Unbiased Estimation Finite sample estimation of non-linear equations

2,500 Monte Carlo runs

2500 is not a good number of runs for a Monte Carlo experiment. It's interesting as it shows why by the lack of precision.

On other pages, you can see results with 6,250,000 runs, 125,000 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)		 0.51223
(0.10112)		 0.50209
(0.09912)		 0.51170
(0.10102)		 0.50204
(0.09912)		 0.50229
(0.10056)
1.0
(0.2)		 1.01978
(0.20574)		 0.99959
(0.20166)		 1.01873
(0.20553)		 0.99956
(0.20167)		 1.00053
(0.20543)
1.5
(0.3)		 1.53578
(0.31398)		 1.50537
(0.30776)		 1.53420
(0.31366)		 1.50534
(0.30776)		 1.50463
(0.31506)
2.0
(0.4)		 2.03347
(0.40345)		 1.99321
(0.39546)		 2.03138
(0.40304)		 1.99321
(0.39550)		 1.99565
(0.40288)
2.5
(0.5)		 2.54231
(0.51420)		 2.49197
(0.50402)		 2.53970
(0.51368)		 2.49199
(0.50398)		 2.49466
(0.50914)
Return to top

Table 2.1: yt = θ exp{σ εt }, σ 0 = 0.5 unknown.
θ0 θML θAML θFS4 θFS5
0.5
(0.05)		 0.50276
(0.05129)		 0.49827
(0.04958)		 0.50014
(0.05107)		 0.50022
(0.05123)
1.0
(0.10)		 1.00535
(0.10089)		 1.00284
(0.09952)		 1.00034
(0.10039)		 1.00060
(0.10059)
1.5
(0.15)		 1.50963
(0.14955)		 1.49751
(0.15227)		 1.50206
(0.14880)		 1.50237
(0.14968)
2.0
(0.20)		 2.00606
(0.20355)		 1.99358
(0.20040)		 1.99595
(0.20241)		 1.99603
(0.20238)
2.5
(0.25)		 2.51146
(0.25647)		 2.50793
(0.24968)		 2.49883
(0.25521)		 2.49896
(0.25580)
Return to top

Table 2.2: yt = θ exp{σ εt }, σ 0 = 1.0 unknown.
θ0 θML θAML θFS4 θFS5
0.5
(0.10)		 0.50840
(0.10250)		 0.49821
(0.10108)		 0.49822
(0.10057)		 0.49838
(0.10144)
1.0
(0.20)		 1.02444
(0.20183)		 0.99999
(0.20163)		 1.00425
(0.19776)		 1.00415
(0.20033)
1.5
(0.30)		 1.53199
(0.31068)		 1.50652
(0.29934)		 1.50211
(0.30519)		 1.50203
(0.30798)
2.0
(0.40)		 2.03803
(0.40401)		 1.99906
(0.41070)		 1.99749
(0.39614)		 1.99867
(0.40139)
2.5
(0.50)		 2.54391
(0.52659)		 2.49770
(0.52327)		 2.49338
(0.51614)		 2.49334
(0.52003)
Return to top

Table 2.3: yt = θ exp{σ εt }, σ 0 = 1.5 unknown.
θ0 θML θAML θFS4 θFS5
0.5
(0.15)		 0.52065
(0.15704)		 0.50009
(0.15542)		 0.49739
(0.15004)		 0.49743
(0.15776)
1.0
(0.30)		 1.03803
(0.30413)		 0.98368
(0.29753)		 0.99246
(0.29098)		 0.99000
(0.30252)
1.5
(0.45)		 1.56771
(0.48856)		 1.50034
(0.47046)		 1.49896
(0.46704)		 1.49560
(0.48407)
2.0
(0.60)		 2.10651
(0.66585)		 1.99064
(0.62739)		 2.01334
(0.63729)		 2.01417
(0.66116)
2.5
(0.75)		 2.61897
(0.81643)		 2.50838
(0.77502)		 2.50432
(0.78221)		 2.50331
(0.82052)
Return to top

Table 2.4: yt = θ exp{σ εt }, σ 0 = 2.0 unknown.
θ0 θML θAML θFS4 θFS5
0.5
(0.20)		 0.54464
(0.22501)		 0.49919
(0.20731)		 0.50315
(0.20858)		 0.50459
(0.23298)
1.0
(0.40)		 1.08059
(0.45786)		 0.99890
(0.43066)		 0.99778
(0.42498)		 0.99809
(0.46686)
1.5
(0.60)		 1.60321
(0.66159)		 1.48929
(0.62482)		 1.48095
(0.61393)		 1.47558
(0.67438)
2.0
(0.80)		 2.14718
(0.92240)		 2.01366
(0.87174)		 1.98080
(0.85270)		 1.97896
(0.93936)
2.5
(1.00)		 2.72250
(1.11931)		 2.50334
(1.02916)		 2.51330
(1.03418)		 2.51107
(1.17331)
Return to top

Table 2.5: yt = θ exp{σ εt }, σ 0 = 2.5 unknown.
θ0 θML θAML θFS4 θFS5
0.5
(0.25)		 0.56794
(0.30818)		 0.49902
(0.26829)		 0.50159
(0.27315)		 0.49787
(0.38587)
1.0
(0.50)		 1.11136
(0.57032)		 1.00467
(0.55106)		 0.98117
(0.50560)		 0.98139
(0.68912)
1.5
(0.75)		 1.69056
(0.91670)		 1.50245
(0.80672)		 1.49076
(0.81242)		 1.46686
(1.07566)
2.0
(1.00)		 2.27930
(1.24934)		 1.98035
(1.07199)		 2.01314
(1.11184)		 2.01675
(1.45688)
2.5
(1.25)		 2.78886
(1.47016)		 2.52604
(1.38458)		 2.46253
(1.29890)		 2.49240
(1.78469)
Return to top

Table 2.6: yt = θ exp{σ εt }, θ0 = 0.5 unknown.
σ0 σML σAML σFS4 σFS5
0.5
(0.07071)		 0.48335
(0.06930)		 0.50532
(0.07245)		 0.49847
(0.07148)		 0.49860
(0.07368)
1.0
(0.14142)		 0.97113
(0.13903)		 1.01180
(0.15099)		 1.00155
(0.14338)		 1.00175
(0.14723)
1.5
(0.21213)		 1.46455
(0.21401)		 1.52110
(0.22503)		 1.51041
(0.22072)		 1.50986
(0.22626)
2.0
(0.28284)		 1.93671
(0.27623)		 2.02728
(0.29317)		 1.99735
(0.28486)		 1.99673
(0.29230)
2.5
(0.35355)		 2.41842
(0.34756)		 2.51949
(0.35470)		 2.49405
(0.35836)		 2.49631
(0.36957)
Return to top

Table 2.7: yt = θ exp{σ εt }, θ0 = 1.0 unknown.
σ0 σML σAML σFS4 σFS5
0.5
(0.07071)		 0.48451
(0.07032)		 0.50124
(0.07374)		 0.49966
(0.07251)		 0.49938
(0.07401)
1.0
(0.14142)		 0.96835
(0.14419)		 1.01577
(0.14177)		 0.99864
(0.14870)		 0.99832
(0.15302)
1.5
(0.21213)		 1.45397
(0.21041)		 1.50981
(0.22335)		 1.49944
(0.21696)		 1.50142
(0.22389)
2.0
(0.28284)		 1.93898
(0.27880)		 2.01702
(0.29545)		 1.99955
(0.28747)		 2.00223
(0.29672)
2.5
(0.35355)		 2.41844
(0.35049)		 2.53427
(0.38289)		 2.49402
(0.36149)		 2.49637
(0.37300)
Return to top

Table 2.8: yt = θ exp{σ εt }, θ0 = 1.5 unknown.
σ0 σML σAML σFS4 σFS5
0.5
(0.07071)		 0.48254
(0.06861)		 0.50381
(0.07320)		 0.49763
(0.07075)		 0.49762
(0.07292)
1.0
(0.14142)		 0.96510
(0.13979)		 1.01012
(0.14608)		 0.99531
(0.14418)		 0.99570
(0.14655)
1.5
(0.21213)		 1.44954
(0.21023)		 1.51340
(0.22600)		 1.49492
(0.21678)		 1.49689
(0.22312)
2.0
(0.28284)		 1.92994
(0.28759)		 2.01791
(0.28875)		 1.99026
(0.29656)		 1.99154
(0.30386)
2.5
(0.35355)		 2.43020
(0.34709)		 2.52702
(0.35964)		 2.50623
(0.35806)		 2.50681
(0.36701)
Return to top

Table 2.9: yt = θ exp{σ εt }, θ0 = 2.0 unknown.
σ0 σML σAML σFS4 σFS5
0.5
(0.07071)		 0.48415
(0.07037)		 0.50395
(0.07219)		 0.49928
(0.07257)		 0.49944
(0.07449)
1.0
(0.14142)		 0.97146
(0.14227)		 1.00896
(0.14395)		 1.00190
(0.14675)		 1.00155
(0.15117)
1.5
(0.21213)		 1.45833
(0.20851)		 1.51507
(0.22738)		 1.50399
(0.21501)		 1.50296
(0.22197)
2.0
(0.28284)		 1.94861
(0.28514)		 2.01463
(0.29041)		 2.00963
(0.29403)		 2.00908
(0.30248)
2.5
(0.35355)		 2.42512
(0.36102)		 2.52457
(0.37163)		 2.50087
(0.37234)		 2.50085
(0.38166)
Return to top

Table 2.10: yt = θ exp{σ εt }, θ0 = 2.5 unknown.
σ0 σML σAML σFS4 σFS5
0.5
(0.07071)		 0.48448
(0.07073)		 0.50656
(0.07334)		 0.49962
(0.07295)		 0.49959
(0.07505)
1.0
(0.14142)		 0.97055
(0.13983)		 1.01135
(0.14838)		 1.00096
(0.14421)		 1.00026
(0.14712)
1.5
(0.21213)		 1.45362
(0.20808)		 1.50972
(0.22176)		 1.49902
(0.21452)		 1.49997
(0.22037)
2.0
(0.28284)		 1.94169
(0.28657)		 2.01326
(0.30084)		 2.00253
(0.29559)		 2.00463
(0.30191)
2.5
(0.35355)		 2.41770
(0.36625)		 2.51833
(0.37409)		 2.49333
(0.37834)		 2.49182
(0.38075)
Return to top


Table 3: zt = 1[yt > 0], yt = α + xt β + &epsilont
β0 βOLS / σOLS &betaFS4
-1.0
(0.46906)		 -0.94988
(0.27061)		 -1.00669
(0.34704)
-0.5
(0.42959)		 -0.56031
(0.33883)		 -0.49936
(0.31079)
0.0
(0.41713)		 -0.01472
(0.37046)		 -0.00942
(0.29448)
0.5
(0.42959)		 0.56639
(0.33136)		 0.51314
(0.30572)
1.0
(0.46906)		 0.94951
(0.26512)		 0.99237
(0.27362)
Return to top