We note that the existence of the maximum likelihood estimates for Poisson regression depends on the data configuration. Because standard software does not check for this problem, the practitioner may be surprised to find that in some applications estimation of the Poisson regression is unusually difficult or even impossible. More seriously, the estimation algorithm may lead to spurious maximum likelihood estimates. We identify the signs of the non-existence of the maximum likelihood estimates and propose a simple empirical strategy to single out the regressors causing this type of identification failure.Poisson estimation, gravity equation

J. M. C. Santos Silva

Silvana Tenreyro

English

Research Papers in Economics

 
 
 
 
 
CEP Discussion Paper No 932 
May 2009 
On the Existence of the Maximum Likelihood Estimates 
for Poisson Regression 
J. M. C. Santos-Silva and Silvana Tenreyro 
  
   
Abstract 
We note that the existence of the maximum likelihood estimates for Poisson regression 
depends on the data configuration. Because standard software does not check for this 
problem, the practitioner may be surprised to find that in some applications estimation of the 
Poisson regression is unusually difficult or even impossible. More seriously, the estimation 
algorithm may lead to spurious maximum likelihood estimates. We identify the signs of the 
non-existence of the maximum likelihood estimates and propose a simple empirical strategy 
to single out the regressors causing this type of identification failure. 
 
Keywords:  Poisson estimation, gravity equation 
JEL Classifications:  C13; C50; F10 
 
 
 
This paper was produced as part of the Centre’s Globalisation Programme.  The Centre for 
Economic Performance is financed by the Economic and Social Research Council. 
 
 
 
 
Acknowledgements 
We are grateful to Ines Buono, Virginia Di Nino, Dave Donaldson, Doireann Fitzgerald, 
Lissandra Flach and Randi Hjalmarsson for posing the problem and for providing examples 
of data sets where it occurs. Special thanks are due to Markus Baldauf for help with the code 
used to bypass the problem. Santos Silva also gratefully acknowledges partial financial 
support from Fundação para a Ciência e Tecnologia (FEDER/POCI 2010).  
  João M. C. Santos-Silva is Professor of Economics at the University of Essex. Silvana 
Tenreyro is an Associate at the Centre for Economic Performance and Reader in the 
Economics Department, London School of Economics. 
 
 
 
 
Published by 
Centre for Economic Performance 
London School of Economics and Political Science 
Houghton Street 
London WC2A 2AE 
 
All rights reserved.  No part of this publication may be reproduced, stored in a retrieval 
system or transmitted in any form or by any means without the prior permission in writing of 
the publisher nor be issued to the public or circulated in any form other than that in which it 
is published. 
 
Requests for permission to reproduce any article or part of the Working Paper should be sent 
to the editor at the above address. 
 
© J. M. C. Santos-Silva and S. Tenreyro, submitted 2009 
 
ISBN 978-0-85328-385-0 1. INTRODUCTION
The Poisson regression model is deﬁned by
Pr(yi = j|xi)=
exp(−λ)λ
j
j!
,j =0 ,1,2,...
where λ is generally speciﬁed as λ =e x p ( x0
iβ)=e x p ( β0 + β1x1i + ...).1 With this
formulation, β, the vector of parameters of interest, can be estimated my maximizing the
log-likelihood function given by
lnL(β)=
n X
i=1
[−exp(x
0
iβ)+( x
0
iβ)yi − ln(yi!)]. (1)
Poisson regression is not only the most widely used model for count data (see Winkel-
mann, 2008, and Cameron and Trivedi, 1997), but it is also becoming increasingly popular
to estimate multiplicative models for other kinds of data (see, among others, Manning and
Mullahy, 2001, and Santos Silva and Tenreyro, 2006).
The reasons that make this estimator popular can be clearly understood by inspecting
the corresponding score vector and Hessian matrix, given respectively by
s(β)=
n X
i=1
[yi − exp(x
0
iβ)]xi, (2)
and
H (β)=−
n X
i=1
exp(x
0
iβ)xix
0
i.
The form of the score vector makes clear that β will be consistently estimated as long
as E(yi|xi)=e x p ( x0
iβ), i.e., the only condition required for consistency is the correct
speciﬁcation of the conditional mean. This is the well known pseudo-maximum likelihood
result of Gourieroux, Monfort and Trognon (1984).
Besides this robustness property, the estimator also has the advantage of being very
well behaved. Indeed, it is easy to see that the Hessian is negative deﬁnite for all x and β,
which facilitates the estimation and ensures the uniqueness of the maximum, if it exists.
1See Winkelmann (2008) and Cameron and Trivedi (1997) for further details and background on the
Poisson regression model and its properties.
2Consequently, estimation of β is relatively simple and generally the estimation algorithm
converges in a handful of iterations, even for relatively large problems.
In spite of this general result, for certain data conﬁgurations, some of the parameters in β
are not identiﬁed by the (pseudo) maximum likelihood estimator described above. That is,
for certain data conﬁgurations, the maximum likelihood estimates do not exist. Because
this type of identiﬁcation failure has not been recognized as a problem in count data
models, standard software does not check for its presence and therefore the practitioner
may be surprised to ﬁnd that estimation of the Poisson regression is unusually diﬃcult,
even in some apparently simple problems. The next section provides details on when this
problem arises and on how it can be detected.
2. THE PROBLEM
To better see the nature of the problem, it is useful to start by considering the case where
a regressor, say xi2,i sz e r ow h e nyi is positive, otherwise being non-negative with at least
one positive observation. The leading example of a regressor with these characteristics is
ad u m m yv a r i a b l et h a ti se q u a lt oz e r of o ra l lo b s e r v a t i o n sw i t hp o s i t i v eyi,h a v i n gs o m e
positive values for yi =0 .F r o me q u a t i o n( 2 ) ,t h eﬁrst order condition for a maximum of
(1) corresponding to the parameter associated with xi2 c a nb ew r i t t e na s
s(β2)=
X
x2i>0
−exp(x
0
iˆ β)x2i =0 ,
which can never be satisﬁed. Therefore, when regressors such as x2 are present, the
(pseudo) maximum likelihood estimate of β does not exist.
More generally, this problem can occur whenever two regressors are perfectly collinear
for the sub-sample with positive observations of yi.2 To see this, write (2) as
s(β)=
X
yi>0
[yi − exp(x
0
iβ)]xi −
X
yi=0
exp(x
0
iβ)xi,
2Notice that the problem identiﬁed here is very diﬀerent from the one resulting from perfect collinearity
between regressors. Perfect collinearity leads to the existence on an inﬁnite number of solutions to the
likelihood equations, whereas here we are concerned wi t ht h es i t u a t i o nw h e r et h el i k e l i h o o de q u a t i o n s
have no solution.
3and notice that the ﬁrst order conditions for a maximum corresponding to β0, β1 and β2
imply
X
yi>0
h
yi − exp(x
0
iˆ β)
i
=
X
yi=0
exp(x
0
iˆ β), (3a)
X
yi>0
h
yi − exp(x
0
iˆ β)
i
x1i =
X
yi=0
exp(x
0
iˆ β)x1i, (3b)
X
yi>0
h
yi − exp(x
0
iˆ β)
i
x2i =
X
yi=0
exp(x
0
iˆ β)x2i, (3c)
where ˆ β denotes the maximum likelihood estimates of β.
Suppose now that x1 and x2 are perfectly collinear for the sub-sample with positive
observations of yi. In particular, let x2i = α0 + α1x1i for yi > 0. Then, writing x2i as a
function of x1i on the left hand side of (3c) and using equalities (3a) and (3b), it is possible
to obtain
α0
n X
yi=0
exp(x
0
iˆ β)+α1
n X
yi=0
exp(x
0
iˆ β)x1i =
n X
yi=0
exp(x
0
iˆ β)x2i. (4)
Whether or not (4) has a solution depends on the values of α0 and α1,a n do nt h e
ranges of x1 and x2 for the observations with yi =0 . For instance, in the illustrative
example presented before, α = β =0and for (4) to have a solution it is necessary, but
not suﬃcient, that x2 has positive and negative values for yi =0 . Heuristically, (4) will
have a solution when there is a reasonable overlap between the ranges of x2i for yi =0and
yi > 0. However, it is not possible to provide a sharp criterion determining the existence
of a ˆ β that solves (4). Therefore, the existence of this sort of identiﬁcation problem has
to be investigated on a case-by-case basis.
Of course, Newton-type algorithms used to maximize the likelihood function may achieve
convergence even when (3) has no solution, leading to spurious maximum likelihood esti-
mates, say b. It is easy to see that for b to provide an approximate solution for (4) it has
to be such that exp(x0
ib) i sc l o s et oz e r of o rt h eo b s e r v a t i o n sw i t hyi =0 . Therefore, these
spurious solutions can be easily identiﬁed because they are characterized by a “perfect”
ﬁt for the observations with yi =0 .
4This situation is analogous to what happens in binary choice models when there is
complete separation or quasi-complete separation, as described by Albert and Anderson
(1984) and Santner and Duﬀy (1986). Moreover, it is clear that it can also occur in any
other regression model where the conditional mean is speciﬁed in such a way that its image
does not include all the points in the support of the dependent variable. Therefore, this
problem can occur not only in the Poisson regression model but whenever y is non-negative
and the conditional mean is speciﬁed as E(yi|xi)=e x p ( x0
iβ).
3. DISCUSSION
The results of the previous section make clear that the non existence of the (pseudo)
maximum likelihood estimates of the Poisson regression models is more likely when the
data has a large number of zeros. For example, this problem is likely to arise when
modelling the number of crimes committed, the number of instances of substance abuse,
or the volume of trade between pairs of countries. Therefore, in these cases, even if the
estimation algorithm converges and delivers a set of estimates, it is recommended that the
researcher checks whether or not the results obtained actually correspond to a maximum of
the (pseudo) log-likelihood function. This can be easily done by checking for the overﬁtting
of the observations with yi =0 , for example by computing descriptive statistics for the
ﬁtted values of y for the relevant sub sample.
When the researcher identiﬁes a situation where the (pseudo) maximum likelihood esti-
mates do not exist, either because the algorithm does not converge or because convergence
is achieved by overﬁtting the zeros, it is useful to have a simple strategy to single out the
regressors causing the problem. Because these regressors are characterized by their perfect
collinearity with the others for the sub-sample with yi > 0,t h e yc a nb ei d e n t i ﬁed using
the following procedure, which explores the fact that statistical software generally drop
perfectly collinear explanatory variables in ordinary least squares regression:
Step 1: For the observations with yi > 0, estimate the ordinary least squares regression of
ln(yi) on xi;
5Step 2: Construct a subset of explanatory variable, say ˜ xi, comprising only the regressors
whose coeﬃcients were estimated in Step 1;
Step 3: Using the observations with yi ≥ 0, i.e., the full sample, run the Poisson regression
of yi on ˜ xi.3
This procedure eliminates all potentially problematic regressors, even those that actually
do not lead to the non-existence of the maximum likelihood estimates. Therefore, the
researcher should then investigate one-by-one all the variables that were excluded, to see
if any of them can be included in the model.
REFERENCES
Albert, A. and Anderson, J.A. (1984). “On the existence of maximum likelihood esti-
mates in logistic models,” Biometrika, 71, 1-10.
Cameron, A.C. and Trivedi, P.K. (1998). Regression analysis of count data,C a m b r i d g e ,
MA: Cambridge University Press.
Gourieroux, C., Monfort, A. and Trognon, A. (1984). “Pseudo maximum likelihood
methods: Applications to Poisson models,” Econometrica, 52, 701-720.
Manning, W.G. and Mullahy, J. (2001). “Estimating Log Models: To Transform or Not
to Transform?,” Journal of Health Economics 20, 461-494.
Santner T.J. and Duﬀy, E.D. (1986). “A note on A. Albert and J.A. Anderson’s condi-
tions for the existence of maximum likelihood estimates in logistic regression models,”
Biometrika, 73, 755-758.
Santos Silva, J.M.C. and Tenreyro, S. (2006), “The log of gravity,” The Review of Eco-
nomics and Statistics, 88, 641-658.
Winkelmann, R. (2008). Econometric analysis of count data,5 th ed., Berlin: Springer-
Verlag.
3The parameters estimated by the OLS regression in Step 1 can be used as starting values for this
Poisson regression.
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On the Existence of the Maximum Likelihood Estimates for Poisson Regression

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