Cross section dependence is a rapidly growing …eld of study in panel data analysis. In this paper we have introduced the notions of weak and strong cross section dependence, and have shown that these are more general and more widely applicable than other characterizations of cross section dependence provided in the existing econometric literature. We have also investigated how our notions of CWD and CSD relate to the properties of common factor models that are widely used for modelling of contemporaneous correlation in regression models. Finally, we have provided further extensions of the CCE procedure advanced in Pesaran (2006) that allow for a large number of weak or semi-weak factors. Under this framework, we have shown that the CCE method still yields consistent estimates of the slope coe¢ cients and the asymptotic normal theory continues to be applicable.
Table1:MCresultsforFEestimator. ExperimentA:m1=3strongfactorsandm2weakfactors. Bias(x100)RMSE(x100)Size(x100)Power(x100) m2N/T203050100200203050100200203050100200203050100200 02020.4320.1720.2120.1220.3022.5821.8221.5421.1621.2285.993.597.199.3100.094.698.399.6100.0100.0 03019.2819.3919.2919.3619.3820.8720.6020.2820.1320.0694.297.699.299.8100.098.599.999.9100.0100.0 05019.7120.0619.9219.9619.7920.9920.9720.5820.4920.2097.299.6100.0100.0100.099.7100.0100.0100.0100.0 010019.2920.0920.0820.0019.9520.2520.7320.5620.3220.1999.5100.0100.0100.0100.0100.0100.0100.0100.0100.0 020019.8819.8019.7519.8119.8120.6720.3420.1220.0319.97100.0100.0100.0100.0100.0100.0100.0100.0100.0100.0 42020.2420.1920.2919.9719.9622.3521.9221.6520.9820.8585.292.697.599.499.994.798.199.6100.0100.0 63020.2119.6219.9819.8519.8321.6520.8920.9420.6220.4695.597.299.5100.0100.099.199.8100.0100.0100.0 105017.3517.5717.3217.4717.3018.8318.6318.1218.0517.7894.798.399.699.9100.098.799.8100.0100.0100.0 2010021.3221.1021.4921.3821.4322.1821.7721.9521.6821.6399.899.9100.0100.0100.0100.0100.0100.0100.0100.0 4020019.1719.0719.2919.6519.5319.9919.6619.6819.9019.6999.7100.0100.0100.0100.0100.0100.0100.0100.0100.0 122021.8821.9922.1221.8421.9323.8123.6423.3822.8822.7790.795.598.399.5100.097.299.299.8100.0100.0 183016.8816.8916.9816.8216.9118.6218.2618.0617.6517.6389.394.898.299.699.897.699.299.799.9100.0 305020.3820.4120.1420.5520.5721.5521.3120.8121.0520.9399.099.7100.0100.0100.099.9100.0100.0100.0100.0 6010021.0620.9320.9021.0120.9921.9421.5721.3421.3221.2199.7100.0100.0100.0100.0100.0100.0100.0100.0100.0 12020019.7220.3219.9820.0220.0020.4920.8220.3420.2520.15100.0100.0100.0100.0100.0100.0100.0100.0100.0100.0 202022.8422.5322.6722.6622.5624.8224.0023.8423.6323.3791.196.899.099.6100.097.299.399.8100.0100.0 303017.3116.9717.2017.2717.2418.9918.3318.2518.0917.9589.895.498.699.6100.097.699.299.9100.0100.0 505021.6621.7021.5821.5921.4822.7922.6022.2522.0821.8798.899.8100.0100.0100.099.8100.0100.0100.0100.0 10010019.9620.2520.1220.3020.2720.8820.9320.6020.6120.5299.7100.0100.0100.0100.0100.0100.0100.0100.0100.0 20020020.8520.9320.9921.1021.0621.5821.4721.3621.3321.21100.0100.0100.0100.0100.0100.0100.0100.0100.0100.0
Table2:MCresultsfortheFEestimator. ExperimentB:m1=3strongfactorsandm2semi-weakfactors. Bias(x100)RMSE(x100)Size(x100)Power(x100) m2N/T203050100200203050100200203050100200203050100200 42022.7922.7123.1023.0423.3924.6324.1324.2523.9524.1491.396.899.299.9100.097.499.399.9100.0100.0 63020.0020.0720.0119.9920.2721.6321.3320.9820.7020.8993.597.599.4100.0100.098.299.6100.0100.0100.0 105016.8717.0916.9617.2316.9318.3618.2117.8617.8017.4293.897.599.2100.0100.098.499.7100.0100.0100.0 2010018.7318.6318.7918.7818.7719.8119.4019.3419.1419.0398.8100.0100.0100.0100.099.9100.0100.0100.0100.0 4020019.5119.8719.9419.8719.9620.3520.4720.3220.1220.1399.9100.0100.0100.0100.0100.0100.0100.0100.0100.0 122020.9120.4220.5120.5320.7023.0422.1621.8221.5821.5487.293.597.399.599.895.698.099.4100.0100.0 183020.1320.2820.2620.3420.1921.9721.6421.3121.1220.8491.696.698.999.999.997.499.699.8100.0100.0 305020.7521.0021.0620.9121.0422.0522.0221.8121.4321.4498.299.6100.0100.0100.099.7100.0100.0100.0100.0 6010020.8520.4420.6220.7120.6821.9721.2321.2021.0820.9499.599.9100.0100.0100.0100.0100.0100.0100.0100.0 12020020.2820.6020.8820.7520.7521.2621.2821.3421.0220.9299.7100.0100.0100.0100.0100.0100.0100.0100.0100.0 202020.8320.8620.7720.8621.0223.3522.8122.2121.9521.8983.591.796.899.299.792.597.599.699.9100.0 303020.8521.6621.5921.5621.5922.8323.1222.7122.3722.2391.496.799.299.9100.097.999.399.9100.0100.0 505021.1421.3621.6321.2721.3322.5322.3322.3921.7821.7197.799.699.9100.0100.099.7100.0100.0100.0100.0 10010019.9920.0920.1220.1820.1321.2921.0020.7720.5820.4198.699.8100.0100.0100.099.7100.0100.0100.0100.0 20020019.2619.2319.3519.3119.2320.3820.0619.8819.6219.4299.499.7100.0100.0100.0100.0100.0100.0100.0100.0
Table3:MCresultsforCCEMGestimator. ExperimentA:m1=3strongfactorsandm2weakfactors. Bias(x100)RMSE(x100)Size(x100)Power(x100) m2N/T203050100200203050100200203050100200203050100200 0200.04-0.13-0.030.080.018.917.136.135.445.196.806.707.107.107.8011.7013.0515.2019.0019.70 0300.370.110.000.04-0.157.465.885.014.484.137.155.857.307.506.2514.4516.0519.2522.7023.70 0500.040.160.030.10-0.095.864.683.933.423.235.956.455.706.305.1516.3520.2526.8032.3534.30 0100-0.050.060.130.05-0.054.063.342.732.492.345.355.856.056.205.6524.6035.3046.5054.6558.20 0200-0.01-0.040.00-0.060.013.022.322.001.751.656.354.605.355.255.6041.3055.9571.3080.7087.05 4200.140.13-0.16-0.08-0.118.657.136.155.505.036.157.306.906.857.4510.4013.6515.8518.2018.75 6300.20-0.160.04-0.10-0.097.165.875.124.444.286.006.257.656.608.1013.3514.1020.0022.3525.60 10500.08-0.05-0.07-0.06-0.065.824.673.963.473.336.055.955.806.006.2515.1519.9525.9032.4034.80 20100-0.05-0.070.12-0.050.004.053.292.792.482.295.004.655.456.304.8022.4032.2544.4553.1559.35 402000.01-0.12-0.050.040.042.832.351.951.721.634.505.355.105.304.6540.5054.3571.2583.7087.30 12200.040.15-0.02-0.070.069.277.616.205.515.057.758.607.208.157.9011.5014.3515.6519.0521.15 1830-0.14-0.01-0.08-0.200.037.036.024.994.514.205.355.455.956.856.9011.3516.2518.3521.7525.65 30500.17-0.17-0.03-0.04-0.115.694.563.903.433.165.456.055.605.705.0016.3519.5025.3031.5034.50 601000.060.08-0.03-0.030.013.973.302.742.522.244.905.155.056.605.2523.7534.1544.8553.9558.70 120200-0.07-0.01-0.07-0.040.042.902.332.011.711.635.455.155.654.505.4040.0056.8071.1582.0086.65 20200.22-0.210.03-0.09-0.118.947.216.125.495.157.506.907.357.807.5511.1012.4516.1018.2020.25 30300.100.090.050.10-0.037.175.895.034.344.205.706.406.556.457.0012.6516.3019.9522.1024.85 50500.200.00-0.03-0.01-0.055.774.703.973.503.246.405.505.705.655.3017.2020.8526.1033.1536.25 1001000.080.11-0.03-0.01-0.064.133.282.822.442.345.955.205.605.555.9024.9034.2045.0054.2059.10 2002000.120.06-0.050.010.002.932.372.011.721.635.105.705.655.055.2042.1557.1071.6582.3086.05
Table4:MCresultsforCCEMGestimator. ExperimentB:m1=3strongfactorsandm2semi-weakfactors. Bias(x100)RMSE(x100)Size(x100)Power(x100) m2N/T203050100200203050100200203050100200203050100200 420-0.10-0.33-0.07-0.010.118.736.996.025.455.026.306.956.107.957.5511.7512.7515.5519.4020.15 630-0.18-0.06-0.13-0.15-0.087.165.775.034.524.126.456.906.357.356.4512.2015.9019.1522.8023.90 10500.090.090.040.100.005.624.723.933.483.234.606.105.656.305.3515.1521.6027.7532.6035.70 20100-0.19-0.06-0.08-0.01-0.063.933.332.832.432.385.006.006.005.206.2022.9033.8543.0555.5558.70 40200-0.040.07-0.020.00-0.052.832.361.891.741.665.155.454.304.806.2042.1559.5072.1082.5586.40 1220-0.13-0.25-0.30-0.11-0.028.597.196.285.595.016.556.958.258.456.9511.4012.8015.8518.5020.65 18300.040.020.020.05-0.117.315.945.104.474.267.055.856.156.657.1512.0515.2519.0523.5024.90 30500.100.05-0.040.040.065.824.583.883.423.216.755.555.305.656.2016.2022.0527.0531.7035.25 60100-0.110.05-0.04-0.010.054.113.352.782.422.265.406.055.555.104.4524.1534.5543.7054.4560.40 120200-0.07-0.040.01-0.030.002.802.361.931.731.604.906.004.854.705.0040.3556.7074.1581.7086.80 2020-0.12-0.12-0.20-0.120.028.897.386.165.385.066.657.357.307.706.7510.1512.9514.4517.1020.70 3030-0.410.200.030.09-0.017.496.115.134.514.336.306.406.456.457.0010.7515.5019.1523.5025.05 50500.020.10-0.010.060.035.474.493.813.453.165.705.855.356.355.2515.9521.1526.1533.0036.80 1001000.17-0.01-0.09-0.07-0.024.053.272.832.462.265.855.305.255.054.3526.2034.0542.3052.2558.20 2002000.03-0.07-0.05-0.050.012.812.291.951.771.624.755.355.356.154.4044.4056.9571.7582.0086.70
Table5:MCresultsforCCEPestimator. ExperimentA:m1=3strongfactorsandm2weakfactors. Bias(x100)RMSE(x100)Size(x100)Power(x100) m2N/T203050100200203050100200203050100200203050100200 0200.13-0.110.010.040.038.437.126.275.665.397.507.607.907.257.8013.0013.6016.4518.2018.85 0300.240.13-0.010.02-0.156.875.875.224.694.296.456.557.506.856.8514.8516.1519.5022.4023.30 0500.080.080.100.05-0.085.394.593.983.523.306.406.305.755.555.3517.6019.9526.5030.7532.10 0100-0.150.120.130.05-0.043.803.352.782.572.406.006.004.905.455.0525.8536.7044.7051.5556.15 02000.05-0.030.02-0.050.002.822.262.031.801.705.804.605.155.455.6046.1557.8070.2079.0084.80 4200.000.06-0.13-0.13-0.168.137.196.325.675.236.606.657.407.957.5012.5514.2516.0016.9017.45 6300.23-0.190.10-0.13-0.056.685.805.194.634.386.455.957.757.257.3014.0514.8520.8021.3024.05 10500.08-0.02-0.09-0.06-0.065.444.574.013.613.436.905.855.056.405.9017.1521.7024.8030.7032.70 20100-0.11-0.100.15-0.020.003.763.292.822.552.345.005.755.505.455.0026.3033.1043.2051.3056.80 40200-0.03-0.12-0.070.040.032.652.342.001.771.704.605.105.205.305.0546.0057.0569.1581.3084.75 12200.100.130.05-0.100.048.747.496.375.695.228.658.457.507.657.4013.7513.5516.1518.3520.05 1830-0.27-0.02-0.07-0.230.006.715.885.124.654.346.306.706.556.956.7512.1516.2518.7520.4524.25 30500.19-0.200.00-0.05-0.095.274.483.983.563.245.655.655.706.005.1017.9019.5524.6029.8032.00 601000.060.09-0.04-0.030.023.773.232.822.582.335.305.505.155.854.9526.6034.2042.2550.9555.25 120200-0.050.00-0.07-0.030.032.722.292.061.761.685.905.405.704.005.3045.1557.1569.0579.1584.70 20200.22-0.100.00-0.09-0.118.387.136.315.705.327.256.757.658.157.7011.3012.5516.2518.3019.90 30300.240.030.030.100.006.735.835.164.544.366.406.956.906.507.6514.2516.5019.2021.1524.50 50500.260.05-0.040.02-0.025.274.684.073.583.345.655.855.955.606.1517.7021.1525.4532.2034.65 100100-0.040.110.000.01-0.063.823.242.872.532.435.705.555.655.456.5027.4036.1043.1052.1556.00 2002000.150.06-0.030.020.002.702.332.041.791.685.755.105.605.454.8549.0559.2069.2080.1583.75
Table6:MCresultsforCCEPestimator. ExperimentB:m1=3strongfactorsandm2semi-weakfactors. Bias(x100)RMSE(x100)Size(x100)Power(x100) m2N/T203050100200203050100200203050100200203050100200 4200.06-0.27-0.080.050.158.307.106.205.675.268.607.456.857.507.1512.3013.8515.4018.8519.80 630-0.20-0.06-0.18-0.19-0.086.685.725.164.664.356.206.607.307.157.0013.6515.5518.7021.0522.60 10500.050.090.050.14-0.035.314.664.043.633.345.556.555.856.105.5017.3522.1026.4031.8533.80 20100-0.12-0.07-0.07-0.01-0.063.773.232.912.502.465.405.056.104.955.6527.2533.5042.5551.4555.70 402000.040.070.01-0.01-0.042.682.321.911.801.725.305.754.105.005.5547.9061.2571.2579.9583.10 12200.08-0.26-0.34-0.10-0.018.197.106.525.835.286.557.258.708.957.1012.8513.9016.5019.2020.85 1830-0.11-0.110.010.05-0.096.965.945.304.654.397.306.356.556.657.5013.1015.4518.1523.4523.30 30500.080.07-0.020.070.085.264.493.953.583.385.956.055.956.006.4016.5520.8025.9530.1534.20 60100-0.070.02-0.05-0.010.033.773.262.782.562.375.905.955.055.604.8026.3534.7041.1050.6556.60 120200-0.10-0.06-0.02-0.020.002.612.331.961.781.654.705.654.454.554.6545.3558.9070.9079.1084.25 2020-0.12-0.15-0.18-0.12-0.048.377.366.415.715.447.007.207.607.857.3511.1513.8014.9517.4018.70 3030-0.270.210.050.09-0.037.006.175.234.744.486.307.106.356.707.0011.7516.9519.6022.9523.65 5050-0.030.190.100.040.015.024.513.993.653.335.456.455.607.305.2017.4522.7027.5030.9533.95 1001000.09-0.03-0.07-0.04-0.023.813.252.912.572.396.405.505.955.005.5027.6534.9041.2550.1054.10 200200-0.01-0.03-0.03-0.030.012.642.322.011.831.705.555.654.956.104.9547.4559.6070.7578.8083.55
Table7:MCresultsforMGPCestimator. ExperimentA:m1=3strongfactorsandm2weakfactors. Bias(x100)RMSE(x100)Size(x100)Power(x100) m2N/T203050100200203050100200203050100200203050100200 020-15.93-10.98-8.33-5.97-4.5521.3414.5411.188.286.7122.0024.3026.1022.4517.7512.9012.0512.108.106.85 030-9.09-6.08-4.67-3.43-2.8514.619.547.195.594.8314.4015.4515.7513.3012.408.007.057.008.708.95 050-4.63-3.01-2.48-1.86-1.599.546.204.763.773.4410.559.259.759.308.956.157.2510.0516.7520.25 0100-2.16-1.42-1.14-0.89-0.795.874.073.052.542.306.657.457.557.256.908.5518.2028.9043.4050.50 0200-1.04-0.66-0.66-0.56-0.363.992.692.121.751.565.955.106.307.006.1019.2538.1057.1575.8087.90 420-15.33-11.03-8.35-6.07-4.6321.0314.7110.978.286.7521.6024.5024.3021.8017.0513.4012.4510.857.656.60 630-8.99-5.98-4.58-3.51-2.8214.089.497.185.614.8913.6015.3515.6014.5012.356.506.956.957.0510.25 1050-4.64-3.21-2.49-1.86-1.499.556.194.733.803.4510.158.809.209.358.806.457.4011.0016.6522.30 20100-1.99-1.58-1.02-1.01-0.755.764.122.982.592.266.157.306.808.756.209.1016.5529.5541.8050.75 40200-0.98-0.81-0.65-0.44-0.354.022.832.081.691.555.856.956.506.056.0020.0037.1559.3079.9586.75 1220-15.62-11.38-8.52-6.22-4.6921.5315.1711.278.456.7622.7525.6527.2024.2017.7014.1512.5511.407.856.10 1830-9.44-5.91-4.59-3.68-2.7314.629.427.185.774.8615.3514.5515.0514.7012.557.906.706.908.2010.80 3050-4.83-3.41-2.72-2.07-1.729.636.394.933.923.439.3010.5511.5510.459.155.857.1010.4015.8018.70 60100-2.21-1.41-1.25-0.99-0.786.073.973.072.582.287.406.357.358.107.259.1518.3026.1041.8050.90 120200-1.03-0.68-0.69-0.54-0.344.082.712.151.721.546.505.456.906.205.2019.8537.6558.7077.3587.00 2020-15.69-11.09-8.36-6.14-4.8621.5014.7110.998.366.9323.3024.5025.9522.8518.1513.4512.6510.208.457.15 3030-9.19-6.19-4.78-3.56-2.8914.519.677.335.634.9115.3016.3016.7514.5512.007.557.506.807.009.90 5050-4.72-3.31-2.65-2.03-1.659.286.314.843.923.478.4510.6010.3010.009.755.106.8010.1516.4520.60 100100-1.89-1.49-1.32-0.99-0.825.994.003.172.532.358.006.708.407.257.4010.0016.1527.5041.4050.65 200200-0.96-0.69-0.63-0.50-0.384.132.782.141.711.576.656.507.505.906.0521.7039.7059.9578.3086.05
Table8:MCresultsforMGPCestimator. ExperimentB:m1=3strongfactorsandm2semi-weakfactors. Bias(x100)RMSE(x100)Size(x100)Power(x100) m2N/T203050100200203050100200203050100200203050100200 420-16.39-11.94-8.72-6.38-4.9822.0115.3511.348.486.9322.6526.2528.1024.2519.0015.0014.1511.807.806.30 630-10.27-7.17-5.43-4.28-3.4715.5010.467.796.195.2417.5517.9018.3018.9514.209.757.757.106.757.20 1050-5.35-4.11-3.33-2.61-2.209.856.865.264.243.7810.7512.3513.7013.0512.805.806.457.7012.1015.50 20100-3.17-2.43-2.06-1.70-1.516.484.563.612.882.698.3510.2012.0511.6512.256.9012.0519.2029.8038.35 40200-1.99-1.54-1.41-1.24-1.094.343.152.432.101.907.759.9510.2512.1511.6012.4026.1043.3061.4572.30 1220-18.36-13.84-11.09-8.17-6.8923.3817.1713.4910.088.4326.6533.7537.6032.7529.0516.5519.2517.7511.857.30 1830-11.76-8.69-7.16-5.79-5.0616.6611.599.187.386.4919.8522.2527.0528.0027.3010.309.658.707.356.45 3050-7.46-5.67-4.97-4.21-3.7411.338.156.585.384.8315.2519.0022.8523.7523.056.807.005.906.256.75 60100-4.82-4.02-3.59-3.12-2.777.615.634.643.903.5214.8518.1023.1526.3024.706.206.308.0511.7018.15 120200-3.61-3.17-2.93-2.65-2.415.434.283.643.172.8414.9021.8029.4534.9034.657.0011.6518.2029.7038.85 2020-20.66-15.49-12.81-10.18-8.2725.8218.6715.0211.829.6731.8037.3544.4544.0539.2021.3520.9023.0517.8511.85 3030-13.53-9.92-8.45-7.04-6.2518.0712.7610.308.427.4823.7527.4034.0035.5534.9513.2511.4010.558.956.85 5050-8.71-7.20-6.46-5.58-5.1312.349.307.816.626.0317.9525.1532.6537.9539.157.757.557.306.605.45 100100-5.88-5.39-5.02-4.54-4.128.546.755.925.194.6717.4027.6039.8045.7547.156.855.655.856.256.35 200200-4.81-4.57-4.26-3.82-3.636.575.484.834.233.9523.2038.0051.2059.4064.306.807.458.2011.7014.65
Table9:MCresultsforPPCestimator. ExperimentA:m1=3strongfactorsandm2weakfactors. Bias(x100)RMSE(x100)Size(x100)Power(x100) m2N/T203050100200203050100200203050100200203050100200 020-14.29-10.00-7.59-5.62-4.4618.3413.3910.398.006.6624.3024.3023.3021.3017.0513.3012.7510.706.856.70 030-8.16-5.54-4.32-3.32-2.9012.538.686.865.574.9516.6014.8014.2514.0513.408.756.357.158.759.00 050-4.25-2.77-2.30-1.89-1.628.105.744.593.813.4911.0010.1010.509.058.157.208.859.9515.7519.15 0100-1.93-1.28-1.08-0.90-0.814.883.722.972.562.326.807.157.107.256.7512.5520.5530.7040.7050.25 0200-0.82-0.63-0.60-0.55-0.393.322.502.071.761.597.055.355.856.406.3528.5544.4559.6074.5586.45 420-14.10-10.09-7.63-5.71-4.5918.4413.3510.267.986.7523.9023.1023.3019.5017.0513.8012.309.806.806.75 630-8.15-5.57-4.26-3.44-2.8412.138.766.915.614.9316.2516.0514.7514.2012.406.757.707.657.1010.00 1050-4.16-2.92-2.34-1.85-1.558.075.724.563.833.5110.609.558.858.809.307.208.0010.9016.5522.10 20100-1.86-1.44-0.93-0.98-0.784.823.782.902.602.286.757.155.907.855.9511.1519.5531.9040.6050.20 40200-0.83-0.70-0.60-0.44-0.373.232.552.051.701.586.106.756.006.055.8527.3042.7060.7578.8085.55 1220-14.05-10.13-7.63-5.79-4.5218.6313.5710.448.126.6524.4024.1524.7022.5516.6013.9012.0010.857.656.35 1830-8.60-5.52-4.30-3.60-2.7712.488.686.955.764.8916.2514.9515.1514.9511.807.507.207.108.059.70 3050-4.29-3.12-2.50-2.04-1.767.985.844.733.923.4710.409.6510.0510.359.055.858.1511.0514.9518.10 60100-1.93-1.25-1.17-0.99-0.824.973.683.032.592.327.306.756.957.956.9012.0019.6528.1040.7549.10 120200-0.90-0.61-0.64-0.53-0.363.252.442.111.741.566.004.857.455.905.8026.4543.2061.1076.8585.75 2020-13.80-10.15-7.64-5.81-4.7118.1413.4610.278.146.8722.9025.4523.5021.5518.2513.4012.108.957.306.60 3030-8.54-5.59-4.48-3.47-2.8912.448.887.035.614.9616.8016.2515.6514.1511.858.307.257.007.1010.25 5050-4.09-2.90-2.46-2.00-1.687.705.834.693.913.509.059.559.959.709.205.509.3011.0515.8020.25 100100-1.68-1.33-1.22-0.97-0.864.843.693.142.552.387.707.007.806.758.1012.3019.7028.3039.5048.90 200200-0.80-0.59-0.58-0.49-0.403.242.522.081.731.595.906.456.555.905.7027.6045.1560.8077.3083.95
Table10:MCresultsforPPCestimator. ExperimentB:m1=3strongfactorsandm2semi-weakfactors. Bias(x100)RMSE(x100)Size(x100)Power(x100) m2N/T203050100200203050100200203050100200203050100200 420-14.67-10.83-8.13-6.12-4.9318.9313.9910.778.316.9524.8526.4525.4523.5018.5015.0012.5511.257.306.25 630-9.44-6.54-5.16-4.20-3.5113.239.557.516.135.3119.2518.3018.2018.1014.108.907.507.006.007.40 1050-4.85-3.62-3.08-2.55-2.248.306.255.034.263.8511.8512.2011.7013.1013.206.006.807.7512.2015.60 20100-2.57-2.19-1.92-1.67-1.515.304.093.512.892.719.309.6011.5510.2511.609.9013.8521.6529.0037.25 40200-1.52-1.32-1.27-1.22-1.083.502.822.342.101.917.658.159.2512.0511.3020.0033.1547.5560.9072.50 1220-16.80-12.67-10.18-7.81-6.7520.8015.5912.529.748.3130.2533.3034.6030.4028.0518.8516.7014.9010.456.80 1830-10.55-7.89-6.70-5.59-5.0414.2410.648.797.236.4922.2023.0525.7526.1526.6511.159.008.706.556.15 3050-6.39-5.06-4.60-4.07-3.779.557.266.225.284.8717.2017.6021.6521.8523.456.855.906.106.256.75 60100-3.86-3.43-3.30-3.01-2.756.064.964.333.853.5213.5516.7020.2025.6523.905.707.908.3512.2017.90 120200-2.79-2.69-2.61-2.56-2.394.283.743.363.092.8414.3519.5526.0032.0532.9512.3016.3023.2529.9538.50 2020-18.49-14.19-11.79-9.70-8.1622.4017.0413.8811.369.5833.1537.1040.8542.4037.6520.5020.3519.5016.1511.50 3030-12.15-8.99-7.79-6.82-6.2215.4611.519.658.237.4926.2527.2530.6032.8034.7012.8510.359.358.607.45 5050-7.35-6.24-5.95-5.45-5.1610.138.257.336.556.0719.5023.9529.6535.1538.856.607.107.106.855.45 100100-4.74-4.55-4.50-4.40-4.136.815.885.425.074.6917.8525.2033.6543.8546.656.355.755.756.306.80 200200-3.61-3.73-3.79-3.68-3.615.004.644.374.113.9521.2030.6044.6555.9063.058.059.7510.7513.4014.95
0%
20%
40%
60%
80%
100%
0.9 0.95 1 1.05 1.1
0 20 40
60 80 100
3 strong factors and varying number of weak factors
0%
20%
40%
60%
80%
100%
0.9 0.95 1 1.05 1.1
20 40 60
80 100
3 strong factors and varying number of semi-weak factors
Figure 1: Power curves for the CCEPt-tests in experiments withN = 100; T = 100; 3strong factors, and a varying numberm2 of weak factors (left chart) and semi-weak factors (right chart).
Table 11: MC results for Bai estimator.12
Experiment A and B: : m1= 3strong factors and m2 weak or semi-weak factors.
Bias (x100) RMSE (x100) Size (x100) Power (x100)
m2 N/T 20 100 20 100 20 100 20 100
Weak factor structure f 0igtg
0 20 0.47 -0.30 9.78 5.72 37.90 48.00 45.60 61.40
0 100 -0.01 0.02 3.57 2.50 21.50 47.20 58.70 91.10
4 20 0.62 -0.15 9.80 5.83 40.10 50.50 48.30 63.20
20 100 0.07 -0.09 3.48 2.47 21.40 44.90 56.20 91.50
20 20 0.30 0.09 9.91 6.07 37.90 52.40 46.50 64.20
100 100 0.10 0.03 3.47 2.42 21.10 45.30 59.80 91.90
Semi-weak factor structure f 0igtg
4 20 0.45 -0.23 9.40 6.08 35.50 52.10 42.70 65.10
20 100 -0.09 -0.17 3.70 2.60 23.60 46.80 58.30 88.70
20 20 1.28 -0.28 10.47 6.27 41.70 52.40 49.40 60.50
100 100 0.02 0.03 3.50 2.46 20.90 44.50 56.20 90.20
1 2Based on R = 1000 replications.
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Appendix
convergence in L1 norm. We now provide some lemmas useful for proving Theorem 2.Lemma 1 Consider the panel data model (37) and (39) and suppose that Assumptions 8-12 hold, and m1 does not vary with N . Then as m2; T; N! 1, such that Nj Pm2
(a) Consider now the following two-dimensional vector array f N t; Ftg1t= 1 1
Consider now
N =1is L1-mixingale with respect to the constant array fcN tg.
Equation (58) established that f N t=cN tg is uniformly bounded in L2norm, which implies uniform integrability.1 3 Finally, note that the constant array fcN tg satisfy the following conditions
lim
N =1satis…es conditions of a mixingale weak law (Davidson, 1994, Theorem 19.11)., which impliesPTN
t=1 N t
`< K < 1. This completes the proof of result (49). Results (50)-(51) can be proved in the same way.1 4 Remaining results are proved below using the similar logical arguments.
(b) Next we establish result (52). Let
N t= pN
TN itvt, (59)
and as before consider the two-dimensional vector array f N t; Ftg1t= 1 1
N =1 is L1-mixingale with respect to constant array fcN tg, and a mixingale weak law (Davidson, 1994, Theorem 19.11) implyPTN
t=1 N t
` < K < 1, which concludes the proof of result (52). Proof of result (53) is identical to the proof of result (52), but this time we set N t=pTN
N itet. (c) Next we establish results (54) and (55) in a similar way. De…ne
N t= N
1 3Su¢ cient condition for uniform integrability is L1+" uniform boundedness for any " > 0.
1 4De…ne N t= pTN teitto prove result (50) and N t=pTN tft to prove result (51)
Using the same arguments as before, f N t; Ftg1t= 1 1
N =1is L1-mixingale with respect to constant array fcN tg, and a mixingale weak law (Davidson, 1994, Theorem 19.11) establishes result (54). Result (55) easily follows by noting that V ar (et)and kV ar (vt)k are both of order O N 1 .
(d) Next we prove equation (56). Set
N t= 1
N =1is L1-mixingale with respect to constant array fcN tg, and a mixingale weak law (Davidson, 1994, Theorem 19.11) implyPTN
t=1 N t
`< K < 1. This completes the proof of result (56).
(e) To establish result (57), de…ne
N t= 1
N =1is L1-mixingale with respect to constant array fcN tg, and a mixingale weak law (Davidson, 1994, Theorem 19.11) imply result (57).
The following lemma collects several results presented in Pesaran (2006), Kapetanios Pesaran and Yamagata (2009), and Pesaran and Tosetti (2009).
Lemma 2 Consider the panel data model (37) and (39) and suppose that Assumptions 8-12 hold, and m1 does not vary with N . Then as T; N! 1 (at any rate) we have:j
pNVi0V
Proof. Lemma 2 follows directly from Pesaran (2006), Kapetanios Pesaran and Yamagata (2009), and Pesaran and Tosetti (2009). These results can also be established in the same way as Lemma 1 by using a mixingale weak law.
Lemma 3 Consider the panel data model (37) and (39) and suppose that Assumptions 8-12 hold, and m1 does not vary with N . Then as m2; T; N! 1, such that Nj Pm2
Proof. Throughout this proof we consider asymptotics m2; T; N! 1, such that Nj Pm2
` ` i< K < 1. We start by
We examine each of the three terms below. We have pN X0i H Q
T =
pN (G i+ Vi)0U
T .
Equation (49) of Lemma 1 and equation (62) of Lemma 2 establish pNV0iU
T
L1
! 0: (70)
In addition, equation (51) of Lemma 1 and equation (61) of Lemma 2 establish pNG0U
T
L1
! 0: (71)
Equations (70) and (71) imply
pN X0i H Q T
L1
! 0, (72)
and noting that HT0H 1= Op(1), and HT0 i = Op(1), establish
Now we focus on the second term of (69). Equations (52),(53) and (57) of Lemma 1 imply pN U0 i
In order to establish the last term of (69), we write pN
Equations (54),(55) and (56) of Lemma 1 and equation (63) of Lemma 2 imply pN U0U
T
L1
! 0.
Similarly, equation (51) of Lemma 1 and equation (61) of Lemma 2 imply pN Q0U
This completes the proof of result (65).
In order to establish result (66), note that MgF= 0and therefore (66) is equivalent with the following statement, pNX0iMF
Using similar steps as in deriving equation (69), we have:
Since HT0F = Op(1), convergence of the …rst and the last term of (75) to zero directly follows from earlier results, in particular equations (72) and (74). Furthermore, equation (73) implies
pN Q0 H0 F which completes the proof of result (66).
Result (67) is established next in a similar fashion. Consider pN
Using equations (72) and (74), and noting that the remaining elements are bounded, we have pN
T X0iMXi X0iMgXi L1 ! 0, which completes the proof of result (67).
Finally, consider
Equation (50) of Lemma 1 and equation (64) of Lemma 2 imply pN Q0 H0 ei
Equations (72), (74) and (78) imply pTN X0iMei X0iMgei L1! 0, which completes the proof of result (68).
Proof of Theorem 2. We prove Theorem 2 in two parts. First, we establish result (46) for the CCE pooled estimation and in the second part we establish result (45) for the CCE mean group estimation.
Let it=Pm2 which is not present in previous studies by Pesaran and Tosetti (2009), Kapetanios Pesaran and Yamagata (2009), and Pesaran (2006). Equation (65) of Lemma 3 implies
p1 and such that Assumption 9.b holds, namely limN!1Pm2;N
`=1 2
i` < K. Consider now the following two-dimensional vector array f N t; Ftg1t= 1
and fFtg denotes an increasing sequence of -…elds (Ft 1 Ft)such that Ftincludes all information available at time tand N tis measurable with respect to Ftfor any N 2 N. Let fcN tg1t= 1
1
N =1be two-dimensional array of constants and set cN t=T1
where the existence of uniform upper bound K2, which does not depend on i; N is assumed in Assumption 9. It follows that
E N t
0N t
c2N t < K < 1, (82)
where the constant K = K1K2 and it does not depend on N . Consider now
N =1is L1-mixingale with respect to the constant array fcN tg. Equation (82) established that f N t=cN tg is uniformly bounded in L2 norm, which implies uniform integrability.1 5 Finally, note that the constant array fcN tg satis…es the following conditions
N =1satis…es conditions of a mixingale weak law (Davidson, 1994, Theorem 19.11)., which establishPTN
t=1 N t
` ` i< K < 1. Convergence results for the remaining terms on the right side of equation (79) can be established in the same way as in Pesaran and Tosetti (2009) or Pesaran (2006). In particular, results (66)-(68) of Lemma 3 imply
1
Next we establish result (45) for the CCE mean group estimation. Let again it=Pm2
`=1 i`g`tand consider
1 5Su¢ cient condition for uniform integrability is L1+" uniform boundedness for any " > 0.
where biT = T 1X0iMXi. Compared to Pesaran (2006), and Pesaran and Tosetti (2009), equation (84) has the extra term p1
N
PN
i=1biT1 X0iMT i , not encountered pepreviously. We focus on this new term …rst. Lemma 3 implies p1
Using the same method as in the …rst part of the proof, we de…ne two-dimensional vector array f N t; Ftg1t= 1 1
which is identical to (81) except that wit is used instead of evit. Following the same steps as in the …rst part of this proof, we have that f N t; Ftg1t= 1
1
N =1is L1-mixingale with respect to constant array fcN tg, and a mixingale weak law (Davidson, 1994, Theorem 19.11) establishesPTN
t=1 N t
` ` i < K < 1. Convergence results for remaining terms on the right side of equation (84) can be established in the same way as in Pesaran and Tosetti (2009) or Pesaran (2006). In particular, we have