Theoretical Economics Letters
Vol. 2 No. 2 (2012) , Article ID: 19296 , 8 pages DOI:10.4236/tel.2012.22034
Hyperbolic Transformation and Average Elasticity in the Framework of the Fixed Effects Logit Model
Faculty of Economics, Kyushu Sangyo University, Fukuoka, Japan
Email: kitazawa@ip.kyusan-u.ac.jp
Received January 13, 2012; revised February 11, 2012; accepted February 20, 2012
Keywords: Fixed Effects Logit; Conditional Logit Estimator; Hyperbolic Transformation; Moment Conditions; GMM; Monte Carlo Experiments; Average Elasticity
ABSTRACT
In this paper, a simple transformation is proposed for the fixed effects logit model, which constructs some valid moment conditions including the first-order condition for one of the conditional MLE proposed by Chamberlain (1980) [1]. Some Monte Carlo experiments are carried out for the GMM estimator based on the transformation. In addition, the average elasticity of the logit probability with respect to the exponential function of explanatory variable is proposed in the framework of the fixed effects logit model, which is computable without the fixed effects.
1. Introduction
Chamberlain (1980) [1] proposes the useful and established estimator for the fixed effects logit model in panel data.1 This estimator is referred to as the conditional logit estimator, which maximizes the likelihood function composed of the probabilities of the (binary) dependent variables conditional on the fixed effects, the (real-valued) explanatory variables, and the intertemporal sums of the dependent variables. The conditional logit estimator is consistent for the situation of small number of time periods and large cross-sectional size, since its conditional likelihood function rules out the fixed effects and accordingly circumvents the incidental parameters problems pointed out by Neyman and Scott (1948) [2].2 This paper advocates another method of consistently estimating the fixed effects logit model for the situation of small number of time periods and large cross-sectional size.3 The procedure of the method is as follows: First, a hyperbolic transformation is applied to the fixed effects logit model with the aim of eliminating the fixed effects. Next, the GMM (generalized method of moments) estimator proposed by Hansen (1982) [20] is constructed by using the moment conditions based on the hyperbolic transformation. It will be seen that these moment conditions include one type of the first-order conditions of the likelihood for the conditional logit estimator. Then, the preferable small sample property of the GMM estimator using the moment conditions based on the hyperbolic transformation is shown by some Monte Carlo experiments.
In addition, this paper presents the calculation formula of the average elasticity of the logit probability with respect to the exponential function of explanatory variable for the fixed effects logit model. The average marginal effect is not obtained due to the incidental parameters problems for the case of the fixed effects logit model with time dimension being strictly fixed, while it seems that no appropriate index measuring the effect of the change of explanatory variable is developed, in author’s best knowledge. Since the average elasticity is able to be calculated using the consistent estimator of the parameter of interest and the average of binary dependent variables without relation to the fixed effects, it can be said that it is a revolutionary index for the fixed effects logit model.
The rest of the paper is as follows: Section 2 presents the implicit form of the fixed effects logit model, the moment conditions based on the hyperbolic transformation, and the GMM estimator. Section 3 illustrates the link between the conditional maximum likelihood estimator (CMLE) mentioned in the first paragraph in this section and the GMM estimator for the case of two periods. Section 4 reports some Monte Carlo results for the GMM estimator. Section 5 presents the average elasticity in the framework of the fixed effects logit model. Section 6 concludes.
2. Fixed Effects Logit Model, Hyperbolic Transformation and GMM Estimator
In this section, the (static) fixed effects logit model is implicitly defined where the error term is of additive form.4 The hyperbolic transformation, which eliminates the fixed effects and then based on which the moment conditions is constructed for estimating the model consistently, is the fruits of the model defined implicitly. The GMM estimator is defined by using the moment conditions constructed. Throughout this paper, the subscripts 
and 
denote the individual and time period respectively, while 
and 
are number of individuals and number of time periods respectively. Since the short panel is supposed, it is assumed that 
and 
is fixed. In addition, it is assumed that the variables in the model are independent among individuals.
The fixed effects logit model is able to be written in the implicit form as follows:
, for
,(2.1)
(2.2)
where the observable variables 
and 
are the binary dependent variable and the real-valued explanatory variable respectively, while the unobservable variables 
and 
are the individual fixed effect and the disturbance respectively.5 Equations (2.1) say that 
take one with probability
, while it is seen from Equations (2.2) that the probability is the logistic cumulative distribution function of
. Allowing for the serially uncorrelated disturbances, the uncorrelatedness between the disturbances and the fixed effect and the strictly exogenous explanatory variables, the assumptions on the disturbances are specified as
, for
,(2.3)
where 
for
, 
is defined as the empty set for convenience and
. The assumptions (2.3) can be derived from the assumption underlying the fixed effects logit model, which is that 
for 
are mutually independent conditional on 
and
.6 From now on, based on the fixed effects logit model composed of (2.1) and (2.2) with (2.3), the moment conditions for estimating the parameter of interest 
consistently are constructed by using a hyperbolic transformation, as stated below. Taking notice of the fact that
(2.4)
and using the formula that
(2.5)
with 
and 
being any real numbers, it follows that
, (2.6)
where 
is the first differencing operator, such as
. Since 
and 
are written as
(2.7)
and
(2.8)
respectively by using (2.1) and (2.3), plugging (2.7) and (2.8) into (2.6) gives
(2.9)
Equations (2.7) and (2.8) are obtained by plugging (2.1) into 
and
and then applying (2.3) to them.
Taking the expectation conditional on 
for both sides of (2.9) and then applying law of iterated expectation and (2.3) dated
, it follows that
(2.10)
Since 
for any positive integer value 
due to the property of binary variable (and accordingly 
and
), Equation (2.10) results in
, for
, (2.11)
where
. (2.12)
The transformation (2.12) is referred to as “the hyperbolic tangent differencing transformation” for the fixed effects logit model in this paper and hereafter abbreviated to “the HTD transformation”.7 It should be noted that as seen from (2.11) and (2.12), observations for which 
and 
make no direct contribution to obtaining the estimates of 
based on the moment conditions (2.11), since 
is invariably zero for these observations.
The conditional moment conditions (2.11) give the following 
vector of unconditional moment conditions:
, (2.13)
where 
is the 
vector and 
is the 
matrix with
. The (transposed) blocks
, for
, (2.14)
are the 
vector-valued functions of
, 
and 
at time
, where 
is number of instruments for time
. By using the empirical counterpart of (2.13):
(2.15)
and the 
inverse of optimal weighting matrix:
, (2.16)
where 
is any initial consistent estimator for
, the GMM estimator is constructed as follows:
, (2.17)
where 
converges in distribution to the normal distribution as follows:
(2.18)
with 
being the true value of
. Taking notice of the assumption that the variables are independent among individuals and adding the assumption that the variables are identically distributed among individuals, 
, which is the (asymptotic) variance-covariance matrix of the moment conditions (2.13), can be written by using 
as follows:
, (2.19)
where it should be noted that (2.16) is the empirical counterpart of (2.19) if 
is replaced by 
and
. Further, the first derivative of (2.13) with respect to 
for 
is as follows:
. (2.20)
It is conceivable that the discussions for the GMM estimator based on the HTD transformation could be permitted to be conducted on the basis of numbers of observations for which 
instead of
, on the grounds that observations except for those for which 
make no direct contribution to estimating
.
In this case, 
is expediently used instead of 
in this section, where 
is number of observations for which 
at time
.
3. Link between CMLE and GMM Estimator
The discussion here is conducted for the case of two periods (i.e. 
and
). It is shown in this section that the GMM estimator opting for an instrument is identical to the CMLE in this case.
First, the GMM estimator is presented. With 
and 
(both of which are scalars), Equation (2.13) turns to
. (3.1)
The moment condition (3.1) says that 
is used as the instrument for the HTD transformation
. The GMM estimator for 
is the just-identified one when using only the moment condition (3.1) for the two periods. This is denoted by 
hereafter.
The first derivative of 
with respect to 
and the square of 
are respectively calculated as follows:
(3.2)
and
(3.3)
where the relationship that 
if 
is even and 
if 
is odd is used since 
is binary. Using (2.19), (2.20), (3.2), and (3.3), 
and 
for (3.1) are respectively calculated as follows:
(3.4)
where 
is usedwhich is obtained from (2.11), and
(3.5)
Looking at (3.4) and (3.5), it can be seen that
. (3.6)
In addition, the relationship (2.18) is also applicable to the just-identified estimator (see pp. 486-487 in Hayashi, 2000, [23]). Therefore, it follows from (2.18) and (3.6) that the following relationship holds for
:
. (3.7)
Lee (2002, pp. 84-87) [24] elucidates the equality conceptually identical to (3.6) in the context of the CMLE to be hereafter described. In addition, Bonhomme (2012) [25] demonstrates that the conditional moment restriction which he proposes for the fixed effects logit model can give birth to the unconditional moment condition identical to (3.1).
Next, the conventional CMLE proposed by Chamberlain (1980) [1] is presented for the two periods as follows:
, (3.8)
where
. Referring to Wooldridge
(2002, pp. 490-492) [26], the logarithm of probability composing the conditional log-likelihood function for the two-periods fixed effects logit model is written as follows, with
:
, (3.9)
where 
if 
and 
otherwise, while 
if 
and 
and 
if 
and
. In (3.9), 
stands for the probability with which 
takes one given
, 
, 
and
, while 
stands for the probability with which 
takes zero given
, 
, 
and
.
The first-order condition of 
is
(3.10)
with
(3.11)
It is corroborated from (3.10) with (3.11) that the first-order condition of 
divided by 
is the empirical counterpart of the moment condition (3.1) for the GMM estimator. The second-order derivative of 
with respect to 
is written as
(3.12)
Taking notice of the fact that
, it is evident that if 
is replaced by
, (3.12) divided by 
is the empirical counterpart of (3.5) and accordingly identical to 
from (3.6). Therefore, the following relationship holds for
:
. (3.13)
Judging from the above, it is ascertained that for the two periods the conventional CMLE for the fixed effects logit model is identical to the GMM estimator selecting 
as the instrument for the HTD transformation.
To make doubly sure, the integration of
with respect to 
is conducted:
(3.14)
where 
is the constant of integration. With 
for (3.14), the logarithm of probability (3.9), which composes the conditional log-likelihood function for the two-periods fixed effects logit model, is compactly rewritten as
(3.15)
The exponential of 
in (3.15), which is equivalent to (3.9), represents the probability density when the restriction 
is imposed. In this case, number of observations for which 
is used instead of 
in this section and therefore
, which is equivalent to
, could be interpreted as being the asymptotically efficient estimator. This is because the CramérRao inequality is applicable in this case.
Incidentally, Abrevaya (1997) [27] shows that for the fixed effects logit model, a scale-adjusted ordinary maximum likelihood estimator is equivalent to the CMLE for the case of two periods.
4. Monte Carlo
In this section, some Monte Carlo experiments are conducted to investigate the small sample performance of the GMM estimator for the fixed effects logit model described in Section 2. The experiments are implemented by using an econometric software TSP version 4.5 (see Hall and Cummins, 2006, [28]).
The data generating process (DGP) is as follows:
,
,
,
,
,
;
.
In the DGP, values are set to the parameters
, 
, 
, 
and
. The experiments are carried out with the cross-sectional sizes
, 
and
, the numbers of time periods
, 
and
, and the number of replications
.
In the experiments, the GMM estimator based on the HTD transformation selects 
as the instruments for the transformation
. That is, the GMM(HTD) estimator uses the vector of moment conditions (2.13) with
, which is able to be written piecewise as follows:
, for
.8 (4.1)
As a control, another GMM estimator is used, which employs the following moment conditions disregarding the unobservable heterogeneity:
, for
. (4.2)
where
. The GMM (LgtLev) estimator (i.e. the level GMM estimator for the logit model) for 
is inconsistent due to the ignorance of the fixed effects.
The Monte Carlo results are exhibited in Table 1. The settings of values of the parameters for the explanatory variables 
are the same as those used by Blundell et al. (2002) [21] for count panel data model. The small sample property of the GMM(HTD) estimator can be said to be preferable and their bias and rmse (root mean squared error) decrease as the cross-sectional size 
increases, which is the reflection of the consistency. In contrast, the sizable downward bias and rmse for the (inconsistent) GMM(LgtLev) estimator remain at virtually constant levels when 
increases. As is seen from comparisons among Simulations (a4), (a8) and (a25), among Simulations (b4), (b8) and (b25), and Simulations (c4), (c8) and (c25) for the GMM(HTD) estimator, the small sample performance of the GMM(HTD) estimator is better off as the number of time periods increases, reflecting the substantive increase of sample size. Furthermore, comparisons among Simulations (a4), (b4) and (c4), among Simulations (a8), (b8) and (c8), and among Simulations (a25), (b25) and (c25) for the GMM(HTD) estimator raise the possibility that more persistent series of the explanatory variables might bring about more deteriorated small sample performance of the GMM(HTD) estimator.9
Table 1. Monte Carlo results for the fixed effects logit model.
5. Average Elasticity
For the fixed effects logit model composed of (2.1) and (2.2), the new index is constructed by using both the consistent estimator for 
described in previous sections and the average of
. The average elasticity of the logit probability with respect to the exponential function of explanatory variable (which is calculated without relation to the fixed effects) is an appropriate index in the framework of the fixed effects logit model with time dimension being strictly fixed, where no (consistent) average marginal effect is available.10 In this section, the assumption that the variables are identically distributed among individuals is unfastened.11
With
, the elasticity of the probability 
with respect to the positive-valued variable 
(with 
being held constant) is defined as follows:
for
. (5.1)
Under the assumption that
, the overall average elasticity of 
with respect to 
is calculated with the following formula:
, (5.2)
where 
is the consistent estimator for 
such that
and
. Since
is the probability and 
(and accordingly variances of 
are finite), it can be seen that
, if
(which is referred to as the average logit probability in this paper).12
In addition, the cross-section average elasticity for a specific time period and the group average elasticity for a group (e.g. a gender) are able to be calculated as follows, respectively: The formula calculating the cross-section average of 
with respect to 
for period 
is
, (5.3)
where
, while that calculating the group average elasticity for group 
in population is
, (5.4)
where 
with subscript 
denoting the member of group
, 
being number of individual units belonging to group
, and 
being the binary dependent variable for the individual 
appertaining to group 
at period
.
6. Conclusion
This paper proposed the hyperbolic tangent differencing (HTD) transformation for the fixed effects logit model, with the intention of ruling out the fixed effects. The consistent GMM estimator was constructed by using the HTD transformation. The equivalence of the GMM estimator opting for an instrument and the CMLE proposed by Chamberlain (1980) [1] was revealed for the case of two periods. Then, the Monte Carlo experiments indicated the desirable small sample property of the GMM estimator based on the HTD transformation. In addition, the average elasticity of the logit probability with respect to the exponential function of explanatory variable was proposed, which is an appropriate index from the point of view that it is able to be calculated without the fixed effects. Both of the simple estimator and index will facilitate empirical researchers exploring the binary choice panel data model.
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NOTES
1The rootstock of this estimator is Rasch (1960) [3], (1961) [4].
2Additionally, Honoré and Kyriazidou (2000) [5] propose an estimator for the fixed effects logit model with the lagged dependent variable (as for details, see also pp. 211-216 in Hsiao, 2003 [6]). Further, Thomas (2006) [7] proposes two estimators for the fixed effects logit model with heterogeneous linear trends.
3It seems that the mainstream of late is the development of the biasadjusted estimators, which is available in nonlinear panel data models and aims at the reduction of time-series finite sample bias (i.e. the approximately unbiased estimation of the incidental parameters as well as the parameters of interest, leading to obtaining the approximate marginal effects). Various approaches are proposed in line with the bias-adjustment: Hahn and Newey (2004) [8], Cox and Reid (1987) [9], Lancaster (2002) [10], Arellano (2003) [11], Arellano and Bonhomme (2009) [12], Carro (2007) [13], Fernández-Val (2009) [14], Severini (1998) [15], Pace and Salvan (2006) [16], Bester and Hansen (2009) [17], etc. Some of the approaches are reviewed in Arellano and Hahn (2007) [18] and Hsiao (2010) [19]. However, author’s policy is to conduct the consistent estimation for the case of small time dimension and therefore this paper is not bent upon the bias-adjusted estimators.
4The regression form defined implicitly is also used by Blundell et al. (2002) [21] for count panel data.
5It is generally assumed that the individual effect 
is correlated with the explanatory variables 
for each
.
6If the underlying assumption holds, the following relationship is obtained:
, where 
is the conditional probability density function and
. Accordingly, it follows that
. As for details, see p. 23 in Cameron and Trivedi (2005) [22]. Taking notice of (2.1) and the fact that
, the assumptions (2.3) are obtained.
7If the much weaker assumptions 
for 
are used instead of (2.3), the moment conditions 
for 
can be obtained instead of (2.11), where 
and 
is defined as the empty set for convenience. It should be noted that under the assumptions 
for
, the (consistent) CMLE proposed by Chamberlain (1980) [1] is no longer obtained for
. The implication of 
is that although the decision 
wields no influence over the explanatory variable 
just behind its decision, it can make some sort of influences on the explanatory variables after
, while that of (2.3) is that the decision 
have no influence on the explanatory variables after its decision. In addition, it is regrettable that at this stage, author is unable to construct the valid moment conditions when 
is endogenous. This would be a task for the future.
8Since the moment conditions (4.1) are valid even under the assumptions 
for
, the usage of the GMM (HTD) estimator using the moment conditions (4.1) is generally more conservative than that of the CMLE proposed by Chamberlain (1980) [1] (see footnote 7 in section 2). The CMLE is inconsistent under the assumptions 
for 
and
.
9This possibility is also pointed out in the framework of ordinary and count panel data models. For example, see Blundell and Bond (1998) [29] and Blundell et al. (2002) [21].
10Frequently, the explanatory variables in the fixed effects logit model are logarithmically transformed.
11In this case, (2.19) and (2.20) in Section 2 are replaced by 
and
, respectively. The same is applied to (3.4) and (3.5) in Section 3.
12Just in case, it is assumed that both 
and 
exist for each 
and
. However, author thinks that it seems that this assumption is satisfied in any case.