TableA1:Randomeffectsworld(REandRobustthree-stage),ρ=0.3,ε=0.5 β11β12β2σ2 εσ2 µλβλb true11110.428571 N=100,T=10REcoef1.0006911.0001691.0002790.9992610.426830 se0.0148580.0148810.014830 rmse0.0148260.0150060.015437 3Scoef0.9990510.9985380.9985840.9965170.4352260.218000<10−6 se0.0073020.0073080.007283 rmse0.0186650.0190040.018925 N=500,T=5REcoef0.9998821.0001490.9993131.0019470.429767 se0.0080230.0080190.008031 rmse0.0080990.0083150.008083 3Scoef0.9987040.9989930.9987741.0009020.4322300.221000<10−6 se0.0043130.0043110.004318 rmse0.0113210.0110510.011187 N=500,T=10REcoef0.9996611.0000771.0001061.0004880.427950 se0.0066380.0066400.006644 rmse0.0069640.0068550.006668 3Scoef0.9995200.9998520.9997950.9999200.4292200.189000<10−6 se0.0032240.0032260.003228 rmse0.0084900.0086080.008463
TableA2:Randomeffectsworld(REandRobustthree-stage),ρ=0.8,ε=0.5 β11β12β2σ2 εσ2 µλβλb true11114 N=100,T=10REcoef1.0003171.0003811.0000500.9992613.977972 se0.0228560.0228480.022866 rmse0.0220940.0228480.023307 3Scoef0.9991180.9992820.9986130.9961523.9890200.117000<10−6 se0.0072820.0072750.007283 rmse0.0226870.0236440.024120 N=500,T=5REcoef1.0001421.0003521.0002841.0019474.007488 se0.0149730.0149660.014970 rmse0.0149730.0146560.014725 3Scoef0.9994740.9996840.9997701.0006334.0100650.110000<10−6 se0.0043050.0043020.004307 rmse0.0163690.0156310.016360 N=500,T=10REcoef0.9996121.0003200.9998171.0004883.998938 se0.0102190.0102200.010223 rmse0.0104700.0099930.00979 3Scoef0.9994030.9999680.9994450.9998513.9991620.123000<10−6 se0.0032230.0032240.003226 rmse0.0107950.0104040.010309
TableA3:Randomeffectsworld(REandRobustthree-stage),N=50,T=20,ρ=0.8,ε=0.5 β11β12β2σ2 εσ2 µλβλb true11114 REcoef1.0000241.0004870.9995311.0015944.014982 se0.0215180.0214870.021501 rmse0.0208230.0209690.022112 3Scoef0.9987450.9990670.9982130.9985704.013042<10−6<10−6 se0.0075680.0075620.007572 rmse0.0212840.0213410.022572 TableA4:Mundlak-typeFE(MundlakandRobustthree-stage),N=50,T=20,ε=0.5 β11β12β2πσ2 εσ2 µλβλb true1110.816.042385 Mundlakcoef0.9995260.9985821.0008590.7987671.0002516.053826 se0.0192080.0191990.0229490.048149 rmse0.0189780.0194870.0237120.049777 3Scoef0.9991840.9983741.0007900.8018380.9971906.073324<10−6 <10−6 se0.0075270.0075300.0223370.024157 rmse0.0202500.0207330.0236980.049073
TableA5:Chamberlain-typefixedeffectsworld(MCSandRobustthree-stage),ε=0.5 N=100T=10β11β12β2σε2σ2 µλβλb true1111164.877911 MCScoef0.9997341.0006410.9998770.991008147.854745 se0.0075570.0075340.007097 rmse0.0280580.0268010.028788 3Scoef0.9955840.9966180.9997100.995767164.127937<10−6<10−6 se0.0076170.0076360.022473 rmse0.0211450.0211870.023026 N=100T=10π1π2π3π4π5π6π7π8π9π10 true0.1342180.1677720.2097150.2621440.3276800.40960.5120.640.81 MCScoef0.1312580.1700690.2126250.2573900.3251590.4172670.5139750.6392430.8000520.994872 se0.0213280.0213700.0214250.0213510.0214340.0214070.0214520.0214640.0214420.021361 rmse0.0785830.0805460.0765780.0799650.0815440.0830820.0791400.0809200.0775190.078152 3Scoef0.1423460.1760480.2156880.2638880.3304490.4060210.5143930.6344830.7964460.987879 se0.0216990.0216620.0216440.0217350.0217270.0216960.0217320.0218300.0216450.021750 rmse0.0737790.0748740.0736880.0753240.0748750.0749410.0738630.0695250.0743270.074024
TableA6:Chamberlain-typefixedeffectsworld(MCSandRobustthree-stage),ε=0.5 N=500T=10β11β12β2σε2σ2 µλβλb true1111166.265383 MCScoef0.9998621.0000520.9997030.988115154.282781 se0.0035090.0035040.003302 rmse0.0109270.0105460.010900 3Scoef0.9989710.9993930.9998750.999579165.371009<10−6<10−6 se0.0032570.0032500.010016 rmse0.0099250.0090060.010399 N=500T=10π1π2π3π4π5π6π7π8π9π10 true0.1342180.1677720.2097150.2621440.3276800.40960.5120.640.81 MCScoef0.1336720.1685540.2087790.2639080.3286630.4088390.5107330.6401100.8000600.999796 se0.0098500.0098590.0098640.0098630.0098570.0098290.0098520.0098540.0098690.009843 rmse0.0312830.0320890.0310670.0324280.0320570.0315660.0310650.0329000.0320300.030432 3Scoef0.1363740.1687310.2105630.2616460.3299550.4099030.5135820.6385420.7966140.997833 se0.0091870.0091780.0091950.0091770.0091850.0091980.0091900.0091890.0091880.009198 rmse0.0315920.0316400.0342280.0320800.0319250.0321240.0329570.0316160.0316920.032376
TableA7:Chamberlain-typefixedeffectsworld(MCSandRobustthree-stage),N=50,T=20,ε=0.5 β11β12β2σ2 εσ2 µλβλb true1111196.385034 MCScoef1.0012091.0012091.0008291.164418178.392318 se0.0064290.0064380.004028 rmse0.0560460.0581380.048501 3Scoef0.9955640.9966891.0003820.998992196.392882<10−6 <10−6 se0.0095130.0095120.022597 rmse0.0213610.0206170.022751 π1π2π3π4π5π6π7π8π9π10 true0.0144120.0180140.0225180.0281470.0351840.0439800.0549760.0687190.0858990.107374 MCScoef0.0190990.0140410.0176200.0347800.0308310.0362220.0543420.0722910.0917050.112630 se0.0194300.0195900.0194470.0195050.0193310.0193190.0194290.0194680.0193500.019408 rmse0.1588690.1650850.1691900.1702400.1626080.1694160.1717410.1662120.1639690.169013 3Scoef0.0204310.0277430.0282570.0402320.0392680.0448440.0626150.0780160.0919760.110463 se0.0287730.0289990.0287590.0288460.0285930.0285840.0287440.0287920.0286290.028723 rmse0.1208760.1307710.1328980.1338620.1271650.1293530.1291070.1294270.1330170.130166 π11π12π13π14π15π16π17π18π19π20 true0.1342180.1677720.2097150.2621440.3276800.40960.5120.640.81 MCScoef0.1400360.1512340.2048680.2649550.3278140.4157800.5208430.6438310.7899281.001918 se0.0194900.0194750.0194090.0193470.0195470.0192830.0195610.0194880.0195130.019495 rmse0.1677700.1717010.1660560.1701180.1668560.1660200.1729920.1674150.1707200.168452 3Scoef0.1389690.1583300.2100690.2658580.3303640.4078560.5062670.6321050.7775250.977140 se0.0288080.0288320.0287340.0286530.0289480.0285270.0289200.0288500.0288740.028834 rmse0.1274810.1298300.1258900.1282290.1320650.1281070.1321030.1352150.1319760.132502
TableA8:Hausman-Taylorworld(HTandRobustthree-stage),ε=0.5 β11β12β2η1η2σ2 εσ2 µλβλb N=500,T=5true1111110.428571 HTcoef1.0001260.9999420.9997491.0015541.0007750.9989950.442810 se0.0152380.0152160.0199450.0360770.057130 ρ=0.3rmse0.0152280.0153610.0198120.0360520.056667 3Scoef0.9936090.9934110.9995641.0009621.0387391.0020630.554927<10−6<10−6 se0.0054880.0054810.0179000.0181300.014502 rmse0.0128740.0132130.0198520.0350990.045958 true1111114 HTcoef0.9998591.0001461.0006410.9987290.9956840.9995884.081871 se0.0176290.0176510.0199510.0927870.079676 ρ=0.8rmse0.0176740.0185190.0200890.0917370.083407 3Scoef0.9924330.9921130.9998960.9985081.0487931.0027743.797021<10−6 <10−6 se0.0054780.0054820.0179100.0188160.013601 rmse0.0139420.0144010.0199950.0866210.055194 N=500,T=10true1111110.428571 HTcoef1.0000260.9995811.0002490.9999991.0018660.9991210.430370 se0.0093600.0093630.0114460.0326810.035614 ρ=0.3rmse0.0090930.0092700.0119170.0336890.034860 3Scoef0.9963810.9963521.0000960.9996791.0264821.0004850.481835<10−6 <10−6 se0.0041170.0041190.0108730.0135410.010913 rmse0.0084120.0085320.0119090.0330950.031538 true1111114 HTcoef1.0000650.9997830.9995991.0002711.0015141.0003354.035335 se0.0108580.0108660.0114550.0911370.064883 ρ=0.8rmse0.0104450.0108170.0115780.0923490.064688 3Scoef0.9961370.9959580.9992530.9997701.0316061.0017293.866142<10−6 <10−6 se0.0041140.0041210.0108830.0139190.010305 rmse0.0085960.0087590.0115830.0895650.035819
TableA9:Hausman-Taylorworld(HTandRobustthree-stage),N=50,T=20,ρ=0.3,0.8,ε=0.5 β11β12β2η1η2σ2 εσ2 µλβλb ρ=0.3true1111110.428571 HTcoef0.9987450.9985721.0012900.9961051.0061610.9946680.426619 se0.0198220.0198720.0225010.0999070.084415 rmse0.0197900.0197780.0221990.0992320.085181 3Scoef0.9971390.9970861.0002380.9925741.0157581.0112380.460064<10−6<10−6 se0.0098210.0098680.0219970.0331540.026932 rmse0.0170180.0171720.0220410.0971420.043981 ρ=0.8true1111114 HTcoef0.9989310.9991361.0016830.9961531.0018150.9983844.066213 se0.0220070.0220130.0225040.2929260.201864 rmse0.0223160.0228780.0233750.2860150.215722 3Scoef0.9976450.9981170.9999450.9920221.0157580.9948943.930940<10−6 <10−6 se0.0023550.0018830.0000550.0079780.025283 rmse0.0176580.0179690.0230180.2725870.043631
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TableA10:Randomeffectsworld(Robustthree-stage),N=100,T=5,ρ=0.8 β11β12β2σ2 εσ2 µλβλb true11114 ε=0.1coef0.9964840.9969610.9977660.9950324.0006690.116000<10−6 se0.0097460.0097250.009743 rmse0.0367110.0431640.037201 ε=0.3coef0.9964860.9969620.9977680.9950324.0006700.116000<10−6 se0.0097460.0097250.009743 rmse0.0367110.0369590.037201 ε=0.5coef0.9964860.9969620.9977680.9950324.0006700.116000<10−6 se0.0097460.0097250.009743 rmse0.0367110.0369590.037201 ε=0.7coef0.9964860.9969630.9977680.9950324.0006700.116000<10−6 se0.0097460.0097250.009743 rmse0.0367110.0369590.037201 ε=0.9coef0.9964860.9969630.9977680.9950324.0006700.116000<10−6 se0.0097460.0097250.009743 rmse0.0367110.0369590.037201
TableA11:Hausman-Taylorworld(Robustthree-stage),N=100,T=5,ρ=0.8 β11β12β2η1η2σ2 εσ2 µλβλb N=100,T=5true1111114 ε=0.13Scoef0.9929700.9924850.9991450.9999191.0455050.9932973.933178<10−6 <10−6 se0.0124930.0124870.0403100.0488660.031320 rmse0.0272370.0268600.0455580.2020910.072753 ε=0.33Scoef0.9919970.9918020.9995920.9906341.0463800.9938123.894530<10−6 <10−6 se0.0125700.0125540.0402600.0492270.031431 rmse0.0277940.0286550.0448640.1999830.072734 ε=0.53Scoef0.9915380.9900871.0015591.0026701.0472910.9926453.975909<10−6 <10−6 se0.0124860.0124990.0402050.0489860.031392 rmse0.0282990.0283940.0433790.0433790.073468 ε=0.73Scoef0.9925650.9924061.0023490.9970651.0457660.9969193.909636<10−6<10−6 se0.0125220.0125670.0403050.0485300.031462 rmse0.0265680.0267790.0443420.1903280.071724 ε=0.93Scoef0.9914700.9911910.9992451.0023721.0469660.9921643.938627<10−6<10−6 se0.0125170.0125530.0402980.0485860.031387 rmse0.0272840.0281200.0453640.1987220.073272
TableA12:Skewedt-distribution-Randomeffectsworld(RE,Robustthree-stage),N=100,T=5,ε=0.5 β11β12β2σ2 εσ2 µλβλb true1116.9894144 REcoef0.9979200.9994340.9987856.9309703.829401 se0.1957550.1963710.195934 rmse0.0529930.0514480.052255 3Scoef0.9945680.9946380.9987616.9454314.0643480.204052<10−6 se0.0250570.0251800.025087 rmse0.0700440.0652930.068485 TableA13:Skewedt-distribution-Chamberlainworld(MCS,Robustthree-stage),N=100,T=5,ε=0.5 β11β12β2π1π2π3π4π5σ2 εσ2 µλβλb true1110.40960.5120.640.816.81961495.995993 MCScoef0.9995680.9985491.0002480.4126060.5104990.6381550.7993740.9934636.57257093.279720 se0.0727120.0728070.2275480.1980720.1984300.1987430.1987190.197763 rmse0.0364030.0366060.0855510.1002380.1008170.1013740.1002600.099848 3Scoef0.9964610.9959220.9982940.4212570.5169240.6416320.7990250.9898156.77778696.3146610.000121<10−6 se0.0253130.0253570.0812240.0694780.0695930.0697350.0696990.069502 rmse0.0568130.0562950.0910280.1042540.1037980.1020570.1041570.102902 TableA14:Skewedt-distribution-Hausman-Taylorworld(HT,Robustthree-stage),N=100,T=5,ε=0.5 β11β12β2η1η2σ2 εσ2 µλβλb true111117.2771044 HTcoef1.0029091.0034511.0057211.0055700.9898247.2109075.558088 se0.0904010.0902000.1139580.2751830.350130 rmse0.0993920.1036280.1153030.2439980.390042 3Scoef0.9914180.9920830.9995410.9890441.0037967.2202984.5715520.003713<10−6 se0.0324280.0324150.1038770.1265180.081135 rmse0.0730330.0764660.1154300.2232650.162827
B Derivations
B.1 The first step of the robust Bayesian estimator in the two-stage hierarchy1
B.1.1 Derivation of eq.(9)
Asy=Xβ+W b+ε,ε∼N(0,Σ) with Σ =τ−1IN T, the joint probability density function (pdf) ofy, given the observables and the parameters, is:
p(y|X, b, τ, β) =τ 2π
N T2
exp
−τ
2(y−Xβ−W b)0(y−Xβ−W b) .
Lety∗ =y−W b. We can write (see Koop (2003), Bauwens et al. (2005) or Hsiao and Pesaran (2008) for instance):
(y∗−Xβ)0(y∗−Xβ) =y∗0y∗−y∗0Xβ−β0X0y∗+β0X0Xβ and
p(y∗|X, b, τ) = τ 2π
N T2
exp
−τ
2(y∗−Xβ)0(y∗−Xβ) .
Letβb(b) = (X0X)−1X0y∗= Λ−1X X0y∗ andv(b) = (y∗−Xbβ(b))0(y∗−Xβb(b)). Then ΛXbβ(b) =X0y∗ andβb0(b)
ΛX =y∗0X(y∗−Xβ)0(y∗−Xβ)
=y∗0y∗−bβ0(b) ΛXβ−β0ΛXbβ(b) +β0ΛXβ
=y∗0y∗−bβ0(b) ΛXβ−β0ΛXbβ(b) +β0ΛXβ +bβ0(b) ΛXβb(b)−βb0(b) ΛXβb(b)
+bβ0(b) ΛXβb(b)−βb0(b) ΛXβb(b) (y∗−Xβ)0(y∗−Xβ) =y∗0y∗+ (β−βb(b))0ΛX
β−bβ(b)
−bβ0(b) ΛXβb(b) +βb0(b) ΛXβb(b)
−bβ0(b) ΛXβb(b). Since
y∗0y∗−βb0(b) ΛXβb(b) +βb0(b) ΛXbβ(b)−bβ0(b) ΛXbβ(b) = y∗0y∗−y∗0Xbβ(b)−bβ0(b)X0y∗ +bβ0(b)X0Xβb(b)
=
y∗−Xbβ(b)0
(y∗−Xβb(b))
= v(β)
then
(y∗−Xβ)0(y∗−Xβ) = (β−bβ(b))0ΛX
β−βb(b)
+v(β). So the joint pdf can be written as:
p(y∗|X, b, τ) = τ 2π
N T2
exp
−τ 2
n
v(β) + (β−bβ(b))0ΛX
β−βb(b)o .
1Derivations of the second step of the robust Bayesian estimator in the two-stage hierarchy are not reported here since they follow strictly those of the first step.
The base prior ofβ is given by: β ∼N
β0ιK1,(τ g0ΛX)−1
with ΛX =X0X. Combining the pdf
ofyand the pdf of the base prior, we get the predictive density corresponding to the base prior:
m(y∗|π0, b, g0) =
First, we will simplify the expression inside the exponential:
z= (β−βb(b))0ΛX(β−βb(b) +g0(β−β0ιK1)0ΛX(β−β0ιK1).
As the Bayes estimate ofβ is given by2 (see Bauwenset al. (2005)):
β∗(b|g0) = βb(b) +g0β0ιK1
We can then write m(y∗|π0, b, g0) =
2Derivation of this estimator is presented below.
The multiple integral
can be written as IRK1 =
and using the Gauss integral formula
∞ Hence we can write
m(y|π0, b, g0) =
B.1.2 Derivation of eq.(15) and eq.(16)
Next we derive the above expression with respect toβq andgq to obtain the first order conditions:
∂logm(y∗|q, b, g0)
The second term of the first order conditions is
∂logm(y∗|q, b)
or equivalently
B.1.3 Derivation of eq.(18), eq.(19), eq.(20) and eq.(22)
If π0∗(β, τ |g0) denotes the posterior density of (β, τ) for the prior π0(β, τ) and if q∗(β, τ |g0) denotes the posterior density of (β, τ) for the prior q(β, τ), then the ML-II posterior density of (β, τ) is given by
bλβ,g0 =
Similarly, the expression ofq∗(β, τ |g0) is defined as:
and wherebβEB(b|g0) is the empirical Bayes estimator ofβfor the contaminated prior distribution q(β, τ) (see the derivation below):
with
Soπ∗0(β|g0) is the pdf of a multivariatet-distribution with mean vectorβ∗(b), variance-covariance matrix
Notice thatq∗(β|g0) is the pdf of a multivariatet-distribution with mean vectorβbEB(b), variance-covariance matrix
ξ1,βM1,β−1 N T−2
and degrees of freedom (N T) with
ξ1,β1= 1 +
B.1.4 Derivation of eq.(21) and eq.(23).
To prove equation (23), start from Bayes’s theorem:
p(β|y∗)∝p(y∗|β)p(β). Asy∗ ∼N Xβ, τ−1IN T
andβ ∼N
βbqιK1,(τbgΛX)−1
, then the productp(y∗|β)p(β) is
pro-portional to exp
−12Q∗ whereQ∗ is given by (see Koop (2003), Bauwenset al. (2005) or Hsiao and Pesaran (2008) for instance):
Q∗ = τ(y∗−Xβ)0(y∗−Xβ) +τbg 1. Therefore, the marginal distribution ofβgiveny∗is proportional to exp
−12Q∗1 . Consequently,
Hence
Using Bayes’s theorem once again:
p(β|y∗)∝p(y∗|β)p(β). Asy∗∼N Xβ, τ−1IN T
andβ∼N
β0ιK1,(τ g0ΛX)−1
, then, following the previous derivations, we can show thatβ∗(b|g0) is the Bayes estimate ofβ for the prior distributionπ0(β, τ |g0) :
β∗(b|g0) = bβ(b) +g0β0ιK1
g0+ 1 . (21)
Q.E.D
B.2 The second step of the robust Bayesian estimator in the three-stage hierarchy
B.2.1 Derivation of eq.(40) and eq.(42)
In the second step of the two-stage hierarchy, we have derived the predictive density corresponding to the base prior conditional onh0:
m(ey|π0, β, h0) = He
Then, the unconditional predictive density corresponding to the base prior is given by m(ye|π0, β) =
where2F1 is the Gaussian hypergeometric function.3 Following the lines of the second step of the robust estimator in the two-stage hierarchy, we have
bhq = min (h0, h∗),
and the predictive density corresponding to the contaminated prior conditional onh0is:
m(y|q, b, hb 0) =
Then the unconditional predictive density corresponding to the contaminated prior is given by:
m(ye|q, β)b =
3The Euler integral formula is given by (see Abramovitz and Stegun (1970)):
1 is a special function represented by the hypergeometric series. For|z|<1, the hypergeometric function is defined by the power series:
2F1(a1;a2;a3;z) =∞j=0 (a1)j(a2)j
(a3)j
zj j!
where (a1)jis the Pochhammer symbol defined by (a1)j=
( 1 if j= 0
a1(a1+ 1)...(a1+j−1) = Γ(aΓ(a1+j)
1) if j >0
Letϕ= hh0
so we get two incomplete Gaussian hypergeometric functions. Letϕ=
h∗ h∗+1
t. The solution of the first one is given by:
h∗
where F1(.) is the Appell hypergeometric function.4 The second incomplete Gaussian
hypergeo-metric function can be written as:
1
4The Appell hypergeometric function (see Appell (1882), Abramovitz and Stegun (1970), Slater (1966)) is a formal extension of the hypergeometric function to two variables:
F1(a;b1;b2;c;x;y) = ∞j=0∞k=0(a)j+k(b1)j(b2)k
where (a1)jis the Pochhammer symbol.
Then the unconditional predictive density corresponding to the contaminated prior is given by:
B.2.2 Derivation of eq.(44)
Under the contamination class of prior, the empirical Bayes estimator ofbin the two-stage hierarchy (conditional onh0) can be written as:
bbEB(β|h0) = bb(β) +bhqbbqιK2
The (unconditional) empirical Bayes estimator of b for the three-stage hierarchy model is thus
given by
We get three incomplete Gaussian hypergeometric functions. Let ϕ=
h∗ h∗+1
t. The solution of the first one is given by
h∗
The solution of the second one is:
and the solution of the third one is:
1 It follows the empirical Bayes estimator ofbfor the three-stage hierarchy model is given by:
bbEB(β) = 1
C.1 Laplace approximation of the predictive density based on the base prior
The unconditional predictive density corresponding to the base prior is given by m(ye|π0, β) =
As shown by Lianget al. (2008), numerical overflow is problematic for moderate to largeN T and
largeR2b
0in Gaussian hypergeometric functions. As the Laplace approximation involves an integral
with respect to a normal kernel, we follow the suggestion of Lianget al. (2008) to develop the
expansion after a change of variables toφ= log
h0