Revisiting the Neyman-Scott model: an Inconsistent MLE or an Ill-defined Model?

0.09 MB·English

by  Aris Spanos

Revisiting the Neyman-Scott model:
an Inconsistent MLE or an Ill-defined Model?
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0 Aris Spanos
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Department of Economics,

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Virginia Tech, USA
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January 2013
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] Abstract
E
M The Neyman and Scott (1948) model is widely used to demonstrate
a serious weakness of the Maximum Likelihood (ML) method: it can
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t give rise to inconsistent estimators. The primary objective of this paper
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t is to revisit this example with a view to demonstrate that the culprit
s for the inconsistent estimation is not the ML method but an ill-defined
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statistical model. It is also shown that a simple recasting of this model

1 renders it well-defined and the ML method gives rise to consistent and
v asymptotically efficient estimators.
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1 Introduction
Despite claims for priority by a number of different authors, Maximum Likeli-
hood(ML)wasfirstarticulatedasageneralmethodofestimationinthecontext
of a parametric statistical model by Fisher (1922). It took several decades to
establish the regularity conditions needed to ensure the key asymptotic prop-
erties of ML Estimators (MLE), such as efficiency and consistency (Cramer,
1946, Wald, 1949), but since then ML has dominated estimation in frequentist
statistics; see Stigler (2007), Hald (2007) for this history.
Severalcounterexampleswereproposedinthe1940sand1950sraisingdoubts
aboutthegeneralityoftheMLmethod. ThesecounterexamplesincludeHodges’s
localsuperefficientestimator(LeCam,1953),themixtureoftwoNormaldistri-
butions (Cox, 2006)) and the inconsistent MLE example proposed by the Ney-
manandScott (1948)model. Commenting onthese examplesStigler (2007),p.
613,arguedthatnoneareconsideredseriousenoughtounderminethecredibility
of the ML method, and singled out the last example:
“TheWald-Neyman-Scottexamplewasofmorepracticalimport,andstillserves
as a warning of what might occur in modern highly parameterized problems,where
the information in the data may be spread too thinly to achieve asymptotic consis-
tency.”
The primary objective of this note is to revisit the Neyman-Scott example
withaviewtounpackStigler’sassessmentbydemonstratingthattherealculprit
for the inconsistentestimator is not the ML method, as such, but an ill-defined
statistical model. It is also shown that a simple recasting of this model renders
it well-defined and the ML method gives rise to consistent and asymptotically
efficient estimators.
2 The Neyman-Scott model
The quintessential example used to demonstrate that ML might give rise to
inconsistent estimators is the Neyman-Scott model:
X =µ +ε ,
it t it
i=1,2, t=1,2,...,n,...
εit ∽NIID(0,σ2)=0, )
which can be viewed as a simple time effects panel data model.
Theunderlyingdistributionoftheobservablerandomvariables(X ,X )is
1t 2t
bivariate Normal Independent, but not Identically Distributed:
X µ σ2 0
X := 1t ∽NI t , , t=1,2,...,n,...
t X µ 0 σ2
(cid:18) 2t (cid:19) (cid:18)(cid:18) t (cid:19) (cid:18) (cid:19)(cid:19)
In light of the fact that the non-ID assumption implies that this model
suffers from the incidental parameter problem, in the sense that the unknown
parameters (µ ,µ ,...,µ ) increase with the sample size, the latter are viewed
1 2 n
as nuisance parameters, and σ2 as the only parameter of interest.
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3 Maximum Likelihood Estimation?
DespitetheincidentalparameterproblemtheNeyman-Scottmodelcontinuesto
be used as a counterexample to the ML method. That is, the literature ignores
the incidentalparameter problem,and defines the distribution of the sample to
be:
f(x;θ) = n 2 σ√12πe{−2σ12(xit−µt)2}= n 2π1σ2e{−2σ12[(x1t−µt)2+(x2t−µt)2]}
t=1i=1 t=1
This gives risQe toQthe ‘log-likelihood functioQn’:
lnL(θ;x) =−nlnσ2− 1 n [(x −µ )2+(x −µ )2].
2σ2 t=1 1t t 2t t
The ‘Maximum Likelihood EstimatorsP’ (MLE) are supposed to be derived by
solving the first-order conditions:
∂lnL(θ;x) = 1 [(x −µ )+(x −µ )]=0,
∂µt σ2 1t t 2t t
∂lnL(θ;x) =− n + 1 n [(x −µ )2+(x −µ )2]=0,
∂σ2 σ2 2σ4 t=1 1t t 2t t
giving rise to:
P
µ =1(X +X ), t=1,2,...,n,
t 2 1t 2t
n n
σ2= 1 [(X −µ )2+(X −µ )2]=1 s2,
b 2n 1t t 2t t n t
t=1 t=1
P P
where s2=1[(Xb−µ )2+(X −µb )2]. b
t 2 1t t 2t t
Notice that for lnL(θ;x), θ :=(µ ,σˆ2,t=1,2,...,n) is a maximum since
MLE t
the second derivativbes at θ =θ areb:
∂2ln∂Lµ2t(θ;x) θ=θbMbbLE= −bσ22 θ=θbMLE=− σˆ22 <0,
(cid:12)(cid:12) (cid:0) (cid:1)(cid:12)(cid:12)
(cid:12)
∂2∂lnσ2L∂(µθt;x) θ=θbMLE= −σ14 nt=1[(x1t−µt)+(x2t−µt)] θ=θbMLE=0
(cid:12)(cid:12) P (cid:12)(cid:12)
(cid:12)
∂2ln∂Lσ4(θ;x) θ=θbMLE= σn4−σ16 nt=1[(x1t−µt)2+(x2t−µt)2] θ=θbMLE=− σˆn4 <0,
(cid:12)
and thus(cid:12): P (cid:12)
(cid:12) (cid:12)
∂2lnL(θ;x) ∂2lnL(θ;x) − ∂2lnL(θ;x) >0.
∂µ2t ∂σ4 ∂σ2∂µt θ=θbMLE
(cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17)(cid:12)
The commonly used argument against the ML met(cid:12)hod is that since:
(cid:12)
E(µ )=µ and E(s2)=1σ2,
t t t 2
it follows that the ‘MLE’ σ2 is both biased and inconsistent because:
b
E(σ2)=1 n E(s2)=1σ2, and σ2 a→.s. 1σ2,
n bt=1 t 2 2
since the bias E(σ2)−σ2=P−1σ2 does not go to 0 as n→∞.
b 2 b
b 3
A moment’sreflectionrevealsthatthe inconsistency argumentisill-thought
out. Theis becausethe incidentalparameterproblemrendersµ =1(X +X )
t 2 1t 2t
an inconsistent estimatorofµ ,fort=1,2,...,n;there areonlytwoobservations
t
foreachµ ,andthustheirvarianceVar(µ )=1σ2 doesnotgotozeroasn→∞.
t t 2 b
WhatthecriticsoftheMLmethoddonotappreciateenoughisthefactthat
treating the unknown parameters (µ ,µ ,...,µ ) as incidental and designating
b 1 b2 n
σ2 the only parameter of interest, does not let the statistician ‘off the hook’.
Thisisbecausetheparameterofinterestσ2=E(X −µ )2,definingthevariation
it t
around µ , invokes the incidental parameters. Put more intuitively, when the
t
data come in the form of Z :=(z ,z ,...,z ), z :=(x ,x ), one can get to σ2
0 1 2 n t 1t 2t
via µ , and using µ leads to problems because it is an inconsistent estimator.
t t
In that sense (Severini, 2000):
“... this modelbfalls outside of the general framework we are considering since
the dimension of the parameter (µ ,µ ,...,µ ,σ2) depends on the sample size.”
1 2 n
(p. 108)
n
That is, calling σ2= 1 [(X −µ )2+(X −µ )2] a MLE is highly misleading
2n 1t t 2t t
t=1
sincetheMLmethodwaPsnevermeanttobeappliedtostatisticalmodelswhose
number of unknobwn parameters inbcreases withbthe sample size n.
In truth, one should be very skeptical of any method of estimation which
yields consistent estimators in cases where the statistical model in question is
ill-defined, asinthe caseofthe incidentalparameterproblem. Hence, the more
interesting question should be:
why would the ML method yield a consistent estimator of σ2?
ThefactthattheMLmethoddoesnot yieldaconsistentestimatorofσ2 should
count in its favor not against it! To paraphrase Stigler’s quotation: the ML
method ‘warns the modeler that the information in the data has been spread too
thinly’.
4 Recasting the original Neyman-Scott model
The question that naturally arises is: can one respecify the above statistical
model to render it well-defined but retaining the parameter of interest? The
answer is surprisingly straightforward. Since the incidental parameter problem
arises because of the unknown but t-varying means (µ ,µ ,...,µ ), one can
1 2 n
re-specify the original bivariate model into a univariate simple Normal (one
parameter), using the transformation:
Y = 1 (X −X )∽NIID 0,σ2 , t=1,2,...,n,...,
t √2 1t 2t
(cid:0) (cid:1)
E(Y )= 1 E(X −X )= 1 (µ −µ )=0,
t √2 1t 2t √2 t t
Var(Y )= 1[Var(X )+Var(X )]=σ2.
t 2 1t 2t
This is a sensible thing to do because taking the difference eliminates the nui-
sance parameters (µ ,µ ,...,µ ), without affecting the parameter of interest.
1 2 n
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For this simple Normal model, the MLE for σ2 is: σ2 =1 n Y2,
MLE n t=1 t
which it is unbiased, fully efficient and strongly consistent: P
b
E(σ2 )=σ2, Var(σ2 )=2σ4, σ2 a→.s.σ2.
MLE MLE n MLE
Notes:
(i) The above rbecasting of the Nebyman-Scott mbodel can be easily extended
to the case (X ,X ,...,X ), 2≤m<n.
1t 2t mt
(ii) Hald (2007), p. 182-3 offers an alternative, highly original, way to
sidesteptheincidentalparameterproblemusingFisher’stwostageMLmethod.
5 Conclusion
The main conclusion from the above discussion is that when the ML method
gives rise to inconsistent estimators, the modeler should take a closer look at
the assumed statistical model; chances are, it is ill-defined. This is particularly
trueinthe casewherethe assumedmodelsuffersfromthe incidentalparameter
problem. Insuchcasesthewayforwardistorecasttheoriginalmodeltorender
it well-defined and then apply the ML method. This argument is illustrated
above using the Neyman-Scott (1948) model.
References
[1] Cox,D. R. (2006), Principles of Statistical Inference, CambridgeUniversity
Press, Cambridge.
[2] Cramer,H.(1946),MathematicalMethodsofStatistics,PrincetonUniversity
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[4] Hald,A.(2007),AHistoryofParametricStatisticalInferencefromBernoulli
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[5] Le Cam,L.(1953),“Onsomeasymptotic propertiesofmaximumlikelihood
estimates and related Bayesian estimates,” University of California Publi-
cations in Statistics, 1: 277-330.
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consistent observations,” Econometrica, 16: 1-32.
[7] Severini, T. A. (2000), Likelihood Methods in Statistics, Oxford University
Press, Oxford.
[8] Stigler, S. M. (2007), “The Epic Story of Maximum Likelihood,” Statistical
Science, 22: 598-620.
[9] Wald, A. (1949), “Note on the consistency of the maximum likelihood esti-
mate,” Annals of Mathematical Statistics, 20: 595-601.
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