Heterogeneity in Multinomial Choice Models, with an Application to a Study of Employment Dynamics

Transcription

1 , with an Application to a Study of Employment Dynamics Victoria Prowse Department of Economics and Nuffield College, University of Oxford and IZA, Bonn This version: September 2006 Abstract In the absence of at least one alternative specific explanatory variable, single period multinomial choice models with heteroscedastic and/or correlated unobserved heterogeneity are known to suffer from a fragile identification problem. This paper establishes a necessary requirement for the use of repeated choice data to solve this fragile identification problem: all parameters describing the within period distribution of unobservables must be identified by their effect on the intertemporal distribution of unobservables. A series of simulations is used to demonstrate that several different specifications of multiperiod mixed multinomial logit models with heteroscedastic and correlated unobservables satisfying this requirement yield well behaved and reliable parameter estimates. This result is applied to the problem of estimating a dynamic multinomial labor supply model using data on a panel of British women. The results of the analysis reveal that estimates of the marginal effects of individual characteristics on employment behavior are sensitive to the assumed distribution of unobservables. In particular, when the multinomial logit model is extended to include heteroscedastic and correlated within period unobservables, the magnitude of estimated income effects falls by half, and the effect of having a child on a woman s employment behavior decreases substantially. Key Words: Discrete Labor Supply; Heterogeneity; Mixed Logit; Multinomial Choice. Address for correspondence: Department of Economics, Manor Road Building, Manor Road, Oxford, OX1 3UQ, UK. victoria.prowse@economics.ox.ac.uk. Telephone: I express my thanks for useful comments and discussion of this paper and earlier drafts to the following people: Steve Bond; David Gill; Kathryn Graddy; Valérie Lechene; and Bent Nielson. I have also benefited from discussions with participants at seminars in Oxford and at the 2006 annual meeting of the Econometric Society in Vienna. Any remaining errors are my own. This work has been supported by the ESRC, grant number PAT

2 JEL Classification: C15; C25. 2

3 1 Introduction Multinomial choice situations are ubiquitous in economics. Examples include the choice of living arrangements by the elderly, the destinations of school leavers, occupational choice, choice of transportation mode, brand choice, individual and household labor supply, the retirement decisions of couples and voting behavior. Analysis of multinomial choice situations generally proceeds by expressing an individual s payoffs from choosing each of the available alternatives as the sum of a component attributed to the observed characteristics of the individual and the alternative, sometimes referred to as the systematic component, and a component due to unobservables (see, for example, McFadden, 1976, 1981). 1 The individual is assumed to be a utility maximizer and thus chooses the alternative with the highest payoff. Different multinomial choice models are obtained depending on the assumptions made about the unobservables. The multinomial logit model is the simplest model of multinomial choice and has been used extensively to analyze a variety of multinomial choice situations. However, it is well known that the multinomial logit model imposes implausible restrictions on the distribution of unobservables. Specifically, the distributional assumptions underlying the multinomial logit model require the unobserved components of an individual s payoffs from each of the alternatives to be identically and independently distributed. These assumptions exclude heteroscedasticity and correlations in the unobserved components of the individual s payoffs. The restrictive distribution of unobservables translates into empirically unappealing restrictions on the structure of choice probabilities and substitution patterns. Specifically, the multinomial logit model suffers from the independence from irrelevant alternatives property meaning that ratio of probabilities with which an individual chooses any two alternatives is entirely unaffected by the systematic component of the individual s payoffs from any other alternatives (see Ben-Akiva and Lerman, 1985; McFadden, 1974). Given the nonrobustness of parameter estimates and estimates of quantities of interest including marginal effects, elasticities and responses to policy variables to the distribution of unobservables it is desirable to use a multinomial modelling framework which permits a more general distribution of unobservables (Chintagunta, 1992; Hausman and Wise, 1979). Given repeated observations, a fixed effects approach can be employed to generalize the multinomial logit model to permit a more general distribution of unobservables. Specifically, the fixed effects binary logit model, also termed the conditional logit model, discussed by Chamberlain (1980) can be extended to a multinomial choice setting. The resulting fixed effects multinomial logit model allows time invariant individual effects with an arbitrary distribution. The fixed effects multinomial 1 Throughout this paper the term unobservables is used to refer to the collection of all unobserved factors including unobserved preference and taste variation and the unobserved attributes of the alternatives. The terms unobserved heterogeneity and errors have been used for this purpose in some related work. 3

4 logit model has however a number of drawbacks. Firstly, the fixed effects multinomial logit model is inefficient in the sense that data collected from individuals who do not choose each of the available alternatives at least once are not used. In many applications there is a high level of persistence in individual s choices over time and thus a large amount of data may be discarded. Moreover the fixed effects multinomial logit can not be extended to dynamic models with explanatory variables nor to models with random coefficients on time varying explanatory variables. Also for a moderately large number of time periods estimation of the model becomes highly cumbersome. Lastly, the results are not informative about the effects of time invariant explanatory variables or the distribution of the fixed effects. Consequently estimates of marginal effects, elasticities and response probabilities can not be obtained. The drawbacks of the fixed effects approach have lead the attention of researchers in this area to be focused mainly on models which generalize the multinomial logit model by making assumptions about the distribution of unobservables. Models of this form, unlike the fixed effects multinomial logit model, allow the researcher to use all available data, are readily applicable to dynamic models and to models with random coefficient and yield estimates of marginal effects, elasticities and response probabilities. The multinomial probit model is one example of a model which permits a more general distribution of unobservables than the multinomial logit model (see, amongst others, Bunch, 1991; Dansie, 1985; Keane, 1992; Heckman and Sedlacek, 1985). In the multinomial probit model the unobserved components of an individual s payoffs are assumed to follow a multivariate normal distribution. The multinomial probit model permits both heteroscedasticity in the unobserved components of an individual s payoffs and correlations in unobservables across the alternatives, and thus imposes less structure on choice probabilities and substitution patterns than the multinomial logit model. The multinomial probit model is however still restrictive is the sense that it constrains the unobservables to be normally distributed. The mixed multinomial logit model provides a more flexible framework for generalizing the distribution of unobservables in the multinomial logit model; in the mixed multinomial logit model unobservables can have any distribution. Moreover McFadden and Train (2000) show that under mild regularity conditions the mixed multinomial logit model can approximate to an arbitrary degree of accuracy any multinomial choice model generated from utility maximization. Due to its flexibility, the multinomial logit model has become a popular modelling framework for multinomial choice analysis (for application see Ben-Akiva et al., 2001; Bhat, 2001; Brownstone and Train, 1998; Train, 2001). This paper focuses on the performance of various mixed multinomial logit models in applications where there are no explanatory variables which vary across alternatives. In some applied settings alternative specific explanatory variables are usually available. For example, in models of brand 4

5 choice price and advertising variables differ across products, and in studies of travel mode choice price and travel time vary across the different modes. Similarly, in studies of voting behavior data on individuals opinions of each of the parties or candidates are sometimes available. However in other applications, including many of the multinomial choice problems encountered in labor economics, education and health economics, alternative specific variables are not observed. Consider, for example, a model of occupational choice. Typically, available explanatory variables consist of various socio-demographic variables including educational qualification and household structure variables. However, variables specific to occupations such as wage rates or training opportunities are unobserved or are observed only for the occupation chosen by the individual. Unfortunately multinomial choice models, including the mixed multinomial logit model, with flexible distributions of unobservables perform poorly in applications where all explanatory variables are individual specific. The first evidence of this was provided by Keane (1992). Using a series of simulations, Keane (1992) showed that multinomial probit models with heteroscedastic and/or correlated unobservables and only individual specific explanatory variables suffer from a fragile identification problem. The symptoms of fragile identification include instability in parameter estimates and a close to singular hessian which results in huge standard errors. Harris and Keane (1998) present further Monte Carlo evidence on this issue. Based on a mixed multinomial logit model including alternative specific explanatory variables Harris and Keane (1998) show that the standard errors of all parameter increase considerably as the variance of the alternative specific variables decreases. The above results imply that the parameters describing the distribution of unobservables in single period multinomial choice models are effectively not identified, although formally all of the parameters of these models are identified. As shown by Keane (1992) and illustrated below, the fragile identification problem is sever enough to render single period multinomial choice models with flexible specifications of unobservables and without alternative specific explanatory variables useless for empirical work. The above result has had a large and important impact on the methodology of empirical studies employing multinomial choice frameworks. Many researches cite the result as a reason for excluding heteroscedastic and correlated unobservables from multinomial choice models when alternative specific explanatory variables are unavailable (for examples see Adams et al., 1999; Bound et al., 1998; Dancer and Fiebig, 2004; Forman, 2005; Sturm and Wells, 1998). As explained above, imposing these restrictions on the structure on unobservables is likely to produce misleading results. Other researchers, unwilling to accept the restrictions implicit in the multinomial logit model, have sort to construct alternative specific explanatory variables from the available individual specific variables. For example, in a model of schooling choice Giannelli and Monfardini (2003) construct a measure of alternative specific expected future earnings while Keane (1992) and Heckman and Sedlacek (1985) 5

6 use predicted occupation specific earnings in models of occupational choice. The empirical success of this method relies on two criteria. Firstly, there must be sufficient variation across alternatives in the constructed alternative specific variables. This requirement is of practical relevance: Keane (1992) found that in one particular application constructed occupation specific income did not have sufficient variation to solve the fragile identification problem, although in other applications this method appears to perform well. Secondly, formal identification requires an exclusion restriction. Thus the researcher must find at least one individual specific variable which affects the alternative specific variable but which does not enter the utility function directly. This paper presents a methodology for estimating multinomial choice models with general specifications of unobservables in situations where alternative specific explanatory variables are unavailable. This method does not require the construction of alternative specific regressors. Instead it is argued that the extra information obtained for repeated observations of individuals choices can be used to obtain multinomial choice models which do not suffer from the fragile identification problem encountered in a single period setting. A necessary requirement for the use of repeated choice data to solve the fragile identification problem is established: all parameters describing the within period distribution of unobservables must be identified by their effect on the intertemporal distribution of unobservables. A series of simulations are used to demonstrate that several different specifications of multiperiod multinomial choice models, specifically mixed multinomial logit models, with heteroscedastic and correlated unobservables satisfying this requirement yield well behaved and reliable parameter estimates. Using the above described methodology, the common binary dynamic model of labor force participation is extended to incorporate a distinction between full-time employment and part-time employment. The resulting multiperiod multinomial labor supply model is estimated with five different specifications of unobserved heterogeneity ranging from identically and independently distributed unobservables to random intercept and random coefficient models which include heteroscedastic and correlated unobservables. The results of this application are of interest in two respects. Firstly, the parameter estimates show no signs of fragile identification, thus confirming the simulation results. Secondly, a comparison of the results across the five different specifications of unobservables reveals that estimates of the effects of individual characteristics on employment choices and estimates of the dynamic structure of employment choices are sensitive to the distribution of unobservables. The remainder of this paper proceeds as follows. Section 2 reviews the general set-up for modelling potentially repeated multinomial choice situations. Section 3 presents the results of a simple simulation experiment designed to illustrate the severity of the fragile identification problem which occurs in single period multinomial choice models. Section 4 outlines the necessary requirement for repeated choice data to solve the fragile identification problem encountered in a single period setting, 6

7 and Section 5 presents the results of simulation experiments designed to investigate the empirical performance of several multiperiod multinomial choice models which satisfy the requirement outlined in Section 4. Section 6 applies the methodology developed in this paper to a study of women s employment dynamics, and Section 7 concludes. 2 An Overview of Multinomial Choice Modelling This section proceeds by outlining the general set-up for modelling potentially repeated multinomial choice situations. Attention is then focused on one particular class of multinomial choice models: mixed multinomial logit models. As explained in the introduction, the mixed multinomial logit framework is a flexible tool for generalizing the distribution of unobservables beyond that permitted by the multinomial logit model. Within the context of the mixed multinomial logit model the requirements for identification are discussed, both for single period models and for repeated choice models. 2.1 General Set-up In each time period t = 1,..., T individual i must choose one of J distinct alternatives or choice possibilities. The individual receives a payoff V i,j,t if they choose alternative j at time t. 2 The payoff V i,j,t depends on a vector of explanatory variables, X i,t, consisting of individual characteristics and possibly the individual s past choices, and an unobserved factor ε i,j,t which varies over individuals, alternatives and time. Attributes of the alternatives enter the model through the unobservables; alternative specific variables are assumed to be unavailable. For the purpose of this analysis V i,j,t takes the following form V i,j,t = β j X i,t + ε i,j,t for j = 1,..., J; t = 1,..., T. (1) In the above β j for j = 1,..., J are suitably dimensioned alternative specific coefficient vectors. X i,t may include both characteristics of the individual at time t and at other time periods. In particular, in the spirt of Chamberlain (1980, 1984), X i,t may include selected individual specific sample means of time varying characteristics. The individual specific sample means can be interpreted as proxies for unobserved elements of preferences, in which case their inclusion permits a correlation between observed and unobserved characteristics or, in other words, preference endogeneity. Utility maximizing behavior implies that the individual chooses the alternative with the highest payoff. Thus 2 Throughout this paper the discussion is made in the context of repeated choice data however it is noted that the results are equally applicable to grouped data, for example children grouped by school or individuals within households. 7

8 the individual chooses alternative j at time t if and only if V i,j,t > V i,k,t for all k j. (2) Equations (1) and (2) combined imply that the individual chooses alternative j at time t with the following probability P i,j,t = P ( ) β j X i,t + ε i,j,t > max {β kx i,t + ε i,k,t }, (3) k=1,...,j, k j where the form of this probability function depends on the distribution of the unobservables. 2.2 The Mixed Multinomial Logit Model Throughout this paper attention is restricted to one particular class of multinomial choice models: mixed multinomial logit models. A mixed multinomial logit model, also called a logit kernel model (Ben-Akiva et al., 2001), is obtained by decomposing ε i,j,t as follows ε i,j,t = η i,j,t + ɛ i,j,t. (4) The error components ɛ i,j,t have the same distribution as the errors in a standard multinomial logit model. Specifically A1 ɛ i,j,t for all i, j and t are mutually independent. A2 For all i, ɛ i,j,t is independent of X i,s for all j and s, t = 1,..., T. A3 ɛ i,j,t for all i, j and t have type I extreme value distributions implying P (ɛ i,j,t q) = exp( exp( q)). To simplify notation let η i,t denote η i,j,t stacked over j, and η i denote η i,t stacked over t Within period heteroscedasticity and correlations in unobservables enter the model through the distribution of η i,j,t denoted F (η i ). 3 In a repeated choice setting the distribution function F (η i ) also determines the nature of any intertemporal dependencies in unobservables. follows Assumptions A1-A3 imply that, conditional on η i, the individual s choice probabilities are as P i,j,t (η i ) = exp(β j X i,t + η i,j,t ) k=1,...,j exp(β kx i,t + η i,k,t ) for j = 1,..., J; t = 1,..., T. (5) Note that equation (5) holds irrespective of the relationship between X i,t and η i,j,t, and thus X i,t may include lagged dependent variables. Conditional on η i, the individual s choices are independent over time and thus the conditional joint probability of the individual s observed sequence of choices 3 The model parametrization is assumed to be such that all deterministic elements of preferences enter the model through the indices β j X i,t for j = 1,..., J. 8

9 takes the following form P i (η i ) = ( ) Yi,j,t T exp(β j X i,t + η i,j,t ) k=1,...,j exp(β, (6) kx i,t + η i,k,t ) t=1 where Y i,j,t is an indicator variable taking the value one if individual i chose alternative j at time t and zero otherwise. The unconditional joint probability of the individual s observed sequence of choices is obtained by integrating with respect to F (η i ). P i = ( T S t=1 exp(β j X i,t + η i,j,t ) k=1,...,j exp(β kx i,t + η i,k,t ) ) Yi,j,t df (η i ), (7) where S denotes the support of the density of η i. The likelihood function for an independent sample of N individuals each observed for T time periods is given by L = N i=1 ( T S t=1 exp(β j X i + η i,j,t ) k=1,...,j exp(β kx i + η i,k,t ) ) Yi,j,t df (η i ). (8) If F (η i ) is degenerate the above model reduces to a multinomial logit model complete with the undesirable independence from irrelevant alternatives (IIA) property. In particular if F (η i ) is degenerate at zero then the following equality holds for any pair of alternatives j and k and any t which does not depend on β l for all l j, k. P i,j,t = exp(β jx i,t ) P i,k,t exp(β k X i,t ) = exp((β j β k )X i,t ), (9) Intuitively, the IIA property means that the ratio of the probabilities with which an individual chooses alternatives j and k is unaffected by the effect of characteristics on the individual s preference for alternative l. One implication of the IIA property is that a change in one of the coefficients in β l does not affect the ratio of probabilities with which the individual chooses any other alternatives. It is however more realistic to suppose that a change in one of the coefficients in β l might increase or decrease the relative probabilities with which the individual chooses the other alternatives. Specifying a non-degenerate distribution for η i yields a more flexible model possibly incorporating correlations and/or heteroscedasticity in the within period and between unobservables affecting the individual s payoffs from each of the alternatives. This generalization of the multinomial logit model provides additional flexibility and thus allows the model to break away from the IIA property. 9

10 2.3 Identification in the Mixed Multinomial Logit Model Prior to estimation identifying normalizations must be imposed on both the coefficients associated with the explanatory variables and the mixing distribution F (η i ). Identification is considered first for the single period mixed multinomial logit model, where the requirements for identification are most stringent. In terms of normalizations on the coefficients, it is clear from examining the likelihood function that the location of the coefficient vectors β j for j = 1,..., J is not identified. Intuitively this situation arises as individuals choices are based on utility differences and therefore adding a common increment to each of an individual s payoffs does not affect his or her choice problem. Location identification is obtained by normalizing all elements of one of these coefficient vectors to zero. In the following the normalization β J = 0 is imposed. Turning to normalizations on the mixing distribution, it is clear that only parameters describing the distribution of the J 1 of the elements of η i,1 differenced with respect to the remaining element of η i are identified. If, for example, F (η i,1 ) is a multivariate normal distribution function then only the parameters of the covariance matrix of J 1 of the elements of η i,1 differenced with respect to the remaining element of η i,1 are identified. Thus, in the case of a multivariate normal mixing distribution, a model with a J dimensional multivariate normal distribution function is observationally equivalent to a model where one element of η i,1 is normalized to zero and the remaining J 1 elements of η i,1 have a multivariate normal distribution. In additional to location normalizations, multinomial choice models require a scale normalization (see Bolduc, 1992; Bunch and Kitamura, 1991; Bunch, 1991; Dansie, 1985). A scale normalization is required as multiplying the payoff associated with each of the alternatives by a common positive constant leaves the individual s choice problem unaffected. However, in the mixed multinomial logit model, a set of scale normalizations have already been imposed: assumption A3 restrict the second component of the unobservables to have a fixed variance. Scale identification is therefore ensured provided that the distribution of η i,1 differs from that of ɛ i,1. 4 The introduction of further parameters into the multinomial logit model through the distribution function F (η i ) imposes an extra requirement for identification: in the presence of a distribution of unobservables possessing unknown parameters the model must contain an explanatory variable which varies sufficiently over individuals. This requirement is discussed by Heckman and Sedlacek (1985) in the context of a multinomial probit model, and Teicher (1963) discusses identification in the presence of a discrete mixing distribution. To understand this requirement consider a multinomial choice model where the only explanatory 4 Indeed the mixed multinomial logit model with a multivariate normal mixing distribution is formally identified without a scale normalization. However, without further restrictions, scale identification relies on the slight difference in the shapes of the appropriately scaled logistic and normal distribution functions (see Ben-Akiva et al., 2001). 10

11 variables are (J 1) alternative specific intercepts. The only information contained in a sample generated by this model is the proportion of individuals choosing each alternative. Hence the only parameters which are identified are the (J 1) alternative specific intercepts which map directly to the proportions of individuals choosing each alternative; any additional parameters, including parameters of F (η i ), are not identified. However, if the model is extended to include a trichomatous explanatory variable it may be possible to identify some or all of the parameters of F (η i ). In the trichomatous case it is possible to estimate for each of the three types of individual the probability of choosing each of the J alternatives. The sample is thus informative about 3 (J 1) separate quantities. Hence, in addition to the (J 1) alternative specific intercepts and (J 1) coefficients on the trichomatous explanatory variable, a maximum of (J 1) additional parameters are identified. In particular, in a three alternative model with a trichomatous explanatory variable and a multivariate normal mixing distribution with mean zero and covariance matrix Σ with Σ 1,1 normalized to one the two alternative specific intercepts and the two alternative specific coefficients are identified together with Σ 2,2 and Σ 1,2 = Σ 2,1. Similarly, in a three alternative model with a discrete mixing distribution with two mass points the two alternative specific intercepts and the two alternative specific coefficients are identified together with the location of one of the mass points and the mixing probability. Identification of the mixed multinomial logit model for repeated choices requires the same location normalization on the coefficients on the explanatory variables as the single period model. Also, at each time period t only the parameters describing the distribution of J 1 of the elements of η i,t differenced with respect to the remaining element of η i,t are identified. However, in a repeated choice model multiple observations on the same individuals together with some restrictions on the distribution of F (η i ) may relax the requirements concerning the variation in X i,t over individuals. Indeed, for the structures of unobservables considered below, identification often does not require an explanatory variable which varies over individuals. 3 Single Period Mixed Multinomial Logit Model The single period mixed multinomial logit model is obtained by setting T = 1 in the above. As explained in the Introduction, the single period multinomial choice model with a flexible specification of unobservables suffers from a fragile identification problem in the absence of an alternative specific explanatory variable. A simple simulation experiment based on a single period three alternative mixed multinomial logit model is conducted in order to illustrate the severity of this problem. The 11

12 payoffs associated with each of the three alternatives are as follows V i,1,1 = β 1,1 + β 2,1 W i,1 + η i,1,1 + ɛ i,1,1, (10) V i,2,1 = β 1,2 + β 2,2 W i,1 + η i,2,1 + ɛ i,2,1, (11) V i,3,1 = ɛ i,3,1. (12) The individual specific explanatory variable W i,1 has an independent standard normal distribution. The intercept, explanatory variable and η i,3,1 are excluded from V i,3,1 for identification purposes. The unobservables, ɛ i,j,1 for all i and j, obey assumptions A1-A3. Correlations and heteroscedasticity in the unobservables enter the model through the distribution of η i,1,1 and η i,2,1. The design of the simulation is such that η i,1,1 and η i,2,1 have a bivariate normal distribution with zero mean and covariance matrix Σ. As described above, without imposing further restrictions, scale identification relies on the slight difference in the shapes of the appropriately scaled logistic and normal distribution functions (see Ben-Akiva et al., 2001) and therefore seems prudent to normalize the variance of one of the η i,j,1 terms. For the purpose of this simulation experiment, Σ 1,1 is fixed at one. All six parameters β 1,1, β 2,1, β 1,2, β 2,2, ρ and Σ 2,2 are formally identified. 5 The simulation experiment involves generating two hundred data sets with N = 3000 and obtaining maximum simulated likelihood estimates for each of these data sets. The motivation for using maximum simulated likelihood estimation is discussed in Section 5. Table 1 shows the true parameter values together with the mean maximum simulated likelihood estimates, E[ θ], average asymptotic standard errors (obtained from the outer products of gradients), E[σ( θ)], and the standard deviation of the estimated parameters, SD( θ). presented together with the associated standard errors. The parameter estimates for four selected data sets are also The results confirm the findings of Keane (1992). Estimates of all of the parameters display clear signs of fragile identification: parameter estimates bear little resemblance to the true parameter values, are unstable over the four trials and many cases standard errors are several times higher than the corresponding parameter estimates. For example, in the first three trials the estimates of Σ 2,2 were much higher than the true value of two and also were insignificant. However in the fourth trial the estimate of Σ 2,2 was close to zero but with a standard error over seven times higher than the parameter estimate. 5 This simulation differers from that conducted by Keane (1992) as it is based on a mixed multinomial logit model with a normal mixing distribution while Keane (1992) uses a multinomial probit model. These two specifications are expected to perform similarly as the distribution of unobservables in the mixed multinomial logit model with a normal distribution of correlated unobservables is, except for scale differences, very similar to the distribution of the unobservables in the multinomial probit model. 12

13 PARAMETER TRUTH TRIAL 1 TRIAL 2 TRIAL 3 TRIAL 4 E[ b θ] E[σ( b θ)] SD( b θ) β 1, β 2, (0.23) β 1, (1.01) β 2, (0.35) Σ 2, (1.91) 0.58 (0.10) (1.49) 0.42 (0.53) 3.47 (2.53) 0.83 (0.11) 0.49 (0.15) 2.78 (2.27) 0.33 (1.04) 1.99 (1.38) 0.99 (0.21) 0.22 (0.33) (0.27) 0.21 (0.27) Σ 2, (9.47) (17.50) (32.60) 0.05 (0.39) Notes: At the chosen parameter values 37.4% of individuals choose alternative 1, 39.9% choose alternative 2 and 22.7% choose alternative 3. The likelihood function was evaluated using 50 Halton draws based on primes 2 and 3. Table 1: Summary simulation results for a one period mixed multinomial logit model with a bivariate normal mixing distribution. 4 Repeated Multinomial Choice and Fragile Identification A necessary requirement for the use of repeated choice data to solve the fragile identification problem encountered in single period multinomial choice models with flexible structures of unobservables is that all parameters describing the within period distribution of unobservables be identified by their effect on the intertemporal distribution of unobservables. Intuitively, repeated observations yield a larger sample size and additional information concerning the intertemporal structure of unobservables. A larger sample size provides more information about within period behavior. However, the additional information concerning the parameters of the distribution of within period unobservables that arises from a larger sample is of the same type as the information about these parameters which is available from a single period of data. Moreover since the fragile identification problem is not a small sample problem it will not be eliminated by increasing the sample size. However, information concerning the intertemporal structure of unobservables may be useful in eliminating the fragile identification problem. In particular, if the parameters describing the within period distribution of unobservables also affect the intertemporal structure of unobservables then repeated observation provide additional identifying power with respect to the parameters of the distribution of within period unobservables. Given that the parameters of the distribution of within period unobservables are (effectively) not identified by their effect on the distribution of within period unobservables, a minimum requirement for these parameters to be (effectively) identified in a multiperiod setting is that they be formally identified solely by their effect on the the intertemporal structure of unobservables. Whether this requirement is satisfied depends on the estimation method. Full information estimation methods such as full information maximum likelihood and efficient method of moments estimation use all available information efficiently and thus utilize any information contained in the 13

14 repeated observations. Thus efficient estimation methods maximize the likelihood of this requirement being satisfied. In contrast pooled estimation methods ignore any information contained in the repeated observations and thus the availability of repeated observations will not solve the fragile identification problem if pooled estimation is employed. All estimations in this paper are conducted using full information maximum likelihood estimation as this method has been the main method previously used to estimate multinomial choice models and it utilizes the maximum amount of information provided by repeated observations. Issues surrounding the implementation of the maximum likelihood estimator are discussed in Section 5. Specific examples of distributions of unobservables satisfying the above requirement are presented in Sections 5 and 6. However it is noted here that unobservables η i taking the form of time invariant normally distributed alternative specific individual effects, perhaps the most obvious first pass attempt at generalizing the distribution of unobservables, satisfy the above requirement provided that an efficient estimation method is used. Similarly, random coefficient models where the random coefficients are alternative specific time invariant normally distributed individual specific effects also satisfy the above requirement. 5 Simulation Experiments The above requirement is a necessary but not sufficient requirement for repeated choice data to eliminate the fragile identification problem. It is practical question as to which specifications, if any, of the intertemporal structure of unobservables produce multinomial choice models which do not suffer from the fragile identification problem encountered in a single period setting. Therefore, simulation experiments are conducted in order to evaluate the empirical performance the several different specifications of the multiperiod mixed multinomial logit model. As in Section 3, these simulations are based on a three alternative model. The payoffs associated with each of the three alternatives are as follows V i,1,t = β 1,1 + β 2,1 W i,t + β 1 W i + γ 1,1 Y i,1,t 1 + γ 2,1 Y i,2,t 1 + η i,1,t + ɛ i,1,t, for t = 1,..., T, (13) V i,2,t = β 1,2 + β 2,2 W i,t + β 2 W i + γ 1,2 Y i,1,t 1 + γ 2,2 Y i,2,t 1 + η i,2,t + ɛ i,2,t, for t = 1,..., T, (14) V i,3,t = ɛ i,3,t, for t = 1,..., T, (15) Y i,3,0 = 1 [Exogenous Initial Conditions]. (16) In the above W i,t is a scalar individual specific explanatory variable and W i is the individual specific sample mean of W i,t. The intercept, explanatory variables and η i,3 are excluded from V i,3,t for identification purposes. The unobservables, ɛ i,j,t for j = 1, 2, 3, obey assumptions A1-A3. Described below are the five different model model specifications for which simulation experiments are conducted. Specification I-IV are static models (obtained by setting γ 1,1, γ 2,1, γ 1,2 and γ 2,2 to 14

15 zero), while Specification V is a dynamic model. For each of these specifications the requirement described in Section 4 is satisfied provided that a sufficient number of repeated observations are available. Specification I (static model; time invariant random intercepts): η i,1,t = ν i,1, for t = 1,..., T, η i,2,t = ν i,2, for t = 1,..., T, ν i,1 ν i,2 N(0, Σ). (17) For this specification the above requirement is satisfied provided T 2: the covariance between the unobservables at t = 1 and t = 2 is Σ, and all parameters Σ are identified from this covariance. In contrast to the single period model, scale identification does not rely of the difference in the shapes of the logistic and normal distribution functions and thus no normalizations, beyond positive definiteness, are placed on the covariance matrix Σ. Specification II (static model with preference endogeneity; time invariant random intercepts): The specification of the unobserved components of preferences is as for Specification I. Individual specific sample means are included to capture preference endogeneity. For this model specification the above requirement is again satisfied provided T 2: adding the individual specific mean of W i,t to the model does not add any complications with respect to the above requirement. However, the coefficient on the individual specific sample mean is only identified separately from the coefficient on the associated explanatory variable if W i,t varies over time and W i varies over individuals. Specification III (static model; time invariant random intercepts and time invariant random coefficients): η i,1,t = ω i,1 W i,t + ν i,1, for t = 1,..., T, η i,2,t = ω i,2 W i,t + ν i,2, for t = 1,..., T, ν i,1 ν i,2 N(0, Σ), ω i,1 ω i,2 N(0, Ω). (18) The number of repeated observations required to satisfy the above requirement depends on the nature of the explanatory variable W i,t. The covariance between the unobservables at time t and time t + j where j > 0 is given by Σ+W i,t W i,t+j Ω. Hence T 2 is sufficient to satisfy the above requirement provided W i,1 W i,2 varies over individuals. If there is no within period variation in W i,t, for example if W i,t is a time trend, then more repeated observations are required. Specifically there must be enough repeated observations such at a minimum of two distinct values of W i,t W i,t+j are available. 15

16 Specification IV (static model; autocorrelated random intercepts): η i,1,t = ν i,1,t, for t = 1,..., T, ν i,1,t ν i,1,t = ρ 1 ν i,1,t 1 + ζ i,1,t. N(0, Σ), η i,2,t = ν i,2,t, for t = 1,..., T, ν i,2,t = ρ 2 ν i,2,t 1 + ζ i,2,t. ν i,2,t (19) In the above ζ i,1,t and ζ i,2,t are mutually independent and independent over time. These assumptions imply the ζ i,1,t is normally distributed with zero mean and variance Σ 1,1 (1 ρ 2 1 ) and, similarly, ζ i,2,t is normally distributed with zero mean and variance Σ 2,2 (1 ρ 2 2 ). Specification IV requires T 3 in order to satisfy the above requirement. covariance matrices Cov(η i,t, η i,t+1 ) = ρ 1Σ 1,1 ρ 2 Σ 2,1 ρ 1 Σ 2,1 ρ 2 Σ 2,2 This can be illustrated by considering the following, Cov(η i,t, η i,t+2 ) = ρ2 1 Σ 1,1 ρ 2 2 Σ 2,1 ρ 2 1 Σ 2,1 ρ 2 2 Σ 2,2. (20) It is not possible to identify the five parameters appearing in Cov(η i,t, η i,t+1 ) from only two periods of observations. However adding a third period of data allows all of the parameters to be identified from the covariances in unobservables between the three periods. Specification V (dynamic model; time invariant random intercepts): This model augments Specification I by including lagged dependent variables. Due to the inclusion of the lagged dependent variables an assumption concerning the initial conditions is required. For the purpose of this simulation experiment all individuals choose alternative 3 at t = 0. This assumption allows the initial conditions to be treated as exogenous. 6 The introduction of the lagged dependent variables does not pose any additional complications with respect to the above requirement and thus, as for Specification I, T 2 is required. Each simulation experiment is based on two hundred data sets where each data set is constructed such that it contains 3000 individuals each observed T times. W i,t is constructed to be identically and independently distributed over both individuals and time and to have a standard normal distribution. With this choice of W i,t specifications I, II, III and V satisfy the above requirement provided T 2 while Specification IV requires T 3. The simulation experiments are conducted with T = 3, T = 4, and, where appropriate, T = 2. For all five model specifications closed form expressions for the individual likelihood contributions do not exist. For two dimensional problems fast and accurate cubature methods are available 6 In the application considered in Section 6 the exogeneity assumption is relaxed and a more general specification of the initial conditions is used. 16

17 to evaluate the individual likelihood contributions (Geweke, 1996, provides a survey). However numerical methods are unable to evaluate the likelihood contributions with sufficient speed and accuracy to be effective in problems where the dimension of integration is greater than two (see Bhat, 2001; Hajivassiliou and Rudd, 1994); in problems where the dimension of integration is three or more simulation techniques are the most appropriate method of evaluating the likelihood contributions. Simulation methods, of which there are many variants, proceed by sampling R times from the distribution of η i and constructing P i (ηi r ) for r = 1,..., R. Simulated individual likelihood contributions are obtained by averaging P i (η r i ) over the R draws from the distribution of η i. Substituting the simulated individual likelihood contributions into the likelihood function yields the simulated likelihood. The maximum simulated estimates are obtained by maximizing the log simulated likelihood function. By the strong law of large numbers the simulated maximum likelihood estimates converge almost surely to the true parameters as R and N. Moreover, if R increases at a fast enough relative to N maximum simulated likelihood estimation is asymptotically equivalent to maximum likelihood estimation. In particular, with pseudo random draws the rate of convergence is R 0.5 (Hajivassiliou and Rudd, 1994). Halton draws (Halton, 1960), described in Appendix I, provide an alternative method of sampling for the distribution of η i. Halton draws are quasi random draws which achieve better coverage of the support of the density of η i than pseudo random draws and generate simulated probabilities which are negatively correlated over individuals. Maximum simulated likelihood estimation using Halton draws converges to maximum likelihood estimation at a rate of at least R 1 (Bhat, 2001). 7 For the purpose of the simulation experiments maximum simulated likelihood estimation using Halton draws is used throughout. While some of the model specifications involve integration over only two dimensions are thus could be estimated using cubature methods simulation methods are used to facilitate comparisons with more complicated specifications which are not feasible to estimate by cubature methods. Moreover, Bhat (2001) shows that for two dimensional problems satisfactorily accurate estimation using cubature techniques requires as many as ten cubature points making the method very costly in terms of computational time, and equally accurate results can be obtained at a fraction of the computational effort using Halton draws. Specifications I, II and V involve integration over two dimensions which is accomplished using 50 Halton draws based on primes 2 and 3. Specification III requires integration over four dimensions, for 7 Antithetic variates, introduced by Hammersley (1956), provide an alternative method of improving the accuracy of maximum simulated likelihood estimation. Let η i have a symmetric distribution with mean µ. In order to obtain R antithetic draws from the distribution of η i R/2 independent draws are made, denoted {ε r } R/2 r=1. The remaining R/2 draws are given by {ηi r } R/2 r=1 +2µ. Hajivassiliou (1999) shows that the use of antithetic variates in maximum simulated likelihood problems effectively halves the number of draws required to obtain a given level of accuracy. While this is a substantial gain, Halton draws can reduce the required number of draws by an even larger amount (see, for example, Hensher, 2001, and Train, 2001) and thus attention is restricted to Halton draws. 17

18 which 100 Halton draws based on primes 2, 3, 5 and 7 are used. Specification IV involves integration over 2 T dimensions. For this 250 Halton Draws are used, based on the first 2 T prime numbers. For model specifications I, II, III and V analytic derivatives of the log likelihood function with respect to the model parameters are easy to compute and speed up the optimization process considerably. For specification IV obtaining analytic derivatives of the log likelihood function with respect to the parameters of the covariance matrix is very complicated and consequently numerical derivatives for these parameters are used, while derivatives for the remaining parameters are computed analytically. Table 2 report the results of these simulation experiments for T = 2, while Tables 3 and 4 report the results of simulation experiments for T = 3 and T = 4 respectively. In sharp contrast to the single period model the maximum simulated likelihood estimates for all multiperiod models are fairly well behaved. Consider, for example Specification I when T = 2. The average maximum simulated likelihood estimates are very close to the true parameter estimates. Furthermore, for all parameters, the average asymptotic standard errors are close to the standard deviation of the parameter estimates. Thus, for this model specification, the fragile identification problem encountered in the single period case is avoided by using repeated observations. Increasing the number of time periods from two to three increases the precision of the parameter estimates, particularly the parameters of the covariance matrix. Much the same applies to specifications II, III and V: average parameter estimates are close to the true values and average asymptotic standard errors are close to the standard deviation of the parameter estimates. For Specification IV, where correlated and heteroscedastic unobservables take the form of autocorrelated random intercepts, the simulations results are less impressive but still a vast improvement on the single period case. In particular, for T = 3 there is some evidence of upwards bias in the autocorrelation parameters ρ 1 and ρ 2, while for the variance Σ 2,2 the average asymptotic standard error exceeds somewhat the standard deviation of the parameter estimates. 6 Application: A Study of Women s Employment Dynamics The methodology developed above is used to estimate a dynamic multinomial labor supply model with several different distributions of unobservables. The implications of different assumptions regarding the distribution of unobservables for conclusions concerning the effects of individual characteristics and the structure of intertemporal dependencies in employment behavior are investigated. This analysis represents an innovation to the literature on dynamic multinomial models of labor supply (see, for example Blank, 1989; Burdett and Taylor, 1994; Francesconi, 2002) where, due to the complications posed by including flexible distributions of unobservables in multinomial choice models, little attempt has been made assess the robustness of these models to the distribution of 18

19 T = 2 SPECIFICATION PARAMETER TRUTH I II III IV V E[ b θ] E[σ( b θ)] SD( b θ) E[ b θ] E[σ( b θ)] SD( b θ) E[ b θ] E[σ( b θ)] SD( b θ) E[σ( b θ)] E[σ( b θ)] SD( b θ) E[ b θ] E[σ( b θ)] SD( b θ) β1, β2, β3, γ1, γ2, β1, β2, β3, γ1, γ2, β β Σ1, Σ2, Σ2, Ω1, Ω2, Ω2, ρ ρ Table 2: Summary of simulation results for two period models. 19