Estimation and Applications of Quantile Regression for Binary Longitudinal Data
aa r X i v : . [ ec on . E M ] S e p Estimation and Applications of Quantile Regression forBinary Longitudinal Data
Mohammad Arshad Rahman Department of Economic Sciences, Indian Institute of Technology Kanpur
Angela Vossmeyer Robert Day School of Economics and Finance, Claremont McKenna College
Abstract
This paper develops a framework for quantile regression in binary longitudinal data settings.A novel Markov chain Monte Carlo (MCMC) method is designed to fit the model and itscomputational efficiency is demonstrated in a simulation study. The proposed approach isflexible in that it can account for common and individual-specific parameters, as well asmultivariate heterogeneity associated with several covariates. The methodology is appliedto study female labor force participation and home ownership in the United States. Theresults offer new insights at the various quantiles, which are of interest to policymakers andresearchers alike.
Keywords:
Bayesian inference, Binary outcomes, Female labor force participation, Homeownership, Limited dependent variables, Panel data.
1. Introduction
The proliferation of panel data studies is well-documented and much of it has beenattributed to data availability and challenging methodology (Hsiao, 2007). While panel datahas been attractive for understanding behavior and dynamics, the modeling complexities
Acknowledgements:
The authors thank the anonymous referees, Ivan Jeliazkov, David Brownstone,John Geweke, K.L. Krishna, Antonio Galvao, Michael Guggisberg, and Editor Justin Tobias for their helpfulcomments. Discussions and suggestions from the participants at the Winter School, Delhi School of Eco-nomics (2017) and Advances in Econometrics Conference (2018) are appreciated. A special thanks to DalePoirier for sharing the Reverend’s insights and teaching us the controversy. Room 672, Faculty Building, IIT Kanpur, Kanpur 208016, India; email: [email protected] . Robert Day School of Economics and Finance, Claremont McKenna College, 500 E. Ninth St., Clare-mont, CA 91711; email: [email protected] . Preprint submitted to Advances in Econometrics September 13, 2019 nvolved in it have moved attention away from its unique capacities. Modeling features suchas a binary outcome variable or a quantile analysis, which are relatively straightforwardto implement with cross-sectional data, are challenging and computationally burdensomefor panel data. However, these features are important as they allow for the modeling ofprobabilities and lead to a richer view of how the covariates influence the outcome variable.Motivated by these difficulties, this paper adds to the methodological advancements for paneldata by developing quantile regression methods for binary longitudinal data and designinga computationally efficient estimation algorithm. The approach is applied to two empiricalstudies, female labor force participation and home ownership.The paper touches on three growing econometric literatures – discrete panel data, quan-tile regression for panel data, and quantile regression for discrete data. In reference tothe latter, quantile regression has been implemented in binary data models (Kordas, 2006;Benoit and Poel, 2012), ordered data models (Rahman, 2016; Alhamzawi and Ali, 2018),count data models (Machado and Silva, 2005; Harding and Lamarche, 2015), and censoreddata models (Portnoy, 2003; Harding and Lamarche, 2012). For limited dependent variables,the concern is modeling the latent utility differential in the quantile framework, since theresponse variable takes limited values and does not yield continuous quantiles. Our paperfollows the work in this literature by using the latent utility setting and interpreting theutility as a “propensity” or “willingness” that underlie the latent scale, thus increasing ourunderstanding of the impact of the covariates on the binary outcomes.The literature on quantile regression in panel data settings includes (but is notlimited to) Koenker (2004), Geraci and Bottai (2007), Liu and Bottai (2009), Galvao(2010), Galvao and Kato (2016), Lamarche (2010), Harding and Lamarche (2009) andHarding and Lamarche (2017). The latter of these papers discusses the issues associatedwith solely focusing on fixed effects estimators and highlights the usefulness of allowing fora flexible specification of individual heterogeneity associated with covariates, also of interestin the present paper. In a recent Bayesian paper, Luo et al. (2012) develop a hierarchicalmodel to estimate the parameters of conditional quantile functions with random effects. Theauthors do so by adopting an Asymmetric Laplace (AL) distribution for the residual errorsand suitable prior distributions for the parameters. However, directly using the AL distri-bution does not yield tractable conditional densities for all of the parameters and hence acombination of Metropolis-Hastings (MH) and Gibbs sampling is required for model estima-tion. The use of the MH algorithm may require tuning at each quantile. To overcome thislimitation, Luo et al. (2012) also present a full Gibbs sampling algorithm that utilizes the2ormal-exponential mixture representation of the AL distribution. This mixture represen-tation is also followed in our work, with important computational improvements.Finally, for discrete panel data, recent work by Bartolucci and Nigro (2010) intro-duces a quadratic exponential model for binary panel data and utilizes a conditionallikelihood approach, which is computationally simpler than previous classical estimators.Bayesian approaches to binary panel data models include work by Albert and Chib (1996),Chib and Carlin (1999), Chib and Jeliazkov (2006), and Burda and Harding (2013). Thesework influence the estimation methods designed in our quantile approach to binary paneldata.This paper contributes to the three literatures by extending the various methodologies toa hierarchical Bayesian quantile regression model for binary longitudinal data and proposinga Markov chain Monte Carlo (MCMC) algorithm to estimate the model. The model handlesboth common (fixed) and individual-specific (random) parameters (commonly referred tomixed effects in statistics). The algorithm implements a blocking procedure that is com-putationally efficient and the distributions involved allow for straightforward calculations ofcovariate effects. The framework is implemented in two empirical applications. The first ap-plication examines female labor force participation, which has been heavily studied in panelform. The topic became of particular interest in the state dependence versus heterogeneitydebate (Heckman, 1981a). We revisit this question and implement our panel quantile ap-proach, which has been otherwise unexplored for this topic. The results offer new insightsregarding the determinants of female labor force participation and how the ages of childrenhave different effects across the quantiles and utility scale. The findings suggest that policyshould be focused on women’s transitions into the labor force after child birth and the fewyears after.The second application considers the probability of home ownership during the GreatRecession. Micro-level empirical analyses on individuals moving into and out of housingmarkets are lacking in the recent literature. Past studies include Carliner (1974) and Poirier(1977), but the recent housing crisis offers a new opportunity to reevaluate the topic. Fur-thermore, a full quantile analysis of home ownership is yet to be explored. Since homeownership is a choice that requires years of planning, individual characteristics may rangedrastically across the latent utility scale. The analysis presented in this paper controls formultivariate heterogeneity in individuals and wealth, and investigates the determinants ofhome ownership, state dependence in home ownership, and how the shock to housing mar-kets affected these items. The results provide an understanding as to how individuals of3articular demographics and socioeconomic status fared during the collapse of the housingmarket.The rest of the paper is organized as follows. Section 2 reviews quantile regression and theAL distribution, Section 3 introduces the quantile regression model for binary longitudinaldata, presents a simulation study, and discusses methods for covariate effects. Section 4considers the two applications and concluding remarks are offered in Section 5.
2. Quantile Regression and Asymmetric Laplace Distribution
The p -th quantile of a random variable Y is the value y such that the probability that Y will be less than y equals p ∈ (0 , Q ( · ) denotes the inverse of thecumulative distribution function ( cdf ) of Y , the p -th quantile is defined as Q Y ( p ) ≡ F − Y ( p ) = inf { y : F ( y ) ≥ p } . Quantile regression implements the idea of quantiles within the regression framework with Q ( · ) modified to denote the inverse cdf of the dependent variable given the covariates.The objective is to estimate conditional quantile functions and to this purpose, regressionquantiles are estimated by minimizing the quantile objective function which is a sum ofasymmetrically weighted absolute residuals.To formally explain the quantile regression problem, consider the following linear model, y i = x ′ i β p + ε i , with Q ε i ( p | x i ) = 0 , (1)where y i is a scalar response variable, x i is a k × β p is a k × p , and ε i is the error term such that its p -thquantile equals zero. Henceforth, we will drop the subscript p for notational simplicity. Inclassical econometrics, the error ε does not (or is not assumed to) follow any distributionand estimation requires minimizing the following objective function,min β ∈ R k " X i : y i
3. The Quantile Regression Model for Binary Longitudinal Data
This section presents the quantile regression model for binary longitudinal data (QBLD)and an estimation algorithm to fit the model. The performance of the proposed algorithm isillustrated in a simulation study. The last part of this section considers methods for modelcomparison and covariate effects.
The proposed model looks at quantiles of binary longitudinal data expressed as a func-tion of covariates with common effects and individual-specific effects. The individual-specificeffects offer additional flexibility in that both intercept and slope heterogeneity can be cap-tured, which are important to avoid biases in the parameter estimates. The QBLD modelcan be conveniently expressed in the latent variable formulation (Albert and Chib, 1993) as6ollows, z it = x ′ it β + s ′ it α i + ε it , ∀ i = 1 , · · · , n, t = 1 , · · · , T i ,y it = z it > , , (4)where the latent variable z it denotes the value of z at the t -th time period on the i -thindividual, x ′ it is a 1 × k vector of explanatory variables, β is k × s ′ it is a 1 × l vector of covariates that have individual-specific effects, α i isan l × ε it is the error term assumed to beindependently distributed as AL (0 , , p ) with Q ε it ( p | x it , α i ) = 0. This implies that theconditional density of z it | α i is an AL ( x ′ it β + s ′ it α i , , p ) for i = 1 , · · · , n , and t = 1 , · · · , T i , with Q z it ( p | x it , α i ) = x ′ it β + s ′ it α i . Note that s it may contain a constant for intercept heterogeneity,as well as other covariates (which are often a subset of those in x it ) to account for slopeheterogeneity of those variables. The variable z it is unobserved and represents the latentutility associated with the observed binary choice y it . The latent variable formulation servesas a convenient tool in the estimation process (Albert and Chib, 1993). Furthermore, latentutility underlies the interpretation of the results at the various quantiles.While working directly with the AL density is an option, the resulting posterior will notyield the full set of tractable conditional distributions necessary for a Gibbs sampler. Thus,we utilize the normal-exponential mixture representation of the AL distribution, presentedin Kozumi and Kobayashi (2011), and express the error as follows, ε it = θw it + τ √ w it u it , ∀ i = 1 , · · · , n ; t = 1 , · · · , T i , (5)where u it ∼ N (0 ,
1) is mutually independent of w it ∼ E (1) with E representing an exponentialdistribution and the constants θ = − pp (1 − p ) and τ = q p (1 − p ) . The mixture representation givesaccess to the appealing properties of the normal distribution.Longitudinal data models often involve a moderately large amount of data, so it is im-portant to take advantage of any opportunity to reduce the computational burden. Onesuch trick is to stack the model for each individual i (Hendricks et al., 1979). We de-fine z i = ( z i , · · · , z iT i ) ′ , X i = ( x ′ i , · · · , x ′ iT i ) ′ , S i = ( s ′ i , · · · , s ′ iT i ) ′ , w i = ( w i , · · · , w iT i ) ′ , D τ √ w i = diag( τ √ w i , · · · , τ √ w iT i ), and u i = ( u i , · · · , u iT i ) ′ . Building on equations (4) and75), the resulting hierarchical model can be written as, z i = X i β + S i α i + θw i + D τ √ w i u i ,y it = z it > , ,α i | ϕ ∼ N l (0 , ϕ I l ) , w it ∼ E (1) , u it ∼ N (0 , ,β ∼ N k ( β , B ) , ϕ ∼ IG ( c / , d / , (6)where we assume that α i are identically distributed as a normal distribution. The last rowrepresents the prior distributions with N and IG denoting the normal and inverse-gammadistributions, respectively. Here, we note that the form of the prior distribution on β holdsa penalty interpretation on the quantile loss function (Koenker, 2004). A normal prior on β implies a ℓ penalty and has been used in Yuan and Yin (2010), and Luo et al. (2012).One may also employ a Laplace prior distribution on β that imposes ℓ penalization, asused in several articles such as Alhamzawi and Ali (2018). While Alhamzawi and Ali (2018)also work with quantile regression for discrete panel data (ordered, in particular), our workcontributes by considering multivariate heterogeneity (not just intercept heterogeneity), andintroducing computational improvements outlined below.By Bayes’ theorem, we express the “complete joint posterior” density as proportional tothe product of likelihood function and the prior distributions as follows, π ( β, α, w, z, ϕ | y ) ∝ ( n Y i =1 f ( y i | z i , β, α i , w i , ϕ ) π ( z i | β, α i , w i ) π ( w i ) π ( α i | ϕ ) ) π ( β ) π ( ϕ ) , ∝ ( n Y i =1 " T i Y t =1 f ( y it | z it ) π ( z i | β, α i , w i ) π ( w i ) π ( α i | ϕ ) ) π ( β ) π ( ϕ ) , (7)where the first line uses independence between prior distributions and second line followsfrom the fact that given z it , the observed y it is independent of all parameters because thesecond line of (6) determines y it given z it with probability 1. Substituting the distributionof the variables associated with the likelihood and the prior distributions in (7) yields thefollowing expression, π ( β, α, w, z, ϕ | y ) ∝ ( n Y i =1 T i Y t =1 (cid:20) I ( z it > I ( y it = 1) + I ( z it ≤ I ( y it = 0) (cid:21)) exp (cid:20) − n X i =1 (cid:26) ( z i − X i β − S i α i − θw i ) ′ D − τ √ w i ( z i − X i β − S i α i − θw i ) (cid:27)(cid:21) × ( n Y i =1 | D τ √ w i | − ) × exp − n X i =1 T i X t =1 w it !(cid:16) πϕ (cid:17) − nl exp (cid:20) − ϕ n X i =1 α ′ i α i (cid:21) (8) × (2 π ) − k | B | − exp (cid:20) −
12 ( β − β ) ′ B − ( β − β ) (cid:21) × ( ϕ ) − ( c +1) exp (cid:20) − d ϕ (cid:21) . The joint posterior density (8) does not have a tractable form, and thus simulationtechniques are necessary for estimation. Bayesian methods are increasing in popularity(Poirier, 2006), and this paper takes the approach for a couple of reasons. First, with discretepanel data, working with the likelihood function is complicated because it is analyticallyintractable. The inclusion of individual-specific effects makes matters worse. Second, whilenumerical simulation methods are available for discrete panel data, they are often slow anddifficult to implement (Burda and Harding, 2013). The availability of a full set of conditionaldistributions (which are outlined below) makes Gibbs sampling an attractive option that willbe simpler to implement, both conceptually and computationally.We can derive the conditional posteriors of the parameters and latent variables by astraightforward extension of the estimation technique for the linear mixed-effects modelpresented in Luo et al. (2012). This is presented as Algorithm 2 in Appendix A, which showsthe conditional posterior distributions for the parameters and latent variables necessary fora Gibbs sampler. While this Gibbs sampler is straightforward, there is potential for poormixing properties due to correlation between ( β, α i ) and ( z i , α i ). The correlation often arisesbecause the variables corresponding to the parameters in α i are often a subset of those in x it . Thus, by conditioning these items on one another, the mixing of the Markov chain willbe slow.To avoid this issue, we develop an alternative algorithm which jointly samples ( β, z i ) inone block within the Gibbs sampler. This blocked approach significantly improves the mixingproperties of the Markov chain. The success of these blocking techniques can be found in Liu(1994), Chib and Carlin (1999), and Chib and Jeliazkov (2006). The details of our blockedsampler are described in Algorithm 1. In particular, β is sampled marginally of α i from amultivariate normal distribution. The latent variable z i is sampled marginally of α i from atruncated multivariate normal distribution denoted by T M V N B i , where B i is the truncationregion given by B i = ( B i × B i × . . . × B iT i ) such that B it is the interval (0 , ∞ ) if y it = 1 and The derivation of the conditional posterior densities are presented in Appendix B. lgorithm 1 (Blocked Sampling)
1. Sample ( β, z i ) in one block. The objects ( β, z i ) are sampled in the following two sub-steps.(a) Let Ω i = (cid:16) ϕ S i S ′ i + D τ √ w i (cid:17) . Sample β marginally of α from β | z, w, ϕ ∼ N ( ˜ β, ˜ B ), where,˜ B − = (cid:18) n X i =1 X ′ i Ω − i X i + B − (cid:19) and ˜ β = ˜ B n X i =1 X ′ i Ω − i ( z i − θw i ) + B − β ! . (b) Sample the vector z i | y i , β, w i , ϕ ∼ T M V N B i ( X i β + θw i , Ω i ) for all i = 1 , · · · , n , where B i = ( B i × B i × . . . × B iT i ) and B it is the interval (0 , ∞ ) if y it = 1 and the interval( −∞ ,
0] if y it = 0. This is done by sampling z i at the j -th pass of the MCMC iterationusing a series of conditional posterior distribution as follows: z jit | z ji , · · · , z ji ( t − , z ji ( t +1) , · · · , z jiT i ∼ T N B it ( µ t |− t , Σ t |− t ) , for t = 1 , · · · , T i , where T N denotes a truncated normal distribution. The terms µ t |− t and Σ t |− t are theconditional mean and variance, respectively, and are defined as, µ t |− t = x ′ it β + θw it + Σ t, − t Σ − − t, − t (cid:0) z ji, − t − ( X i β + θw i ) − t (cid:1) , Σ t |− t = Σ t,t − Σ t, − t Σ − − t, − t Σ − t,t , where z ji, − t = ( z ji , · · · , z ji ( t − , z j − i ( t +1) , · · · , z j − iT i ), ( X i β + θw i ) − t is a column vector with t -th element removed, Σ t,t denotes the ( t, t )-th element of Ω i , Σ t, − t denotes the t -th rowof Ω i with element in the t -th column removed and Σ − t, − t is the Ω i matrix with t -th rowand t -th column removed.2. Sample α i | z, β, w, ϕ ∼ N (˜ a, ˜ A ) for i = 1 , · · · , n , where,˜ A − = (cid:18) S ′ i D − τ √ w i S i + 1 ϕ I l (cid:19) and ˜ a = ˜ A (cid:16) S ′ i D − τ √ w i (cid:0) z i − X i β − θw i (cid:1)(cid:17) .
3. Sample w it | z it , β, α i ∼ GIG (0 . , ˜ λ it , ˜ η ) for i = 1 , · · · , n and t = 1 , · · · , T i , where,˜ λ it = (cid:18) z it − x ′ it β − s ′ it α i τ (cid:19) and ˜ η = (cid:18) θ τ + 2 (cid:19) .
4. Sample ϕ | α ∼ IG (˜ c / , ˜ d / c = (cid:16) nl + c (cid:17) and ˜ d = (cid:16) n X i =1 α ′ i α i + d (cid:17) . −∞ ,
0] if y it = 0. To draw from a truncated multivariate normal distribution,we utilize the method proposed in Geweke (1991). This involves drawing from a series ofconditional posteriors which are univariate truncated normal distributions. Previous workusing this approach include Chib and Greenberg (1998) and Chib and Carlin (1999). Therandom effects parameter α i is sampled conditionally on β, z i from another multivariate nor-mal distribution. The variance parameter ϕ is sampled from an inverse-gamma distributionand finally the latent weight w is sampled element-wise from a generalized inverse Gaussian(GIG) distribution (Dagpunar, 1988, 1989; Devroye, 2014).We end this section with a cautionary note on sampling from a truncated multivari-ate normal distribution, with the hope that it will be useful to researchers on quantileregression. In our algorithm above, we sample z i from a T M V N B i ( X i β + θw i , Ω i ) us-ing a series of conditional posteriors which are univariate truncated normal distributions.This method is distinctly different and should not be confused with sampling from arecursively characterized truncation region typically related to the Geweke-Hajivassiliou-Keane (GHK) estimator (Geweke, 1991; Börsch-Supan and Hajivassiliou, 1993; Keane, 1994;Hajivassiliou and McFadden, 1998). The difference between the two samplers have been ex-hibited in Breslaw (1994) and carefully discussed in Jeliazkov and Lee (2010).
This subsection evaluates the performance of the algorithm in a simulation study, wherethe data are generated from a model that has common effects and individual-specific effectsin both the intercept and slopes. We estimate the quantile regression model for binarylongitudinal data (QBLD) using our proposed blocked sampler (Algorithm 1) and the non-blocked sampler (Algorithm 2).The data are simulated from the model z it = x ′ it β + s ′ it α i + ε it where t = 1 , . . . ,
10 and i = 1 , . . . , β = ( − , , ′ , α i ∼ N (0 , I ), x ′ it =(1 , x it , x it ) with x it ∼ U (0 ,
1) and x it ∼ U (0 , s ′ it = (1 , s it ) with s it ∼ U (0 , ε it ∼ AL (0 , , p ) for p = 0 . , . , . In the latter scenario, the model z i ∼ N ( X i β + θw i , Ω i ) can be written as z i = X i β + θw i + L i η i ,where L i is a lower triangular Cholesky factor of Ω i such that L i L ′ i = Ω i . To be general, let the lower andupper truncation vectors for z i be a i = ( a i , . . . , a iT i ) and b i = ( b i , . . . , b iT i ), respectively. Then the randomvariable η it is sampled from T N (cid:0) , , ( a it − x ′ it β − θw it − P t − j =1 l tj η ij ) /l tt , ( b it − x ′ it β − θw it − P t − j =1 l tj η ij ) /l tt (cid:1) ,where l tj are the elements of L i . This is a recursively characterized truncation region, since the range of η it depends on the draw of η ij for j = 1 , . . . , t −
1. The vector z i can be obtained by substituting the recursivelydrawn η i into z i = X i β + θw i + Lη i . However, the draws so obtained are not the same as drawing z i from amultivariate normal distribution truncated to the region a i < z i < b i . able 1: Posterior means ( mean ), standard deviations ( std ) and inefficiency factors ( if ) of the parametersin the simulation study from the QBLD model. The first panel presents results from Algorithm 1 and thesecond panel presents results from Algorithm 2.Blocked Sampling β − .
33 0 .
22 4 . − .
06 0 .
18 4 . − .
08 0 .
24 4 . β .
16 0 .
28 4 .
38 5 .
96 0 .
22 3 .
87 6 .
16 0 .
27 4 . β .
34 0 .
24 3 .
86 3 .
88 0 .
19 3 .
66 3 .
88 0 .
23 3 . ϕ .
95 0 .
16 4 .
68 0 .
66 0 .
11 4 .
60 0 .
81 0 .
15 4 . β − .
32 0 .
22 5 . − .
05 0 .
20 6 . − .
07 0 .
23 6 . β .
15 0 .
27 6 .
05 5 .
95 0 .
23 6 .
57 6 .
15 0 .
26 6 . β .
35 0 .
24 5 .
52 3 .
88 0 .
20 5 .
40 3 .
88 0 .
23 5 . ϕ .
95 0 .
16 5 .
58 0 .
66 0 .
11 5 .
26 0 .
81 0 .
14 6 . Here, the notation U (0 ,
1) denotes a standard uniform distribution. The binary responsevariable y it is constructed by assigning 1 to all positive values of z it and 0 to all negativevalues of z it . Since the values generated from an AL distribution are different at eachquantile, the number of 0s and 1s are also different at each quantile. In the simulation, thenumber of observations corresponding to 0s and 1s for the 25th, 50th and 75th quantiles are(1566 , , , β ∼ N (0 k , I k ), and ϕ ∼ IG (10 / , / ,
000 MCMC iterations after a burn-in of 3 ,
000 iterations. The inefficiency factorsare calculated using the batch-means method discussed in Greenberg (2012). The simulationexercise was repeated for various covariates, sample sizes, common and individual-specificparameters, and the results do not change from this baseline case; hence they are not pre-sented.The posterior mean for regression coefficients for both the samplers (blocked and non-blocked methods) are near the true values, β = ( − , , ′ . Additionally, the standarddeviations are small. Across each quantile, the number of 0s and 1s varies, and the samplers12 able 2: Autocorrelation in MCMC draws at Lag 1, Lag 5 and Lag 10.Blocked Sampling Lag 1 Lag 5 Lag 10 Lag 1 Lag 5 Lag 10 Lag 1 Lag 5 Lag 10 β .
86 0 .
59 0 .
41 0 .
85 0 .
54 0 .
35 0 .
88 0 .
61 0 . β .
89 0 .
61 0 .
43 0 .
87 0 .
53 0 .
34 0 .
89 0 .
60 0 . β .
86 0 .
50 0 .
31 0 .
83 0 .
44 0 .
25 0 .
84 0 .
45 0 . ϕ .
93 0 .
73 0 .
54 0 .
92 0 .
70 0 .
51 0 .
93 0 .
75 0 . Lag 1 Lag 5 Lag 10 Lag 1 Lag 5 Lag 10 Lag 1 Lag 5 Lag 10 β .
96 0 .
84 0 .
71 0 .
97 0 .
85 0 .
73 0 .
97 0 .
85 0 . β .
96 0 .
81 0 .
68 0 .
96 0 .
82 0 .
68 0 .
96 0 .
80 0 . β .
95 0 .
77 0 .
61 0 .
95 0 .
77 0 .
60 0 .
94 0 .
75 0 . ϕ .
92 0 .
76 0 .
63 0 .
92 0 .
74 0 .
59 0 .
93 0 .
79 0 . perform well in each case. Furthermore, starting the algorithm at different values appearsinconsequential, which is a benefit of the full Gibbs sampler.Turning attention to the differences between the two algorithms, it is clear that the in-efficiency factors from the blocked algorithm are much lower, suggesting better samplingperformance and a nice mixing of the Markov chain. The advantages of the blocking pro-cedure are more apparent from the autocorrelation in the MCMC draws at different lags.Table 2 presents the autocorrelation in MCMC draws at lag 1, lag 5, and lag 10. Lookingat lag 10, the autocorrelation for the β s are between 0 . − .
43 in the blocked algorithm,which is nearly half of 0 . − .
73, obtained from the non-blocked sampler. Recall that inour data generation process, we did not make the covariates in s it a subset of those in x it .Whereas in real-data exercises, it is typical for s it to be a subset. Therefore, we expect thebenefits of the blocked sampler to be even more pronounced in real data settings.Finally, Figure 2 presents the trace plots of the parameters at the 25th quantile forthe blocked algorithm, which graphically demonstrate the appealing sampling. Given thecomputational efficiency with the blocking procedure, it is our preferred way for estimatingQBLD models and will be used in the subsequent real data applications.13 Trace of β Trace of β Trace of β Trace of ϕ Figure 2: Trace plots of the MCMC draws at the 25th quantile from Algorithm 1.
In this section, we briefly discuss methods for model comparison and computation ofcovariate effects. For model comparison, we follow standard techniques for longitudinal datamodels. Specifically, in the application sections we provide the log-likelihood, conditionalAIC (Greven and Kneib, 2010), and conditional BIC (Delattre et al., 2014). This is a bitunusual for a Bayesian analysis, however, we want the results in our empirical applications toalign with the classical work on the topics, such as Bartolucci and Farcomeni (2012). Thus,we follow the approaches so as to allow for better comparisons and cross references.For covariate effects, in general terms, we are interested in the average difference in theimplied probabilities between the case when x it is set to the value x † it and x ‡ it . Given thevalues of the other covariates denoted x − it , s it and those of the model parameters θ , one canobtain the probabilities Pr( y it = 1 | x † it , x − it , s it , θ ) and Pr( y it = 1 | x ‡ it , x − it , s it , θ ). Followingfrom Jeliazkov et al. (2008) and Jeliazkov and Vossmeyer (2018), if one is interested in thedistribution of the difference { Pr( y it = 1 | x † it ) − Pr( y it = 1 | x ‡ it ) } marginalized over { x − it , s it } and θ given the data y , a practical procedure is to marginalize out the covariates using theirempirical distribution, while the parameters are integrated out with respect to their posterior14istribution. Formally, the goal is to obtain a sample of draws from the distribution, { Pr( y it = 1 | x † it ) − Pr( y it = 1 | x ‡ it ) } = Z { Pr( y it = 1 | x † it , x − it , s it , θ ) − Pr( y it = 1 | x ‡ it , x − it , s it , θ ) }× π ( x − it , s it ) π ( θ | y ) d ( x − it , s it ) dθ. The computation of these probabilities is straightforward because the differences betweenthe probabilities of success is related to differences in AL cdf , marginalized over { x − it , s it } and the posterior distribution of θ . Also, the procedure handles uncertainty stemming fromthe sample and estimation strategy. This approach is demonstrated in each of the followingapplications.
4. Applications
Modeling female labor force participation has been an important area of work in theeconomics and econometric literature for decades. The list of work is vast, but a partiallist includes Heckman and Macurdy (1980), Heckman and Macurdy (1982), Mroz (1987),Hyslop (1999), Arellano and Carrasco (2003), Chib and Jeliazkov (2006), Kordas (2006),Carro (2007), Bartolucci and Nigro (2010), and Eckstein and Lifshitz (2011).Within the literature, several pertinent questions have been analyzed including the rela-tionship between participation and age, education, fertility, and permanent and transitoryincomes. However, serial persistence in the decision to participate and its two competingtheories – heterogeneity and state dependence – have been of substantive interest. Hetero-geneity implies that females may differ in terms of certain unmeasured variables that affecttheir probability of labor force participation. If heterogeneity is not properly controlled, thenpast decisions may appear significant to current decisions leading to what is called spuriousstate dependence. In contrast, pure state dependence implies that dynamic effects of pastparticipation genuinely affect current employment decisions. Consideration of heterogeneityand state dependence is important in modeling female labor force participation and can haveeconomic implications as discussed in Heckman (1981a), Heckman (1981b) and Hsiao (2014,pp. 261-270). We re-examine the above mentioned aspects using our proposed Bayesianquantile regression model for binary longitudinal data. To our knowledge, this is the first at-tempt to analyze female labor force participation within a longitudinal quantile framework.So, what can we learn from a panel quantile approach? Of particular interest are the impacts15f infants and children across the various quantiles. Understanding the differential effectsacross the latent utility scale can help shape female labor force policies, such as maternityleave and child care.Before proceeding forward, we draw attention to Kordas (2006) who evaluated female la-bor force participation using cross sectional data and smoothed binary regression quantiles.His results offer interesting insights across the quantiles, which further motivate our appli-cation and extension to transitions into and out of the labor force in the panel setting. Wealso follow his interpretation where the latent utility differential between working and notworking may be interpreted as a “propensity” or “willingness-to-participate” (WTP) index.The data for this study are taken from Bartolucci and Farcomeni (2012), which wereoriginally extracted from the Panel Study of Income Dynamics (PSID) conducted by theUniversity of Michigan. The data consist of a sample of n = 1446 females who were followedfor the period 1987 to 1993 with respect to their employment status and a host of demo-graphic and socio-economic variables. The dependent variable in the model is employment status (= 1 if the individual is employed, = 0 otherwise) and the covariates include age (in1986), education (number of years of schooling), child 1-2 (number of children aged 1 to 2,referred to the previous year), child 3-5 , child 6-13 , child 14- , Black (indicator for Black race), income of the husband (in US dollars, referred to the previous year), and fertility (indicatorvariable for birth of a child in a certain year). Lagged employment status is also included asa covariate to examine state dependence of female labor force participation decision.Table 3 presents summary statistics for the variables. The presentation of the tablefollows from Hyslop (1999), where statistics are broken up into subgroups of women thathave worked 0 years, 7 years, or transitioned during the period. As one can see from thetable, the average age in the sample is roughly 30, about 40% of the sample is employedthroughout the entire period, 10% are not in the labor force throughout the entire period,20% transition into or out of the labor force once, and 30% transition multiple times. Lookingclosely at the different variables for children, there is a decent amount of variation across thesubgroups. For mothers who are employed 0 years, the average values for child 1-2 and child3-5 are 0.46 and 0.56, respectively. These numbers are more than double compared to thatof mothers who are employed for all the 7 years. Further, as children age ( child 6-13 ) moremothers have a single transition to work. While these differences demonstrate some observedheterogeneity, unobserved heterogeneity still plays a role, which motivates further analysis.Particularly, a quantile setting will reveal information not available in the raw observed databy utilizing the latent scale as the willingness-to-participate index.16he data are modeled following equations (4) and (5) and the model (QBLD) is specifiedwith a random intercept (i.e., s it only includes a constant). We also estimate the probitmodel for binary longitudinal data (PBLD) using the algorithm presented in Koop et al.(2007) and Greenberg (2012) and identical priors for relevant parameters. The results forthe QBLD and PBLD models are presented in Table 4 and are based on data for the years Table 3: Sample characteristics of the female labor force participation data – The first panel presents themean/proportion and standard deviations (in parenthesis) of the variables in the full and the sub-samples.The second panel displays the column percentages for the number of years worked and the third panel (i.e.,last row) presents the number of observations in the full and the sub-samples.FullSample Employed7 Years Employed0 Years SingleTransitionfrom Work SingleTransitionto Work MultipleTransitions(1) (2) (3) (4) (5) (6)Age 29 .
55 30 .
44 29 .
18 29 .
21 29 .
23 28 . .
61) (4 .
34) (4 .
51) (4 .
77) (4 .
62) (4 . .
14 13 .
33 12 .
68 13 .
20 13 .
01 13 . .
06) (1 .
98) (2 .
15) (2 .
13) (2 .
19) (2 . .
31 0 .
22 0 .
46 0 .
31 0 .
34 0 . .
53) (0 .
45) (0 .
60) (0 .
53) (0 .
57) (0 . .
37 0 .
27 0 .
56 0 .
32 0 .
50 0 . .
57) (0 .
49) (0 .
65) (0 .
54) (0 .
65) (0 . .
75 0 .
71 0 .
92 0 .
55 0 .
99 0 . .
92) (0 .
87) (1 .
00) (0 .
81) (1 .
03) (0 . .
32 0 .
39 0 .
31 0 .
29 0 .
26 0 . .
67) (0 .
72) (0 .
71) (0 .
69) (0 .
61) (0 . .
24 0 .
27 0 .
26 0 .
19 0 .
21 0 . .
43) (0 .
44) (0 .
44) (0 .
39) (0 .
40) (0 . ,
000 3 .
04 2 .
82 3 .
81 3 .
43 2 .
99 2 . .
60) (1 .
82) (5 .
28) (3 .
14) (2 .
04) (1 . .
07 0 .
04 0 .
08 0 .
10 0 .
05 0 . .
25) (0 .
21) (0 .
28) (0 .
29) (0 .
22) (0 . . − − − − . − − .
00 9 .
03 8 .
252 6 . − − .
86 12 .
26 14 .
393 6 . − − .
14 10 .
97 13 .
684 8 . − − .
43 17 .
42 19 .
345 9 . − − .
57 23 .
23 18 .
876 13 . − − .
00 27 .
10 25 .
477 39 .
97 100 − − − −
Observations 1446 578 149 140 155 424 able 4: Results from the female labor force participation study – Posterior means ( mean ), standard devia-tions ( std ) and inefficiency factors ( if ) of the parameters from the QBLD and PBLD models are provided. QBLD25th quantile 50th quantile 75th quantile PBLDmean std if mean std if mean std if mean std if
Intercept − .
11 0 .
21 4 . − .
31 0 .
18 4 .
45 1 .
35 0 .
23 4 . − .
08 0 .
07 2 . † .
03 0 .
01 2 .
27 0 .
01 0 .
01 2 . − .
01 0 .
02 2 .
72 0 .
01 0 .
01 1 . † ) / − .
23 0 .
26 1 . − .
19 0 .
25 2 . − .
13 0 .
33 2 . − .
08 0 .
10 1 . † .
17 0 .
03 2 .
29 0 .
21 0 .
03 2 .
57 0 .
28 0 .
05 3 .
18 0 .
08 0 .
01 1 . − .
22 0 .
11 2 . − .
28 0 .
11 2 . − .
38 0 .
13 2 . − .
12 0 .
04 1 . − .
55 0 .
10 2 . − .
52 0 .
10 3 . − .
56 0 .
12 2 . − .
21 0 .
04 1 . − .
17 0 .
07 2 . − .
18 0 .
07 2 . − .
18 0 .
08 2 . − .
07 0 .
02 1 . − .
05 0 .
10 2 . − .
02 0 .
10 2 . − .
01 0 .
13 3 . − .
01 0 .
04 1 . .
20 0 .
15 2 .
02 0 .
24 0 .
15 2 .
24 0 .
26 0 .
19 2 .
69 0 .
09 0 .
06 1 . † / , − .
13 0 .
03 3 . − .
14 0 .
02 3 . − .
18 0 .
03 3 . − .
05 0 .
01 1 . − .
91 0 .
20 2 . − .
06 0 .
20 2 . − .
60 0 .
33 3 . − .
72 0 .
07 1 . .
89 0 .
16 3 .
75 3 .
88 0 .
13 4 .
47 6 .
71 0 .
20 5 .
24 1 .
49 0 .
05 3 . ϕ .
42 0 .
35 6 .
36 1 .
39 0 .
33 6 .
16 2 .
12 0 .
50 7 .
12 0 .
33 0 .
05 4 . − . − . − . − . .
45 6280 .
77 6319 .
36 5801 . .
82 6378 .
14 6416 .
74 5899 . † denotes variable minus the sample average. β ∼ N (0 k , I k ) and ϕ ∼ IG (10 / , / able 5: Covariate effects in the female labor force participation study. QBLD25th 50th 75th PBLD
Education 0 . . . . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . The results for the education variable are positive, statistically different from zero, andshow various incremental differences across the quantiles. Education is found to have strongereffects in the upper part of the latent index, which is expected since these are women whohave a high utility for working and thus have obtained the requisite education. Regarding thestate dependence versus heterogeneity debate, we find that employment is serially positivelycorrelated, which is a consequence of state dependence. The effect gets incrementally largeras one moves up the latent utility scale. While we are controlling for individual heterogeneitywith the random intercept, we still find evidence of state dependence. This result agrees withBartolucci and Farcomeni (2012), who investigate the question with a latent class model.Other papers that find empirical evidence of strong state dependence effects include Heckman(1981a), Hyslop (1999), and Chib and Jeliazkov (2006).To further understand the results, covariate effects are computed for several variables forthe 3 quantiles and the PBLD model. The covariate effect calculations follow from Section 3.3and the results are displayed in Table 5. Note that the 50th quantile results are similar tothat of the PBLD, which is to be expected. The covariate effect for education is calculatedon the restricted sample of individuals with a high school degree (12 years of schooling).The effect that is computed is 4 additional years of schooling, implying a college degree.The effect for income is a discrete change by $10,000, the effect for children is increasing thecount by one, and for fertility it is a discrete change to the indicator variable.The results show that the birth of a child in that year ( fertility ), reduces the probabilitythat a woman works by 16.7 percentage points at the 25th quantile, 17.4 percentage pointsat the 50th quantile, and 13.3 percentage points at the 75th quantile. For individuals in thelower part of the latent index, having children ages 1-2 impacts their employment decisionless than those at the upper quantiles. Perhaps, women with a low utility for working are19ess impacted by infants and toddlers because it is often a desire to stay home with thechild for a few years. Whereas, women with a high utility for working face negative impactsbecause of the desire to enter the work force.The most pronounced negative effect of children occurs when the child is ages 3-5. Oftenwomen temporarily exit the work force until children are ready for pre-school and this resultprovides evidence of the difficulty mothers faces re-entering the work force after a severalyear leave of absence (Drange and Rege, 2013). The finding is interesting from a policystandpoint. If policy is focused on increasing participation, offering more support in theyears when the child is likely not breastfeeding but before kindergarten would be beneficial.The covariate effect of a college degree is 5.2 to 7.1 percentage points across the quantiles,while husband’s income is approximately − The recent financial crisis had major implications for home ownership in the UnitedStates. Figure 3 displays the home ownership rates for the United States from the 1960sto 2017. These data were taken from the FRED website provided by the Federal ReserveBank of St. Louis. The rate of home ownership rose in the late 1990s and early 2000s, butstarted to decline after 2007. The determinants of home ownership was reviewed in the 1970s(Carliner, 1974; Poirier, 1977). However, the recent crisis offers a unique event and shock tohousing markets to reevaluate this topic.The literature on home ownership has examined racial gaps (Charles and Hurst,2002; Turner and Smith, 2009), wealth accumulation and income (Turner and Luea, 2009),mobility and the labor market (Ferreira et al., 2010; Fairlie, 2013), and tax policy(Hilber and Turner, 2014). However, unlike the labor force context, state dependence hasonly been lightly examined with regard to housing tenure. Given the large down paymentsand extensive mortgage processes typical in home ownership, state dependence is likely tobe a key factor, as well as individual heterogeneity. Chen and Ost (2005) control for state dependence in a study of housing allowance in Sweden. willingness or utility of owning a home. Owning a home in theUnited States usually requires an individual to produce a large upfront investment, a promis-ing credit history, and a willingness to engage in 30 year mortgages, resulting in less liquidity.Given these requirements, interest lies in how the determinants of home ownership variesacross the latent utility scale. Therefore, this paper adds to the literature on the probabilityof home ownership by employing the QBLD model. The approach has several advantages,namely that we can control for multivariate heterogeneity, visit the state dependence versusheterogeneity argument in the housing context, and analyze willingness of home ownershipacross the quantiles.The dataset is constructed from the Panel Study of Income Dynamics (PSID) and con-sists of a balanced panel of 4092 individuals observed for the years 2001, 2003, 2005, 2007,2009, 2011, and 2013. The sample is restricted to individuals aged 25-65 who answered therelevant questions for the 7 years and captures the period before, during, and after the GreatRecession. The dependent variable is defined as follows: y it = , (9)for i = 1 , . . . , t = 2003 , , , , , Figure 3: Home ownership rates in the United States. Data taken from FRED, provided by the FederalReserve Bank of St. Louis.
JobCat1 is an indicator for jobs in construction, manufacturing, agriculture, and wholesale.
JobCat2 is an indicator for jobs in business, finance, and real estate.
JobCat3 is an indicator forjobs in the military and public services. The omitted category (
JobCat4 ) consists of jobs inprofessional and technical services, entertainment and arts services, health care, and other.Education is broken up into categories: less than high school (omitted), high school degreeor some college (
Below Bachelors ), and college or advanced degree (
Bachelors & Above ).Race is broken up into white/asian (omitted), black , and other . Marital status is discretizedinto married , single , and divorced/widowed (omitted). Region is discretized to west , south , northeast , and midwest (omitted). We have two income measures, including income-to-needsratio and net wealth . We employ an inverse hyperbolic sine (IHS) transformation for netwealth because it adjusts for skewness and retains negative and 0 values, which is a commonfeature of data on net wealth (Friedline et al., 2015).The table demonstrates some drastic differences across the subgroups. As expected, the“owned 6 years” group is older and wealthier than the others. Families that transition tendto have more children, and a higher proportion of females and singles are in the “owned 0years” group. These differences in the raw data motivate our question of interest – with somuch state dependence in home ownership and heterogeneity among individuals and income, This measure of net wealth excludes home equity and housing assets, so as to not conflate with theoutcome of interest. able 6: Sample characteristics of the home ownership data – The first panel presents the mean and standarddeviations (in parenthesis) of the continuous variables and proportions of the categorical variables in the fulland the sub-samples. The second panel displays the column percentages for the number of years home isowned and the third panel (i.e., last row) presents the number of observations in the full and the sub-samples.FullSample Owned 6Years Owned 0Years SingleTransitionfromOwnership SingleTransitiontoOwnership MultipleTransitions(1) (2) (3) (4) (5) (6)Age of Head (2003) 45 .
74 48 .
78 42 .
31 46 .
71 38 .
91 39 . .
51) (12 .
56) (13 .
82) (15 .
48) (11 .
63) (12 . .
78 0 .
71 0 .
82 0 .
83 0 .
85 1 . .
15) (1 .
08) (1 .
25) (1 .
21) (1 .
12) (1 . ,
000 7 .
71 9 .
66 3 .
17 6 .
54 6 .
74 6 . .
53) (11 .
18) (3 .
33) (16 .
28) (5 .
71) (10 . .
89 6 .
07 2 .
17 4 .
25 4 .
37 4 . .
38) (7 .
51) (2 .
30) (14 .
01) (3 .
74) (5 . ,
000 23 .
05 35 .
76 1 .
98 12 .
40 8 .
20 10 . .
08) (167 .
12) (16 .
79) (54 .
30) (32 .
70) (45 . .
23 0 .
14 0 .
49 0 .
26 0 .
22 0 . .
61 0 .
78 0 .
23 0 .
49 0 .
53 0 . .
15 0 .
06 0 .
42 0 .
11 0 .
20 0 . .
54 0 .
52 0 .
53 0 .
58 0 .
57 0 . .
27 0 .
34 0 .
11 0 .
19 0 .
26 0 . .
58 0 .
61 0 .
54 0 .
61 0 .
53 0 . .
11 0 .
12 0 .
07 0 .
10 0 .
12 0 . .
06 0 .
06 0 .
03 0 .
06 0 .
08 0 . .
92 0 .
97 0 .
83 0 .
90 0 .
92 0 . .
30 0 .
20 0 .
58 0 .
32 0 .
31 0 . .
04 0 .
04 0 .
06 0 .
05 0 .
04 0 . .
06 0 .
03 0 .
12 0 .
07 0 .
06 0 . .
23 0 .
24 0 .
28 0 .
29 0 .
11 0 . .
19 0 .
19 0 .
18 0 .
20 0 .
19 0 . .
41 0 .
38 0 .
44 0 .
45 0 .
45 0 . .
14 0 .
15 0 .
14 0 .
10 0 .
16 0 . . − − − − . − − .
10 14 .
89 17 .
342 4 . − − .
40 14 .
10 21 .
393 4 . − − .
94 14 .
63 13 .
874 5 . − − .
55 23 .
67 19 . able 6 – Continued from previous page . − − .
01 32 .
71 27 .
466 55 .
96 100 − − − −
Observations 4092 2290 754 326 376 346 what are the determinants of home ownership through an economic downturn? The resultsshould provide insights into discrepancies across subgroups of the population and shouldbetter inform policy aiming to assist home owners during downturns. Standard methods forinvestigating a binary panel dataset of this sort do not capture the extensive heterogeneityproblem, nor do they offer quantile analyses, which highlights the usefulness of our approach.The results for the home ownership application are presented in Table 7. Posterior means,standard deviations, and inefficiency factors calculated using the batch-means method arepresented for the 25th, 50th, and 75th quantiles, as well as for the binary longitudinal probitmodel (PBLD). The results are based on 12,000 MCMC draws with a burn in of 3,000draws. The priors on the parameters are: β ∼ N (0 k , I k ), and ϕ ∼ IG (10 / , / utility for home ownership, age ofthe head is not statistically different from 0. Number of children, on the other hand, has apositive impact across the quantiles. Family growth seems to play a role in owning a home.The coefficient for female is positive which implies that females relative to males are morein favor of home ownership. Given that housing was previously thought of as a safe invest-ment, this finding aligns with Croson and Gneezy (2009), who investigate gender differencesin preferences and find that women are more risk averse than men. Furthermore, relative todivorced/widowed individuals, being single has a positive effect only at the lower quantile.Interestingly, health insurance has a positive effect at the lower and middle quantiles andis not statistically different from zero at the higher willingness. Thus, if one has a highutility for home ownership, potential costs related to health do not play into the the decision24 able 7: Posterior means ( mean ), standard deviations ( std ) and inefficiency factors ( if ) of the parametersin the QBLD model and PBLD model for the home ownership application. QBLD25th quantile 50th quantile 75th quantile PBLDmean std if mean std if mean std if mean std if
Intercept − .
25 0 .
85 3 . − .
26 0 .
80 4 . − .
17 0 .
88 3 . − .
15 0 .
31 2 . .
63 0 .
21 3 .
40 0 .
97 0 .
20 3 .
76 0 .
09 0 .
23 3 .
47 0 .
54 0 .
08 2 . .
14 0 .
05 3 .
46 0 .
18 0 .
05 4 .
32 0 .
22 0 .
05 3 .
65 0 .
08 0 .
02 2 . .
48 0 .
03 8 .
42 0 .
45 0 .
03 8 .
10 0 .
55 0 .
04 6 .
68 0 .
23 0 .
01 6 . .
25 0 .
03 4 .
05 0 .
32 0 .
03 4 .
56 0 .
41 0 .
04 5 .
11 0 .
10 0 .
01 2 . .
95 0 .
14 3 .
21 0 .
82 0 .
14 3 .
84 0 .
59 0 .
18 3 .
79 0 .
25 0 .
05 1 . .
28 0 .
14 3 .
42 2 .
18 0 .
15 4 .
05 1 .
70 0 .
17 4 .
35 0 .
74 0 .
05 2 . .
32 0 .
15 3 .
37 0 .
17 0 .
15 4 . − .
27 0 .
17 3 .
49 0 .
01 0 .
05 1 . .
17 0 .
12 3 .
71 0 .
28 0 .
11 3 .
80 0 .
35 0 .
14 3 .
89 0 .
11 0 .
04 2 . .
28 0 .
18 3 .
81 0 .
37 0 .
16 4 .
11 0 .
51 0 .
20 4 .
52 0 .
13 0 .
06 2 . .
31 0 .
13 4 .
23 0 .
39 0 .
12 4 .
34 0 .
55 0 .
13 4 .
07 0 .
16 0 .
04 2 . .
06 0 .
20 3 .
92 0 .
21 0 .
19 4 .
52 0 .
35 0 .
21 3 .
86 0 .
08 0 .
07 2 . .
03 0 .
24 3 .
54 0 .
08 0 .
23 3 .
82 0 .
11 0 .
26 3 .
74 0 .
01 0 .
08 2 . .
46 0 .
16 3 .
78 0 .
46 0 .
15 3 .
90 0 .
23 0 .
19 4 .
49 0 .
09 0 .
05 2 . − .
40 0 .
12 3 . − .
54 0 .
12 3 . − .
52 0 .
14 3 . − .
18 0 .
04 1 . − .
15 0 .
22 3 . − .
51 0 .
21 3 . − .
85 0 .
25 4 . − .
19 0 .
07 1 . − .
91 0 .
18 3 . − .
89 0 .
19 4 . − .
76 0 .
23 5 . − .
23 0 .
06 2 . − .
40 0 .
14 4 . − .
28 0 .
14 4 . − .
06 0 .
16 4 . − .
10 0 .
05 2 . − .
46 0 .
15 3 . − .
49 0 .
15 3 . − .
52 0 .
18 3 . − .
16 0 .
06 1 . .
15 0 .
13 3 .
29 0 .
22 0 .
13 3 .
88 0 .
26 0 .
14 3 .
97 0 .
10 0 .
05 2 . − .
28 0 .
18 3 . − .
43 0 .
17 3 . − .
59 0 .
20 3 . − .
17 0 .
06 2 . − .
32 0 .
23 6 . − .
53 0 .
13 4 . − .
44 0 .
11 2 . − .
09 0 .
04 2 . .
46 0 .
17 5 .
93 5 .
90 0 .
12 4 .
25 9 .
55 0 .
20 7 .
04 2 .
19 0 .
04 2 . .
47 0 .
25 5 .
99 0 .
72 0 .
18 4 .
94 0 .
76 0 .
29 8 .
53 0 .
11 0 .
06 2 . ϕ .
13 0 .
01 9 .
88 0 .
11 0 .
01 8 .
67 0 .
16 0 .
02 8 .
94 0 .
04 0 .
01 8 . − . − . − . − . .
14 10110 .
24 10221 .
27 8942 . .
69 10327 .
79 10438 .
83 9160 . to invest in a home. While race-black is negative across the quantiles, which is consistentwith findings in Charles and Hurst (2002), race-other is meaningful and negative only at themiddle and upper quantiles. Thus, policy interested in race disparities in home ownership,should focus on high willingness individuals, because low willingness race-other individualsare not statistically different from whites. 25he coefficient for Post-Recession (2009-2013) is negative across all of the quantiles.This finding is expected given the major collapse in housing markets. The state dependencevariable ( lag-Home Own ) is very large and positive for all of the quantiles. Even with ashock to housing markets and heterogeneity in the intercept and income controlled for, statedependence is a key element of home ownership. Interestingly, the interaction term betweenthe state dependence variable and the post-recession indicator has a credibility interval thatincludes 0 for the PBLD model, but is positive across the quantiles. This finding is intriguingbecause the positive state dependence effect offsets the negative effect from the recession.Perhaps individuals who did not own a home prior to the recession had trouble transitioningto ownership as a result of the tightened lending and credit channels. This reasoning falls inline with the work of Hilber and Turner (2014) in that mortgage policies can effect subgroupsof home owners, but not in aggregate. The aggregate finding in PBLD shows the result isnot statistically different from 0, but we find new results at the quantiles.Covariate effect calculations, which follow from the discussion in Section 3.3, are com-puted for several variables in both of the models, QBLD and PBLD. The results are displayedin Table 8, and show that being a female increases the probability of home ownership by 2.9to 1.6 percentage points, for the 25th and 75th quantiles, respectively. The size of the effectis roughly halved at the 75th quantile. This is useful for understanding the differences inpreferences between males and females, in particular, that at a higher willingness, they aremore similar than at a lower willingness. Similar differing effects are found for the variablemarried, where being married increases the probability of home ownership by 8.7 percentagepoints at the 25th quantile and 5.4 percentage points at the 75th quantile. Furthermore,health insurance increases the probability of home ownership by 1.5 percentage points at alow willingness and 0.06 percentage points at the high willingness (although the basic resultat the 75th quantile was not different from 0).The aforementioned results find smaller effects at the higher willingness, however, thisis not the case for education and wealth. Wealth and education have a greater impact forthose with a high utility. Increasing net wealth by $50,000 increases the probability of homeownership by 2.0 percentage points, and achieving a bachelors degree or more increasesthe probability by 1.5 percentage points. Understanding how these effects differ across thequantiles is important from a policy standpoint. For instance, if policymakers are lookingto push more people into home ownership, they can consider the various types of people(high utility - low utility), and focus policy on the variables that have a greater impact onthe subgroups. Additionally, when downturns occur, there are clear difficulties transitioning26 able 8: Covariate effects in the home ownership study. Age is increased by 10 years and the untransformednet wealth is increased by $50,000. The rest of the variable are indicators. QBLD25th 50th 75th PBLD log Age of Head 0 . . . . . . . . . . . . . . . . . . . . . . . . − . − . − . − . − . − . − . − . into or out of housing markets, which is clear from the results of the interaction term.These results, along with those of the demographic variables, shed light on findings that areunavailable or different than those produced from modeling the mean (PBLD).
5. Conclusion
This paper presents quantile regression methods for binary longitudinal data that accom-modate various forms of heterogeneity, and designs an estimation algorithm to fit the model.The framework developed in this paper contributes to literatures on quantile regression fordiscrete data, panel data models for quantile regression, and discrete panel data models.A simulation study is performed, which demonstrates the computational efficiency of theestimation algorithm and blocking approach.The model is first applied to examine female labor force participation. Although this isa heavily studied topic, the panel quantile approach offers a new perspective to understandthe impact of the covariates, while controlling for heterogeneity and state dependence. Theresults show that particular attention needs to be paid to women with newborns and childrenages 3-5 as the impacts of these variables on female labor force participation are large anddispersed across the quantiles. The model is also applied to investigate the determinantsof home ownership before, during, and after the Great Recession. The state dependenceeffect in home ownership is strong (even when controlling for multivariate heterogeneity),however, after the recession the effect differs nontrivially from mean regression. Other re-sults, including race, number of children, gender, health insurance, and location, also offerunique findings across the quantiles, which are unavailable in other modeling settings. The27pproach provided in this paper leads to a richer view of how the covariates influence theoutcome variables, which better informs policy on female labor force participation and homeownership. 28 eferencesReferences
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The algorithm below presents the sampler for non-blocked sampling in the QBLD model.
Algorithm 2 (Non-blocked sampling)
1. Let Ψ i = D τ √ w i . Sample β | α, ϕ , z, w ∼ N ( ˜ β, ˜ B ), where,˜ B − = (cid:18) n X i =1 X ′ i Ψ − i X i + B − (cid:19) and ˜ β = ˜ B n X i =1 X ′ i Ψ − i ( z i − S i α i − θw i ) + B − β ! .
2. Sample α i | β, ϕ , z, w ∼ N (˜ a, ˜ A ) for i = 1 , · · · , n , where,˜ A − = (cid:18) S ′ i D − τ √ w i S i + 1 ϕ I l (cid:19) and ˜ a = ˜ A (cid:16) S ′ i D − τ √ w i (cid:0) z i − X i β − θw i (cid:1)(cid:17) .
3. Sample w it | β, α i , z it ∼ GIG (0 . , ˜ λ it , ˜ η ) for i = 1 , · · · , n and t = 1 , · · · , T i , where,˜ λ it = (cid:18) z it − x ′ it β − s ′ it α i τ (cid:19) and ˜ η = (cid:18) θ τ + 2 (cid:19) .
4. Sample ϕ | α ∼ IG (˜ c / , ˜ d / c = (cid:16) nl + c (cid:17) and ˜ d = (cid:16) P i α ′ i α i + d (cid:17) .5. Sample the latent variable z | y, β, α, w for all values of i = 1 , · · · , n and t = 1 , · · · , T i from anunivariate truncated normal (TN) distribution as follows, z it | y, β, w ∼ T N ( −∞ , (cid:18) x ′ it β + s ′ it α i + θw it , τ w it (cid:19) if y it = 0 ,T N (0 , ∞ ) (cid:18) x ′ it β + s ′ it α i + θw it , τ w it (cid:19) if y it = 1 . Appendix B. The Conditional Densities for Blocked Sampling in QBLD Model
This appendix presents a derivation of the conditional posterior densities for blockedsampling in the QBLD model. Specifically, the parameters β and latent variable z i aresampled marginally of the random effects parameter α i , from an updated multivariate normaland a truncated multivariate normal distribution, respectively. The parameter α i is sampledconditional on ( β, z i ) from an updated multivariate normal distribution. The latent weights w are sampled element wise from a generalized inverse Gaussian (GIG) distribution and thevariance ϕ is sampled from an updated inverse-gamma distribution.36 . The mean and variance of the QBLD model, z i = X i β + S i α i + θw i + D τ √ w i u i for i = 1 , . . . , n , (marginally of α i ) can be shown to have the following expressions, E ( z i ) = X i β + θw i ,V ( z i ) = ϕ S i S ′ i + D τ √ w i = Ω i . First, we derive the conditional posterior of β and z i , marginally of α i , but conditional onother variables in the model. . Starting with β , the conditional posterior density π ( β | z, w, ϕ ) can be derived as, π ( β | z, w, ϕ ) ∝ ( n Y i =1 f ( z i | β, w i , ϕ ) ) π ( β ) ∝ exp " − ( n X i =1 ( z i − X i β − θw i ) ′ Ω − i ( z i − X i β − θw i )+ ( β − β ) ′ B − ( β − β ) ) ∝ exp " − ( β ′ n X i =1 X ′ i Ω − i X i + B − ! β − β ′ n X i =1 X ′ i Ω − i ( z i − θw i ) + B − β ! − n X i =1 ( z i − θw i ) ′ Ω − i X i + β ′ B − !) ∝ exp " − ( β ′ ˜ B − β − β ′ ˜ B − ˜ β − ˜ β ′ ˜ B − β ) , where the third line only keeps terms involving β and the fourth line introduces the terms˜ β and ˜ B , which are defined as,˜ B − = n X i =1 X ′ i Ω − i X i + B − ! and ˜ β = ˜ B X ′ i Ω − i ( z i − θw i ) + B − β ! . Adding and subtracting ˜ β ′ ˜ B − ˜ β and absorbing the term exp[ − {− ˜ β ′ ˜ B − ˜ β } ] into the pro-portionality constant, the square can be completed as follows, π ( β | z, w, ϕ ) ∝ exp " −
12 ( β − ˜ β ) ′ ˜ B − ( β − ˜ β ) . The above expression is recognized as the kernel of a Gaussian or normal distribution andhence β | z, w, ϕ ∼ N ( ˜ β, ˜ B ). 37 (b) . The conditional posterior density of the latent variable z marginally of α can beobtained from the joint posterior density (8) as, π ( z | β, w, ϕ , y ) ∝ n Y i =1 ( π ( z i | β, w i , ϕ , y i ) ) ∝ n Y i =1 ( T i Y t =1 (cid:20) I ( z it > I ( y it = 1) + I ( z it ≤ I ( y it = 0) (cid:21) × exp " −
12 ( z i − X i β − θw i ) ′ Ω − i ( z i − X i β − θw i ) . The expression inside the curly braces corresponds to a truncated multivariate normal dis-tribution, so z i | y i , β, w i , ϕ ∼ T M V N B i ( X i β + θw i , Ω i ) for all i = 1 , · · · , n . Here, B i is thetruncation region such that B i = ( B i × B i × . . . × B iT i ), where B it is the interval (0 , ∞ ) if y it = 1 and the interval ( −∞ ,
0] if y it = 0 for t = 1 , . . . , T i . Sampling directly from a TMVNis not possible, hence we resort to the method proposed in Geweke (1991), which utilizesGibbs sampling to make draws from a TMVN.Let z ji denote the values of z i at the j -th pass of the MCMC iteration. Then sampling isdone from a series of conditional posterior distribution as follows: z jit | z ji , · · · , z ji ( t − , z ji ( t +1) , · · · , z jiT i ∼ T N B it ( µ t |− t , Σ t |− t ) , for t = 1 , · · · , T i , where T N denotes a truncated normal distribution. The terms µ t |− t and Σ t |− t are theconditional mean and variance, respectively, and are defined as, µ t |− t = x ′ it β + θw it + Σ t, − t Σ − − t, − t (cid:16) z ji, − t − ( X i β + θw i ) − t (cid:17) , Σ t |− t = Σ t,t − Σ t, − t Σ − − t, − t Σ − t,t , where z ji, − t = ( z ji , · · · , z ji ( t − , z j − i ( t +1) , · · · , z j − iT i ), ( X i β + θw i ) − t is a column vector with t -thelement removed, Σ t,t denotes the ( t, t )-th element of Ω i , Σ t, − t denotes the t -th row of Ω i with element in the t -th column removed and Σ − t, − t is the Ω i matrix with t -th row and t -thcolumn removed. (2) . The conditional posterior density of the random effects parameters α i for i = 1 , . . . , n is derived from the joint posterior density (8) as follows, π ( α i | z i , β, w i , ϕ ) ∝ f ( z i | β, α i , w i ) π ( α i | ϕ ) 38 exp " − ( ( z i − X i β − S i α i − θw i ) ′ D − τ √ w i ( z i − X i β − S i α i − θw i )+ α ′ i α i ϕ ) ∝ exp " − ( α ′ i (cid:18) S ′ i D − τ √ w i S i + ϕ − I l (cid:19) α i − α ′ i (cid:18) S ′ i D − τ √ w i (cid:16) z i − X i β − θw i (cid:17)(cid:19) − (cid:18)(cid:16) z i − X i β − θw i (cid:17) ′ D − τ √ w i S i (cid:19) α i ) ∝ exp " −
12 ( α i − ˜ a ) ′ ˜ A − ( α i − ˜ a ) , where the third line omits all terms not involving α i and the fourth line introduces the terms,˜ A − = S ′ i D − τ √ w i S i + 1 ϕ I l ! and ˜ a = ˜ A (cid:16) S ′ i D − τ √ w i (cid:16) z i − X i β − θw i (cid:17)(cid:17) , as the posterior precision and posterior mean, respectively, and completes the square. Theresult is a kernel of a normal distribution, hence, α i | z i , β, w i , ϕ ∼ N (˜ a, ˜ A ) for i = 1 , . . . , n . (3) . The conditional posterior density of w is obtained from the joint posterior density(8) by collecting terms involving w . Each term in w is updated element-wise as follows: π ( w it | z it , β, α i ) ∝ (cid:16) πτ w it (cid:17) − / exp " − τ w it (cid:16) z it − x ′ it β − s ′ it α i − θw it (cid:17) − w it ∝ w − / it exp " − ( z it − x ′ it β − s ′ it α i τ ! w − it + θ τ + 2 ! w it ) ∝ w − / it exp " − ( ˜ λ it w − it + ˜ ηw it ) , where the second line omits all terms not involving w it and the third line introduces theterms defined below, ˜ λ it = z it − x ′ it β − s ′ it α i τ ! and ˜ η = θ τ + 2 ! . The expression in the third line is recognized as the kernel of a generalized inverse Gaussian(GIG) distribution. Hence, we have w it | z it , β, α i ∼ GIG (0 . , ˜ λ it , ˜ η ) for t = 1 , . . . , T i and i = 1 , . . . , n . (4) . The conditional posterior density of ϕ is obtained from the joint posterior density398) by collecting terms involving ϕ conditional on the remaining model parameters. This isdone below. π ( ϕ | α ) ∝ (2 π ) − nl/ (cid:16) ϕ (cid:17) − nl/ exp " − ϕ n X i =1 α ′ i α i ϕ (cid:17) − ( c / exp " − d ϕ ∝ (cid:16) ϕ (cid:17) − ( nl/ c / exp " − ϕ ( n X i =1 α ′ i α i + d ) ∝ (cid:16) ϕ (cid:17) (˜ c / exp " − ϕ ˜ d , where ˜ c = nl + c and ˜ d = (cid:16) P ni =1 α ′ i α i + d (cid:17) . The expression in the last line is recognizedas the kernel of an inverse gamma (IG) distribution and consequently, we have ϕ | α ∼ IG (˜ c //
12 ( α i − ˜ a ) ′ ˜ A − ( α i − ˜ a ) , where the third line omits all terms not involving α i and the fourth line introduces the terms,˜ A − = S ′ i D − τ √ w i S i + 1 ϕ I l ! and ˜ a = ˜ A (cid:16) S ′ i D − τ √ w i (cid:16) z i − X i β − θw i (cid:17)(cid:17) , as the posterior precision and posterior mean, respectively, and completes the square. Theresult is a kernel of a normal distribution, hence, α i | z i , β, w i , ϕ ∼ N (˜ a, ˜ A ) for i = 1 , . . . , n . (3) . The conditional posterior density of w is obtained from the joint posterior density(8) by collecting terms involving w . Each term in w is updated element-wise as follows: π ( w it | z it , β, α i ) ∝ (cid:16) πτ w it (cid:17) − / exp " − τ w it (cid:16) z it − x ′ it β − s ′ it α i − θw it (cid:17) − w it ∝ w − / it exp " − ( z it − x ′ it β − s ′ it α i τ ! w − it + θ τ + 2 ! w it ) ∝ w − / it exp " − ( ˜ λ it w − it + ˜ ηw it ) , where the second line omits all terms not involving w it and the third line introduces theterms defined below, ˜ λ it = z it − x ′ it β − s ′ it α i τ ! and ˜ η = θ τ + 2 ! . The expression in the third line is recognized as the kernel of a generalized inverse Gaussian(GIG) distribution. Hence, we have w it | z it , β, α i ∼ GIG (0 . , ˜ λ it , ˜ η ) for t = 1 , . . . , T i and i = 1 , . . . , n . (4) . The conditional posterior density of ϕ is obtained from the joint posterior density398) by collecting terms involving ϕ conditional on the remaining model parameters. This isdone below. π ( ϕ | α ) ∝ (2 π ) − nl/ (cid:16) ϕ (cid:17) − nl/ exp " − ϕ n X i =1 α ′ i α i ϕ (cid:17) − ( c / exp " − d ϕ ∝ (cid:16) ϕ (cid:17) − ( nl/ c / exp " − ϕ ( n X i =1 α ′ i α i + d ) ∝ (cid:16) ϕ (cid:17) (˜ c / exp " − ϕ ˜ d , where ˜ c = nl + c and ˜ d = (cid:16) P ni =1 α ′ i α i + d (cid:17) . The expression in the last line is recognizedas the kernel of an inverse gamma (IG) distribution and consequently, we have ϕ | α ∼ IG (˜ c // , ˜ d //