Table 2 Average of the estimators using...

Table 2

Average of the estimators using simulation

	(k_M,k_i)	True	Imp	Hist	Conv-Real	Prac-Real	Real
Panel A. Average of the joint cumulants
2nd joint	(2,0)	0.202	0.273	0.203	0.203	0.203	0.203
cumulants		(0.126)	(0.096)	(0.101)	(0.203)	(0.203)	(0.203)
( × 100)	(1,1)	0.216	0.268	0.217	0.217	0.217	0.217
		(0.142)	(0.107)	(0.109)	(0.207)	(0.207)	(0.207)
	(0,2)	0.251	0.307	0.252	0.254	0.254	0.254
		(0.164)	(0.130)	(0.122)	(0.230)	(0.230)	(0.230)
3rd joint	(3,0)	−0.021	−0.128	−0.021	−0.01	−0.014	−0.021
cumulants		(0.023)	(0.018)	(0.149)	(0.082)	(0.125)	(0.142)
( × 1,000)	(2,1)	−0.016	−0.09	−0.017	−0.006	−0.01	−0.017
		(0.027)	(0.020)	(0.160)	(0.064)	(0.109)	(0.125)
	(1,2)	−0.012	−0.058	−0.012	−0.004	−0.005	−0.012
		(0.032)	(0.023)	(0.177)	(0.058)	(0.102)	(0.117)
	(0,3)	−0.009	−0.023	−0.009	−0.003	−0.002	−0.009
		(0.040)	(0.032)	(0.202)	(0.061)	(0.104)	(0.117)
4th joint	(4,0)	0.032	0.597	0.051	−0.231	0.015	0.035
cumulants		(0.011)	(0.009)	(0.232)	(0.810)	(0.174)	(0.238)
( × 10,000)	(3,1)	0.03	0.586	0.053	−0.242	0.016	0.034
		(0.013)	(0.008)	(0.254)	(0.862)	(0.168)	(0.233)
	(2,2)	0.03	0.599	0.056	−0.261	0.017	0.033
		(0.016)	(0.007)	(0.282)	(0.939)	(0.168)	(0.237)
	(1,3)	0.031	0.634	0.061	−0.288	0.019	0.034
		(0.020)	(0.009)	(0.316)	(1.040)	(0.174)	(0.248)
	(0,4)	0.034	0.754	0.07	−0.335	0.025	0.038
		(0.025)	(0.020)	(0.359)	(1.191)	(0.198)	(0.273)
Panel B. Average of coskewness and cokurtosis
Coskewness	(3,0)	−0.384	−1.039	−0.122	−0.009	−0.106	−0.188
		(0.223)	(0.355)	(0.917)	(0.196)	(0.369)	(0.321)
	(2,1)	−0.287	−0.710	−0.082	−0.005	−0.098	−0.171
		(0.173)	(0.262)	(0.881)	(0.178)	(0.324)	(0.297)
	(1,2)	−0.220	−0.455	−0.053	−0.002	−0.083	−0.153
		(0.145)	(0.192)	(0.873)	(0.171)	(0.302)	(0.288)
	(0,3)	−0.179	−0.196	−0.039	−0.002	−0.087	−0.155
		(0.127)	(0.110)	(0.898)	(0.173)	(0.305)	(0.291)
Cokurtosis	(4,0)	1.399	9.742	0.478	−2.854	0.166	0.282
		(1.044)	(3.901)	(1.454)	(0.077)	(2.111)	(0.723)
	(3,1)	1.184	9.150	0.432	−2.748	0.150	0.267
		(0.890)	(3.780)	(1.402)	(0.246)	(1.544)	(0.651)
	(2,2)	1.051	8.964	0.404	−2.729	0.162	0.259
		(0.804)	(3.861)	(1.375)	(0.254)	(1.206)	(0.621)
	(1,3)	0.966	9.104	0.376	−2.752	0.185	0.252
		(0.758)	(4.097)	(1.370)	(0.240)	(0.993)	(0.615)
	(0,4)	0.972	10.413	0.395	−2.863	0.261	0.277
		(0.776)	(4.885)	(1.388)	(0.047)	(0.906)	(0.648)

	(k_M,k_i)	True	Imp	Hist	Conv-Real	Prac-Real	Real
Panel A. Average of the joint cumulants
2nd joint	(2,0)	0.202	0.273	0.203	0.203	0.203	0.203
cumulants		(0.126)	(0.096)	(0.101)	(0.203)	(0.203)	(0.203)
( × 100)	(1,1)	0.216	0.268	0.217	0.217	0.217	0.217
		(0.142)	(0.107)	(0.109)	(0.207)	(0.207)	(0.207)
	(0,2)	0.251	0.307	0.252	0.254	0.254	0.254
		(0.164)	(0.130)	(0.122)	(0.230)	(0.230)	(0.230)
3rd joint	(3,0)	−0.021	−0.128	−0.021	−0.01	−0.014	−0.021
cumulants		(0.023)	(0.018)	(0.149)	(0.082)	(0.125)	(0.142)
( × 1,000)	(2,1)	−0.016	−0.09	−0.017	−0.006	−0.01	−0.017
		(0.027)	(0.020)	(0.160)	(0.064)	(0.109)	(0.125)
	(1,2)	−0.012	−0.058	−0.012	−0.004	−0.005	−0.012
		(0.032)	(0.023)	(0.177)	(0.058)	(0.102)	(0.117)
	(0,3)	−0.009	−0.023	−0.009	−0.003	−0.002	−0.009
		(0.040)	(0.032)	(0.202)	(0.061)	(0.104)	(0.117)
4th joint	(4,0)	0.032	0.597	0.051	−0.231	0.015	0.035
cumulants		(0.011)	(0.009)	(0.232)	(0.810)	(0.174)	(0.238)
( × 10,000)	(3,1)	0.03	0.586	0.053	−0.242	0.016	0.034
		(0.013)	(0.008)	(0.254)	(0.862)	(0.168)	(0.233)
	(2,2)	0.03	0.599	0.056	−0.261	0.017	0.033
		(0.016)	(0.007)	(0.282)	(0.939)	(0.168)	(0.237)
	(1,3)	0.031	0.634	0.061	−0.288	0.019	0.034
		(0.020)	(0.009)	(0.316)	(1.040)	(0.174)	(0.248)
	(0,4)	0.034	0.754	0.07	−0.335	0.025	0.038
		(0.025)	(0.020)	(0.359)	(1.191)	(0.198)	(0.273)
Panel B. Average of coskewness and cokurtosis
Coskewness	(3,0)	−0.384	−1.039	−0.122	−0.009	−0.106	−0.188
		(0.223)	(0.355)	(0.917)	(0.196)	(0.369)	(0.321)
	(2,1)	−0.287	−0.710	−0.082	−0.005	−0.098	−0.171
		(0.173)	(0.262)	(0.881)	(0.178)	(0.324)	(0.297)
	(1,2)	−0.220	−0.455	−0.053	−0.002	−0.083	−0.153
		(0.145)	(0.192)	(0.873)	(0.171)	(0.302)	(0.288)
	(0,3)	−0.179	−0.196	−0.039	−0.002	−0.087	−0.155
		(0.127)	(0.110)	(0.898)	(0.173)	(0.305)	(0.291)
Cokurtosis	(4,0)	1.399	9.742	0.478	−2.854	0.166	0.282
		(1.044)	(3.901)	(1.454)	(0.077)	(2.111)	(0.723)
	(3,1)	1.184	9.150	0.432	−2.748	0.150	0.267
		(0.890)	(3.780)	(1.402)	(0.246)	(1.544)	(0.651)
	(2,2)	1.051	8.964	0.404	−2.729	0.162	0.259
		(0.804)	(3.861)	(1.375)	(0.254)	(1.206)	(0.621)
	(1,3)	0.966	9.104	0.376	−2.752	0.185	0.252
		(0.758)	(4.097)	(1.370)	(0.240)	(0.993)	(0.615)
	(0,4)	0.972	10.413	0.395	−2.863	0.261	0.277
		(0.776)	(4.885)	(1.388)	(0.047)	(0.906)	(0.648)

Note(s): We generate 10,000 paths of daily returns for 20 years according to the stock price process defined in Eqs. (24)–(27), and estimate joint cumulants using the data from the last month of each path. In the first row of each panel, k_M and k_i denote the orders for the market and the stock, respectively. Panel A reports the average joint cumulant values of the index and a security, with numbers in parentheses showing the standard deviations of the estimates. Panel B reports the average values of (co)skewness and (co)kurtosis, which are defined as cumulants divided by the k_M and k_i orders of standard deviations, $σ_{M}^{k_{M}} σ_{i}^{k_{i}}$

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