Estimation of Dynamic Nonlinear Random E ects Models with Unbalanced Panels.

Transcription

1 Estimation of Dynamic Nonlinear Random E ects Models with Unbalanced Panels. Pedro Albarran y Raquel Carrasco z Jesus M. Carro x June 2014 Preliminary and Incomplete Abstract This paper presents and evaluates estimation methods for dynamic nonlinear correlated random e ects (CRE) models. Heckman (1981) and Wooldridge (2005) have proposed solutions to deal with the initial conditions problem typically present in dynamic nonlinear models with unobserved e ects, but these solutions are developed for balanced panels. Accounting for the unbalancedness is crucial in dynamic non-linear models and it cannot be ignored even if the process that produces it is completely at random. One solution typically applied by practitioners it to make the unbalanced panel balanced and then to use the available methods. Nonetheless, this approach is in some cases not feasible because the constructed balanced panel might not contain enough number of common periods across individuals, making the estimation unfeasible. Moreover, when feasible, reducing the data set to make the panel balanced will discard useful information, which might imply important e ciency losses. The solutions we propose in this paper can be implemented using both the Heckman s and Wooldridge s solutions to the initial conditions problems extensively used by practitioners and can be easily implemented in the context of commonly used models, such as dynamic binary choice models. We present several scenarios in which the sample selection process can be arbitrarily correlated with the unobserved heterogeneity, although uncorrelated with the unobserved shocks. For each of these scenarios we propose an estimator, using either the Heckman s or the Wooldridge s solution to the initial conditions problem, but accounting for the unbalanced structure of the data. The authors gratefully acknowledge that this research was supported by grants ECO and ECO from the Spanish Minister of Education. y Universidad de Alicante. albarran@ua.es z Department of Economics, Universidad Carlos III de Madrid. rcarras@eco.uc3m.es x Department of Economics, Universidad Carlos III de Madrid. jcarro@eco.uc3m.es

2 1 Introduction Many available methods in econometrics are restricted to balanced panels. Nonetheless, unbalanced panels are the norm. For example, in large panel data sets like the PSID, there are always individuals who drop out (non-randomly) of the sample. In other cases, like in the so called rotating panels, the unbalancedness is generated by the sample design and, therefore, the missingness is completely at random (for instance, the Monthly Retail Trade Survey in the US, and the Household Budget Continuous Survey in Spain). It is well known that parameter estimates from unbalanced panel data in linear models with individual e ects are not biased when certain assumptions about selection hold. This situation includes the xed-e ects (FE) approach, which leaves the correlation between the selection and the heterogeneity parameters unrestricted, and the random e ects (RE) approach which is less robust because the selection must be assumed to be uncorrelated with the heterogeneity. However, fewer results are available on the estimation of non-linear models for unbalanced panel data. The purpose of this paper is to present and evaluate estimation methods of dynamic nonlinear correlated random e ects (CRE) models for unbalanced panels. The CRE approach represents a simple method to estimate this type of models. As opposed to the FE approach 1, it is not subject to the incidental parameters problem although at a cost of imposing restrictive parametric assumptions on the conditional distribution of the heterogeneity parameters. Other strand of the literature has considered bias-corrected versions of xed-e ects estimators, but they usually require a greater number of periods for the bias adjustments to work than the available in many data sets. Under these circumstances, correlated random e ects methods can be regarded as an useful alternative. To the best of our knowledge only Wooldridge (2010) addresses this issue, but for static models. Speci cally, he proposes several strategies for allowing unobserved heterogeneity to be correlated with observed covariates and selection mechanism for unbalanced 1 For the purpose of this paper, FE methods are those that treat the heterogeneity as parameters to be estimated, while RE methods are those that impose a certain amount of structure in the dependence between the individual e ects and the endogenous variables. 1

3 panels. But the assumption of strict exogeneity of the covariates is very restrictive. In this paper we extend CRE approaches for models with predetermined lagged dependent variables and unbalanced data. In a dynamic setting the main drawback of the CRE approach is that it gives rise to the initial conditions problem. Heckman (1981) and Wooldridge (2005) have proposed solutions to deal with this problem, but these solutions are developed for balanced panels and, in general they cannot be directly implemented with unbalanced panel data. Accounting for the unbalancedness is crucial in dynamic non-linear models and it cannot be ignored even if the process that produces it is completely at random (i.e. independent of the process of the observables and the unobservables). One solution typically applied by practitioners it to make the unbalanced panel balanced and then to use the available methods (see Wooldridge, 2005, pp. 44). Nonetheless, this approach is in some cases not feasible because the constructed balanced panel might not contain enough number of common periods across individuals, making the estimation unfeasible. Moreover, when feasible, reducing the data set to make the panel balanced will discard useful information, which mayimply important e ciency losses. The solutions we propose in this paper can be implemented using both the Heckman s and Wooldridge s solutions to the initial conditions problems extensively used by practitioners. Therefore, they can be easily implemented in the context of commonly used models, such as dynamic binary choice models. We present several scenarios in which the sample selection process can be arbitrarily correlated with the unobserved heterogeneity, although uncorrelated with the unobserved shocks. For each of these scenarios we propose an estimator, using either the Heckman s or the Wooldridge s solution to the initial conditions problem, but accounting for the unbalanced structure of the data. The paper is organized as follows. Section 2 presents the model and. In Section 3 we study the nite sample properties of the proposed estimators in a binary choice model with a single lagged dependent variable by means of Monte Carlo simulations. In Section 4, we present an empirical illustration. Finally, Section 5 concludes. 2

4 2 Model framework We present a general approach that can be applied to dynamic non-linear panel data models. Let us denote Y i = (y i1 ; :::; y it ) 0 ; X i = X 0 i1; :::; X 0 it 0 ; S i = (s i1 ; :::; s it ) 0 ; where i = 1; :::; N represents cross-sectional units, y it is the potentially observed outcome, and X it are potentially observed covariates. The possibility of having an unbalanced panel is captured through a set of selection indicators, s it ; which take the value 1 if the corresponding observation can be used in the estimation, that is 1 if yit and X s it = it are observed 0 otherwise We only consider cases in which either both y it and X it are observed or both are not observed. We de ne t i as the rst period in which unit i is observed, i.e. t i = ft: s it = 1 and s ij = 0 8 j < tg, and T i = TX t=1 is the number of periods we observe for unit i: Another characteristic of the panels considered is that all the observations for unit i are consecutive. This means that s it s it = 1 8 t 2 [t i ; t i + T i ] s it = 0 8 t < t i or t > t i + T i We are interested in the conditional distribution P (y it j y it 1 ; X i ; i ; S i ), where i denotes the vector of unobserved heterogeneity. As in Wooldridge (2010) we make the following assumption: 3

5 Assumption 1: The sample selection process s it is strictly exogenous with respect to the idiosyncratic shocks to y it, although it is allowed to be correlated with i and the observed covariates. Therefore, we can express P (y it j y it 1 ; X i ; i ; S i ) = P (y it j y it 1 ; X i ; i ) Let f(y it j y it 1 ; X i ; i ; S i ; ) be the correctly speci ed density for the conditional distribution on previous equation and h( i jx i ; S i ; ) a correctly speci ed model for the density of P ( i jx i ; S i ). The density of (s i1 y i1 ; : : : ; s it y it ) for a given individual is f (s i1 y i1 ; : : : ; s it y it jx i ; S i ) = = TY f (y it jy it 1 ; X i ; S i ) s its it 1 f (y it jx i ; S i ) s it(1 s it 1 ) t=1 t i Y+T i t=t i +1 f (y it jy it 1 ; X i ; S i ) f (y iti jx i ; S i ) (1) Given that previous equation depends on unobservables, i, and in the absence of the start of the sample coinciding with the start of the stochastic process, the rst observation will not be independent of i. Heckman (1981) proposes to integrate out the unobserved e ect in the log-likelihood function by specifying the density for the rst observation conditional on the unobserved e ect and the density of the unobserved e ect, and then integrate with respect to the latter. He suggested using for the rst observation the same parametric model as the conditional density for the rest of the observations. Following this strategy in the unbalanced panel case, we get the following function: Z f (s i1 y i1 ; : : : ; s it y it jx i ; S i ) = t i Y+T i i t=t i +1 f (y it jy it 1 ; X i ; S i ; i ) f (y iti jx i ; S i ; i ) h( i jx i ; S i )d i (2) Another possibility, proposed by Wooldridge (2005) to the balanced case is to use the density of (y i2 ; : : : ; y it ) conditional on (y i1 ; i ). In our unbalanced case, this would lead to the 4

6 following function: Z f (s i1 y i1 ; : : : ; s it y it jx i ; S i ) = t i Y+T i i t=t i +1 f (y it jy it 1 ; X i ; S i ; i ) h( i jy iti ; X i ; S i )d i f (y iti jx i ; S i ) As Wooldridge (2005) points out a solution to the unbalancedness under independence is to balance the sample and then apply the standard Heckman s and Wooldridge s solutions to the initial conditions problem to the balanced sample. Nonetheless, this has two limitations: (i) it discards useful information leading to an e ciency loss, and (ii) the balanced sample might not contain enough number of common periods across individuals, making the estimation unfeasible. We present other estimation strategies under di erent assumptions about the correlation between the selection mechanism and the individual e ect. In all cases, we present estimators that can be implemented using the available solutions to the initial conditions problem. 2.1 Unbalancedness independent of the individual e ect In this case, in addition to Assumption 1, we assume, Assumption 2: S i is indepent of i The model under the Heckman s solution to the initial conditions problem To adapt the Heckman s approach to this unbalanced panel data case we need to make a distributional assumption about f (y iti jx i ; S i ; i ; ) and the density h( i jx i ; S i ; ) in (2). Notice that we have a vector of parameters di erent for each starting period in the unbalanced panel. This implies that P (y iti j X i ; i ; S i ) 6= P (y jtj j X j ; j ; S j ) whenever t i 6= t j, and they will be the same whenever t i = t j = t. 2 For h( i jx i ; S i ) we follow Chamberlain (1984) to allow for correlation between the individual e ect and the explanatory variables. Given that Assumption 2 establishes independence 2 Even under Assumption 2, P (y iti j X i; i ; S i) is di erent for each t i just because the process has been running a di erent number of periods until that rst observation, and we are not assuming that the process is on steady state. (3) 5

7 betwwen S i and i, then, h( i jx i ; S i ) = h( i jx i ): i jx i ; S i N X 0 i ; 2 (4) where X i contains the within-means of the time-varying explanatory variables The model under the Wooldridge s solution to the initial conditions problem Another possibility is to implement the Wooldridge s approach to the unbalanced framework. As previously pointed out, Wooldridge (2005) proposed to consider the distribution conditional on the initial period observation, thus, specifying an approximation for the density of i conditional on the initial observation. In our case this means specifying h( i jy iti ; X i ; S i ) in (3) and discarding f (y iti jx i ; S i ) since that term is outside the integral. Continuing with the Normal case, under Assumption 2: i jy iti ; X i ; S i N 0ti + 1ti y iti + X 0 i 2ti ; 2 ti (5) Notice that even though we have assumed that the sample selection process S i is independent of i, h( i jy iti ; X i ; S i ) will be di erent for each t i as in (5), unless the process is not dynamic or it is in its steady state since period t = The model when the unbalancenss is independent of i, and y it is in steady state. When we add the steady state assumption to the independence between S i and i we have a model of particular interest since we can obtain consistent estimates even when ignoring the unbalancedness. Assumption 3: y it is in steady state for t = 1; :::; T. In this case, h( i jy iti = y; X i ; S i ) = h( i jy iti = y; X i ) = h( i jy i1 = y; X i ; S i ) for all t i 1. Thus, following the Wooldridge s approach we can write i jy iti = y; X i ; S i N y + X 0 i 2 ; 2 (6) 6

8 and P (y iti = 1jX i ; S i ; i ) = P (y iti = 1jX i ; i ) P (y i1 = 1jX i ; i ) (7) together with (4) in the Heckman s approach. In this particular situation, and keeping in mind that in our model the selection process is exogenous, using the standard Heckman s or Wooldridge s approaches for balanced panels but ignoring the unbalancedness i.e. treating all units as if they all were observed the same periods will produce a consistent estimator where no observations from the sample will be discarded (except the initial observations in the Wooldridge s approach). 2.2 Unbalancedness correlated wtih the individual e ect We can consider situations in which the selection mechanisim S i is correlated with the individual e ect, i. If Assumption 2 is not correct, adapting Heckman s solution the initial conditions problem requires a di erent distribution of the initial condition for each subpanel, which implies having a vector of parameters di erent for each subpanel. For h( i jx i ; S i ) we follow Chamberlain (1984) to allow for correlation between the individual e ect and the explanatory variables: i jx i ; S i N X 0 i Si ; 2 Si (8) where X i contains the within-means of the time-varying explanatory variables. Previous speci cation allows for correlation between the sample selection process and the permanent unobserved heterogeneity. In the case of the Wooldridge s solution to the initial conditions problem and continuing with the Normal case what we need to assume is: i jy iti ; X i ; S i N 0 + 1Si y iti + X 0 i 2Si ; 2 Si (9) 7

9 3 Finite-sample Performance In this section we use Monte Carlo techniques to illustrate the behavior of the our proposed solutions/approaches. We are particularly interested in the nite sample performance of the estimator under di erent degrees of unbalancedness. 3.1 Monte Carlo Design and Unbalancedness We simulate the following model speci cation: y it = 1fy it 1 + i + " it 0g t = 1; :::; T ; i = 1; :::; N (10) " it iid N(0; 1) i iid N( ; 2 ) (11) Moreover, the initial condition is given by y i0 = 1f i + v i0 0g; v i0 iid N (0; 1) (12) 3.2 Monte Carlo Results Table 1 and Table 2 shows the simulation results about the nite-sample performance of several of the approaches discussed in this paper in our baseline speci cation. Here we have simulated the model (10)-(12) for = 0:75 and with an exogenous initial condition. Under this setting, and irrespectively of the unbalancedness, it is know that all the proposed approaches give consistent estimates. We actually observe that all the ve approaches considered here provide estimated values of the parameter very close to its true value. However, there exists some other relevant points that are worth noting. On the one hand, solution approaches à -la-wooldridge and à -la-heckman have similar performance in terms of Root Mean Square Error (RMSE), independently of the sample size (both N and T ) and of the unbalancedness. On the other hand and more importantly, the usual solutions that 8

10 employ standard methods after balancing the sample, namely A1H and A1W, have two important drawbacks compared to any of the other approaches. First, those solutions cannot be employed in many cases, including some where the unbalancedness is moderate: for J = 4 with T = 6 or J = 6 with T = 8. It is important to note that these settings are often found in empirical analyses. Second, those solutions imply an important loss of e ciency in terms of RMSE when they can be employed compared to the newly proposed approaches in this paper. Our new proposals always dominates the usual solutions in terms of RMSE and they can have as less as one half of its RMSE. This is true both if we consider double unbalancedness (Table 1) or only left-side unbalancedness (Table 2) and, again, losses are remarkable even for moderate unbalancedness. For instance, Panel A of Table 1, shows that for T = 8 and J = 4 the RMSE of A1H and A1W is around 0:26 compared with around 0:13 for A3W, A3Wb and A4H; although the RMSE of all the solutions is reduced when the sample size increases from N = 200 (Panel A) to N = 500 (Panel B) the relative loss of e ciency remains: around 0:16 for A1H and A1W compared with around 0:08 for A3W, A3Wb and A4H. Table 2 displays a similar picture: for T = 8 and J = 5, the RMSE of A1H and A1W is around 0:44 and 0:28 in Panels A and B, respectively, compared with around 0:18 and 0:12 for A3W, A3Wb and A4H. Given that we have found no di erences in performance between solution approaches à -la-wooldridge and à -la-heckman, we considered only the former ones in the rest of our Monte Carlo experimentes. In Table 3 and 4, we have the same baseline speci cation as before but with an even larger simple size N = This simply con rms our previous ndings and warns that increasing the sample size does not help much to attenuate the relative ine ciency of the standard solution. This approach cannot be employed very often, and its performance quickly deteriorates even with moderate unbalancedness. For instance, Table 3 shows small di erences for T = 8 and J = 2 in RMSE (0:06 of A1W compared with 0:05 of A3W and A3Wb), but if unbalancedness is just a bit more intense, J = 4, the RMSE of A1W almost doubles to 0:11, whereas the RMSE of A3W and A3Wb barely changes. 9

11 These results remain unchanged when we consider in Table 5 and 6 a speci cation with lower state dependence, = 0:50. So far we have discussed only how well our proposed approaches perform to estimate the parameter. However, practitioners estimating non-linear models are ultimately interested in marginal e ects. Therefore, we nally consider the nite-sample performance of the estimated Average Marginal E ect (AME) in the model speci cation of Table 8. Since the true AME (slightly) varies with the sample drawn in each Monte Carlo simulation, Table 9 reports this true expected AME along with the estimated AME and the RMSE of the estimator. 4 References Heckman, J.J. (1981), The incidental parameters problem and the problem of initial conditions in estimating a discrete time discrete data stochastic process, in Structural Analysis of Discrete Data with Econometric Applications, Manski, C., McFadden, D. (eds). MIT Press: Cambridge, MA, Wooldridge, J.M. (2005), Simple Solutions to the Initial Conditions Problem for Dynamic, Nonlinear Panel Data Models with Unobserved Heterogeneity, Journal of Applied Econometrics 20, Wooldridge, J.M. (2010), Correlated Random E ects Models with Unbalanced Panels, unpublished manuscript. 10

12 Table 1: Monte Carlo Simulation results. Baseline Specification for α = 0.75 with Double Unbalancedness Panel A: N=200 A1H A1W A3W A3Wb A4H A1H A1W A3W A3Wb A4H α RMSE T=4 J= J= T=6 J= J= J= J= T=8 J= J= J= J= T=10 J= J= T=15 J= Panel B: N=500 A1H A1W A3W A3Wb A4H A1H A1W A3W A3Wb A4H α RMSE T=4 J= J= T=6 J= J= J= J= T=8 J= J= J= J= J= J= T=10 J= J= J= J= J= J= T=15 J= J= Note: In the baseline specification, first observation Y0 is exogenous (µ1 = 0), µη = 0, ση = 1, π0 = 1.25, µ2 = 1 and σ1 = σ2 = 0. 1

13 Table 2: Monte Carlo Simulation results. Baseline Specification for α = 0.75 with Left-side Unbalancedness Panel A: N=200 A1H A1W A3H A3W A3Wb A4H A1H A1W A3H A3W A3Wb A4H α RMSE T=4 J= T=6 J= J= T=8 J= J= J= J= T=10 J= J= J= J= T=15 J= Panel B: N=500 A1H A1W A3W A3Wb A4H A1H A1W A3W A3Wb A4H α RMSE T=4 J= T=6 J= J= T=8 J= J= J= J= T=10 J= J= J= J= J= J= T=15 J= J= J= J= J= J= Note: See note in Table 1. 2

14 Table 3: Monte Carlo Simulation results. Baseline Specification for α = 0.75 with Double Unbalancedness. N=1000 α RMSE T=4 J= J= T=6 J= J= J= J= T=8 J= J= J= J= J= J= T=10 J= J= J= J= J= J= T=15 J= J= J= J= J= J= Note: See note in Table 1. 3

15 Table 4: Monte Carlo Simulation results. Baseline Specification for α = 0.75 with Left-side Unbalancedness. N=1000 α RMSE T=4 J= T=6 J= J= T=8 J= J= J= J= T=10 J= J= J= J= J= J= T=15 J= J= J= J= J= J= Note: See note in Table 1. 4

16 Table 5: Monte Carlo Simulation results. Baseline Specification for α = 0.50 with Double Unbalancedness Panel A: N=500 α RMSE T=4 J= J= T=6 J= J= J= J= T=8 J= J= J= J= J= J= T=10 J= J= J= J= J= J= T=15 J= J= Panel B: N=1000 α RMSE T=4 J= J= T=6 J= J= J= J= T=8 J= J= J= J= J= J= T=10 J= J= J= J= J= J= T=15 J= J= J= J= J= J= Note: See note in Table 1.

17 Table 6: Monte Carlo Simulation results. Baseline Specification for α = 0.50 with Left-side Unbalancedness Panel A: N=500 α RMSE T=4 J= T=6 J= J= T=8 J= J= J= J= T=10 J= J= J= J= J= J= T=15 J= J= J= J= J= J= Panel B: N=1000 α RMSE T=4 J= T=6 J= J= T=8 J= J= J= J= T=10 J= J= J= J= J= J= T=15 J= J= J= J= J= J= Note: See note in Table 1. 6

18 Table 7: Monte Carlo Simulation results. Specification with Endogenouse Initial Condition for α = 0.75 and with Double Unbalancedness Panel A: N=500 α RMSE T=4 J= J= T=6 J= J= J= J= T=8 J= J= J= J= J= J= T=10 J= J= J= J= J= J= T=15 J= J= Panel B: N=1000 α RMSE T=4 J= J= T=6 J= J= J= J= T=8 J= J= J= J= J= J= T=10 J= J= J= J= J= J= T=15 J= J= J= J= J= J= Note: In this specification, µ 1 = 1 (so, the initial condition is endogenous) and the remaining parameters are as in Table 1.

19 Table 8: Monte Carlo Simulation results. Specification with Endogenouse Initial Condition for α = 0.75 and with Left-side Unbalancedness Panel A: N=500 α RMSE T=4 J= T=6 J= J= T=8 J= J= J= J= T=10 J= J= J= J= J= J= T=15 J= J= J= J= J= J= Panel B: N=1000 α RMSE T=4 J= T=6 J= J= T=8 J= J= J= J= T=10 J= J= J= J= J= J= T=15 J= J= J= J= J= J= Note: See note in Table 7. 8

20 Table 9: Monte Carlo Simulation results for Average Marginal Effects. Specification with Endogenouse Initial Condition and with Left-side Unbalancedness Panel A: N=500 AME ÂM E AME ÂM E AME ÂM E RMSE T=4 J= T=6 J= J= T=8 J= J= J= J= T=10 J= J= J= J= J= J= T=15 J= J= J= J= J= J= Panel B: N=1000 AME ÂM E AME ÂM E AME ÂM E RMSE T=4 J= T=6 J= J= T=8 J= J= J= J= T=10 J= J= J= J= J= J= T=15 J= J= J= J= J= J= Note: See note in Table 7. 9