The specification of dynamic discrete-time two-state panel data models

Gørgens, Tue; Hyslop, Dean Robert Article The specification of dynamic discrete-time two-state panel data models Econometrics Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Gørgens, Tue; Hyslop, Dean Robert (2019) : The specification of dynamic discretetime two-state panel data models, Econometrics, ISSN 2225-1146, MDPI, Basel, Vol. 7, Iss. 1, pp. 1-16, https://doi.org/10.3390/econometrics7010001 This Version is available at: https://hdl.handle.net/10419/247502 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ econometrics Article The Specification of Dynamic Discrete-Time Two-State Panel Data Models Tue Gørgens 1,* and Dean Robert Hyslop 2,* 1Research School of Economics, The Australian National University, Acton ACT 2601, Australia 2Motu Economic and Public Policy Research, P.O. Box 24390, Wellington 6142, New Zealand *Correspondence: [email protected] (T.G.); [email protected] (D.R.H.) Received: 1 November 2018; Accepted: 18 December 2018; Published: 24 December 2018   Abstract: This paper compares two approaches to analyzing longitudinal discrete-time binary outcomes. Dynamic binary response models focus on state occupancy and typically specify low-order Markovian state dependence. Multi-spell duration models focus on transitions between states and typically allow for state-specific duration dependence. We show that the former implicitly impose strong and testable restrictions on the transition probabilities. In a case study of poverty transitions, we show that these restrictions are severely rejected against the more flexible multi-spell duration models. Keywords: panel data; transition data; binary response; duration analysis; event history analysis; dynamic models; censored data; initial conditions; random effects JEL Classification: C33; C35; C41; C51 1. Introduction This paper is about modeling discrete-time two-state panel data, where outcomes indicate which of two states an individual is occupying in each period, and where transitions between states occur between periods. Often individuals’ outcomes are characterized by a degree of persistence, and an important part of the analysis is to discover the extent to which persistence is due to heterogeneity across individuals or to true state dependence (e.g., Heckman 1978,1981c;Heckman and Borjas 1980). Analysis of such data is central to many empirical studies in economics and other social sciences.1 There are two conceptually distinct approaches to analyzing such two-state panel data in the literature. First, dynamic binary response (DBR) approaches focus on the probability of occupying one of the two states in each period, and usually assume Markovian state dependence, in which the current period’s occupancy depends on the occupancy of previous periods. In contrast, multi-spell duration (MSD) approaches focus on the probability of a transition between states occurring in each period, and usually assume current spell duration dependence (or semi-Markovian state dependence), in which the transition probability depends on the elapsed duration in the current state. This paper compares the DBR and MSD approaches. In the first part of the paper, we present prototype DBR and MSD models and discuss the relationship between them. The DBR models are simpler and more restrictive than the MSD models, involving fewer equations and parameters, and require less data and information about past outcomes. We show that typical DBR models embody 1 Typical topics include employment (e.g., Heckman 1981a;Hyslop 1999), unemployment (e.g., Arulampalam et al. 2000), poverty (e.g., Stevens 1999;Cappellari and Jenkins 2004), welfare dependency (e.g., Bane and Ellwood 1983), health (e.g., Halliday 2008), and peace and conflict between national states (e.g., Beck and Katz 1997;Beck et al. 2001). Econometrics 2019,7, 1; doi:10.3390/econometrics7010001 www.mdpi.com/journal/econometrics Econometrics 2019,7, 1 2 of 16 strong restrictions on the corresponding probabilities of transitioning between states (the hazard rates). First, they restrict the effects of observed and unobserved heterogeneity to have the same magnitude but opposite signs on the implied transition probabilities. Second, r th order DBR models restrict the implied transition probabilities to be constant after r periods into a spell. In fact, first and second-order DBR models are special cases of a particularly simple MSD model, where duration dependence is limited to one or two periods. In the second part of the paper, we use an empirical case study to illustrate the two approaches. We analyze data from the US Panel Study of Income Dynamics (PSID) on individual poverty experiences, previously analyzed by Stevens (1999) using an MSD approach. We fit a range of DBR and MSD model specifications. The estimation results show MSD models dominate the more restrictive DBR models on several dimensions. In particular, the patterns of state dependence in these data are more complicated than allowed for in simple DBR models, and the restriction of opposite effects of heterogeneity on poverty entry and exit is also strongly rejected. Consequently, the MSD models provide better within-sample predictions than do the DBR models. Consistent with recent literature (e.g., Bhuller et al. 2017), we conclude that the standard dynamic binary response model is unacceptably restrictive in this context. In practice, simple first- or at most second-order DBR models are far more widely used than MSD models. Our theoretical and empirical results suggest that data analysis can benefit from considering less restrictive models and, in particular, from considering the implications of model specifications on both probabilities of occupying a particular state and probabilities of transitioning between states. To the best of our knowledge, the relationship between the DBR and MSD models has not been formally recognized, and very few studies have discussed the implications of the DBR models for transition probabilities and spell durations. Cappellari et al. (2007) compared a duration and a Markov model for employment transitions, and Bhuller et al. (2017) analyzed the adequacy of first-order dynamic binary response models against more general models that allow for duration and occurrence dependence. Gørgens and Hyslop (2018) showed that the DBR and MSD approaches are equivalent in a nonparametric context, and that nonparametric DBR models of order r≤ 2 are nested within a simple MSD model. Among other things, this implies that either model can be used to estimate both probabilities of occupying a state and probabilities of transitioning between states. The present paper complements that analysis by showing that the equivalence and the nesting property carries over to commonly used parametric model specifications, and by documenting that the DBR restrictions are rejected in an empirical case study. The paper is organized as follows. Section 2introduces the two approaches and present prototypical DBR and MSD models. Section 3presents the empirical analysis. The paper concludes with a discussion in Section 4. 2. Modeling Discrete-Time Two-State Panel Data In this section we provide context for our analysis, present prototype DBR and MSD models, and discuss how they are related. 2.1. Context, Data, and Likelihood Our interest in this paper is processes that are well represented in discrete time. In a typical application, time is divided into periods of equal length, an individual occupies one of two states during each period, and transitions between states occur between periods. This framework is particularly well suited for studies where a time scale is determined by convention or by law. For example, in some countries receiving welfare is determined on a weekly or monthly basis. The framework is also applicable when an outcome indicates the state an individual is occupying at a point in time and transitions take place during the period between these observations, provided it is reasonable to assume that at most one transition takes place in each period and that the precise timing of the transition within this period can be ignored. The framework is not suitable for data where an outcome Econometrics 2019,7, 1 3 of 16 indicates state occupancy at a point in time and multiple transitions are likely between the observation times. For example, an analysis of employment status at the time of an annual interview must deal with unobserved transitions between interviews. The data available for analysis are indicators of state occupancy, indicators of transition between states, and covariates for N individuals observed over T periods. We assume the sample is an independent random sample from a given population. The indicators of the state occupied by individual i at time t are denoted Yit with Yit ∈ { 0, 1 } . The indicators of whether or not individual i makes a transition between times t− 1 and t are denoted Cit with Cit ∈ { 0, 1 } . The covariates for individual i at time t are denoted Xit with Xit ∈Rdim(x) . The covariate time reference is for modeling purposes, and Xit may include contemporaneous values, lags and leads, of underlying variables. Let Hit = (Yi1. . . , Yit) denote the outcome history up to time t . Lower case letters with a subscript i represent observed values of the corresponding upper case random variables. For simplicity, we assume the data constitute a balanced panel, and the data for each individual are both left- and right-censored. For identification and estimation purposes, we also assume that the data censoring is independent of the underlying process. Following the literature (e.g., Heckman and Singer 1984), we allow for unobserved heterogeneity in the form of random effects in the equations which make up the DBR and MSD models discussed below. Let Vi denote a random vector representing unobserved heterogeneity for individual i . We assume that Vi has a discrete distribution with support {ν1 , . . . , νK} and probability distribution π1 , . . . , πK with ∑K k=1πk= 1. Each νk is a vector with as many element as there are equations in the model. As mentioned, the DBR approach focuses on the probabilities of occupying one of the states in each period given previous history while the MSD approach focuses on the probabilities of making a transition between states in each period. For convenience, define θI k(xi1) = P(Yi1=1|Xi1=xi1,Vi=νk), (1) θY k(hit−1,xit) = P(Yit =1|Hit−1=hit−1,Xit =xit,Vi=νk),t=2, . . . , T, (2) θC k(hit−1,xit) = P(Cit =1|Hit−1=hit−1,Xit =xit,Vi=νk),t=2, . . . , T. (3) Assuming independent sampling and independent censoring, the conditional likelihood function for the outcome data given covariates has the form LW= N ∏ i=1K ∑ k=1 πkθI k(xi1,νk)yi1[1−θI k(xi1,νk)]1−yi1× T ∏ t=2 θW k(hit−1,xit)wit [1−θW k(hit−1,xit)]1−wit ,(4) where W∈(Y , C) , and (θW k , wit) is either (θY k , yit) or (θC k , cit) . Typical modeling involves parameterizing the probabilities and estimating the unknown parameters by maximizing this likelihood function. Gørgens and Hyslop (2018) showed that the DBR and MSD approaches are equivalent in a nonparametric context. In particular, one can transform data on state occupancy, (Yi1 , Yi2 , . . . , Yit) , to data on transitions between states, (Yi1 , Ci2 , . . . , Cit) , and vice versa, and one can transform probabilities of state occupancy, θY k , to probabilities of transition between states, θC k , and vice versa. In the present paper, we show that prototype parametric DBR and MSD models are also equivalent (barring initial conditions and left-censoring as explained later). Econometrics 2019,7, 1 4 of 16 2.2. Prototype DBR Models The DBR approach assumes that the probabilities of being in a given state depend on the history only through the r most recent outcomes (Markovian state dependence of order r ). Formally, for r≥ 1 it is assumed that θY k(hit−1,xit) = P(Yit =1|Yit−r=yit−r, . . . , Yit−1=yit−1,Xit =xit,Vi=νk),t=r+1, . . . , T, (5) where (yit−r , . . . , yit−1) are the r -most recent outcomes in hit−1 . Equation (5) is the DBR model’s main equation of interest, which we refer to as the “structural” equation. The simplest and most common DBR model used empirically adopts r= 1, although r= 2 is sometimes used in cases of either higher-frequency and/or longer-period data (e.g., Chay et al. 1999;Card and Hyslop 2005,2009; Andrén and Andrén 2013). Equation (5) does not restrict the probability distribution for the initial r outcomes, (Yi1 , . . . , Yir) , referred to as the “initial conditions” of the process. When the data are left-censored, there is little interest in the probabilities associated with the initial conditions, but it is important they are dealt with unless they can be considered to be exogenous. Adapting the ideas of Heckman (1981b), we shall model the initial conditions using r“approximate reduced form” equations. In the empirical case study in Section 3we consider models with r= 1 and r= 2, labeled DBR1 and DBR2 respectively. The DBR1 model has two equations: θI k(xi1) = G(νk1+β0 1xi1), (6) θY k(hit−1,xit) = G(νk2+β0 2xit +γ2yit−1),t=2, . . . , T, (7) where Gdenotes the logistic function. In the DBR2 model we allow second-order Markovian state dependence. This model extends the first-order model to include two equations for the first two outcomes, while the structural equation includes two lags of the outcome variable as well as their interaction term. Thus, the DBR2 model has three equations: θI k(xi1) = G(νk1+β0 1xi1), (8) θY k(hi1,xi2) = G(νk2+β0 2xi2+γ2yi1), (9) θY k(hit−1,xit) = G(νk3+β0 3xit +γ31yit−1+γ32yit−2+γ33yit−1yit−2),t=3, . . . , T. (10) The prototype DRB models can be generalized. In addition to allowing for higher-order state dependence, several authors have pointed out the possibility of interacting covariates with the lagged outcome variables (e.g., Heckman 1981c;Barmby 1998;Beck et al. 2001). Such an extension leads to a intermediate specification between the prototypical DBR and MSD models, discussed in Section 2.4 and in the case study in Section 3. 2.3. Prototype MSD Models The MSD approach assumes that the probabilities of a transition between states depends only on the state currently occupied and the elapsed time spent in the current state (duration dependence, or semi-Markovian state dependence). Let Fi denote the time of the first observed transition for individual i , with Fi>T if no transitions are observed. Also, let Dit denote the observed elapsed time in the spell observed in period t : Di1= 1 for the left-censored spells, and DiFi= 1 when Fi≤T . Then the prototype MSD model assumes that the transition probabilities for the “fresh” (i.e., non-left-censored) spells that begin during the observation period satisfy θC k(hit−1,xit) = P(Cit =1|Yit−1=yit−1,Dit−1=dit−1,Xit =xit,Vi=νk),t=fi+1, . . . , T. (11) Econometrics 2019,7, 1 5 of 16 Conditioning on Yit−1=yit−1 captures first-order Markovian state dependence, and conditioning on Dit−1=dit−1 captures duration dependence. We refer to (11) as the MSD model’s “structural” equation of interest. In practice, the usefulness of assumption (11) will depend on the number of transitions observed within the observation period. In applications with substantial persistence in state occupancy and few transitions, there may be relatively few fresh-spell observations for which Equation (11) applies. This is alleviated if the effect of elapsed duration becomes constant after some time. For this purpose, we consider the additional assumption that the effect of elapsed duration is constant after m periods within the spell. (For example, if m= 1, the transition probabilities are constant throughout each spell.) To write this compactly, define Dm it =min(Dit,m). Then, for m≥1, θC k(hit−1,xit) = P(Cit =1|Yit−1=yit−1,Dm it−1=dm it−1,Xit =xit,Vi=νk),t=min(fi,m) + 1, . . . , T.(12) The main advantage of assumption (12) is that only parameters for times 1, . . . , m depend on unobserved outcome variables. In other words, Equation (12) implies that all data after the earlier of the first observed transition at fi or time m contribute to identifying and estimating the structural parameters of interest. Neither (11) nor (12) restrict the probabilities of making a transition out of the left-censored initial spells, θC k(hit−1,xit),t=1, . . . , min(fi,m). (13) As in the DBR case, these are typically nuisance parameters, and again we use Heckman’s (1981b) ideas in modeling them. A fully flexible specification of the approximate reduced form would involve separate equations for each probability in (13). Depending on the extent of left-censoring, this may be prohibitive in practice. We adopt a more parsimonious approach and specify three equations for the probability of the initial outcome, and the initial spell transitions from each state. In the empirical study in Section 3we consider two models which both assume (11) and (12), and capture duration dependence through a flexible specification with separate parameters for the first m potential transition times in each state ( m= 6). The first model, MSD1, uses all the data available but does not fully exploit Assumption (12). Specifically, the duration dependence is assumed to be constant after duration m , but this restriction is not imposed across the structural equations and probabilities for left-censored spells. This model has five equations: a reduced form equation for the initial state, two reduced-form equations for modeling transitions from the initial spells, and two structural equations for modeling the transitions from the fresh spells. The model specification (for m>1) is θI k(xi1) = G(νk1+β0 1xi1), (14) θC k(hit−1,xit) = G(νk2+β0 2xit +∑m j=2λ2j1(dm it−1≥j)),yi1=0, t=2, . . . , min(fi,m), (15) θC k(hit−1,xit) = G(νk3+β0 3xit +∑m j=2λ3j1(dm it−1≥j)),yi1=1, t=2, . . . , min(fi,m), (16) θC k(hit−1,xit) = G(νk4+β0 4xit +∑m j=2λ4j1(dm it−1≥j)),yit−1=0, t=min(fi,m) + 1, . . . , T, (17) θC k(hit−1,xit) = G(νk5+β0 5xit +∑m j=2λ5j1(dm it−1≥j)),yit−1=1, t=min(fi,m) + 1, . . . , T. (18) The second model, MSD2, utilizes assumption (12) together with a second implication, namely that we can ignore observed transitions in the first m periods, and still obtain consistent estimates of the structural equation parameters of interest. This may entail some loss of precision in estimating the structural parameters, but simplifies modeling the nuisance parameters. Specifically, for dit−1≥m we restrict Equations (15) and (17) to be the same and Equations (16) and (18) to be the same. Furthermore, we ignore the likelihood contributions for the first m− 1 time periods. This allows us to estimate a three-equation model, which represent the approximate reduced form specification for the probability Econometrics 2019,7, 1 6 of 16 distribution of the state at time m , and the two transition probabilities out of the state-specific spells. The MSD2 model specification (for m>1) is θC k(him−1,xim) = G(νk1+β0 1xim), (19) θC k(hit−1,xit) = G(νk2+β0 2xit +∑m j=2λ2j1(dm it−1≥j)),yit−1=0, t=m+1, . . . , T, (20) θC k(hit−1,xit) = G(νk3+β0 3xit +∑m j=2λ3j1(dm it−1≥j)),yit−1=1, t=m+1, . . . , T. (21) The prototype MSD models can be generalized, for example, by allowing occurrence dependence (i.e., the effects of the number of previous transitions), and/or lagged duration dependence (i.e., completed durations of previous spells) (e.g., Heckman and Borjas 1980; Doiron and Gørgens 2008 ). In practice, the higher data demands mean that estimation of such models may only be feasible when the data are non-left-censored for all individuals. 2.4. The Relationship between DBR and MSD Models As mentioned, the data representations for occupancies and transitions are equivalent. The probabilities for state occupancy and for transitions between states are also equivalent in a nonparametric context. Furthermore, Gørgens and Hyslop (2018) also showed that the Markov assumption (5) with r= 1 or r= 2 implies the semi-Markov assumption (11). That is, the low-order DBR model is a special case of the MSD model in nonparametric setting. The prototype parametric models differ in the details of how they deal with initial conditions and left-censoring. However, in this section we show that both the DBR1 and the DBR2 structural equations are special cases of the structural MSD1/MSD2 equations. That is, the nesting property survives typical parameterizations. To see this for the DBR1 model, note that by symmetry of the logistic function the DBR1 Equation (7) implies θC k(hit−1,xit) = (G(νk2+β0 2xit)if yit−1=0, G(−νk2−β0 2xit −γ2)if yit−1=1, t=2, . . . , T, (22) whereas the MSD1 equations can be written (the MSD2 model is similar) θC k(hit−1,xit) = (G(νk4+β0 4xit +∑m j=2λ4j1(dm it−1≥j)) if yit−1=0, G(νk5+β0 5xit +∑m j=2λ5j1(dm it−1≥j)) if yit−1=1, t=m+1, . . . , T. (23) Matching coefficients shows that the DBR1 model arises as a special case of the MSD1 models when there is no duration dependence ( m= 1), and the effects of observed and unobserved heterogeneity are opposite in the two transition probabilities. Specifically, the restrictions are λ4j= 0 and λ5j= 0 for j= 2, . . . , m ; β4+β5= 0; and νk4−ν14 +νk5−ν15 = 0 for k= 2, . . . , K . Under these restrictions, the state dependence parameter in the DBR1 model can be recovered from γ2=−ν14 −ν15 , with νk2=νk4. To show that the DBR2 model is also a special case, note that the DBR2 structural Equation (10) implies θC k(hit−1,xit) =              G(νk3+β0 3xit +γ32)if yit−1=0, yit−2=1, G(νk3+β0 3xit)if yit−1=0, yit−2=0, G(−νk3−β0 3xit −γ31)if yit−1=1, yit−2=0, G(−νk3−β0 3xit −γ31 −γ32 −γ33)if yit−1=1, yit−2=1, t=3, . . . , T. (24) Econometrics 2019,7, 1 7 of 16 Notice that unequal values of yit−1 and yit−2 means that the spell has lasted exactly one period at time t , while equal values means that the spell has lasted two or more periods. Since dm it−1≥ 2 if and only if yit−1=yit−2 , the MSD1 structural equations for m≥ 2 can be re-written (the MSD2 model is similar) θC k(hit−1,xit) = (G(νk4+β0 4xit +λ42(1−yit−2) + ∑m j=3λ4j1(dm it−1≥j)) if yit−1=0, G(νk5+β0 5xit +λ52yit−2+∑m j=3λ5j1(dm it−1≥j)) if yit−1=1, t=m+1, . . . , T. (25) Matching coefficients shows that the DBR2 model arises as a special case when there is no duration dependence after one period ( m= 2), and the effects of observed and unobserved heterogeneity are opposite in the two transition probabilities. Specifically, the restrictions are λ4j= 0 and λ5j= 0 for j= 3, . . . , m ; β4+β5= 0; and νk4−ν14 +νk5−ν15 = 0 for k= 2, . . . , K . Under there restrictions, the state dependence parameters in the DBR1 model can be recovered from γ31 =−ν14 −ν15 −λ42 , γ32 =−λ42, and γ33 =λ42 −λ52, with νk3=νk4+λ42. Higher-order DBR models are generally not nested within the MSD framework. However, it is easy to show that a r th-order DBR model has transition probabilities that are constant after r periods in a spell. Moreover, prototype DBR models implicitly assume that the effects of covariates and unobserved heterogeneity on the probability of being in state 1 at time t are the same whether or not the individual is in state 0 or state 1 at time t− 1. In other words, the effects on the entry and exit transition probabilities have the same magnitude but opposite signs. From the perspective of duration analysis, the restrictions on the transition probabilities embodied in typical DBR models appear strong. Several authors working with DBR models have considered the possibility of interacting the covariates with the lagged occupancy indicator (e.g., Heckman 1981c;Barmby 1998;Beck et al. 2001; Card and Hyslop 2009;Browning and Carro 2010;Cappellari and Jenkins 2014). Although the intent was to allow for heterogeneity in state dependence rather than a specific consideration of transition probabilities, Equation (25) shows that the extension to permit β46=−β5 is an intermediate case with the restrictions on duration dependence and the effects of unobserved heterogeneity maintained. The nesting of DBR1 and DBR2 models within the MSD model means that it is possible to test the former against the latter using a simple Wald statistic. Obviously, it is also possible to test intermediate models against the MSD model to investigate separately the validity of the restrictions on duration dependence and on the heterogeneity effects. In the case study below, we investigate whether rejection of the DBR models is mainly due to inadequate modeling of duration dependence or to restricting the heterogeneity effects to be opposite by estimating and testing a hierarchy of model specifications. 3. Case Study We now apply each of the methods discussed above to an analysis of poverty persistence, previously analyzed using multi-spell duration models by Stevens (1999). Our focus here is to estimate and compare the DBR and MSD models, rather than to replicate or critique Stevens’ original analysis. So, for example, we select a different analytical extract from the data provided to us than that used by Stevens. 3.1. Data Our analytical sample consists of a balanced panel of 5248 individuals over the 20 years 1970–89 from the Panel Study of Income Dynamics (PSID). 2 Each individual’s poverty status is determined by 2 The data we use come from the PSID survey years 1970–89, with the income and poverty observations corresponding to calendar years 1969–88. Years mentioned in the text refer to survey years. We use a balanced panel in order to abstract from attrition issues that may differentially affect the estimation methods. Econometrics 2019,7, 1 8 of 16 whether their family’s annual income is below or above a needs threshold which depends on family size and composition, so that all individuals in a family have the same poverty status in that year. 3 The main sample selection criteria we apply is that all individuals experience at least one year in poverty over the extended period 1968–89, and are observed and have no missing outcome or covariate information over the analysis period 1970–89. 4 The covariates include dummy variables for age groups 0–5, 6–17, 18–24, and 55+, and dummy variables for whether the household head is female and/or black. The sample characteristics are summarized in Table 1. Table 1. Descriptive statistics. Mean (Standard Error) Person-years Aged 0–5 0.025 (0.0005) Aged 6–17 0.225 (0.001) Aged 18–24 0.204 (0.001) Aged 25–54 0.420 (0.002) Aged 55+ 0.126 (0.001) Female head 0.336 (0.001) Black head 0.582 (0.002) Poor (yit) 0.353 (0.001) Transition (cit) 0.177 (0.001) No. person-years 104,960 Persons Transitions 3.35 (0.032) No. persons 5248 Spells Duration of all spells 4.59 (0.033) Duration of initial spells 7.11 (0.084) Duration of fresh spells 3.84 (0.032) No. spells 22,849 Notes: No adjustments for censoring. 3.2. Estimation Results We present the estimates of the DBR and MSD models in Tables 2and 3. Table 2contains estimates for the first- and second-order DBR models with two discrete points of unobserved heterogeneity, DBR1 and DBR2. Similarly, Table 3contains the estimates for two MSD models with two random effects mass points: a five-equation model, MSD1, estimated using the full sample; and a three-equation model, MSD2, which exploits the assumption of constant transition probabilities after 6 years to estimate common equations for initial and fresh spells, excluding the first five years of initial spells. As the data do not constitute a random sample, we report robust standard errors with clustering at the level of the households originally selected for the survey. Note that, although the initial state equation is the same in each model, since each model is jointly estimated as a system of different equations, the initial state equation parameter estimates will vary across the models. The MSD models relax restrictions implied by the DBR models in two important respects. First, the DBR model specifications imply that covariate coefficients should be equal in magnitude and opposite in sign in the entry and exit equations. In contrast, although the covariate coefficients are predominantly positive in the entry equation and negative in the exit equation (in line with the DBR 3See Stevens (1999) for more details of this and other data issues. 4This criteria is used as a proxy to identify the poverty at-risk population, and follows Stevens (1999). Econometrics 2019,7, 1 15 of 16 as Markovian state dependence and allows the transition probabilities to vary flexibly. Generally, DBR models are tightly specified and parsimonious, while MSD models are comparatively flexible and more demanding. In this paper we formally show that typical (first and second order) DBR models are nested within an MSD model specification, so that the two approaches should not be viewed as separate. The case study analysis of poverty experiences demonstrates the two approaches. We conclude that the estimated MSD models fit the data far better than the first- and second-order DBR models, with each of the DBR model restrictions being severely rejected. Given that the MSD models are more flexibly specified than the parsimonious DBR models, it is perhaps not surprising that the log quasi-likelihood values are much higher for MSD models and that the DBR models are formally rejected in statistical tests. However, this conclusion also holds in terms of the model predictions: the MSD model’s within-sample predictions were substantially better than the predictions from the DBR model. Finally, we showed that the choice of model specification also matters for deriving policy implications from the fitted models. These findings underscore the potential limitations of the popular DBR model approach in this case. Of course the DBR model may be sufficient in some empirical situations, but the results here emphasize the importance of considering more flexible alternatives. While all commonly used DBR model specifications are special cases of a relatively simple MSD model specification, it is of course possible to consider more general specifications. As mentioned, phenomena such as occurrence dependence and lagged duration dependence may be important in practice. Furthermore, it is straightforward to allow for covariate effects to vary with elapsed time spent in the current state. Lastly, the influence of unobserved heterogeneity can be extended by allowing for random coefficients more generally. Author Contributions: Both authors contributed equally to the paper. Funding: T.G.’s research was supported in part by Australian Research Council Grant DP1096862. D.R.H.’s research was supported in part by the Royal Society of New Zealand Marsden Fund Grant MEP1301. 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