Kernel regression utilizing heterogeneous datasets

Figures & data

Table 1. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains only X, with S = 200 under $\gamma =0$, $n/N\approx 13\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{C1}$	${\hat{\mu }}_1^{C2}$	${\hat{\mu }}_1^{C3}$	IMP %	Mean of $\hat{\gamma }$
Normal	M1	Sample	Best	0.040	0.017	0.020	0.022	56.96	−0.003
			CV	0.057	0.043	0.042	0.051	27.16	−0.007
		Grid	Best	0.026	0.009	0.013	0.014	66.49	−0.012
			CV	0.037	0.022	0.022	0.028	40.53	−0.003
	M2	Sample	Best	0.042	0.021	0.027	0.026	50.58	−0.042
			CV	0.074	0.056	0.061	0.060	23.52	−0.039
		Grid	Best	0.030	0.015	0.025	0.023	50.36	−0.026
			CV	0.051	0.033	0.040	0.036	35.62	−0.035
	M3	Sample	Best	0.039	0.020	0.025	0.023	48.55	−0.023
			CV	0.063	0.050	0.053	0.055	21.16	−0.020
		Grid	Best	0.028	0.014	0.020	0.021	51.73	−0.010
			CV	0.046	0.029	0.033	0.033	36.33	−0.024
	M4	Sample	Best	0.041	0.018	0.020	0.023	56.28	−0.007
			CV	0.064	0.050	0.047	0.056	25.83	−0.017
		Grid	Best	0.026	0.009	0.014	0.015	65.54	−0.006
			CV	0.036	0.021	0.020	0.026	44.18	−0.007
Bounded	M1	Sample	Best	0.010	0.002	0.006	0.007	79.58	−0.007
			CV	0.016	0.006	0.007	0.010	63.65	0.001
		Grid	Best	0.013	0.002	0.006	0.007	82.93	−0.000
			CV	0.041	0.012	0.013	0.017	70.86	0.009
	M2	Sample	Best	0.011	0.003	0.007	0.008	74.88	−0.031
			CV	0.037	0.008	0.011	0.013	77.46	−0.018
		Grid	Best	0.016	0.004	0.009	0.010	74.00	−0.023
			CV	0.086	0.018	0.019	0.021	79.20	−0.019
	M3	Sample	Best	0.011	0.003	0.006	0.007	73.85	−0.003
			CV	0.037	0.009	0.011	0.014	76.48	−0.008
		Grid	Best	0.016	0.004	0.008	0.009	73.94	−0.004
			CV	0.083	0.018	0.020	0.023	78.46	0.004
	M4	Sample	Best	0.009	0.002	0.005	0.006	77.43	−0.007
			CV	0.018	0.006	0.008	0.011	68.73	0.002
		Grid	Best	0.013	0.002	0.006	0.007	82.23	−0.007
			CV	0.036	0.010	0.011	0.015	71.59	0.003

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.005 and 0.006.

Table 2. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains only X, with S = 200 under $\gamma =0$, $n/N\approx 50\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{C1}$	${\hat{\mu }}_1^{C2}$	${\hat{\mu }}_1^{C3}$	IMP %	Mean of $\hat{\gamma }$
Normal	M1	Sample	Best	0.012	0.007	0.006	0.007	44.75	−0.017
			CV	0.033	0.027	0.021	0.028	36.37	−0.005
		Grid	Best	0.006	0.004	0.005	0.005	34.29	−0.009
			CV	0.017	0.012	0.008	0.012	53.01	−0.007
	M2	Sample	Best	0.013	0.008	0.010	0.009	36.08	−0.057
			CV	0.052	0.027	0.027	0.026	48.95	−0.062
		Grid	Best	0.010	0.005	0.007	0.006	45.83	−0.058
			CV	0.036	0.016	0.018	0.016	57.27	−0.063
	M3	Sample	Best	0.014	0.009	0.010	0.010	37.36	−0.032
			CV	0.050	0.027	0.027	0.027	46.48	−0.041
		Grid	Best	0.008	0.005	0.006	0.005	43.13	−0.027
			CV	0.030	0.013	0.015	0.014	55.91	−0.031
	M4	Sample	Best	0.014	0.009	0.008	0.009	44.00	−0.020
			CV	0.041	0.033	0.026	0.033	36.63	−0.014
		Grid	Best	0.006	0.004	0.005	0.005	37.33	−0.008
			CV	0.017	0.012	0.008	0.013	53.05	−0.023
Bounded	M1	Sample	Best	0.002	0.001	0.001	0.002	40.67	−0.008
			CV	0.009	0.005	0.004	0.005	52.83	0.009
		Grid	Best	0.004	0.002	0.002	0.002	55.17	−0.010
			CV	0.021	0.008	0.007	0.008	67.74	−0.000
	M2	Sample	Best	0.004	0.002	0.002	0.002	58.55	−0.035
			CV	0.027	0.007	0.006	0.007	77.51	−0.039
		Grid	Best	0.006	0.002	0.003	0.003	64.61	−0.033
			CV	0.056	0.011	0.011	0.011	80.27	−0.036
	M3	Sample	Best	0.004	0.002	0.002	0.002	60.82	−0.006
			CV	0.024	0.006	0.006	0.006	76.60	−0.008
		Grid	Best	0.006	0.002	0.002	0.003	66.02	−0.011
			CV	0.050	0.011	0.011	0.011	78.56	−0.010
	M4	Sample	Best	0.002	0.001	0.001	0.002	43.16	−0.011
			CV	0.010	0.005	0.005	0.005	54.67	−0.003
		Grid	Best	0.004	0.002	0.002	0.002	53.45	−0.007
			CV	0.024	0.009	0.008	0.009	68.53	−0.002

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.004 and 0.005.

Table 3. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains only X, with S = 200 under $\gamma =0.5$, $n/N\approx 13\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{C1}$	${\hat{\mu }}_1^{C2}$	${\hat{\mu }}_1^{C3}$	IMP %	Mean of $\hat{\gamma }$
Normal	M1	Sample	Best	0.042	0.181	0.033	0.028	32.15	0.449
			CV	0.055	0.193	0.061	0.055	−0.91	0.449
		Grid	Best	0.022	0.164	0.019	0.013	40.36	0.453
			CV	0.030	0.193	0.048	0.030	2.13	0.435
	M2	Sample	Best	0.041	0.295	0.047	0.026	37.01	0.429
			CV	0.077	0.324	0.103	0.066	15.26	0.424
		Grid	Best	0.030	0.279	0.040	0.022	27.11	0.428
			CV	0.059	0.315	0.087	0.043	28.20	0.421
	M3	Sample	Best	0.048	0.335	0.047	0.028	40.81	0.452
			CV	0.076	0.370	0.104	0.066	12.50	0.448
		Grid	Best	0.031	0.300	0.035	0.021	31.59	0.441
			CV	0.053	0.355	0.082	0.041	23.25	0.447
	M4	Sample	Best	0.048	0.193	0.035	0.030	37.27	0.452
			CV	0.067	0.205	0.067	0.061	8.73	0.433
		Grid	Best	0.023	0.177	0.023	0.014	39.42	0.437
			CV	0.036	0.204	0.049	0.031	15.80	0.452
Bounded	M1	Sample	Best	0.009	0.146	0.011	0.007	23.90	0.482
			CV	0.017	0.150	0.026	0.011	35.66	0.483
		Grid	Best	0.013	0.152	0.016	0.007	44.41	0.470
			CV	0.037	0.159	0.030	0.017	52.24	0.485
	M2	Sample	Best	0.012	0.196	0.018	0.008	33.06	0.468
			CV	0.044	0.206	0.036	0.017	60.76	0.468
		Grid	Best	0.016	0.205	0.022	0.010	39.95	0.463
			CV	0.088	0.214	0.043	0.024	72.45	0.469
	M3	Sample	Best	0.011	0.209	0.016	0.007	34.46	0.479
			CV	0.040	0.216	0.034	0.016	60.92	0.475
		Grid	Best	0.018	0.218	0.019	0.011	39.73	0.486
			CV	0.089	0.228	0.042	0.026	70.77	0.483
	M4	Sample	Best	0.010	0.152	0.012	0.007	28.69	0.484
			CV	0.019	0.157	0.025	0.012	36.41	0.485
		Grid	Best	0.014	0.164	0.015	0.007	46.99	0.485
			CV	0.040	0.170	0.029	0.019	52.63	0.492

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.005 and 0.006.

Table 4. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains only X, with S = 200 under $\gamma =0.5$, $n/N\approx 50\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{C1}$	${\hat{\mu }}_1^{C2}$	${\hat{\mu }}_1^{C3}$	IMP %	Mean of $\hat{\gamma }$
Normal	M1	Sample	Best	0.013	0.035	0.008	0.009	39.89	0.431
			CV	0.037	0.055	0.027	0.034	25.63	0.434
		Grid	Best	0.007	0.032	0.005	0.004	34.86	0.439
			CV	0.014	0.046	0.014	0.012	14.98	0.437
	M2	Sample	Best	0.015	0.061	0.014	0.010	33.07	0.408
			CV	0.063	0.086	0.041	0.030	52.47	0.406
		Grid	Best	0.009	0.061	0.011	0.007	28.26	0.406
			CV	0.043	0.078	0.032	0.018	59.17	0.400
	M3	Sample	Best	0.015	0.070	0.014	0.010	33.24	0.441
			CV	0.056	0.093	0.037	0.029	48.27	0.440
		Grid	Best	0.008	0.070	0.011	0.006	30.34	0.439
			CV	0.034	0.095	0.032	0.017	51.53	0.438
	M4	Sample	Best	0.015	0.041	0.010	0.010	35.19	0.423
			CV	0.048	0.063	0.033	0.040	31.52	0.414
		Grid	Best	0.006	0.033	0.005	0.005	21.67	0.422
			CV	0.017	0.054	0.016	0.014	17.89	0.423
Bounded	M1	Sample	Best	0.003	0.022	0.003	0.002	20.26	0.477
			CV	0.010	0.027	0.008	0.006	39.55	0.474
		Grid	Best	0.004	0.024	0.003	0.002	37.63	0.480
			CV	0.022	0.029	0.009	0.009	57.30	0.480
	M2	Sample	Best	0.004	0.034	0.004	0.002	45.43	0.462
			CV	0.030	0.042	0.011	0.008	72.70	0.462
		Grid	Best	0.007	0.037	0.006	0.003	56.85	0.457
			CV	0.057	0.041	0.014	0.012	78.52	0.468
	M3	Sample	Best	0.004	0.037	0.004	0.002	47.13	0.474
			CV	0.026	0.045	0.010	0.007	73.40	0.487
		Grid	Best	0.006	0.043	0.005	0.002	56.60	0.476
			CV	0.050	0.046	0.014	0.011	77.10	0.483
	M4	Sample	Best	0.003	0.024	0.002	0.002	26.50	0.483
			CV	0.010	0.030	0.007	0.006	43.47	0.475
		Grid	Best	0.004	0.027	0.003	0.002	45.69	0.480
			CV	0.023	0.032	0.009	0.009	61.42	0.485

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.004 and 0.005.

Table 5. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains both X and Z, with S = 200 under $\gamma =0$, $n/N\approx 13\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{E1}$	${\hat{\mu }}_1^{E2}$	${\hat{\mu }}_1^{E3}$	IMP %	Mean of $\hat{\gamma }$
Normal	M1	Sample	Best	0.055	0.020	0.035	0.025	64.46	0.020
			CV	0.072	0.032	0.052	0.036	55.51	0.015
		Grid	Best	0.047	0.010	0.027	0.017	78.44	0.013
			CV	0.060	0.019	0.034	0.024	68.49	0.014
	M2	Sample	Best	0.066	0.026	0.044	0.032	60.99	−0.054
			CV	0.088	0.035	0.052	0.040	60.73	−0.058
		Grid	Best	0.072	0.019	0.032	0.025	74.04	−0.051
			CV	0.091	0.027	0.041	0.034	70.63	−0.058
	M3	Sample	Best	0.065	0.027	0.047	0.033	58.73	−0.014
			CV	0.078	0.035	0.053	0.041	54.98	−0.009
		Grid	Best	0.055	0.016	0.031	0.023	70.04	−0.004
			CV	0.067	0.026	0.038	0.033	60.79	0.002
	M4	Sample	Best	0.055	0.020	0.037	0.025	63.83	0.018
			CV	0.075	0.034	0.053	0.040	54.39	0.012
		Grid	Best	0.045	0.010	0.026	0.019	77.21	0.015
			CV	0.057	0.017	0.031	0.023	71.13	0.012
Bounded	M1	Sample	Best	0.016	0.005	0.009	0.009	69.52	0.003
			CV	0.023	0.011	0.015	0.015	53.87	−0.003
		Grid	Best	0.036	0.017	0.023	0.022	54.15	0.009
			CV	0.075	0.043	0.050	0.050	42.12	0.005
	M2	Sample	Best	0.026	0.006	0.011	0.010	78.31	−0.024
			CV	0.030	0.015	0.020	0.018	49.71	−0.027
		Grid	Best	0.084	0.018	0.028	0.025	77.95	−0.021
			CV	0.100	0.055	0.064	0.059	44.75	−0.016
	M3	Sample	Best	0.025	0.006	0.011	0.011	78.40	0.000
			CV	0.037	0.014	0.022	0.019	63.38	0.010
		Grid	Best	0.073	0.017	0.024	0.023	76.96	0.006
			CV	0.107	0.048	0.060	0.061	55.27	0.004
	M4	Sample	Best	0.016	0.005	0.009	0.010	69.74	0.002
			CV	0.027	0.011	0.017	0.015	59.30	0.001
		Grid	Best	0.038	0.016	0.021	0.021	59.45	0.011
			CV	0.063	0.042	0.048	0.044	33.36	0.011

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.005 and 0.007.

Table 6. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains both X and Z, with S = 200 under $\gamma =0$, $n/N\approx 50\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{E1}$	${\hat{\mu }}_1^{E2}$	${\hat{\mu }}_1^{E3}$	IMP %	Mean of $\hat{\gamma }$
Normal	M1	Sample	Best	0.027	0.017	0.026	0.017	36.71	0.024
			CV	0.033	0.020	0.024	0.020	39.59	0.023
		Grid	Best	0.017	0.007	0.013	0.008	57.00	0.021
			CV	0.019	0.010	0.014	0.011	46.34	0.023
	M2	Sample	Best	0.041	0.022	0.025	0.023	46.91	−0.089
			CV	0.043	0.023	0.025	0.024	46.51	−0.093
		Grid	Best	0.022	0.013	0.016	0.014	42.47	−0.095
			CV	0.027	0.013	0.016	0.015	49.76	−0.090
	M3	Sample	Best	0.039	0.022	0.025	0.023	43.14	−0.034
			CV	0.042	0.022	0.025	0.023	48.01	−0.026
		Grid	Best	0.023	0.013	0.016	0.014	43.63	−0.032
			CV	0.024	0.013	0.015	0.014	44.94	−0.028
	M4	Sample	Best	0.029	0.017	0.026	0.018	40.57	0.014
			CV	0.039	0.021	0.026	0.022	45.54	0.013
		Grid	Best	0.018	0.008	0.013	0.009	54.08	0.005
			CV	0.020	0.011	0.014	0.011	46.79	0.015
Bounded	M1	Sample	Best	0.006	0.004	0.005	0.005	31.24	0.007
			CV	0.008	0.005	0.006	0.006	28.27	0.008
		Grid	Best	0.019	0.012	0.012	0.012	39.18	−0.003
			CV	0.025	0.017	0.018	0.017	30.05	0.004
	M2	Sample	Best	0.007	0.005	0.006	0.006	32.97	−0.035
			CV	0.012	0.012	0.013	0.012	2.38	−0.040
		Grid	Best	0.021	0.013	0.015	0.014	38.04	−0.036
			CV	0.042	0.036	0.040	0.036	14.28	−0.030
	M3	Sample	Best	0.007	0.005	0.006	0.005	31.15	−0.006
			CV	0.010	0.010	0.011	0.010	6.03	−0.010
		Grid	Best	0.022	0.013	0.015	0.015	37.83	−0.007
			CV	0.032	0.027	0.030	0.029	13.81	−0.002
	M4	Sample	Best	0.007	0.004	0.005	0.005	32.99	−0.004
			CV	0.008	0.006	0.007	0.006	26.81	−0.004
		Grid	Best	0.019	0.012	0.012	0.013	36.25	0.007
			CV	0.025	0.017	0.019	0.018	32.72	0.001

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.004 and 0.006.

Table 7. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains both X and Z, with S = 200 under $\gamma =0.5$, $n/N\approx 13\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{E1}$	${\hat{\mu }}_1^{E2}$	${\hat{\mu }}_1^{E3}$	IMP %	Mean of $\hat{\gamma }$
	M1	Sample	Best	0.055	0.180	0.060	0.034	37.57	0.436
			CV	0.064	0.184	0.064	0.039	38.64	0.434
		Grid	Best	0.040	0.177	0.035	0.018	54.10	0.431
			CV	0.053	0.196	0.045	0.022	59.51	0.432
	M2	Sample	Best	0.076	0.170	0.064	0.041	46.31	0.380
			CV	0.092	0.177	0.071	0.045	51.02	0.377
		Grid	Best	0.071	0.179	0.050	0.032	54.78	0.386
			CV	0.082	0.190	0.056	0.038	53.31	0.385
	M3	Sample	Best	0.072	0.188	0.064	0.043	39.63	0.425
			CV	0.091	0.192	0.067	0.049	45.96	0.421
		Grid	Best	0.053	0.191	0.042	0.033	38.08	0.436
			CV	0.074	0.202	0.048	0.035	52.56	0.425
	M4	Sample	Best	0.060	0.187	0.066	0.033	44.90	0.435
			CV	0.070	0.197	0.070	0.040	42.84	0.427
		Grid	Best	0.041	0.180	0.037	0.018	57.34	0.421
			CV	0.059	0.197	0.041	0.021	64.72	0.430
Bounded	M1	Sample	Best	0.016	0.155	0.019	0.010	39.49	0.478
			CV	0.022	0.160	0.022	0.013	41.00	0.494
		Grid	Best	0.037	0.193	0.036	0.021	44.14	0.478
			CV	0.061	0.212	0.047	0.037	38.34	0.489
	M2	Sample	Best	0.030	0.162	0.022	0.015	48.90	0.460
			CV	0.036	0.164	0.031	0.022	37.67	0.464
		Grid	Best	0.087	0.202	0.040	0.049	53.27	0.473
			CV	0.105	0.228	0.086	0.083	20.55	0.471
	M3	Sample	Best	0.029	0.171	0.020	0.015	48.81	0.485
			CV	0.039	0.174	0.029	0.022	43.49	0.492
		Grid	Best	0.076	0.205	0.036	0.045	52.75	0.490
			CV	0.107	0.240	0.076	0.073	31.97	0.492
	M4	Sample	Best	0.016	0.163	0.019	0.009	39.27	0.479
			CV	0.025	0.169	0.025	0.015	37.49	0.479
		Grid	Best	0.039	0.193	0.034	0.023	41.60	0.484
			CV	0.072	0.230	0.062	0.052	27.74	0.483

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.005 and 0.007.

Table 8. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains both X and Z, with S = 200 under $\gamma =0.5$, $n/N\approx 50\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{E1}$	${\hat{\mu }}_1^{E2}$	${\hat{\mu }}_1^{E3}$	IMP %	Mean of $\hat{\gamma }$
Normal	M1	Sample	Best	0.027	0.061	0.032	0.019	29.74	0.430
			CV	0.034	0.061	0.033	0.022	36.98	0.422
		Grid	Best	0.015	0.041	0.015	0.008	47.14	0.423
			CV	0.018	0.049	0.018	0.010	45.47	0.417
	M2	Sample	Best	0.043	0.060	0.033	0.027	36.46	0.353
			CV	0.044	0.062	0.035	0.030	33.60	0.360
		Grid	Best	0.023	0.053	0.022	0.016	27.23	0.367
			CV	0.024	0.052	0.022	0.016	31.83	0.353
	M3	Sample	Best	0.043	0.068	0.031	0.029	32.86	0.417
			CV	0.042	0.067	0.030	0.029	30.68	0.424
		Grid	Best	0.022	0.058	0.018	0.015	30.01	0.416
			CV	0.023	0.059	0.019	0.017	27.17	0.417
	M4	Sample	Best	0.033	0.062	0.034	0.022	34.75	0.413
			CV	0.039	0.064	0.033	0.023	40.31	0.414
		Grid	Best	0.017	0.046	0.017	0.010	42.46	0.409
			CV	0.020	0.056	0.018	0.011	44.86	0.421
Bounded	M1	Sample	Best	0.006	0.028	0.007	0.006	8.96	0.481
			CV	0.007	0.029	0.008	0.006	13.22	0.482
		Grid	Best	0.018	0.047	0.016	0.017	13.08	0.482
			CV	0.023	0.054	0.023	0.021	8.56	0.478
	M2	Sample	Best	0.008	0.033	0.008	0.006	20.59	0.465
			CV	0.015	0.040	0.017	0.013	14.82	0.465
		Grid	Best	0.023	0.059	0.021	0.020	12.38	0.460
			CV	0.045	0.081	0.050	0.048	-6.84	0.473
	M3	Sample	Best	0.008	0.037	0.008	0.006	16.16	0.493
			CV	0.010	0.040	0.012	0.009	2.05	0.497
		Grid	Best	0.022	0.060	0.018	0.020	17.75	0.495
			CV	0.033	0.078	0.041	0.040	-21.18	0.498
	M4	Sample	Best	0.007	0.032	0.007	0.005	21.96	0.476
			CV	0.008	0.032	0.008	0.006	21.82	0.489
		Grid	Best	0.019	0.052	0.016	0.019	14.56	0.481
			CV	0.023	0.059	0.022	0.023	2.61	0.485

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.004 and 0.006.

Table 9. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains only normally distributed $(X_1,X_2)$, with S = 200.

				Estimator
Test data	γ	n/N	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{C1}$	${\hat{\mu }}_1^{C2}$	${\hat{\mu }}_1^{C3}$	IMP %	Mean of $\hat{\gamma }$
Sample	0	13%	Best	0.090	0.046	0.054	0.051	49.496	−0.006
			CV	0.080	0.056	0.063	0.062	29.115	0.007
		50%	Best	0.027	0.020	0.023	0.020	25.701	−0.006
			CV	0.031	0.025	0.027	0.026	20.040	−0.008
	0.5	13%	Best	0.085	0.198	0.067	0.055	35.178	0.421
			CV	0.070	0.190	0.072	0.062	11.429	0.406
		50%	Best	0.025	0.049	0.025	0.021	17.000	0.389
			CV	0.031	0.052	0.031	0.026	15.798	0.391

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.004 and 0.006.