Robust Estimation of Large Panels with Factor Structures

Figures & data

Table 1 DGP1 with heterogeneous AR(1) idiosyncratic error ${ϵ}_{it}$

	absolute bias (per cent)					rmse (per cent)					10%	5%	1%
(N, T)	FWLS	WLS	GLS	OLS	CCE	FWLS	WLS	GLS	OLS	CCE	FWLS
(60,30)	18.17	11.13	11.23	47.71	14.73	21.86	13.35	13.49	50.54	17.71	0.103	0.056	0.015
(200,30)	12.77	11.16	11.27	44.75	14.15	15.62	13.68	13.82	48.00	17.29	0.112	0.062	0.018
(200,100)	9.14	5.82	5.84	42.18	8.01	11.13	7.08	7.10	42.67	9.65	0.103	0.054	0.012
(600,30)	11.71	11.11	11.24	45.30	14.04	14.23	13.53	13.69	49.16	17.25	0.116	0.065	0.019
(600,100)	6.76	5.80	5.82	43.13	7.86	8.16	7.00	7.02	44.24	9.56	0.106	0.055	0.013
(600,300)	5.18	3.30	3.30	42.48	4.63	6.25	3.98	3.98	42.36	5.61	0.103	0.053	0.011

NOTE: Columns 2–6 report the absolute bias (in percentage: $100\times \text{abs}({\tilde{\beta }}_{i0}/{\beta }_{i0}-1)$) of the FWLS (iterated), WLS, GLS, OLS, and CCE estimators of ${\beta }_{i0}$ of model (5.1), columns 7–11 report the root mean square error (in percentage) for the same estimators, and columns 12–14 report the size of the $t-\mathit{ratios}$ for the FWLS (iterated) estimator, corresponding to a nominal size of 10%, 5%, and 1%, respectively. We report averages, across 1000 Monte Carlo iterations and across the N units, for all the above quantities, that is $N^{-1}{1000}^{-1}{\sum }_{i=1}^N{\sum }_{m=1}^{1000}{\Delta }_{m,i}$ where ${\Delta }_{m,i}$ refers to any of the above quantities (absolute bias, rmse, and $t-\mathit{ratios}$ sizes) for the ith unit and mth Monte Carlo iteration for a given (N, T) and a given estimator. The true ${\beta }_{i0}$ are generated as NID$(1,0.04)$.

Table 2 DGP2 with heterogeneous AR(1) idiosyncratic error ${ϵ}_{it}$

	absolute bias (per cent)					rmse (per cent)					10%	5%	1%
(N, T)	FWLS	WLS	GLS	OLS	CCE	FWLS	WLS	GLS	OLS	CCE	FWLS
(60,30)	19.28	11.93	11.93	68.88	21.74	22.03	13.50	13.49	65.70	24.13	0.102	0.055	0.015
(200,30)	13.59	11.66	11.54	66.54	20.97	16.26	13.93	13.82	65.07	24.06	0.124	0.07	0.021
(200,100)	9.34	5.99	5.97	67.53	17.17	11.13	7.12	7.09	64.64	18.50	0.103	0.053	0.012
(600,30)	12.35	11.33	11.23	65.31	20.81	14.99	13.79	13.70	65.20	24.11	0.133	0.078	0.024
(600,100)	6.85	5.83	5.81	66.74	17.18	8.28	7.05	7.02	65.26	18.56	0.11	0.057	0.014
(600,300)	5.17	3.30	3.30	66.87	15.82	6.25	3.99	3.98	64.95	16.23	0.103	0.053	0.011

NOTE: See the note to Table 1.

Table 3 DGP3 with heterogeneous AR(1) idiosyncratic error ${ϵ}_{it}$.

	absolute bias (per cent)					rmse (per cent)					10%	5%	1%
(N, T)	FWLS	WLS	GLS	OLS	CCE	FWLS	WLS	GLS	OLS	CCE	FWLS
(60,30)	17.41	11.11	11.56	46.44	21.85	21.20	13.49	14.06	50.56	24.80	0.113	0.063	0.018
(200,30)	12.50	11.31	11.65	47.90	20.11	15.29	13.92	14.38	52.39	23.23	0.131	0.074	0.022
(200,100)	9.03	5.83	5.87	44.02	17.00	11.05	7.13	7.18	45.22	18.32	0.106	0.055	0.013
(600,30)	11.53	11.44	11.74	49.86	20.65	13.95	13.87	14.27	53.81	23.73	0.128	0.073	0.022
(600,100)	6.74	5.86	5.89	47.14	17.79	8.11	7.05	7.09	47.97	19.02	0.109	0.058	0.014
(600,300)	5.18	3.32	3.32	46.08	16.88	6.23	3.99	3.99	45.49	17.13	0.104	0.053	0.011

NOTE: See the note to Table 1.

Table 4 DGP4 with heterogeneous AR(1) idiosyncratic error ${ϵ}_{it}$.

	absolute bias (per cent)					rmse (per cent)					10%	5%	1%
(N, T)	FWLS	WLS	GLS	OLS	CCE	FWLS	WLS	GLS	OLS	CCE	FWLS
(60,30)	17.77	10.52	11.23	21.94	17.04	21.32	12.59	13.49	24.33	20.01	0.107	0.058	0.016
(200,30)	12.42	10.53	11.27	21.05	20.05	15.16	12.82	13.82	23.80	23.01	0.118	0.066	0.019
(200,100)	9.08	5.70	5.84	18.36	16.42	11.04	6.92	7.10	19.39	17.60	0.104	0.054	0.013
(600,30)	11.46	10.49	11.24	20.92	18.35	13.87	12.70	13.69	23.74	21.52	0.128	0.074	0.023
(600,100)	6.72	5.69	5.82	18.19	14.45	8.11	6.86	7.02	19.32	15.89	0.111	0.059	0.014
(600,300)	5.17	3.28	3.30	17.26	13.06	6.24	3.95	3.98	17.45	13.56	0.104	0.054	0.012

NOTE: See the note to Table 1.

Fig. 1 The left panel presents the cross-section across N = 151 countries of the p-values for the Augmented Dickey–Fuller test for log GDP, in level (black line) and in first-difference (red line). The blue line indicate the 5% level. The right panel presents the largest eigenvalues of the sample covariance matrix for the panel of log GDP (black line) and for the panel of log investment (red line), normalized by setting the maximum eigenvalue equal to unity.

Fig. 2 The top three panels present the estimated cross-sectional density, based on the nonparameteric Kernel estimator, of the growth effect (${\theta }_{i1}$) corresponding to all countries (N = 151—left panel), OECD countries (N = 31—middle panel), and Non-OECD countries (N = 120—right panel), and the bottom three panels present the estimated cross-sectional density, based on the nonparameteric Kernel estimator, of the level effect (${\beta }_{i1}+{\beta }_{i2}$) corresponding to all countries (N = 151—left panel), OECD countries (N = 31—middle panel), and Non-OECD countries (N = 120—right panel).

Table 5 Capital accumulation and GDP growth.

	ALL			OECD			Non-OECD
	$\stackrel{25\%\mathrm{ }\mathit{perc}}{(t¯\mathit{ratio})}$	$\stackrel{50\%\mathrm{ }\mathit{perc}}{(t¯\mathit{ratio})}$	$\stackrel{75\%\mathrm{ }\mathit{perc}}{(t¯\mathit{ratio})}$	$\stackrel{25\%\mathrm{ }\mathit{perc}}{(t¯\mathit{ratio})}$	$\stackrel{50\%\mathrm{ }\mathit{perc}}{(t¯\mathit{ratio})}$	$\stackrel{75\%\mathrm{ }\mathit{perc}}{(t¯\mathit{ratio})}$	$\stackrel{25\%\mathrm{ }\mathit{perc}}{(t¯\mathit{ratio})}$	$\stackrel{50\%\mathrm{ }\mathit{perc}}{(t¯\mathit{ratio})}$	$\stackrel{75\%\mathrm{ }\mathit{perc}}{(t¯\mathit{ratio})}$
	FWLS
$\stackrel{{\tilde{\theta }}_{1i}}{(\text{growth}\hspace{0.17em}\text{effect})}$	$\stackrel{¯0.036}{(¯4.293)}$	$\stackrel{0.024}{(2.485)}$	$\stackrel{0.079}{(13.267)}$	$\stackrel{0.083}{(3.416)}$	$\stackrel{0.185}{(3.073)}$	$\stackrel{0.271}{(12.220)}$	$\stackrel{¯0.042}{(¯6.783)}$	$\stackrel{0.009}{(0.893)}$	$\stackrel{0.068}{(11.061)}$
$\stackrel{({\tilde{\beta }}_{1i}+{\tilde{\beta }}_{2i})}{(\text{level}\hspace{0.17em}\text{effect})}$	$\stackrel{¯0.071}{(¯7.844)}$	$\stackrel{0.022}{(1.618)}$	$\stackrel{0.105}{(11.221)}$	$\stackrel{¯0.186}{(¯7.216)}$	$\stackrel{¯0.049}{(¯1.240)}$	$\stackrel{0.008}{(0.373)}$	$\stackrel{¯0.065}{(¯5.936)}$	$\stackrel{0.033}{(1.998)}$	$\stackrel{0.127}{(11.282)}$
	OLS
$\stackrel{{\tilde{\theta }}_{1i}}{(\text{growth}\hspace{0.17em}\text{effect})}$	$\stackrel{¯0.023}{(¯4.373)}$	$\stackrel{0.029}{(3.584)}$	$\stackrel{0.075}{(14.415)}$	$\stackrel{¯0.038}{(¯1.779)}$	$\stackrel{0.029}{(0.743)}$	$\stackrel{0.215}{(5.166)}$	$\stackrel{¯0.025}{(¯4.227)}$	$\stackrel{0.028}{(3.316)}$	$\stackrel{0.067}{(14.125)}$
$\stackrel{({\tilde{\beta }}_{1i}+{\tilde{\beta }}_{2i})}{(\text{level}\hspace{0.17em}\text{effect})}$	$\stackrel{¯0.086}{(¯7.065)}$	$\stackrel{0.029}{(1.276)}$	$\stackrel{0.194}{(12.708)}$v	$\stackrel{¯0.017}{(¯0.445)}$	$\stackrel{0.147}{(2.643)}$	$\stackrel{0.261}{(8.394)}$	$\stackrel{¯0.097}{(¯7.602)}$	$\stackrel{0.011}{(0.477)}$	$\stackrel{0.151}{(9.824}$
	CCE
$\stackrel{{\tilde{\theta }}_{1i}}{(\text{growth}\hspace{0.17em}\text{effect})}$	$\stackrel{¯0.031}{(¯6.950)}$	$\stackrel{0.006}{(0.840)}$	$\stackrel{0.055}{(9.563)}$	$\stackrel{¯0.036}{(¯2.375)}$	$\stackrel{0.036}{(1.693)}$	$\stackrel{0.078}{(6.171)}$	$\stackrel{¯0.025}{(¯5.414)}$	$\stackrel{0.011}{(1.439)}$	$\stackrel{0.050}{(9.254)}$
$\stackrel{({\tilde{\beta }}_{1i}+{\tilde{\beta }}_{2i})}{(\text{level}\hspace{0.17em}\text{effect})}$	$\stackrel{¯0.085}{(¯6.706)}$	$\stackrel{0.041}{(1.950)}$	$\stackrel{0.173}{(12.318)}$	$\stackrel{0.124}{(5.611)}$	$\stackrel{0.228}{(6.053)}$	$\stackrel{0.325}{(13.142)}$	$\stackrel{¯0.112}{(¯8.565)}$	$\stackrel{¯0.011}{(¯0.487)}$	$\stackrel{0.131}{(8.112)}$

NOTE: This table is made by three panels, corresponding to the FWLS estimates (top panel), the OLS estimates (middle panel), and the CCE estimate (bottom panel). Each panel presents the empirical quantiles (25%, median, and 75%) of the cross-sectional empirical distribution of the estimated parameters (corresponding $t-\mathit{ratio}$ in brackets) of model (5.4), for the full sample (N = 151, columns two-four), for the OECD countries (N = 31, columns five–seven), and for the Non-OECD countries (N = 120, columns eight–ten). The growth effect parameter is defined as ${\theta }_{2i}$ and the level effect parameter is defined as $({\beta }_{1i}+{\beta }_{2i})$.