A Robust Approach to Heteroscedasticity, Error Serial Correlation and Slope Heterogeneity in Linear Models with Interactive Effects for Large Panel Data

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Table 1 Experimental designs.

Design	Model	r	${\sigma }_{\eta }$	${\rho }_{x\eta }$	${\psi }_{\mathcal{l}i}^{(\mathcal{P})}$
1	Homogeneous slopes	2	0	0	–
2	Heterogeneous slopes uncorrelated with $x_{\mathcal{l}it}$	3	0.50	0	–
3	Heterogeneous slopes correlated with $x_{\mathcal{l}it}$	3	0.50	0.50	${\psi }_{\mathcal{l}i}^{(1,2,3,4)}$
4	Heterogeneous slopes correlated with $x_{\mathcal{l}it}$	3	0.50	0.50	${\psi }_{\mathcal{l}i}^{(1)}$
5	Heterogeneous slopes correlated with $x_{\mathcal{l}it}$	3	0.50	0.50	${\psi }_{\mathcal{l}i}^{(2)}$
6	Heterogeneous slopes correlated with $x_{\ell it}$	3	0.50	0.50	${\psi }_{\mathcal{l}i}^{(3)}$
7	Heterogeneous slopes correlated with $x_{\mathcal{l}it}$	3	0.50	0.50	${\psi }_{\mathcal{l}i}^{(4)}$

NOTE: r is the number of factors in the error term. ${\sigma }_{\eta }$ is standard deviation of ${\beta }_{\mathcal{l}i}-{\beta }_{\mathcal{l}}^0(={\sigma }_{\eta }{\eta }_{\mathcal{l}i})$, and ${\rho }_{x\eta }$ is the correlation between ${\eta }_{\mathcal{l}i}$ and a function of the defactored regressor ${\psi }_{\mathcal{l}i}^{(\mathcal{P})}$, where ${\psi }_{\mathcal{l}i}^{(1,2,3,4)}={\sum }_{p=1}^4{\tilde{z}}_{\mathcal{l}i,p}/\sqrt{4},{\psi }_{\mathcal{l}i}^{(p)}={\tilde{z}}_{\mathcal{l}i,p}$ for $p=1,\dots ,4$, and ${\tilde{z}}_{\mathcal{l}i,p}$ is the cross-sectionally standardized within average of the defactored regressor, $T^{-1}{\sum }_{t=1}^T{v}_{\mathcal{l}it}^p$, for $\mathcal{l}=1,\dots ,k$. In the article we report the results of designs 1–3, with the rest in the supplementary materials.

Table 2 Frequency of the estimated number of factors in the error term by the ER and GR methods.

Design 1: homogeneous slopes (r = 2)							Design 2: random coefficients (r = 3)
Freq	ER			GR			ER			GR
T / N	50	100	200	50	100	200	50	100	200	50	100	200
$\widehat{r}=1$
25	1.0	0.3	0.2	3.6	1.5	1.0	3.2	1.4	0.7	13.5	7.9	5.0
50	0.0	0.0	0.0	0.2	0.0	0.0	0.4	0.0	0.0	3.1	0.6	0.1
100	0.0	0.0	0.0	0.0	0.0	0.0	0.1	0.0	0.0	0.6	0.1	0.0
200	0.0	0.0	0.0	0.0	0.0	0.0	0.1	0.0	0.0	0.3	0.0	0.0
$\widehat{r}=2$
25	98.9	99.7	99.8	96.5	98.5	99.1	16.6	10.3	6.7	23.4	17.9	14.4
50	100.0	100.0	100.0	99.9	100.0	100.0	6.5	1.4	0.5	11.7	4.3	1.7
100	100.0	100.0	100.0	100.0	100.0	100.0	2.3	0.1	0.0	6.4	0.6	0.0
200	100.0	100.0	100.0	100.0	100.0	100.0	1.7	0.0	0.0	4.2	0.1	0.0
$\widehat{r}=3$
25	0.1	0.0	0.0	0.0	0.0	0.0	80.1	88.3	92.7	63.1	74.3	80.7
50	0.0	0.0	0.0	0.0	0.0	0.0	93.2	98.7	99.6	85.2	95.2	98.3
100	0.0	0.0	0.0	0.0	0.0	0.0	97.7	100.0	100.0	93.1	99.4	100.0
200	0.0	0.0	0.0	0.0	0.0	0.0	98.3	100.0	100.0	95.5	99.9	100.0
$\widehat{r}=4$
25	0.0	0.0	0.0	0.0	0.0	0.0	0.2	0.0	0.0	0.0	0.0	0.0
50	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
100	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
200	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0

NOTE: We set the maximum number of factors to six to estimate r (and $r_1+r_2$). The frequencies of $\widehat{r}=5$ and 6 were zero. ER and GR stands for the Eigenvalue Ratio and Growth Ratio methods proposed by Ahn and Horenstein (2013). See Table 1 for a summary of experimental designs.

Table 3 Finite sample properties of the IPC estimators of ${\beta }_1^0$ and the associated robust Wald test with interactive effects, heteroscedastic and serially correlated idiosyncratic errors.

Design 1: homogeneous slopes $({\sigma }_{\eta =0},{\rho }_{x\eta }=0)$
	Bias($\times 100$)			SD($\times 100$)			RMSE($\times 100$)			Size(%)
T / N	50	100	200	50	100	200	50	100	200	50	100	200
IPC(ER)
25	–3.599	–1.674	–0.739	6.556	4.418	3.074	7.478	4.725	3.161	49.4	44.8	39.0
50	–4.069	–1.978	–0.992	4.485	2.878	1.974	6.056	3.492	2.209	57.5	46.6	39.6
100	–4.315	–2.085	–1.034	3.122	1.887	1.250	5.326	2.812	1.622	71.6	54.4	41.2
200	–4.721	–2.274	–1.113	2.323	1.424	0.996	5.262	2.683	1.494	87.9	68.5	51.7
BC-IPC(ER)
25	0.228	0.160	0.099	3.042	2.083	1.441	3.050	2.089	1.445	8.2	7.6	6.0
50	0.047	0.033	0.016	1.958	1.367	0.939	1.959	1.368	0.939	7.4	6.9	5.6
100	–0.003	–0.006	0.008	1.497	1.010	0.710	1.497	1.010	0.710	7.7	6.7	5.7
200	0.006	0.000	0.001	1.218	0.825	0.573	1.218	0.825	0.573	7.0	5.9	5.4
JK-IPC(ER)
25	0.096	0.093	0.038	3.506	2.274	1.580	3.507	2.276	1.580	11.8	11.0	9.2
50	0.015	0.007	0.002	2.156	1.480	1.005	2.156	1.480	1.005	9.3	9.6	7.6
100	–0.033	–0.014	0.004	1.590	1.061	0.746	1.590	1.061	0.746	9.4	7.3	6.4
200	–0.022	–0.004	–0.002	1.292	0.856	0.590	1.292	0.856	0.590	9.5	7.2	5.9
Design 2: heterogeneous slopes uncorrelated with $x_{\mathcal{l}it}({\sigma }_{\eta }=0.5,{\rho }_{x\eta }=0)$
	Bias($\times 100$)			SD($\times 100$)			RMSE($\times 100$)			Size(%)
T / N	50	100	200	50	100	200	50	100	200	50	100	200
IPC(ER)
25	–4.294	–2.282	–0.963	10.680	7.490	5.247	11.511	7.830	5.334	16.4	12.3	9.7
50	–4.693	–2.506	–1.143	9.069	6.257	4.322	10.212	6.741	4.471	17.6	11.7	8.7
100	–4.901	–2.611	–1.205	8.114	5.763	3.984	9.479	6.327	4.162	19.2	12.7	8.4
200	–5.371	–2.770	–1.311	7.666	5.487	3.824	9.360	6.147	4.042	20.4	13.1	7.9
BC-IPC(ER)
25	0.163	0.099	0.155	10.455	7.240	4.939	10.457	7.241	4.942	12.6	9.3	6.2
50	–0.202	–0.174	–0.017	9.073	6.150	4.219	9.076	6.153	4.219	12.8	8.6	7.1
100	–0.233	–0.197	–0.049	8.328	5.742	4.004	8.331	5.746	4.005	12.9	8.7	7.3
200	–0.263	–0.171	–0.050	7.933	5.555	3.860	7.937	5.557	3.861	12.8	9.1	6.5
JK-IPC(ER)
25	–0.414	–0.199	–0.077	12.167	7.994	5.314	12.174	7.997	5.315	18.0	11.9	9.0
50	–0.184	–0.204	–0.082	9.763	6.342	4.373	9.765	6.345	4.373	15.1	9.5	8.1
100	–0.100	–0.121	–0.060	8.726	5.865	4.062	8.726	5.867	4.062	14.5	9.7	7.4
200	–0.093	–0.083	–0.027	8.145	5.617	3.874	8.145	5.617	3.874	13.9	9.0	6.4
Design 3: heterogeneous slopes correlated with $x_{\mathcal{l}it}({\sigma }_{\eta }=0.5,{\rho }_{x\eta }=0.5,{\psi }_{\mathcal{l}i}^{(1,2,3,4)})$
	Bias($\times 100$)			SD($\times 100$)			RMSE($\times 100$)			Size(%)
T / N	50	100	200	50	100	200	50	100	200	50	100	200
IPC(ER)
25	1.070	3.570	4.964	11.075	7.527	5.137	11.127	8.331	7.144	15.3	12.8	18.6
50	1.066	3.746	5.217	9.357	6.377	4.164	9.418	7.396	6.675	13.5	13.3	22.6
100	0.882	3.735	5.256	8.228	5.596	3.568	8.275	6.728	6.353	11.5	12.5	23.1
200	0.778	3.876	5.493	7.879	5.248	3.320	7.917	6.524	6.418	11.7	13.5	26.9
BC-IPC(ER)
25	4.567	5.160	5.592	10.452	7.129	4.913	11.406	8.801	7.444	15.3	16.3	22.0
50	5.065	5.823	6.250	9.064	6.149	3.964	10.383	8.469	7.401	16.9	19.6	29.6
100	5.221	5.956	6.356	8.318	5.555	3.565	9.821	8.145	7.287	18.5	20.2	34.4
200	5.540	6.291	6.662	8.050	5.322	3.368	9.772	8.240	7.465	21.4	23.7	40.0
JK-IPC(ER)
25	4.798	5.451	5.789	12.302	8.062	5.462	13.205	9.732	7.959	22.3	20.5	26.0
50	5.937	6.444	6.572	9.981	6.522	4.141	11.613	9.168	7.768	21.7	24.1	33.7
100	6.118	6.548	6.656	8.793	5.734	3.634	10.712	8.704	7.583	23.4	25.1	38.2
200	6.560	6.909	6.981	8.521	5.404	3.398	10.754	8.772	7.764	26.4	27.7	43.3

Table 4 Size and power of the ${\text{LM}}_{\text{CRC}}^{(1)}$ and ${\text{LM}}_{\text{CRC}}^{(2)}$ tests of correlated random coefficient.

	Size (Design 1)			Size (Design 2)			Power (Design 3)
	${\sigma }_{\eta }=0,{\rho }_{x\eta }=0$			${\sigma }_{\eta }=0.5,{\rho }_{x\eta }=0$			${\sigma }_{\eta }=0.5.{\rho }_{x\eta }=0.5$
T,N	50	100	200	50	100	200	50	100	200
${\text{LM}}_{\text{CRC}}^{(1)}$
25	2.6	2.6	3.1	2.7	2.4	2.4	5.9	7.1	9.7
50	3.4	3.3	2.3	2.5	3.0	3.3	8.9	12.0	16.8
100	3.3	2.0	2.6	2.7	2.3	2.3	12.1	16.9	28.8
200	3.4	2.9	2.3	2.7	3.0	2.4	18.8	32.2	51.4
${\text{LM}}_{\text{CRC}}^{(2)}$
25	2.9	3.5	3.9	3.4	3.7	3.6	9.9	18.3	34.2
50	2.9	2.7	3.2	4.4	3.9	3.8	22.7	48.8	77.8
100	2.9	2.8	3.4	4.4	3.2	3.0	45.3	83.2	97.2
200	2.9	3.0	2.9	4.5	3.7	3.7	68.4	97.7	100.0

NOTE: For Designs 1–3, see Table 1. The ${\text{LM}}_{\text{CRC}}^{(g)}$ is the proposed LM test of correlated random effects defined by (20), based on the BC-IPC estimator. The test statistics are referred to the 95% quantile of ${\chi }_g^2$ distribution. All results are based on 2000 replications. The number of factors is estimated by the ER method.

Table 5 Estimation results for the Feldstein–Horioka puzzle based on OECD data covering 1968–1996, (N = 24, T = 29).

Estimated number of factors
$\widehat{r}$	${\widehat{r}}_{,ER}=1$	${\widehat{r}}_{,GR}=1$
LM test for correlated random coefficient
	test statistic	p-value
${\text{LM}}_{\text{CRC}}^{(2)}$	0.5101	0.7749
Estimation results with $\widehat{r}=1$
	BC-IPC	JK-CCEP
Coef.	0.4238	0.3518
S.E.	0.0891	0.1193

NOTE: BC-IPC is the bias-corrected IPC estimator with the PHAC standard error. JK-CCEP is the jackknife bias-corrected CCEP estimator of Westerlund (2018) with the nonparametric standard error of Pesaran (2006) for random coefficients. For the estimation of the number of factors and computation of BC-IPC, data are demeaned in both dimensions while undemeaned data is used for JK-CCEP.

Ahn, S., and Horenstein, A. (2013), “Eigenvalue Ratio Test for the Number of Factors,” Econometrica, 81, 1203–1227.

 Web of Science ®Google Scholar

Westerlund, J. (2018), “CCE in Panels with General Unknown Factors,” Econometrics Journal, 21, 264–276. DOI: 10.1111/ectj.12110.

 Web of Science ®Google Scholar

Pesaran, M. H. (2006), “Estimation and Inference in Large Heterogeneous Panels with a Multifactor Error Structure,” Econometrica, 74, 967–1012. DOI: 10.1111/j.1468-0262.2006.00692.x.

 Web of Science ®Google Scholar