Unbiased Estimation Finite sample estimation of non-linear equations

125,000 Monte Carlo runs

This could be a good number for a Monte Carlo experiment, people often use 10,000. By the difference with 6,250,000 runs, we can see what we lose, wich is not a lot.

On other pages, you can see results with 6,250,000 runs, 2,500 runs and 50 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)		 0.51027
(0.10325)		 0.50016
(0.10120)		 0.50974
(0.10314)		 0.50012
(0.10120)		 0.50018
(0.10269)
1.0
(0.2)		 1.02127
(0.20644)		 1.00105
(0.20236)		 1.02023
(0.20623)		 1.00103
(0.20236)		 1.00100
(0.20545)
1.5
(0.3)		 1.52937
(0.30910)		 1.49908
(0.30298)		 1.52780
(0.30878)		 1.49906
(0.30298)		 1.49906
(0.30722)
2.0
(0.4)		 2.04081
(0.41206)		 2.00040
(0.40390)		 2.03872
(0.41164)		 2.00040
(0.40393)		 2.00083
(0.40995)
2.5
(0.5)		 2.55114
(0.51497)		 2.50063
(0.50477)		 2.54852
(0.51444)		 2.50063
(0.50463)		 2.50076
(0.51184)
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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)		 0.50238
(0.05056)		 0.49986
(0.05018)		 0.49975
(0.05034)		 0.49987
(0.05043)
1.0
(0.10)		 1.00521
(0.10093)		 1.00001
(0.10044)		 1.00020
(0.10043)		 1.00021
(0.10068)
1.5
(0.15)		 1.50726
(0.15191)		 1.49964
(0.15017)		 1.49963
(0.15114)		 1.49972
(0.15153)
2.0
(0.20)		 2.00958
(0.20162)		 2.00018
(0.20087)		 1.99945
(0.20067)		 1.99956
(0.20113)
2.5
(0.25)		 2.51257
(0.25202)		 2.49920
(0.25028)		 2.49993
(0.25072)		 2.50003
(0.25137)
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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)		 0.50966
(0.10280)		 0.49949
(0.10104)		 0.49947
(0.10083)		 0.49954
(0.10212)
1.0
(0.20)		 1.02055
(0.20586)		 1.00022
(0.20256)		 1.00045
(0.20191)		 1.00037
(0.20441)
1.5
(0.30)		 1.52842
(0.30919)		 1.49807
(0.30142)		 1.49824
(0.30314)		 1.49800
(0.30689)
2.0
(0.40)		 2.04193
(0.41298)		 1.99801
(0.40458)		 2.00152
(0.40500)		 2.00139
(0.40969)
2.5
(0.50)		 2.54835
(0.51496)		 2.49831
(0.50661)		 2.49782
(0.50483)		 2.49791
(0.51090)
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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)		 0.52242
(0.16034)		 0.49939
(0.15366)		 0.49942
(0.15349)		 0.49954
(0.15984)
1.0
(0.30)		 1.04560
(0.32269)		 0.99886
(0.30706)		 0.99957
(0.30860)		 1.00000
(0.32158)
1.5
(0.45)		 1.56854
(0.48055)		 1.49715
(0.46009)		 1.49941
(0.45978)		 1.49930
(0.47790)
2.0
(0.60)		 2.09185
(0.64342)		 1.99651
(0.61336)		 1.99992
(0.61601)		 1.99922
(0.64047)
2.5
(0.75)		 2.61487
(0.80101)		 2.49425
(0.76324)		 2.50023
(0.76678)		 2.49998
(0.79750)
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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)		 0.54109
(0.22547)		 0.49843
(0.20871)		 0.49958
(0.20848)		 0.50013
(0.23425)
1.0
(0.40)		 1.08660
(0.45337)		 0.99513
(0.41361)		 1.00279
(0.41896)		 1.00323
(0.47156)
1.5
(0.60)		 1.62897
(0.67979)		 1.49599
(0.62338)		 1.50360
(0.62914)		 1.50285
(0.70363)
2.0
(0.80)		 2.16676
(0.89957)		 1.99014
(0.82959)		 2.00036
(0.83253)		 2.00009
(0.93260)
2.5
(1.00)		 2.70631
(1.12661)		 2.49817
(1.04311)		 2.49869
(1.03980)		 2.49933
(1.17234)
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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)		 0.56499
(0.30003)		 0.49764
(0.26677)		 0.49858
(0.26553)		 0.49974
(0.35933)
1.0
(0.50)		 1.13279
(0.60597)		 0.99523
(0.53029)		 0.99931
(0.53604)		 0.99875
(0.72132)
1.5
(0.75)		 1.69911
(0.90280)		 1.49678
(0.80108)		 1.49951
(0.79929)		 1.50255
(1.08155)
2.0
(1.00)		 2.27264
(1.21181)		 1.99091
(1.06214)		 2.00551
(1.07254)		 2.00936
(1.44940)
2.5
(1.25)		 2.83849
(1.52207)		 2.48531
(1.33057)		 2.50496
(1.34655)		 2.49854
(1.81596)
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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)		 0.48508
(0.07033)		 0.50530
(0.07354)		 0.50025
(0.07255)		 0.50026
(0.07453)
1.0
(0.14142)		 0.96933
(0.14023)		 1.01074
(0.14700)		 0.99969
(0.14462)		 0.99969
(0.14876)
1.5
(0.21213)		 1.45517
(0.21102)		 1.51672
(0.22040)		 1.50074
(0.21763)		 1.50043
(0.22392)
2.0
(0.28284)		 1.93973
(0.28160)		 2.02170
(0.29391)		 2.00048
(0.29040)		 2.00034
(0.29859)
2.5
(0.35355)		 2.42490
(0.35217)		 2.52771
(0.36695)		 2.50073
(0.36309)		 2.50059
(0.37334)
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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)		 0.48484
(0.07005)		 0.50559
(0.07333)		 0.49999
(0.07224)		 0.49992
(0.07429)
1.0
(0.14142)		 0.96926
(0.14060)		 1.01179
(0.14708)		 0.99958
(0.14501)		 0.99967
(0.14913)
1.5
(0.21213)		 1.45503
(0.21157)		 1.51681
(0.21989)		 1.50053
(0.21816)		 1.50051
(0.22443)
2.0
(0.28284)		 1.94083
(0.28213)		 2.02205
(0.29327)		 2.00146
(0.29091)		 2.00142
(0.29912)
2.5
(0.35355)		 2.42594
(0.35274)		 2.52679
(0.36643)		 2.50176
(0.36380)		 2.50195
(0.37379)
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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)		 0.48491
(0.07059)		 0.50575
(0.07360)		 0.50007
(0.07279)		 0.50009
(0.07488)
1.0
(0.14142)		 0.96952
(0.14019)		 1.01104
(0.14649)		 0.99987
(0.14460)		 0.99998
(0.14882)
1.5
(0.21213)		 1.45439
(0.21058)		 1.51666
(0.21963)		 1.49992
(0.21714)		 1.49989
(0.22323)
2.0
(0.28284)		 1.93837
(0.28124)		 2.02273
(0.29396)		 1.99895
(0.29003)		 1.99919
(0.29869)
2.5
(0.35355)		 2.42304
(0.35068)		 2.52751
(0.36684)		 2.49888
(0.36173)		 2.49873
(0.37176)
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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)		 0.48458
(0.07032)		 0.50561
(0.07315)		 0.49973
(0.07251)		 0.49979
(0.07462)
1.0
(0.14142)		 0.96921
(0.14087)		 1.01070
(0.14686)		 0.99958
(0.14531)		 0.99960
(0.14936)
1.5
(0.21213)		 1.45331
(0.21080)		 1.51577
(0.21916)		 1.49880
(0.21737)		 1.49902
(0.22364)
2.0
(0.28284)		 1.93876
(0.28193)		 2.02252
(0.29360)		 1.99938
(0.29080)		 1.99954
(0.29914)
2.5
(0.35355)		 2.42524
(0.35280)		 2.52717
(0.36635)		 2.50122
(0.36405)		 2.50108
(0.37423)
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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)		 0.48459
(0.07014)		 0.50528
(0.07342)		 0.49973
(0.07234)		 0.49977
(0.07435)
1.0
(0.14142)		 0.96980
(0.14022)		 1.01104
(0.14670)		 1.00019
(0.14461)		 1.00011
(0.14870)
1.5
(0.21213)		 1.45343
(0.21117)		 1.51671
(0.22003)		 1.49884
(0.21771)		 1.49886
(0.22409)
2.0
(0.28284)		 1.93896
(0.28078)		 2.02161
(0.29356)		 1.99957
(0.28959)		 1.99957
(0.29736)
2.5
(0.35355)		 2.42540
(0.35129)		 2.52692
(0.36625)		 2.50130
(0.36280)		 2.50164
(0.37266)
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Table 3: zt = 1[yt > 0], yt = α + xt β + &epsilont
β0 βOLS / σOLS &betaFS4
-1.0
(0.46906)		 -0.95726
(0.27352)		 -1.00331
(0.32987)
-0.5
(0.42959)		 -0.56037
(0.33492)		 -0.49457
(0.30620)
0.0
(0.41713)		 -0.00330
(0.35854)		 -0.00124
(0.28671)
0.5
(0.42959)		 0.56087
(0.33782)		 0.49882
(0.30535)
1.0
(0.46906)		 0.95920
(0.27307)		 1.00208
(0.30467)
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