Invariant tests based on M-estimators, estimating functions, and the generalized method of moments

Figures & data

Table 1. Test statistics for H₀:λ = 1 for different measurement units five moment model.

	Two-step GMM				CUP-GMM				Pseudo ML
k	D	Wald	Score	C(α)	D	Wald	Score	C(α)	LR	Mod. score
0.2	0.001	44.750	84.810	33.972	5.771	44.750	5.771	5.066	66.408	31.060
0.4	0.000	44.746	47.692	16.726	5.771	44.746	5.771	0.922	66.408	31.060
0.6	0.001	44.745	42.983	14.106	5.771	44.745	5.771	4.482	66.408	31.060
0.8	0.010	44.744	39.161	12.369	5.771	44.744	5.771	5.282	66.408	31.060
1.0	0.056	44.743	35.676	10.593	5.771	44.743	5.771	5.3838	66.408	31.060
3.0	34.629	44.743	118.876	42.124	5.771	44.743	5.771	0.6720	66.408	31.060
5.0	1.641	44.743	62.195	34.746	5.771	44.743	5.771	2.5545	66.408	31.060
7.0	0.282	44.742	61.766	34.953	5.771	44.742	5.771	3.9336	66.408	31.060
10.0	0.068	44.739	61.147	34.465	5.771	44.739	5.771	4.5010	66.408	31.060

Table 2. Test statistics for H₀:λ = 1 for different measurement units six moment models.

	Two-step GMM				CUP-GMM
k	D	Wald	Score	C(α)	D	Wald	Score	C(α)
0.2	0.016	416.546	106.734	54.462	19.480	359.380	11.107	3.189
0.4	0.036	221.829	108.142	54.852	19.480	83.743	16.296	7.318
0.6	0.248	213.918	107.764	52.818	19.480	40.481	18.637	7.063
0.8	1.068	178.757	106.053	47.539	19.480	34.101	17.678	0.661
1.0	3.562	139.364	103.364	37.915	19.480	35.580	17.769	5.215
3.0	47.490	46.214	110.751	7.960	19.480	45.146	15.250	4.650
5.0	1.651	129.698	48.704	6.518	19.480	59.667	13.367	4.611
7.0	1.511	384.944	49.719	9.978	19.480	118.911	13.937	5.639
10.0	2.031	905.870	50.264	10.747	19.480	406.974	14.162	6.136

Figure 1. Two-step GMM tests based on five moment conditions.

Figure 2. Two-step GMM tests based on six moment conditions.

Figure 3. CUP GMM tests based on six moment conditions.

Table 3. J statistic for the validity of (7.4) under different identifying restrictions (p-values in parentheses).

Two-step GMM				CUP-GMM
Normalization 1		Normalization 2		Normalization 1		Normalization 2
QS	Bartlett	QS	Bartlett	QS	Bartlett	QS	Bartlett
0.218	0.106	17.214	5.670	1.235	1.774	1.759	1.866
(0.897)	(0.949)	(0.000)	(0.059)	(0.539)	(0.412)	(0.415)	(0.393)