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1、1 Electronic copy available at: http:/ A dynamic model for binary panel data with unobserved heterogeneity admitting a n-consistent conditional estimator Francesco Bartolucci and Valentina Nigro Ottobre 2009 Abstract A model for binary panel data is introduced which allows for state dependence and u
2、n- observed heterogeneity beyond the effect of available covariates. The model is of quadratic exponential type and its structure closely resembles that of the dynamic logit model. However, it has the advantage of being easily estimable via conditional likelihood with at least two observations (furt
3、her to an initial observation) and even in the presence of time dummies among the regressors. Key words: longitudinal data; quadratic exponential distribution; state dependence. We thank the Co-Editor and three anonymous Referees for helpful suggestions and insightful comments. We are also grateful
4、to Franco Peracchi and Frank Vella for their comments and suggestions. Francesco Bartolucci acknowledges financial support from the “Einaudi Institute for Economics and Finance” (EIEF), Rome. Most of the article has been developed during the period spent by Valentino Nigro at the University of Rome
5、“Tor Vergata” and is part of her PhD dissertation. Dipartimento di Economia, Finanza e Statistica, Universita di Perugia, 06123 Perugia, Italy, e-mail: bartstat.unipg.it Dipartimento di Studi Economico-Finanziari e Metodi Quantitativi, Universita di Roma “Tor Vergata”, Via Columbia 2, 00133 Roma, It
6、aly, e-mail: Valentina.Nigrouniroma2.it 2 Electronic copy available at: http:/ 1 Introduction Binary panel data are usually analyzed by using a dynamic logit or probit model which includes, among the explanatory variables, the lags of the response variable and has individual-specific intercepts; see
7、 Arellano & Honore (2001) and Hsiao (2005), among others. These models allow us to disentangle the true state dependence (i.e. how the experience of an event in the past can influence the occurrence of the same event in the future) from the propensity to experience a certain outcome in all periods,
8、when the latter depends on unobservable factors (see Heckman, 1981a, 1981b). State dependence arises in many economic contexts, such as job decision, invest- ment choice and brand choice and can determine different policy implications. The parameters of main interest of these models are typically th
9、ose for the covariates and the true state de- pendence, which are referred to as structural parameters. The individual-specific intercepts are referred to as incidental parameters; these are of interest only in certain situations, such as when we need to obtain marginal effects and predictions. In t
10、his paper, we introduce a model for binary panel data which closely resembles the dy- namic logit model and, as such, allows for state dependence and unobserved heterogeneity between subjects, beyond the effect of the available covariates. The model is a version of the quadratic exponential model (C
11、ox, 1972) with covariates in which: (i) the first-order effects depend on the covariates and on an individual-specific parameter for the unobserved hetero- geneity; (ii) the second-order effects are equal to a common parameter when they are referred to pairs of consecutive response variables and to
12、0 otherwise. We show that this parameter has the same interpretation that it has in the dynamic logit model in terms of log-odds ratio, a measure of association between binary variables which is well known in the statistical literature on categorical data analysis (Agresti, 2002, Ch. 8). For the pro
13、posed model we also provide a justification as a latent index model in which the systematic component depends on expectation about future outcomes, beyond the covariates and the lags of the response variable, and the stochastic component has a standard logistic distribution. An important feature of
14、the proposed model is that, as for the static logit model, the inciden- tal parameters may be eliminated by conditioning on sufficient statistics for these parameters, which correspond to the sums of the response variables at individual level. Using a terminology 3 Electronic copy available at: http
15、:/ it it it it derived from Rasch (1961), these statistics will be referred to as total scores. The resulting conditional likelihood allows us to identify the structural parameters for the covariates and the state dependence with at least two observations (further to an initial observation). The est
16、ima- tor of the structural parameters based on the maximization of this function is n-consistent; moreover, it is simpler to compute than the estimator of Honore & Kyriazidou (2000) and may be used even in the presence of time dummies. On the basis of a simulation study we also notice that the estim
17、ator has good finite sample properties in terms of both bias and efficiency. The paper is organized as follows. In the next section we briefly review the dynamic logit model for binary panel data. The proposed model is described in Section 3 where we also show that the total scores are sufficient st
18、atistics for its incidental parameters. Identification of the structural parameters and the conditional maximum likelihood estimator of these parameters are illustrated in Section 4. The simulation study is illustrated in Section 5. Final conclusions are given in Section 6. 2 Dynamic logit model for
19、 binary panel data In the following, we first review the dynamic logit model for binary panel data and then we discuss conditional inference, and related inferential methods, on its structural parameters. 2.1 Basic assumptions Let yit be a binary response variable equal to 1 if subject i (i = 1, . .
20、 . , n) makes a certain choice at time t (t = 1, . . . , T ) and to 0 otherwise; also let xit be a corresponding vector of strictly exogenous covariates. The standard fixed-effects approach for binary panel data assumes that yit = 1y 0, y = i + x0 + yi,t1 + it, i = 1, . . . , n, t = 1, . . . , T, (1
21、) where 1 is the indicator function and y is a latent variable which may be interpreted as utility (or propensity) of the choice. Moreover, the zero-mean random variables it represent error terms. Of primary interest are the vector of parameters for the covariates, , and the parameter measuring the
22、state dependence effect, . These are the structural parameters which 4 it i it i| i i i0 Q it it t are collected in the vector = (0, )0. The individual-specific intercepts i are instead the incidental parameters. The error terms it are typically assumed to be independent and identically distributed
23、conditionally on the covariates and the individual-specific parameters and to have a standard logistic distribution. The conditional distribution of yit given i, Xi = ( xi1 xiT ) and yi0, . . . , yi,t1 may then be expressed as expyit(i + x0 + yi,t1) p(yit|i, Xi, yi0, . . . , yi,t1) = p(yit|i, xit, y
24、i,t1) = 1 + exp( + x0 + yi,t1) , (2) for i = 1, . . . , n and t = 1, . . . , T . This is a dynamic logit formulation which implies the following conditional distribution of the overall vector of response variables yi = (yi1, . . . , yiT )0 given i, Xi and yi0: p(y , X , y ) = exp(yi+i + Pt yitx0 + y
25、i) t1 + exp(i + x0 + yi,t1) , (3) where yi+ = P yit and yi = Pt yi,t1yit, with the sum Pt and the product Qt ranging over t = 1, . . . , T . The statistic yi+ is referred to as the total score of subject i. For what follows it is important to note that log p(yit = 0|i, X i, yi,t 1 = 0)p(yit = 1|i, X
26、 i, yi,t 1 = 1) p(yit = 0|i, Xi, yi,t1 = 1)p(yit = 1|i, Xi, yi,t1 = 0) = , for i = 1, . . . , n and t = 1, . . . , T . Thus, the parameter for the state dependence corresponds to the conditional log-odds ratio between (yi,t1, yit) for every i and t. 2.2 Conditional inference As mentioned in Section
27、1, an effective approach to estimate the model illustrated above is based on the maximization of the conditional likelihood given suitable sufficient statistics. For the static version of the model, in which the parameter is equal to 0, we have that yi is conditionally independent of i given yi0, Xi
28、 and the total score yi+, and then p(yi|i, Xi, yi+) = p(yi|Xi, yi+). The likelihood based on this conditional probability allows us to identify for T 2; by maximizing this likelihood we also obtain a n-consistent estimator of . Even if referred to a simpler context, this result goes back to Rasch (1
29、961) and was developed by 5 Andersen (1970). See also Magnac (2004) who characterized other situations in which the total scores are sufficient statistics for the individual-specific intercepts. Some of the first authors to deal with the conditional approach for the dynamic logit model ( is unconstr
30、ained) were Cox (1958) and Chamberlain (1985). In particular, the latter noticed that when T = 3 and the covariates are omitted from the model, p(yi|i, yi0, yi1 + yi2 = 1, yi3) does not depend on i for every yi0 and yi3. On the basis of this conditional distribution it is therefore possible to const
31、ruct a likelihood function which depends on the response configurations of only certain subjects (those such that yi1 + yi2 = 1) and which allows us to identify and consistently estimate the parameter . The approach of Chamberlain (1985) was extended by Honore & Kyriazidou (2000) to the case where,
32、as in (2), the model includes exogenous covariates. In particular, when these covariates are continuous, they proposed to estimate the vector of structural parameters by maximizing a weighted conditional log-likelihood, with weights depending on the individual covariates through a kernel function wh
33、ich must be defined in advance. Although the weighted conditional approach of Honore & Kyriazidou (2000) is of great interest, their results about identification and consistency are based on certain assumptions on the support of the covariates which rule out, for instance, time dummies. Moreover, th
34、e approach requires careful choice of the kernel function and of its bandwidth, since these choices affect the performance of their estimator. Furthermore, the estimator is consistent as n , but its rate of convergence to the true parameter value is slower than n, unless in the presence of only disc
35、rete covariates. See also Magnac (2004) and Honore & Tamer (2006). Even if not strictly related to the conditional approach, it is worth mentioning that a recent line of research investigates dynamic discrete choice models with fixed-effects proposing bias corrected estimators (see Hahn & Newey, 200
36、4; Carro, 2007). Although these estimators are only consistent when the number of time periods goes to infinity, they have a reduced order of the bias without increasing the asymptotic variance. Monte Carlo simulations have shown their good finite sample performance in comparison to the estimator of
37、 Honore & Kyriazidou (2000) even with not very long panels (e.g. seven time periods). 6 it iT t it iT z t it it it t 3 Proposed model for binary panel data In this section, we introduce a quadratic exponential model for binary panel data and we discuss its main features in comparison to the dynamic
38、logit model. 3.1 Basic assumptions We assume that expyi+i + P yitx0 1 + yiT ( + x0 2) + yi p(yi|i, Xi, yi0) = P t , (4) z expz+i + P ztx0 1 + zT ( + x0 2) + zi where the sum P ranges over all the possible binary response vectors z = (z1, . . . , zT )0; more- over, z+ = P zt and zi = yi0z1 + P t1 zt1
39、zt. The denominator, does not depend on yi and it is simply a normalizing constant that we denote by (i, Xi, yi0). The model may be seen as a version of the quadratic exponential model of Cox (1972) with covariates in which the first-order effect for yit is equal to i + x0 1 (to which we add + x0 2
40、when t = T ) and the second-order effect for (yis, yit) is equal to when t = s + 1 and to 0 otherwise. The need of a different parametrization of the first-order effect when t = T and t 0, y = i + x0 + yi,t1 + e(i, Xi) + it, (8) it it it t 8 t t T t p(y(T 1) where the error terms it are independent
41、and have standard logistic distribution. Assumption (8) is similar to assumption (1) on which the dynamic logit model is based, the main difference being in the correction term e(i, Xi). As is clear from (6), this term may be interpreted as a measure of the effect of the present choice yit on the ex
42、pected utility (or propensity) at the next occasion (t + 1). In the presence of positive state dependence ( 0), this correction term is positive since making the choice today has a positive impact on the expected utility. Also note that the different definition of e(i, Xi) for t 1 yitd0 1 P + yiT (
43、+ x0 2) + yi , (11) Pz(y ) exp t1 ztd0 1 + zT ( + xiT 2) + zi with dit = xitxi1, t = 2, . . . , T . We consequently assume that 1 does not include any intercept common to all time occasions and regression parameters for covariates which are time constant; if included, these parameters would not be i
44、dentified. This is typical of other conditional approaches, such as that of Honore & Kyriazidou (2000), and of fixed-effects approaches in which the individual intercepts are estimated together with the structural parameters. Similarly, 2 must not contain any intercept for the last occasion, since t
45、his is already included through . 4 Conditional inference on the structural parameters In the following, we introduce a conditional likelihood based on (11). We also provide for- mal arguments on the identification of the structural parameters via this function and on the 11 asymptotic properties of
46、 the estimator resulting from its maximization. 4.1 Structural parameters identification via conditional likelihood For an observed sample (Xi, yi0, yi), i = 1, . . . , n, the conditional likelihood has logarithm () = X 10 2, whereas identification of the structural parameters of the dynamic logit m
47、odel is only possible when T 3 (Chamberlain, 1993). See also the discussion provided by Honore & Tamer (2006). 4.2 Conditional maximum likelihood estimator The conditional maximum likelihood estimator of , denoted by = ( 0 , 0 , , )0, is obtained 1 2 by maximizing the conditional log-likelihood ().
48、This maximum may be found by a simple iterative algorithm, of Newton-Raphson type. At the hth step, this algorithm updates the estimate of at the previous step, (h1), as (h) = (h1) + J (h1)1s(h1). Note that the information matrix J () is always non-negative definite since it corresponds to the sum of a series of variance-covariance matrices. Provided E0A(X)A(X)0 is of full rank, J () is also positive definite with pr
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