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.

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)

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

and using the formula that

with and being any real numbers, it follows that

where is the first differencing operator, such as. Since and are written as

and

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

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

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:

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):

and the inverse of optimal weighting matrix:

where is any initial consistent estimator for, the GMM estimator is constructed as follows:

where converges in distribution to the normal distribution as follows:

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:

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:

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.

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

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:

and

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:

where is usedwhich is obtained from (2.11), and

Looking at (3.4) and (3.5), it can be seen that

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:

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:

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:

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

with

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

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:

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:

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

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.

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

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

Under the assumption that, the overall average elasticity of with respect to is calculated with the following formula:

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

where, while that calculating the group average elasticity for group in population is

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.

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.