Unbiased Estimation

50 Monte Carlo runs

We don't normaly do a Monte Carlo experiment with only 50 runs, this is only interesting as it shows how much precise a Monte Carlo experiment gets with the number of runs.

On other pages, you can see results with 6,250,000 runs, 125,000 runs and 2,500 runs.

For further details, e-mail me at gbel@math.creuset.org.


Table 1: yt = θ exp{σ εt }, σ = 1 known.
θ0 θML θAML θFS1 θFS4 θFS5
0.5
(0.1)		 0.52246
(0.10327)		 0.51211
(0.10123)		 0.52192
(0.10317)		 0.51207
(0.10123)		 0.51293
(0.10875)
1.0
(0.2)		 0.99286
(0.20952)		 0.97320
(0.20537)		 0.99184
(0.20930)		 0.97317
(0.20538)		 0.98050
(0.20850)
1.5
(0.3)		 1.51835
(0.29708)		 1.48828
(0.29120)		 1.51679
(0.29678)		 1.48826
(0.29120)		 1.49586
(0.29614)
2.0
(0.4)		 1.99286
(0.34378)		 1.95339
(0.33697)		 1.99081
(0.34342)		 1.95339
(0.33702)		 1.93134
(0.32682)
2.5
(0.5)		 2.46106
(0.45728)		 2.41233
(0.44822)		 2.45854
(0.45681)		 2.41238
(0.44825)		 2.43012
(0.47027)
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Table 2.1: yt = θ exp{σ εt }, σ 0 = 0.5 unknown.
θ0 θML θAML θFS4 θFS5
0.5
(0.05)		 0.50924
(0.05581)		 0.48944
(0.04245)		 0.50641
(0.05539)		 0.50615
(0.05500)
1.0
(0.10)		 1.01622
(0.10732)		 0.99682
(0.10723)		 1.01109
(0.10720)		 1.01214
(0.10655)
1.5
(0.15)		 1.49692
(0.12450)		 1.50905
(0.12993)		 1.48907
(0.12431)		 1.48985
(0.12632)
2.0
(0.20)		 1.99016
(0.18580)		 2.03275
(0.21934)		 1.98003
(0.18505)		 1.97895
(0.18315)
2.5
(0.25)		 2.51132
(0.24892)		 2.47581
(0.23591)		 2.49900
(0.24688)		 2.49789
(0.24570)
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Table 2.2: yt = θ exp{σ εt }, σ 0 = 1.0 unknown.
θ0 θML θAML θFS4 θFS5
0.5
(0.10)		 0.49685
(0.08611)		 0.49090
(0.08430)		 0.48710
(0.08535)		 0.48452
(0.08833)
1.0
(0.20)		 1.02602
(0.24717)		 1.04001
(0.24650)		 1.00672
(0.24290)		 1.01404
(0.25649)
1.5
(0.30)		 1.52772
(0.29620)		 1.42941
(0.33908)		 1.49805
(0.28912)		 1.48924
(0.28929)
2.0
(0.40)		 2.03806
(0.37478)		 2.02323
(0.34432)		 1.99806
(0.37028)		 1.99959
(0.36588)
2.5
(0.50)		 2.53260
(0.61872)		 2.50038
(0.52035)		 2.48257
(0.60262)		 2.49556
(0.64179)
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Table 2.3: yt = θ exp{σ εt }, σ 0 = 1.5 unknown.
θ0 θML θAML θFS4 θFS5
0.5
(0.15)		 0.50538
(0.15301)		 0.46151
(0.10919)		 0.48286
(0.14677)		 0.47705
(0.13638)
1.0
(0.30)		 1.10934
(0.33657)		 1.03631
(0.34661)		 1.05661
(0.32083)		 1.04169
(0.33396)
1.5
(0.45)		 1.64973
(0.61234)		 1.46908
(0.49727)		 1.57590
(0.58507)		 1.53318
(0.56658)
2.0
(0.60)		 2.11486
(0.65302)		 1.95076
(0.54012)		 2.03097
(0.62813)		 2.00357
(0.60436)
2.5
(0.75)		 2.56623
(0.64420)		 2.40107
(0.68499)		 2.44783
(0.61513)		 2.45083
(0.68810)
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Table 2.4: yt = θ exp{σ εt }, σ 0 = 2.0 unknown.
θ0 θML θAML θFS4 θFS5
0.5
(0.20)		 0.55952
(0.24886)		 0.42179
(0.16966)		 0.51729
(0.23150)		 0.52061
(0.32788)
1.0
(0.40)		 1.24947
(0.55949)		 0.95210
(0.38514)		 1.14950
(0.50254)		 1.18976
(0.58978)
1.5
(0.60)		 1.59849
(0.72323)		 1.69841
(0.88511)		 1.47166
(0.67344)		 1.44331
(0.69151)
2.0
(0.80)		 2.17444
(0.89418)		 2.01072
(1.06311)		 2.01880
(0.84056)		 2.02352
(1.00192)
2.5
(1.00)		 2.84527
(1.02714)		 2.60559
(1.01042)		 2.63521
(0.95791)		 2.68182
(1.16116)
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Table 2.5: yt = θ exp{σ εt }, σ 0 = 2.5 unknown.
θ0 θML θAML θFS4 θFS5
0.5
(0.25)		 0.54874
(0.21874)		 0.53739
(0.29794)		 0.48809
(0.19746)		 0.48185
(0.27581)
1.0
(0.50)		 1.22509
(0.59867)		 1.01795
(0.52146)		 1.07628
(0.51359)		 1.26459
(0.91227)
1.5
(0.75)		 1.72498
(0.89761)		 1.50111
(0.65080)		 1.51404
(0.79927)		 1.46993
(0.70157)
2.0
(1.00)		 2.10180
(0.87859)		 2.02131
(0.96383)		 1.85086
(0.77220)		 1.76239
(0.86758)
2.5
(1.25)		 2.94406
(1.39531)		 2.47584
(1.04441)		 2.60137
(1.24491)		 2.66127
(1.67912)
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Table 2.6: yt = θ exp{σ εt }, θ0 = 0.5 unknown.
σ0 σML σAML σFS4 σFS5
0.5
(0.07071)		 0.49833
(0.07363)		 0.50465
(0.05902)		 0.51392
(0.07595)		 0.51615
(0.07917)
1.0
(0.14142)		 0.96747
(0.13718)		 1.01593
(0.15203)		 0.99777
(0.14147)		 1.00530
(0.14921)
1.5
(0.21213)		 1.46416
(0.20018)		 1.57917
(0.20810)		 1.51002
(0.20645)		 1.52565
(0.20178)
2.0
(0.28284)		 1.91182
(0.26826)		 2.02255
(0.30263)		 1.97169
(0.27663)		 1.98045
(0.28373)
2.5
(0.35355)		 2.35182
(0.34431)		 2.55177
(0.31764)		 2.42537
(0.35497)		 2.41618
(0.37428)
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Table 2.7: yt = θ exp{σ εt }, θ0 = 1.0 unknown.
σ0 σML σAML σFS4 σFS5
0.5
(0.07071)		 0.49091
(0.06216)		 0.51377
(0.07846)		 0.50625
(0.06410)		 0.50658
(0.06905)
1.0
(0.14142)		 0.95334
(0.11370)		 1.03313
(0.14335)		 0.98317
(0.11727)		 0.97787
(0.11955)
1.5
(0.21213)		 1.50656
(0.20587)		 1.54830
(0.18891)		 1.55366
(0.21225)		 1.56091
(0.22191)
2.0
(0.28284)		 1.93671
(0.26211)		 2.03908
(0.28731)		 1.99722
(0.27030)		 2.00856
(0.28274)
2.5
(0.35355)		 2.41128
(0.37660)		 2.50508
(0.35421)		 2.48665
(0.38839)		 2.48049
(0.38903)
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Table 2.8: yt = θ exp{σ εt }, θ0 = 1.5 unknown.
σ0 σML σAML σFS4 σFS5
0.5
(0.07071)		 0.49550
(0.06671)		 0.51824
(0.07241)		 0.51099
(0.06879)		 0.50870
(0.07337)
1.0
(0.14142)		 0.95377
(0.16094)		 1.00443
(0.16127)		 0.98363
(0.16601)		 0.98987
(0.17222)
1.5
(0.21213)		 1.46338
(0.21279)		 1.49301
(0.17165)		 1.50916
(0.21942)		 1.51119
(0.23762)
2.0
(0.28284)		 1.98589
(0.25301)		 2.01344
(0.29419)		 2.04787
(0.26088)		 2.04437
(0.24295)
2.5
(0.35355)		 2.49198
(0.34645)		 2.50489
(0.37363)		 2.56991
(0.35699)		 2.55664
(0.34953)
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Table 2.9: yt = θ exp{σ εt }, θ0 = 2.0 unknown.
σ0 σML σAML σFS4 σFS5
0.5
(0.07071)		 0.48575
(0.08078)		 0.50939
(0.06839)		 0.50093
(0.08331)		 0.50093
(0.08477)
1.0
(0.14142)		 0.96877
(0.15031)		 1.02918
(0.13286)		 0.99913
(0.15505)		 1.00187
(0.15961)
1.5
(0.21213)		 1.38442
(0.17513)		 1.52880
(0.22731)		 1.42776
(0.18060)		 1.43016
(0.18721)
2.0
(0.28284)		 1.88710
(0.29317)		 2.00304
(0.24401)		 1.94601
(0.30239)		 1.95049
(0.31349)
2.5
(0.35355)		 2.44820
(0.30613)		 2.49672
(0.38955)		 2.52511
(0.31582)		 2.50859
(0.32579)
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Table 2.10: yt = θ exp{σ εt }, θ0 = 2.5 unknown.
σ0 σML σAML σFS4 σFS5
0.5
(0.07071)		 0.47730
(0.06512)		 0.50518
(0.06330)		 0.49222
(0.06716)		 0.49245
(0.07024)
1.0
(0.14142)		 0.96464
(0.12754)		 1.01922
(0.15870)		 0.99488
(0.13157)		 0.98970
(0.13865)
1.5
(0.21213)		 1.49306
(0.18967)		 1.58642
(0.24435)		 1.53970
(0.19558)		 1.55192
(0.20424)
2.0
(0.28284)		 1.91047
(0.27609)		 1.98177
(0.28417)		 1.97007
(0.28509)		 1.95996
(0.29313)
2.5
(0.35355)		 2.41113
(0.33638)		 2.55803
(0.35014)		 2.48676
(0.34810)		 2.46551
(0.34740)
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Table 3: zt = 1[yt > 0], yt = α + xt β + &epsilont
β0 βOLS / σOLS &betaFS4
-1.0
(0.46906)		 -0.92195
(0.22081)		 -0.93661
(0.26350)
-0.5
(0.42959)		 -0.49762
(0.37841)		 -0.44320
(0.33625)
0.0
(0.41713)		 -0.03724
(0.27804)		 -0.02992
(0.20932)
0.5
(0.42959)		 0.56398
(0.29731)		 0.53985
(0.27691)
1.0
(0.46906)		 0.84478
(0.23113)		 1.01222
(0.20596)
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