075, β =025 108 0 108 0 108 0 54 54 108 0 108 0 BOD0 total 207 9 216 0 216 0 90 126 216 0 216 0 Scheme 2: T =6 BIAS α STD DEV α RMSE α BIAS β STD DEV β RMSE β LEV1 α =025, β =075 9 93 51 51 51 51 61 41 97 5 97 5 LEV1 α =075, β =025 33 75 81 27 81 27 54 54 108 0 108 0 LEV1 total 42 168 132 78 132 78 115 95 205 5 205 5 LEV0 α =025, β =075 22 80 102 0 102 0 36 66 102 0 102 0 LEV0 α =075, β =025 66 42 108 0 108 0 63 45 108 0 108 0 LEV0 total 88 122 210 0 210 0 99 111 210 0 210 0 BOD1 α =025, β =075 84 18 102 0 102 0 15 87 102 0 102 0 BOD1 α =075, β =025 108 0 108 0 108 0 105 3 108 0 108 0 BOD1 total 192 18 210 0 210 0 120 90 210 0 210 0 BOD0 α =025, β =075 48 54 102 0 102 0 31 71 75 27 75 27 BOD0 α =075, β =025 105 3 108 0 108 0 54 54 108 0 108 0 BOD0 total 153 57 210 0 210 0 85 125 183 27 183 27 Scheme 2: T =15 BIAS α STD DEV α RMSE α BIAS β STD DEV β RMSE β LEV1 α =025, β =075 61 42 59 43 59 43 61 41 97 5 97 5 LEV1 α =075, β =025 71 37 72 36 72 36 48 60 108 0 108 0 LEV1 total 132 78 131 78 131 78 108 101 204 5 204 5 LEV0 α =025, β =075 59 43 102 0 102 0 91 11 102 0 102 0 LEV0 α =075, β =025 99 9 108 0 108 0 96 12 108 0 108 0 LEV0 total 158 52 210 0 210 0 187 23 210 0 210 0 BOD1 α =025, β =075 102 0 102 0 102 0 55 47 102 0 102 0 BOD1 α =075, β =025 108 0 108 0 108 0 107 1 108 0 108 0 BOD1 total 210 0 210 0 210 0 162 48 210 0 210 0 BOD0 α =025, β =075 102 0 102 0 102 0 14 88 102 0 102 0 BOD0 α =075, β =025 108 0 108 0 108 0 76 32 108 0 108 0 BOD0 total 210 0 210 0 210 0 90 120 210 0 210 0 8

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 =075 0010 0010 0039 0039 0041 0041 0005 0005 0075 0075 0076 0076 LEV2 α =075, β =025 0052 0052 0081 0081 0098 0098-0014 -0014 0077 0077 0080 0080 LEV2 total 0031 0031 0060 0060 0069 0069-0005 -0005 0076 0076 0078 0078 LEV1 α =025, β =075 0016 0023 0076 0079 0078 0084 0015 0024 0122 0134 0125 0139 LEV1 α =075, β =025 0060 0087 0145 0186 0161 0210-0021 -0038 0121 0142 0126 0152 LEV1 total 0038 0055 0111 0133 0120 0147-0003 -0007 0121 0138 0125 0145 LEV0 α =025, β =075 0003 0003 0059 0061 0059 0061 0003 0004 0111 0118 0111 0118 LEV0 α =075, β =025 0011 0015 0131 0144 0131 0145-0003 -0005 0114 0124 0114 0124 LEV0 total 0007 0009 0095 0103 0095 0103 0000 0000 0112 0121 0113 0121 BOD2 α =025, β =075 0015 0015 0059 0059 0061 0061 0011 0011 0168 0168 0171 0171 BOD2 α =075, β =025 0106 0106 0158 0158 0193 0193-0032 -0032 0169 0169 0175 0175 BOD2 total 0061 0061 0108 0108 0127 0127-0011 -0011 0168 0168 0173 0173 BOD1 α =025, β =075 0019 0027 0090 0095 0093 0101 0025 0043 0276 0313 0281 0324 BOD1 α =075, β =025 0063 0085 0211 0243 0221 0260-0025 -0031 0250 0291 0253 0294 BOD1 total 0041 0056 0150 0169 0157 0180 0000 0006 0263 0302 0267 0309 BOD0 α =025, β =075 0005 0006 0073 0075 0073 0075 0003 0005 0359 0354 0359 0354 BOD0 α =075, β =025 0023 0028 0223 0241 0224 0243-0014 -0012 0369 0395 0369 0395 BOD0 total 0014 0017 0148 0158 0149 0159-0005 -0004 0364 0375 0364 0375 Scheme 1: T =15 BIAS α STD DEV α RMSE α BIAS β STD DEV β RMSE β LEV2 α =025, β =075 0006 0006 0015 0015 0017 0017 0003 0003 0021 0021 0022 0022 LEV2 α =075, β =025 0024 0024 0022 0022 0032 0032-0002 -0002 0021 0021 0022 0022 LEV2 total 0015 0015 0019 0019 0025 0025 0000 0000 0021 0021 0022 0022 LEV1 α =025, β =075 0005 0006 0029 0028 0030 0029 0005 0007 0037 0040 0037 0041 LEV1 α =075, β =025 0014 0023 0038 0054 0041 0059-0001 -0007 0028 0040 0029 0041 LEV1 total 0010 0015 0034 0041 0035 0044 0002 0000 0032 0040 0033 0041 LEV0 α =025, β =075 0001 0001 0020 0025 0020 0025 0000 0002 0028 0040 0028 0040 LEV0 α =075, β =025 0001 0001 0034 0047 0034 0047 0000 0001 0027 0038 0027 0038 LEV0 total 0001 0001 0027 0036 0027 0036 0000 0001 0028 0039 0028 0039 BOD2 α =025, β =075 0008 0008 0018 0018 0020 0020 0002 0002 0032 0032 0033 0033 BOD2 α =075, β =025 0037 0037 0030 0030 0048 0048-0008 -0008 0032 0032 0034 0034 BOD2 total 0022 0022 0024 0024 0034 0034-0003 -0003 0032 0032 0034 0034 BOD1 α =025, β =075 0003 0022 0028 0038 0028 0046 0001 0033 0059 0097 0059 0111 BOD1 α =075, β =025 0011 0035 0042 0063 0044 0073-0008 -0014 0055 0079 0056 0081 BOD1 total 0007 0028 0035 0050 0036 0059-0003 0010 0057 0088 0058 0096 BOD0 α =025, β =075 0000 0001 0021 0025 0021 0025 0002 0001 0079 0100 0079 0100 BOD0 α =075, β =025 0001 0002 0037 0055 0037 0055 0001 0002 0075 0151 0075 0151 BOD0 total 0000 0001 0029 0040 0029 0040 0002 0002 0077 0126 0077 0126 9

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 =075 0009 0009 0037 0037 0038 0038-0001 -0001 0116 0116 0116 0116 LEV2 α =075, β =025 0052 0052 0075 0075 0093 0093-0021 -0021 0138 0138 0140 0140 LEV2 total 0031 0031 0057 0057 0066 0066-0011 -0011 0127 0127 0129 0129 LEV1 α =025, β =075 0015 0008 0066 0055 0069 0056-0001 -0004 0139 0150 0139 0150 LEV1 α =075, β =025 0052 0046 0114 0115 0129 0126-0027 -0031 0166 0179 0170 0184 LEV1 total 0034 0028 0091 0086 0100 0092-0015 -0018 0153 0165 0155 0168 LEV0 α =025, β =075 0001 0001 0049 0052 0049 0052 0004 0003 0140 0148 0140 0148 LEV0 α =075, β =025 0007 0011 0110 0121 0110 0121 0000-0003 0175 0192 0175 0192 LEV0 total 0004 0006 0080 0087 0081 0088 0002 0000 0158 0171 0158 0171 BOD2 α =025, β =075 0009 0009 0053 0053 0054 0054-0006 -0006 0319 0319 0319 0319 BOD2 α =075, β =025 0100 0100 0146 0146 0178 0178-0038 -0038 0358 0358 0362 0362 BOD2 total 0055 0055 0101 0101 0118 0118-0023 -0023 0339 0339 0341 0341 BOD1 α =025, β =075 0002 0003 0056 0064 0056 0064-0014 -0005 0458 0545 0459 0545 BOD1 α =075, β =025 0044 0066 0170 0206 0176 0217-0026 -0034 0536 0641 0536 0643 BOD1 total 0023 0036 0115 0137 0118 0142-0020 -0020 0498 0595 0499 0595 BOD0 α =025, β =075 0002 0002 0066 0069 0066 0069-0008 -0003 0818 0800 0818 0800 BOD0 α =075, β =025 0026 0033 0223 0239 0225 0242-0030 -0024 0907 0965 0907 0965 BOD0 total 0015 0018 0147 0156 0148 0158-0019 -0014 0864 0885 0864 0885 Scheme 2: T =15 BIAS α STD DEV α RMSE α BIAS β STD DEV β RMSE β LEV2 α =025, β =075 0007 0007 0015 0015 0017 0017 0002 0002 0028 0028 0028 0028 LEV2 α =075, β =025 0023 0023 0020 0020 0031 0031-0001 -0001 0030 0030 0030 0030 LEV2 total 0015 0015 0018 0018 0024 0024 0000 0000 0029 0029 0029 0029 LEV1 α =025, β =075 0005 0003 0028 0022 0029 0022 0002 0001 0035 0042 0035 0042 LEV1 α =075, β =025 0014 0012 0035 0037 0038 0039-0002 -0003 0034 0048 0034 0048 LEV1 total 0010 0008 0032 0030 0034 0031 0000-0001 0034 0045 0034 0045 LEV0 α =025, β =075 0001 0000 0020 0022 0020 0022 0000 0002 0033 0045 0033 0045 LEV0 α =075, β =025 0001 0001 0030 0039 0030 0039 0001 0002 0036 0053 0036 0053 LEV0 total 0001 0001 0025 0031 0025 0031 0000 0002 0034 0049 0034 0049 BOD2 α =025, β =075 0008 0008 0017 0017 0019 0019-0004 -0004 0049 0049 0049 0049 BOD2 α =075, β =025 0035 0035 0028 0028 0045 0045-0011 -0011 0057 0057 0058 0058 BOD2 total 0022 0022 0023 0023 0033 0033-0008 -0008 0053 0053 0054 0054 BOD1 α =025, β =075 0001 0004 0018 0024 0018 0024-0006 -0003 0082 0149 0082 0149 BOD1 α =075, β =025 0006 0025 0031 0052 0032 0057-0009 -0021 0098 0171 0098 0173 BOD1 total 0004 0015 0025 0038 0025 0041-0007 -0012 0090 0161 0090 0161 BOD0 α =025, β =075 0000 0001 0020 0022 0020 0022 0003 0003 0170 0225 0170 0225 BOD0 α =075, β =025 0000 0002 0036 0059 0036 0059 0004 0006 0178 0384 0178 0384 BOD0 total 0000 0001 0028 0041 0028 0041 0003 0005 0174 0307 0174 0307 10