Unbiased Estimation Finite sample estimation of non-linear equations

Monte Carlo runs

The following is the result of 6,250,000 Monte Carlo runs of nonlinear models using finite or small sample estimators that are unbiased. It is the 'Results' section of an article that will be posted shortly. The article present the FS1, FS4 and FS5 schemes to estimate without bias when faced with finite sample. Every example uses 25 observations.

The estimated models are specified in the title box of each table. The standard error is in parentheses below each estimate with the Cramér-Rao lower bound below the real value. The last model (Table 3) is a Probit. There is an error in it's (Table 3) calculation of the Cramér-Rao bound; I'll get to it later. Also the FS1 result in Table 1 does not make any sense. This is due to a small programming error (which I can't find). I think anyway that it's possible to show analytically that this estimator should be exactly the same as the adjusted maximum likelihood (AML) one. In any case, the FS1 never was very interesting; hence the little effort I'll make on it.

These results come from the average of 6,250,000 estimates by each estimator for each model. This takes a few hours on today's PCs. Results with less runs are much faster, but a less precise. On other pages, you can see results with 125,000 runs, 2,500 runs and 50 runs.

The estimation was done using the GilLib matrix library in C. It was designed for people who want to program in C without having to learn anything about pointers, memory allocation or object programming. In other words, people who use high level software like MATLAB, GAUSS, SAS, ...

  • Here is a manual for this library in PDF format: manual.pdf
  • Here is the library itself: gillib-0.9.tgz

    • For a description of the FS1 estimator, see here.

    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)    		 0.51012
    (0.10306)    		 0.50002
    (0.10102)    		 0.50959
    (0.10296)    		 0.49997
    (0.10102)    		 0.50002
    (0.10250)
    1.0
    (0.2)    		 1.02024
    (0.20607)    		 1.00004
    (0.20199)    		 1.01919
    (0.20586)    		 1.00001
    (0.20199)    		 1.00005
    (0.20496)
    1.5
    (0.3)    		 1.53030
    (0.30923)    		 1.50000
    (0.30311)    		 1.52873
    (0.30891)    		 1.49998
    (0.30311)    		 1.49998
    (0.30750)
    2.0
    (0.4)    		 2.04036
    (0.41232)    		 1.99996
    (0.40416)    		 2.03826
    (0.41190)    		 1.99996
    (0.40419)    		 1.99997
    (0.41001)
    2.5
    (0.5)    		 2.55048
    (0.51537)    		 2.49998
    (0.50516)    		 2.54786
    (0.51484)    		 2.49998
    (0.50502)    		 2.49999
    (0.51247)
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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)    		 0.50249
    (0.05037)    		 0.49987
    (0.05012)    		 0.49986
    (0.05014)    		 0.49998
    (0.05024)
    1.0
    (0.10)    		 1.00505
    (0.10073)    		 0.99977
    (0.10028)    		 1.00004
    (0.10023)    		 1.00003
    (0.10048)
    1.5
    (0.15)    		 1.50750
    (0.15111)    		 1.49958
    (0.15038)    		 1.49987
    (0.15034)    		 1.49998
    (0.15072)
    2.0
    (0.20)    		 2.01010
    (0.20144)    		 1.99960
    (0.20049)    		 1.99997
    (0.20048)    		 2.00008
    (0.20091)
    2.5
    (0.25)    		 2.51250
    (0.25193)    		 2.49944
    (0.25067)    		 2.49984
    (0.25064)    		 2.49996
    (0.25128)
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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)    		 0.51012
    (0.10313)    		 0.49953
    (0.10096)    		 0.49992
    (0.10115)    		 0.50003
    (0.10234)
    1.0
    (0.20)    		 1.02018
    (0.20608)    		 0.99921
    (0.20189)    		 1.00006
    (0.20212)    		 0.99998
    (0.20452)
    1.5
    (0.30)    		 1.53019
    (0.30919)    		 1.49884
    (0.30289)    		 1.49996
    (0.30313)    		 1.49988
    (0.30685)
    2.0
    (0.40)    		 2.04038
    (0.41238)    		 1.99836
    (0.40399)    		 1.99997
    (0.40436)    		 1.99997
    (0.40926)
    2.5
    (0.50)    		 2.55052
    (0.51498)    		 2.49815
    (0.50514)    		 2.49997
    (0.50497)    		 2.50002
    (0.51109)
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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)    		 0.52294
    (0.16045)    		 0.49906
    (0.15334)    		 0.49993
    (0.15360)    		 0.49995
    (0.15979)
    1.0
    (0.30)    		 1.04618
    (0.32098)    		 0.99795
    (0.30681)    		 1.00018
    (0.30706)    		 1.00010
    (0.31965)
    1.5
    (0.45)    		 1.56917
    (0.48190)    		 1.49745
    (0.46009)    		 1.50000
    (0.46110)    		 1.50011
    (0.47969)
    2.0
    (0.60)    		 2.09204
    (0.64209)    		 1.99624
    (0.61321)    		 1.99996
    (0.61478)    		 2.00001
    (0.63945)
    2.5
    (0.75)    		 2.61486
    (0.80232)    		 2.49527
    (0.76606)    		 2.50002
    (0.76790)    		 2.49969
    (0.79893)
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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)    		 0.54153
    (0.22569)    		 0.49827
    (0.20790)    		 0.50002
    (0.20866)    		 0.49988
    (0.23391)
    1.0
    (0.40)    		 1.08328
    (0.45144)    		 0.99679
    (0.41619)    		 0.99987
    (0.41729)    		 0.99983
    (0.46809)
    1.5
    (0.60)    		 1.62478
    (0.67715)    		 1.49513
    (0.62421)    		 1.49964
    (0.62654)    		 1.49990
    (0.70193)
    2.0
    (0.80)    		 2.16608
    (0.90201)    		 1.99309
    (0.83219)    		 1.99965
    (0.83483)    		 1.99954
    (0.93510)
    2.5
    (1.00)    		 2.70847
    (1.12780)    		 2.49184
    (1.03932)    		 2.50067
    (1.04114)    		 2.50009
    (1.16940)
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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)    		 0.56650
    (0.30175)    		 0.49760
    (0.26596)    		 0.49993
    (0.26696)    		 0.49981
    (0.36009)
    1.0
    (0.50)    		 1.13305
    (0.60372)    		 0.99542
    (0.53254)    		 0.99975
    (0.53437)    		 1.00017
    (0.72065)
    1.5
    (0.75)    		 1.70011
    (0.90615)    		 1.49299
    (0.79819)    		 1.50025
    (0.80261)    		 1.50003
    (1.08057)
    2.0
    (1.00)    		 2.26595
    (1.20751)    		 1.99091
    (1.06413)    		 1.99982
    (1.06860)    		 1.99975
    (1.44155)
    2.5
    (1.25)    		 2.83222
    (1.50961)    		 2.48766
    (1.32938)    		 2.49942
    (1.34786)    		 2.49969
    (1.79898)
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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)    		 0.48481
    (0.07036)    		 0.50553
    (0.07328)    		 0.49997
    (0.07257)    		 0.50000
    (0.07463)
    1.0
    (0.14142)    		 0.96966
    (0.14071)    		 1.01105
    (0.14663)    		 1.00003
    (0.14512)    		 1.00001
    (0.14923)
    1.5
    (0.21213)    		 1.45445
    (0.21096)    		 1.51642
    (0.21996)    		 1.49999
    (0.21757)    		 1.49996
    (0.22375)
    2.0
    (0.28284)    		 1.93910
    (0.28133)    		 2.02181
    (0.29337)    		 1.99983
    (0.29012)    		 1.99981
    (0.29840)
    2.5
    (0.35355)    		 2.42419
    (0.35155)    		 2.52742
    (0.36677)    		 2.50000
    (0.36246)    		 2.50007
    (0.37295)
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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)    		 0.48489
    (0.07036)    		 0.50544
    (0.07334)    		 0.50004
    (0.07256)    		 0.50007
    (0.07462)
    1.0
    (0.14142)    		 0.96969
    (0.14064)    		 1.01082
    (0.14667)    		 1.00002
    (0.14504)    		 1.00004
    (0.14917)
    1.5
    (0.21213)    		 1.45449
    (0.21100)    		 1.51640
    (0.21996)    		 1.49997
    (0.21757)    		 1.50002
    (0.22383)
    2.0
    (0.28284)    		 1.93930
    (0.28133)    		 2.02167
    (0.29337)    		 1.99988
    (0.29008)    		 2.00009
    (0.29844)
    2.5
    (0.35355)    		 2.42397
    (0.35168)    		 2.52763
    (0.36671)    		 2.49973
    (0.36271)    		 2.49982
    (0.37298)
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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)    		 0.48477
    (0.07032)    		 0.50536
    (0.07335)    		 0.49993
    (0.07251)    		 0.49996
    (0.07459)
    1.0
    (0.14142)    		 0.96963
    (0.14065)    		 1.01100
    (0.14664)    		 0.99998
    (0.14507)    		 1.00000
    (0.14920)
    1.5
    (0.21213)    		 1.45455
    (0.21094)    		 1.51637
    (0.22002)    		 1.50008
    (0.21751)    		 1.50006
    (0.22372)
    2.0
    (0.28284)    		 1.93911
    (0.28110)    		 2.02207
    (0.29340)    		 1.99971
    (0.28988)    		 1.99977
    (0.29816)
    2.5
    (0.35355)    		 2.42406
    (0.35165)    		 2.52725
    (0.36664)    		 2.49992
    (0.36273)    		 2.49995
    (0.37303)
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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)    		 0.48483
    (0.07038)    		 0.50546
    (0.07332)    		 0.49998
    (0.07258)    		 0.50000
    (0.07465)
    1.0
    (0.14142)    		 0.96957
    (0.14065)    		 1.01098
    (0.14674)    		 0.99995
    (0.14508)    		 0.99995
    (0.14921)
    1.5
    (0.21213)    		 1.45440
    (0.21107)    		 1.51637
    (0.21993)    		 1.49993
    (0.21765)    		 1.49994
    (0.22390)
    2.0
    (0.28284)    		 1.93937
    (0.28142)    		 2.02176
    (0.29345)    		 2.00000
    (0.29028)    		 2.00006
    (0.29853)
    2.5
    (0.35355)    		 2.42405
    (0.35154)    		 2.52745
    (0.36650)    		 2.50001
    (0.36277)    		 2.49994
    (0.37286)
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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)    		 0.48488
    (0.07033)    		 0.50544
    (0.07335)    		 0.50003
    (0.07253)    		 0.50007
    (0.07461)
    1.0
    (0.14142)    		 0.96969
    (0.14068)    		 1.01091
    (0.14657)    		 1.00008
    (0.14509)    		 1.00005
    (0.14921)
    1.5
    (0.21213)    		 1.45458
    (0.21095)    		 1.51636
    (0.21990)    		 1.50003
    (0.21748)    		 1.50012
    (0.22376)
    2.0
    (0.28284)    		 1.93913
    (0.28125)    		 2.02186
    (0.29327)    		 1.99974
    (0.29012)    		 1.99983
    (0.29841)
    2.5
    (0.35355)    		 2.42393
    (0.35150)    		 2.52730
    (0.36678)    		 2.49985
    (0.36750)    		 2.49983
    (0.37283)
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    Table 3: zt = 1[yt > 0], yt = α + xt β + &epsilont
    β0 βOLS / σOLS &betaFS4
    -1.0
    (0.46906)    		 -0.95792
    (0.27339)    		 -0.99944
    (0.32213)
    -0.5
    (0.42959)    		 -0.56040
    (0.33589)    		 -0.49517
    (0.30735)
    0.0
    (0.41713)    		 -0.00026
    (0.35861)    		 0.00143
    (0.28714)
    0.5
    (0.42959)    		 0.56053
    (0.33560)    		 0.49961
    (0.30393)
    1.0
    (0.46906)    		 0.95768
    (0.27329)    		 1.00216
    (0.31657)
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