Optimal model averaging estimator for multinomial logit models

Figures & data

Table 1. Simulations results of the KL loss for setting 1.

${\gamma }_1$	n		OPT1-KL	OPT2-KL	SAIC	SBIC	AIC	BIC
0.5	100	Mean	1.2854	1.0000	2.6779	2.2431	3.8806	3.0228
		SD	2.1785	1.0000	5.2774	4.5048	4.5793	4.5313
	200	Mean	1.0181	1.0000	1.8427	1.7991	2.4709	2.2732
		SD	1.6082	1.0000	3.3939	3.0055	2.9032	2.8244
1	100	Mean	1.0359	1.0000	1.9150	1.8519	2.4402	2.3318
		SD	1.5007	1.0000	3.0423	2.7406	2.7229	2.5896
	200	Mean	1.0000	1.0373	1.5212	1.6712	1.8182	1.9670
		SD	1.1837	1.0000	2.2872	2.3921	2.3530	2.4111
2	100	Mean	1.0260	1.0000	1.7058	1.7868	1.9758	2.0719
		SD	1.3210	1.0000	2.6279	2.6401	2.7112	2.6516
	200	Mean	1.0000	1.0173	1.3756	1.5121	1.5009	1.6475
		SD	1.1638	1.0000	2.3457	2.8700	2.6478	3.2289

Figure 1. The means of ratio by methods of OPT1-KL and OPT2-KL with ${\gamma }_1=1$.

Table 2. Simulation results of MSFE.

		OPT1-KL	OPT2-KL	SAIC	SBIC	AIC	BIC	MCV
	Mean	1.0000	1.0195	1.0703	1.0728	1.2243	1.2424	1.1675
	SD	1.0858	1.0906	1.1869	1.1656	1.3430	1.2729	1.0000

Table 3. Simulations results of the KL loss for setting 2.

${\gamma }_2$	n		OPT1-KL	OPT2-KL	SAIC	SBIC	AIC	BIC
0.25	100	Mean	1.0000	1.0302	1.0886	1.0955	1.3113	1.2951
		SD	1.1114	1.0000	1.2915	1.0422	1.4790	1.1611
	200	Mean	1.0000	1.1340	1.1028	1.2919	1.2384	1.4722
		SD	1.0000	1.0499	1.1158	1.0986	1.2548	1.2265
0.50	100	Mean	1.0000	1.0189	1.0736	1.0566	1.2941	1.2694
		SD	1.1225	1.0000	1.3122	1.0794	1.4152	1.1858
	200	Mean	1.0000	1.1282	1.1018	1.2484	1.2400	1.4465
		SD	1.0000	1.0259	1.1005	1.0071	1.2501	1.1436
0.75	100	Mean	1.0084	1.0000	1.0614	1.0103	1.2938	1.2220
		SD	1.1939	1.0000	1.3298	1.0451	1.6517	1.1342
	200	Mean	1.0000	1.1333	1.0764	1.2175	1.2037	1.4217
		SD	1.0324	1.0108	1.1369	1.0000	1.3275	1.1295

Figure 2. The relationship between the mean of KL loss and ${\lambda }_n$. The points with the smallest losses are indicated by the filled circle •. (a) n = 100 and (b) n = 200.

Table 4. Simulations results of the KL loss for setting 3.

n		OPT1-KL	OPT2-KL	SAIC	SBIC	AIC	BIC	LASSO
100	Mean	1.0863	1.0000	1.3113	1.1209	1.4165	1.2136	1.0172
	SD	1.7691	1.3568	2.7897	1.6756	3.1682	1.9177	1.0000
200	Mean	1.0322	1.0000	1.1918	1.0406	1.2664	1.0821	1.1053
	SD	1.1986	1.1635	1.7025	1.4451	1.8834	1.5681	1.0000

Table 5. Simulations results of the KL loss for setting 4.

Case	n		OPT1-KL	OPT2-KL	SAIC	SBIC	AIC	BIC
1	100	Mean	1.2060	1.0000	1.4672	1.1862	1.2667	1.2462
		SD	1.5345	1.1268	2.1304	1.4188	1.4251	1.0000
	200	Mean	1.1020	1.0000	1.2307	1.0853	1.0256	1.1022
		SD	1.2962	1.0000	1.7283	1.3664	1.0145	1.1259
2	100	Mean	1.1659	1.0000	1.5795	1.3179	1.3301	1.3501
		SD	1.7378	1.0000	2.9602	1.9637	1.8268	1.4928
	200	Mean	1.0311	1.0000	1.2623	1.2498	1.0708	1.3389
		SD	1.5307	1.1471	2.0867	1.6240	1.0798	1.0000

Table 6. Simulations results of MSFE for setting 4.

Case		OPT1-KL	OPT2-KL	SAIC	SBIC	AIC	BIC	A-OPT	MCV
1	Mean	1.0901	1.0000	1.2252	1.1081	1.2523	1.2838	2.0084	1.0225
	SD	1.1847	1.0764	1.3439	1.1847	1.4777	1.4331	14.1238	1.0000
2	Mean	1.0694	1.0000	1.2367	1.1510	1.4000	1.4735	1.8465	1.0653
	SD	1.1278	1.0278	1.2833	1.1722	1.4611	1.5611	10.6498	1.0000

Figure 3. Assessing the estimation consistency of OPT1-KL and OPT2-KL. (a) case 1 and (b) case 2.

Figure 4. Boxplots of KL-type prediction losses by seven methods in the website phishing data.

Table 7. Out-of-sample performances in the website phishing data.

	OPT1-KL	OPT2-KL	SAIC	SBIC	AIC	BIC	LASSO
HRV	0.8879	0.8921	0.8811	0.8812	0.8799	0.8800	0.8841
OPR	0.2200	0.6100	0.0200	0.0400	0.0100	0.0100	0.0900
WOR	0.0200	0.0200	0.2100	0.2600	0.2600	0.1600	0.0700

Table 8. Diebold–Mariano statistics of hit-rate in the website phishing data.

	$\frac{\mathrm{O}\mathrm{P}\mathrm{T}1-\mathrm{K}\mathrm{L}}{\mathrm{O}\mathrm{P}\mathrm{T}2-\mathrm{K}\mathrm{L}}$	$\frac{\mathrm{O}\mathrm{P}\mathrm{T}1-\mathrm{K}\mathrm{L}}{\mathrm{S}\mathrm{A}\mathrm{I}\mathrm{C}}$	$\frac{\mathrm{O}\mathrm{P}\mathrm{T}1-\mathrm{K}\mathrm{L}}{\mathrm{S}\mathrm{B}\mathrm{I}\mathrm{C}}$	$\frac{\mathrm{O}\mathrm{P}\mathrm{T}1-\mathrm{K}\mathrm{L}}{\mathrm{A}\mathrm{I}\mathrm{C}}$	$\frac{\mathrm{O}\mathrm{P}\mathrm{T}1-\mathrm{K}\mathrm{L}}{\mathrm{B}\mathrm{I}\mathrm{C}}$	$\frac{\mathrm{O}\mathrm{P}\mathrm{T}1-\mathrm{K}\mathrm{L}}{\mathrm{L}\mathrm{A}\mathrm{S}\mathrm{S}\mathrm{O}}$	$\frac{\mathrm{O}\mathrm{P}\mathrm{T}2-\mathrm{K}\mathrm{L}}{\mathrm{S}\mathrm{A}\mathrm{I}\mathrm{C}}$
DM	−7.6502	16.6320	9.7040	14.0080	13.1790	4.9572	16.8250
p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	$\frac{\mathrm{O}\mathrm{P}\mathrm{T}2-\mathrm{K}\mathrm{L}}{\mathrm{S}\mathrm{B}\mathrm{I}\mathrm{C}}$	$\frac{\mathrm{O}\mathrm{P}\mathrm{T}2-\mathrm{K}\mathrm{L}}{\mathrm{A}\mathrm{I}\mathrm{C}}$	$\frac{\mathrm{O}\mathrm{P}\mathrm{T}2-\mathrm{K}\mathrm{L}}{\mathrm{B}\mathrm{I}\mathrm{C}}$	$\frac{\mathrm{O}\mathrm{P}\mathrm{T}2-\mathrm{K}\mathrm{L}}{\mathrm{L}\mathrm{A}\mathrm{S}\mathrm{S}\mathrm{O}}$	$\frac{\mathrm{S}\mathrm{A}\mathrm{I}\mathrm{C}}{\mathrm{S}\mathrm{B}\mathrm{I}\mathrm{C}}$	$\frac{\mathrm{S}\mathrm{A}\mathrm{I}\mathrm{C}}{\mathrm{A}\mathrm{I}\mathrm{C}}$	$\frac{\mathrm{S}\mathrm{A}\mathrm{I}\mathrm{C}}{\mathrm{B}\mathrm{I}\mathrm{C}}$
DM	22.0980	14.1900	16.1360	8.4057	−0.0124	5.7799	4.1535
p-value	0.0000	0.0000	0.0000	0.0000	0.9901	0.0000	0.0000
	$\frac{\mathrm{S}\mathrm{A}\mathrm{I}\mathrm{C}}{\mathrm{L}\mathrm{A}\mathrm{S}\mathrm{S}\mathrm{O}}$	$\frac{\mathrm{S}\mathrm{B}\mathrm{I}\mathrm{C}}{\mathrm{A}\mathrm{I}\mathrm{C}}$	$\frac{\mathrm{S}\mathrm{B}\mathrm{I}\mathrm{C}}{\mathrm{B}\mathrm{I}\mathrm{C}}$	$\frac{\mathrm{S}\mathrm{B}\mathrm{I}\mathrm{C}}{\mathrm{L}\mathrm{A}\mathrm{S}\mathrm{S}\mathrm{O}}$	$\frac{\mathrm{A}\mathrm{I}\mathrm{C}}{\mathrm{B}\mathrm{I}\mathrm{C}}$	$\frac{\mathrm{A}\mathrm{I}\mathrm{C}}{\mathrm{L}\mathrm{A}\mathrm{S}\mathrm{S}\mathrm{O}}$	$\frac{\mathrm{B}\mathrm{I}\mathrm{C}}{\mathrm{L}\mathrm{A}\mathrm{S}\mathrm{S}\mathrm{O}}$
DM	−5.0119	1.9948	4.0979	−4.7816	−0.0955	−5.7478	−6.4553
p-value	0.0000	0.0474	0.0000	0.0000	0.9240	0.0000	0.0000