An LM Test for the Conditional Independence between Regressors and Factor Loadings in Panel Data Models with Interactive Effects

Figures & data

Table 1 The size of the LM_X test under Experiment 1.

	Experiment 1
	Case 1				Case 2				Case 3
T/N	30	50	100	200	30	50	100	200	30	50	100	200
	k = 1
	$r=0$
30	0.034	0.048	0.047	0.049	0.050	0.053	0.054	0.061	0.036	0.039	0.048	0.040
50	0.033	0.039	0.053	0.053	0.038	0.036	0.045	0.064	0.039	0.050	0.035	0.054
100	0.047	0.050	0.051	0.048	0.052	0.044	0.052	0.045	0.028	0.037	0.051	0.051
200	0.038	0.042	0.048	0.039	0.043	0.034	0.051	0.043	0.046	0.027	0.049	0.040
	r = 1
30	0.037	0.050	0.057	0.048	0.045	0.047	0.075	0.044	0.044	0.044	0.053	0.044
50	0.059	0.045	0.054	0.046	0.056	0.048	0.042	0.044	0.061	0.046	0.046	0.040
100	0.043	0.059	0.038	0.049	0.043	0.047	0.055	0.059	0.053	0.050	0.043	0.058
200	0.059	0.054	0.041	0.048	0.049	0.040	0.054	0.053	0.045	0.053	0.054	0.050
	r = 2
30	0.046	0.054	0.047	0.063	0.053	0.049	0.053	0.042	0.057	0.034	0.061	0.035
50	0.052	0.050	0.047	0.043	0.051	0.046	0.066	0.054	0.040	0.054	0.045	0.062
100	0.050	0.037	0.059	0.049	0.047	0.060	0.054	0.050	0.045	0.041	0.048	0.045
200	0.052	0.055	0.045	0.051	0.047	0.063	0.057	0.052	0.051	0.045	0.053	0.044
	r = 3
30	0.050	0.040	0.046	0.055	0.049	0.051	0.046	0.036	0.044	0.055	0.049	0.051
50	0.041	0.045	0.047	0.036	0.044	0.041	0.059	0.048	0.048	0.038	0.042	0.048
100	0.052	0.049	0.056	0.048	0.043	0.041	0.054	0.053	0.054	0.044	0.043	0.049
200	0.042	0.044	0.060	0.051	0.045	0.057	0.051	0.052	0.040	0.051	0.042	0.046
	r = 4
30	0.053	0.055	0.040	0.064	0.053	0.045	0.041	0.058	0.056	0.053	0.056	0.056
50	0.050	0.046	0.066	0.041	0.045	0.039	0.047	0.052	0.056	0.046	0.043	0.053
100	0.044	0.061	0.043	0.042	0.058	0.050	0.050	0.055	0.052	0.051	0.050	0.045
200	0.046	0.048	0.047	0.043	0.044	0.041	0.058	0.053	0.043	0.051	0.036	0.048
	k = 2
	r = 0
30	0.035	0.052	0.050	0.045	0.039	0.053	0.037	0.046	0.041	0.045	0.046	0.049
50	0.044	0.053	0.040	0.058	0.029	0.042	0.056	0.052	0.052	0.033	0.050	0.061
100	0.038	0.046	0.050	0.048	0.043	0.048	0.040	0.054	0.029	0.048	0.050	0.045
200	0.041	0.044	0.051	0.031	0.031	0.040	0.054	0.041	0.042	0.036	0.044	0.045
	r = 1
30	0.035	0.051	0.053	0.057	0.040	0.051	0.042	0.046	0.053	0.044	0.048	0.044
50	0.040	0.039	0.053	0.046	0.028	0.045	0.032	0.045	0.044	0.039	0.052	0.058
100	0.045	0.045	0.040	0.048	0.041	0.047	0.052	0.048	0.045	0.039	0.050	0.047
200	0.041	0.054	0.050	0.051	0.049	0.043	0.046	0.055	0.037	0.037	0.041	0.055
	r = 2
30	0.046	0.037	0.049	0.047	0.039	0.044	0.043	0.049	0.048	0.059	0.057	0.045
50	0.042	0.041	0.045	0.044	0.049	0.042	0.048	0.055	0.045	0.046	0.059	0.033
100	0.032	0.048	0.048	0.058	0.043	0.044	0.045	0.050	0.039	0.045	0.045	0.051
200	0.038	0.058	0.046	0.055	0.035	0.039	0.049	0.044	0.035	0.041	0.045	0.056
	$,r=3$
30	0.050	0.051	0.043	0.050	0.033	0.051	0.059	0.045	0.030	0.037	0.044	0.050
50	0.038	0.055	0.058	0.040	0.045	0.041	0.049	0.059	0.035	0.041	0.050	0.056
100	0.042	0.034	0.045	0.044	0.035	0.037	0.037	0.050	0.028	0.040	0.047	0.047
200	0.041	0.048	0.047	0.053	0.039	0.040	0.054	0.051	0.039	0.040	0.050	0.043
	r = 4
30	0.052	0.028	0.079	0.051	0.046	0.039	0.046	0.062	0.049	0.048	0.041	0.042
50	0.039	0.058	0.049	0.043	0.033	0.050	0.045	0.046	0.036	0.038	0.038	0.049
100	0.038	0.043	0.050	0.053	0.043	0.042	0.038	0.036	0.034	0.036	0.042	0.054
200	0.046	0.040	0.046	0.052	0.036	0.053	0.052	0.052	0.040	0.043	0.052	0.049
	k = 3
	r = 0
30	0.036	0.043	0.037	0.043	0.034	0.042	0.046	0.056	0.038	0.047	0.027	0.044
50	0.042	0.047	0.052	0.047	0.036	0.044	0.050	0.043	0.032	0.034	0.048	0.034
100	0.030	0.042	0.049	0.058	0.032	0.036	0.041	0.045	0.032	0.044	0.043	0.048
200	0.030	0.044	0.060	0.044	0.030	0.049	0.038	0.037	0.032	0.047	0.059	0.057
	r = 1
30	0.032	0.048	0.045	0.035	0.033	0.038	0.035	0.043	0.031	0.038	0.046	0.047
50	0.029	0.038	0.043	0.047	0.035	0.051	0.046	0.046	0.024	0.034	0.041	0.051
100	0.032	0.040	0.050	0.049	0.032	0.047	0.051	0.048	0.026	0.033	0.035	0.037
200	0.050	0.052	0.057	0.048	0.032	0.044	0.047	0.055	0.033	0.042	0.048	0.054
	r = 2
30	0.041	0.039	0.054	0.047	0.026	0.051	0.037	0.049	0.030	0.036	0.045	0.044
50	0.034	0.034	0.043	0.047	0.034	0.034	0.047	0.034	0.039	0.044	0.027	0.043
100	0.033	0.047	0.041	0.052	0.027	0.044	0.045	0.054	0.028	0.038	0.035	0.054
200	0.039	0.047	0.047	0.043	0.035	0.047	0.046	0.046	0.028	0.045	0.045	0.048
	r = 3
30	0.036	0.050	0.046	0.041	0.033	0.039	0.044	0.058	0.033	0.032	0.039	0.045
50	0.037	0.036	0.044	0.041	0.026	0.046	0.049	0.031	0.030	0.040	0.045	0.043
100	0.034	0.048	0.057	0.045	0.047	0.041	0.036	0.051	0.039	0.030	0.039	0.037
200	0.035	0.032	0.060	0.047	0.030	0.044	0.050	0.053	0.035	0.037	0.040	0.046
	r = 4
30	0.023	0.056	0.045	0.053	0.036	0.037	0.041	0.047	0.019	0.044	0.040	0.036
50	0.036	0.046	0.036	0.052	0.029	0.040	0.037	0.044	0.029	0.041	0.041	0.051
100	0.028	0.033	0.056	0.047	0.035	0.039	0.045	0.063	0.019	0.046	0.035	0.038
200	0.039	0.035	0.040	0.046	0.026	0.041	0.054	0.055	0.039	0.035	0.045	0.044

NOTE: LM_X denotes the LM test in (16), where we construct the residuals ${\ddot{u}}_{it}$ in (15) using the individual FE estimator, ${\widehat{\mathit{\beta }}}_{\text{FE},i}$. We assume parameter heterogeneity, ${\beta }_{ik}=1+{\eta }_{ik}$ and examine three cases: Case 1 (weak heterogeneity) with ${\eta }_{ik}\sim \text{iid}N(0,0.04)$; Case 2 (medium heterogeneity) with ${\eta }_{ik}\sim \text{iid}N(0,0.25)$; Case 3 (strong heterogeneity) with ${\eta }_{ik}\sim \text{iid}N(0,1)$. k is the number of regressors and r the number of factors.

Table 2 The power of the LM_X test under Experiment 2.

	Experiment 2
	Case 1							Case 2					Case 3
T/N	30		50	100	200			30		50	100	200	30	50	100	200
	k = 1
	r = 1
30	0.998		1	1	1			0.973		0.998	1	1	0.699	0.894	0.999	1
50	1		1	1	1			0.993		1	1	1	0.730	0.916	0.999	1
100	1		1	1	1			0.998		1	1	1	0.774	0.941	1	1
200	1		1	1	1			1		1	1	1	1	0.956	1	1
	r = 2
30	0.996		1	1	1			0.940		0.996	1	1	0.638	0.863	0.990	1
50	1		1	1	1			0.977		0.998	1	1	0.686	0.882	0.991	1
100	1		1	1	1			0.986		1	1	1	0.701	0.886	0.999	1
200	1		1	1	1			0.995		1	1	1	0.709	0.922	0.999	1
	r = 3
30	0.992		0.999	1	1			0.929		0.991	1	1	0.588	0.807	0.968	0.998
50	1		1	1	1			0.970		0.997	1	1	0.616	0.843	0.983	1
100			1	1	1			0.977		1	1	1	0.664	0.856	0.993	1
200	1		1	1	1			0.991		1	1	1	0.660	0.899	0.993	1
	r = 4
30	0.992		1	1	1			0.880		0.984	1	1	0.539	0.746	0.948	0.997
50	0.999		1	1	1			0.935		0.997	1	1	0.608	0.786	0.967	1
100	1		1	1	1			0.962		0.998	1	1	0.622	0.828	0.982	1
200	1		1	1	1			0.969		1	1	1	0.610	0.842	0.988	1
	k = 2
	r = 1
30	0.993		1	1	1			0.802		0.969	1	1	0.308	0.569	0.873	0.996
50	0.998		1	1	1			0.831		0.978	0.999	1	0.349	0.588	0.911	0.998
100	1		1	1	1			0.885		0.991	1	1	0.364	0.620	0.926	0.998
200	1		1	1	1			0.880		0.992	1	1	0.382	0.609	0.916	1
	r = 2
30	0.973		0.998	1	1			0.711		0.956	0.998	0.999	0.307	0.539	0.861	0.992
50	0.995		0.999	1	1			0.810		0.965	1	1	0.327	0.570	0.898	0.999
100	1		1	1	1			0.860		0.988	1	1	0.357	0.599	0.917	0.997
200	1		1	1	1			0.882		0.988	1	1	0.336	0.592	0.923	0.998
	r = 3
30	0.956		0.993	1		0.999		0.695		0.927	0.996	0.999	0.306	0.511	0.833	0.978
50	0.995		1	1		1		0.804		0.975	1	1	0.330	0.593	0.887	0.992
100	0.999		1	1		1		0.854		0.981	1	1	0.341	0.600	0.899	0.997
200	1		1	1		1		0.896		0.995	1	1	0.352	0.609	0.927	0.999
		r = 4
30	0.935		0.991	0.999		1		0.665		0.901	0.988	0.999	0.278	0.474	0.786	0.962
50	0.991		0.999	1		1		0.744		0.954	0.999	1	0.289	0.523	0.839	0.982
100	0.996		1	1		1		0.810		0.970	1	1	0.293	0.537	0.853	0.988
200	0.999		1	1		1		0.822		0.979	1	1	0.320	0.564	0.871	0.997
	k = 3
	r = 1
30	0.946		0.999	1	1		0.518		0.831		0.993	1	0.167	0.314	0.646	0.935
50	0.987		1	1	1		0.561		0.853		0.997	1	0.194	0.334	0.686	0.945
100	0.993		1	1	1		0.621		0.897		0.997	1	0.177	0.356	0.691	0.954
200	1		1	1	1		0.642		0.899		0.999	1	0.179	0.347	0.693	0.956
	r = 2
30	0.900		0.988	1	1		0.459		0.779		0.988	1	0.144	0.280	0.601		0.918
50	0.967		1	1	1		0.528		0.876		0.997	1	0.182	0.320	0.652		0.946
100	0.995		1	1	1		0.588		0.900		0.996	1	0.159	0.347	0.699		0.958
200	0.992		1	1	1		0.590		0.911		0.999	1	0.159	0.331	0.700		0.967
	r = 3
30	0.863		0.992	0.999	1		0.460		0.754		0.977	0.999	0.143	0.318	0.620		0.918
50	0.942		1	1	1		0.497		0.847		0.988	1	0.157	0.331	0.658		0.930
100	0.989		1	1	1		0.588		0.886		0.998	1	0.180	0.349	0.717		0.955
200	0.997		1	1	1		0.596		0.901		1	1	0.167	0.359	0.712		0.967
	r = 4
30	0.828		0.978	1		1	0.404		0.713		0.963	0.998	0.138	0.287	0.586		0.865
50	0.940		0.996	1		1	0.465		0.805		0.987	0.999	0.147	0.298	0.612		0.897
100	0.975		1	1		1	0.502		0.809		0.992	1	0.155	0.314	0.630		0.927
200	0.995		1	1		1	0.553		0.855		0.998	1	0.151	0.306	0.665		0.945

NOTE: see notes to Table 1.

Table 3 The size and power of the LM_X test with small T and k = 1.

	Experiment 1
	Case A				Case B
T/N	30	50	100	200	30	50	100	200
	r = 1
3	0.047	0.041	0.071	0.046	0.049	0.042	0.047	0.050
5	0.045	0.043	0.048	0.058	0.048	0.050	0.054	0.045
	r = 2
3	0.054	0.055	0.054	0.066	0.049	0.042	0.047	0.050
5	0.042	0.045	0.044	0.051	0.048	0.050	0.054	0.045
	r = 3
3	0.044	0.052	0.056	0.044	0.042	0.039	0.046	0.047
5	0.039	0.056	0.054	0.045	0.042	0.038	0.046	0.055
	r = 4
3	0.033	0.047	0.048	0.061	0.048	0.057	0.041	0.057
5	0.053	0.046	0.053	0.051	0.044	0.064	0.050	0.041
	Experiment 2
	r = 1
3	0.141	0.262	0.413	0.564	0.153	0.248	0.381	0.540
5	0.456	0.640	0.829	0.912	0.408	0.576	0.776	0.876
	r = 2
3	0.138	0.261	0.378	0.567	0.138	0.247	0.358	0.545
5	0.404	0.615	0.809	0.920	0.372	0.526	0.743	0.887
	r = 3
3	0.146	0.242	0.394	0.530	0.140	0.213	0.358	0.546
5	0.376	0.601	0.781	0.919	0.358	0.544	0.754	0.896
	r = 4
3	0.148	0.210	0.384	0.552	0.140	0.202	0.322	0.541
5	0.329	0.549	0.775	0.912	0.304	0.463	0.724	0.881

NOTE: LM_X denotes the LM test defined in (16), where we construct the residuals ${\ddot{u}}_{it}$ in (15) using the individual FE estimator, ${\widehat{\mathit{\beta }}}_{\text{FE},i}$. The performance of the LM_X test is explored under parameter homogeneity, β = 1 and k = 1. We examine two cases. Case A with iid error, ${\epsilon }_{it}\sim \text{iid}N(0,1)$ and Case B with serially and weakly cross-sectionally correlated error generated by ${\epsilon }_{it}={\rho }_{\epsilon }{\epsilon }_{i,t-1}+{\upsilon }_{it}+\theta \sum _{1\le h\le 8}{\upsilon }_{i-h,t}$ with ${\upsilon }_{it}\sim \text{iid}N(0,1),{\rho }_{\epsilon }=0.5$ and $\theta =0.2$.

Table 4 The size of the LM_X and $H_{\text{KSS}}$ tests in the presence of the model misspecification.

	Case 1				Case 2				Case 3
T/N	30	50	100	200	30	50	100	200	30	50	100	200
	r = 0
	LM X
30	0.044	0.041	0.050	0.047	0.046	0.053	0.059	0.059	0.042	0.036	0.048	0.046
50	0.046	0.044	0.067	0.048	0.046	0.048	0.056	0.055	0.045	0.046	0.057	0.047
100	0.041	0.039	0.054	0.052	0.050	0.045	0.040	0.040	0.035	0.037	0.041	0.048
200	0.044	0.041	0.053	0.037	0.036	0.048	0.052	0.053	0.035	0.046	0.038	0.038
	$H_{\text{KSS}}$
30	0.911	0.917	0.945	0.946	0.944	0.948	0.964	0.978	0.953	0.970	0.977	0.987
50	0.934	0.939	0.963	0.975	0.971	0.975	0.989	0.995	0.967	0.980	0.984	0.996
100	0.967	0.968	0.974	0.991	0.980	0.983	0.992	0.995	0.965	0.975	0.997	0.996
200	0.984	0.986	0.992	0.995	0.995	0.993	0.996	0.999	0.972	0.979	0.993	0.998
	r = 1
	LM X
30	0.046	0.042	0.048	0.047	0.055	0.048	0.046	0.048	0.048	0.047	0.053	0.049
50	0.061	0.047	0.047	0.040	0.037	0.052	0.043	0.046	0.051	0.042	0.056	0.046
100	0.045	0.048	0.051	0.055	0.050	0.042	0.046	0.042	0.048	0.061	0.037	0.044
200	0.065	0.054	0.060	0.052	0.055	0.054	0.048	0.055	0.040	0.044	0.061	0.058
	$H_{\text{KSS}}$
30	0.841	0.814	0.809	0.820	0.886	0.872	0.898	0.870	0.909	0.911	0.906	0.910
50	0.846	0.839	0.810	0.823	0.908	0.919	0.897	0.875	0.925	0.933	0.913	0.912
100	0.864	0.861	0.830	0.814	0.935	0.918	0.897	0.906	0.934	0.936	0.941	0.932
200	0.893	0.850	0.824	0.791	0.942	0.945	0.916	0.913	0.948	0.943	0.936	0.940
	r = 2
	LMX
30	0.046	0.047	0.050	0.041	0.050	0.038	0.046	0.061	0.050	0.054	0.055	0.045
50	0.054	0.053	0.047	0.054	0.056	0.057	0.059	0.061	0.039	0.043	0.058	0.052
100	0.050	0.048	0.046	0.055	0.044	0.038	0.040	0.054	0.044	0.035	0.048	0.047
200	0.063	0.052	0.033	0.039	0.053	0.045	0.044	0.042	0.035	0.043	0.046	0.054
	$H_{\text{KSS}}$
30	0.823	0.824	0.794	0.781	0.839	0.838	0.820	0.817	0.833	0.862	0.818	0.844
50	0.867	0.835	0.778	0.767	0.854	0.860	0.830	0.824	0.871	0.858	0.846	0.876
100	0.890	0.857	0.775	0.713	0.894	0.871	0.867	0.828	0.868	0.868	0.885	0.860
200	0.890	0.849	0.769	0.690	0.891	0.877	0.867	0.844	0.884	0.886	0.875	0.884
	r = 3
	LM X
30	0.033	0.036	0.051	0.056	0.053	0.066	0.060	0.051	0.036	0.051	0.051	0.051
50	0.048	0.049	0.054	0.058	0.049	0.041	0.048	0.051	0.047	0.042	0.047	0.057
100	0.050	0.052	0.035	0.043	0.052	0.045	0.041	0.042	0.040	0.044	0.049	0.050
200	0.044	0.041	0.046	0.048	0.052	0.054	0.054	0.060	0.053	0.050	0.038	0.041
	$H_{\text{KSS}}$
30	0.866	0.863	0.835	0.798	0.864	0.828	0.794	0.818	0.799	0.815	0.801	0.798
50	0.898	0.899	0.858	0.756	0.866	0.868	0.813	0.788	0.810	0.837	0.826	0.828
100	0.914	0.902	0.826	0.728	0.873	0.864	0.849	0.805	0.851	0.849	0.849	0.842
200	0.953	0.924	0.824	0.762	0.912	0.878	0.832	0.804	0.866	0.852	0.844	0.852
	r = 4
	LM X
30	0.039	0.045	0.050	0.044	0.050	0.051	0.050	0.050	0.045	0.054	0.040	0.064
50	0.041	0.058	0.063	0.053	0.049	0.050	0.051	0.041	0.034	0.041	0.053	0.053
100	0.058	0.050	0.049	0.060	0.046	0.053	0.056	0.048	0.047	0.054	0.052	0.056
200	0.059	0.044	0.047	0.058	0.056	0.044	0.054	0.053	0.041	0.050	0.046	0.042
	$H_{\text{KSS}}$
30	0.915	0.887	0.839	0.800	0.796	0.795	0.764	0.677	0.716	0.714	0.705	0.637
50	0.951	0.926	0.866	0.787	0.846	0.821	0.777	0.710	0.728	0.730	0.726	0.713
100	0.970	0.944	0.877	0.777	0.862	0.815	0.774	0.747	0.749	0.723	0.734	0.709
200	0.979	0.956	0.909	0.800	0.863	0.846	0.777	0.725	0.776	0.736	0.723	0.704

NOTE: $H_{\text{KSS}}$ is the Hausman test statistic proposed by Kapetanios, Serlenga, and Shin (2023), see Section S6 in the Online Supplement. See also notes to Table 1.

Table 5 The size and power of the LM_X test for the heterogeneous dynamic panel data.

	Experiment 1
	Case 1				Case 2				Case 3
T/N	30	50	100	200	30	50	100	200	30	50	100	200
	r = 0
30	0.051	0.055	0.051	0.051	0.055	0.049	0.043	0.055	0.050	0.048	0.042	0.049
50	0.057	0.047	0.050	0.054	0.053	0.038	0.040	0.056	0.047	0.053	0.041	0.045
100	0.028	0.045	0.050	0.055	0.046	0.051	0.050	0.045	0.036	0.039	0.065	0.046
200	0.044	0.040	0.056	0.053	0.032	0.054	0.051	0.056	0.036	0.045	0.051	0.058
	r = 1
30	0.065	0.047	0.060	0.053	0.054	0.048	0.054	0.047	0.044	0.048	0.050	0.041
50	0.052	0.057	0.049	0.040	0.037	0.051	0.060	0.055	0.037	0.045	0.044	0.061
100	0.052	0.037	0.052	0.054	0.044	0.056	0.054	0.044	0.042	0.049	0.048	0.050
200	0.041	0.057	0.045	0.050	0.047	0.054	0.043	0.050	0.034	0.041	0.062	0.044
	r = 2
30	0.058	0.051	0.045	0.052	0.044	0.059	0.048	0.059	0.032	0.053	0.047	0.053
50	0.045	0.042	0.046	0.047	0.045	0.047	0.045	0.048	0.047	0.042	0.054	0.053
100	0.058	0.055	0.044	0.048	0.060	0.041	0.039	0.048	0.044	0.045	0.043	0.048
200	0.037	0.057	0.048	0.055	0.046	0.055	0.048	0.056	0.038	0.039	0.056	0.043
	r = 3
30	0.058	0.051	0.045	0.052	0.044	0.059	0.048	0.059	0.032	0.053	0.047	0.053
50	0.045	0.042	0.046	0.047	0.045	0.047	0.045	0.048	0.047	0.042	0.054	0.053
100	0.058	0.055	0.044	0.048	0.060	0.041	0.039	0.048	0.044	0.045	0.043	0.048
200	0.037	0.057	0.048	0.055	0.046	0.055	0.048	0.058	0.038	0.039	0.056	0.043
	r = 4
30	0.050	0.054	0.047	0.058	0.039	0.064	0.054	0.048	0.050	0.048	0.041	0.056
50	0.046	0.056	0.062	0.046	0.040	0.057	0.060	0.045	0.050	0.056	0.045	0.047
100	0.052	0.053	0.043	0.065	0.052	0.058	0.055	0.043	0.047	0.051	0.049	0.041
200	0.059	0.046	0.051	0.056	0.049	0.056	0.048	0.047	0.041	0.045	0.055	0.055
	Experiment 2
	Case 1				Case 2				Case 3
T/N	30	50	100	200	30	50	100	200	30	50	100	200
	r = 1
30	0.999	1	1	1	0.962	1	1	1	0.698	0.901	0.999	1
50	1	1	1	1	0.989	1	1	1	0.730	0.919	0.999	1
100	1	1	1	1	0.995	1	1	1	0.748	0.947	0.998	1
200	1	1	1	1	0.994	1	1	1	0.738	0.919	1	1
	r = 2
30	0.99	1	1	1	0.921	0.984	1	1	0.595	0.841	0.975	1
50	1	1	1	1	0.952	0.997	1	1	0.648	0.866	0.989	1
100	1	1	1	1	0.969	0.999	1	1	0.645	0.872	0.990	1
200	1	1	1	1	0.970	1	1	1	0.661	0.871	0.995	1
	r = 3
30	0.946	0.997	1	1	0.777	0.922	0.996	1	0.491	0.698	0.909	0.986
50	0.990	1	1	1	0.845	0.955	0.997	1	0.541	0.719	0.938	0.996
100	0.997	1	1	1	0.885	0.976	1	1	0.595	0.781	0.948	0.998
200	0.999	1	1	1	0.923	0.989	1	1	0.609	0.794	0.960	0.999
	r = 4
30	0.855	0.982	0.999	1	0.650	0.836	0.988	0.999	0.410	0.624	0.836	0.967
50	0.948	0.997	1	1	0.761	0.911	0.989	0.999	0.491	0.705	0.879	0.968
100	0.989	1	1	1	0.844	0.946	0.990	0.999	0.579	0.751	0.888	0.982
200	0.994	1	1	1	0.876	0.949	0.995	1	0.585	0.786	0.916	0.978

NOTE: LMX denotes the LM test defined in (16), where we construct ${\ddot{u}}_{it}$ by the individual ARDL residuals in (26). The performance of the LMX test is explored under parameter heterogeneities—${\rho }_i\sim U({\rho }_L,{\rho }_U)$ and ${\beta }_{ik}=1+{\eta }_{ik}$—and serially correlated factors. We examine three cases of heterogeneities. Case 1 (weak heterogeneity) with ${\rho }_i\sim U(0.4,0.6)$ and ${\eta }_i\sim N(0,0.04)$; Case 2 (medium heterogeneity) with ${\rho }_i\sim U(0.25,0.75)$ and ${\eta }_i\sim N(0,0.25)$; Case 3 (strong heterogeneity) with ${\rho }_i\sim U(0.1,0.9)$ and ${\eta }_i\sim N(0,1)$. Factors are generated as $f_{1t}={\rho }_{f1}{f}_{1,t-1}+{\nu }_{1t}\text{ and }{f}_{2t}={\rho }_{f2}{f}_{2,t-1}+{\nu }_{2t}$ where ${({\nu }_{1t},{\nu }_{2t})}^{\prime }\sim \mathit{iidN}(0,{\mathit{I}}_2)$ and ${\rho }_{f1}={\rho }_{f2}=0.8$. r is the number of factors.

Table 6 Empirical applications for the static heterogeneous panel data.

		βMG	βP	${\beta }_{\text{PC}}$	LMX	HB	HW	CD	CD*
Production function
OECD	${\beta }_{\frac{k}{l}}$	0.534 ***	0.589 ***	0.633 ***	0.81	0.003	0.71	–2.743	–3.725
		(0.015)	(0.015)	(0.012)	(0.368)	(0.954)	(0.399)	(0.006)	(0.000)
ITA	${\beta }_{\frac{k}{l}}$	0.312 ***	0.537 ***	0.343 ***	2.134	0.004	0.707	–2.930	–2.928
		(0.042)	(0.035)	(0.016)	(0.144)	(0.946)	(0.401)	(0.003)	(0.003)
US	${\beta }_{\frac{k}{l}}$	0.134 ***	0.165 ***	0.106 ***	3.607	0.001	0.039	–1.111	–2.316
		(0.011)	(0.011)	(0.004)	(0.057)	(0.978)	(0.841)	(0.266)	(0.021)
UNIDO	${\beta }_{\frac{k}{l}}$	0.441 ***	0.637 ***	0.563 ***	0.095	0.008	4.091	–2.139	–1.976
		(0.028)	(0.022)	(0.036)	(0.757)	(0.929)	(0.043)	(0.032)	(0.048)
Production and $RD$	βl	0.418 ***	0.603 ***	0.487 ***
k = 3		(0.018)	(0.013)	(0.010)
	βk	0.068 ***	0.490 ***	0.462 ***
		(0.024)	(0.020)	(0.012)
	βrd	0.074 ***	0.068 ***	0.097 ***	1.257	2.175	2.476	–2.053	–3.071
		(0.008)	(0.006)	(0.007)	(0.739)	(0.536)	(0.479)	(0.040)	(0.002)
Gravity model	βgdp	2.643 ***	3.299 ***	1.808 ***
k = 6		(0.050)	(0.039)	(0.010)
	βrer	–0.094 ***	0.037 ***	–0.041 ***
		(0.006)	(0.004)	(0.006)
	βsim	1.350 ***	1.362 ***	1.231 ***
		(0.036)	(0.022)	(0.057)
	βrlf	–0.195 ***	0.033 ***	0.014 ***
		(0.006)	(0.002)	(0.005)
	βcee	0.088 ***	0.291 ***	0.369 ***
		(0.006)	(0.004)	(0.006)
	βemu	0.074 ***	0.254 ***	0.179 ***	15.12	11.41	24.48	–2.386	–2.193
		(0.008)	(0.006)	(0.006)	(0.019)	(0.076)	(0.001)	(0.017)	(0.028)
Gasoline demand	βp	–0.138 ***	–0.161 ***	–0.101 ***
k = 2		(0.007)	(0.007)	(0.002)
	βinc	0.536 ***	0.436 ***	0.392 ***	0.838	0.001	0.296	–2.051	–1.066
		(0.022)	(0.020)	(0.012)	(0.657)	(0.999)	(0.862)	(0.040)	(0.286)
Income elasticity
US States	βinc	0.605 ***	0.832 ***	0.634 ***	3.122	0.361	0.738	–1.712	–1.847
		(0.033)	(0.030)	(0.011)	(0.077)	(0.547)	(0.391)	(0.087)	(0.065)
US MSAs	βinc	0.610 ***	0.595 ***	0.558 ***	6.304	0.127	14.21	30.442	2.017
		(0.004)	(0.004)	(0.002)	(0.012)	(0.721)	(0.001)	(0.000)	(0.044)
Technological spillovers
OECD countries	βsd	0.020 **	0.057 ***	0.087 ***
k = 3		(0.008)	(0.006)	(0.003)
	βsf	0.306 ***	0.139 **	0.133 ***
		(0.067)	(0.061)	(0.044)
	βhc	0.036 *	0.112 ***	0.013 ***	0.974	0.002	1.657	–3.066	–3.731
		(0.021)	(0.018)	(0.002)	(0.808)	(0.999)	(0.646)	(0.002)	(0.000)
OECD industries	βl	0.527 ***	0.154 ***	0.185 ***
k = 4		(0.032)	(0.019)	(0.023)
	βnitc	–0.148 ***	0.491 ***	0.512 ***
		(0.041)	(0.026)	(0.040)
	βitc	0.027 ***	0.024 ***	0.084 ***
		(0.008)	(0.008)	(0.006)
	βrd	–0.010 ***	0.027 ***	0.031 ***	2.855	0.974	4.121	–1.029	–2.449
		(0.004)	(0.003)	(0.008)	(0.582)	(0.913)	(0.389)	(0.303)	(0.014)
Health	βgdp	0.871 ***	0.721 ***	0.732 ***
k = 2		(0.008)	(0.006)	(0.007)
	βpe	0.239 ***	0.019 ***	0.021 **	2.610	0.003	3.676	–1.099	0.904
		(0.009)	(0.002)	(0.009)	(0.271)	(0.998)	(0.159)	(0.272)	(0.366)
Volatility	βs	–0.141	–0.617 *	–0.683 *	0.192	0.003	0.161	–0.233	–2.644
		(0.374)	(0.370)	(0.391)	(0.661)	(0.955)	(0.688)	(0.816)	(0.008)
Carbon emission	βgdp	0.393 ***	0.429	0.427 ***
k = 4		(0.023)	(0.023)	(0.020)
	βtsh	–0.169 ***	–0.106	–0.096 ***
		(0.016)	(0.015)	(0.013)
	βish	0.004	–0.123	–0.125 *
		(0.045)	(0.040)	(0.066)
	βfsh	0.880 ***	0.866 ***	0.987 ***	5.954	0.168	2.323	17.314	–2.482
		(0.038)	(0.037)	(0.053)	(0.202)	(0.996)	(0.676)	(0.000)	(0.013)

NOTE: ${\beta }_{\text{MG}}$ and β_P denote the mean group and the pooled FE estimator; ${\beta }_{\text{PC}}$ is the (bias-corrected) iterative principal component estimator (see Cui et al. 2023) applying the $\mathbf{I}{\mathbf{C}}_{p1}$ criterion by Bai and Ng (2002), we extract 4 factors in the gravity dataset and 2 factors in the other datasets. The standard errors inside (.) are computed by the nonparametric variance-covariance estimators. LM_X denotes the LM test defined in (16) using the residuals in (15). H_B is the Hausman test statistic proposed by Bai (2009); H_W the Hausman test statistic proposed by Westerlund (2019b); CD is the Pesaran (2015) test; CD* is the bias corrected CD tests proposed by Pesaran and Xie (2021) with 1 factor. The CD tests are applied to the two way FE residuals.

Table 7 Empirical applications for the dynamic heterogeneous panel data.

	LMX	CD	CD*
Growth	1.607	2.942	–0.22
	(0.447)	(0.003)	(0.826)
Temperature	2.468	–3.564	–4.434
	(0.116)	(0.000)	(0.001)
Debt	3.461	–1.965	–2.843
	(0.063)	(0.049)	(0.001)
Energy intensity	3.839	–2.908	–3.217
	(0.279)	(0.004)	(0.001)
Oil	1.552	–3.474	–3.831
	(0.671)	(0.001)	(0.001)

NOTE: LM_X denotes the LM test defined in (16), where we construct the ARDL residuals from the $\text{ARDL}(p_T,p_T)$ approximation, (25) with p_T = 3. See also notes to Table 6.

Table 8 Empirical applications for heterogeneous panel data with small T.

	LMX
Health expenditure	0.291
	(0.589)
Health care	1.933
	(0.164)
Wage	2.417
	(0.120)

NOTE: See notes to Tables 6.

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