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Flexible Bayesian Quantile Analysis of Residential Rental Rates
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This paper aims to provide complementary methodological and empirical contributions to the quantile regression literature. On the methodological side, we develop a flexible Bayesian approach to random effects quantile regression based on a generalization of the asymmetric Laplace distribution, specify an efficient Markov chain Monte Carlo (MCMC) estimation algorithm, and present methods for formal model comparison of various nested and non-nested models that also enable us to assess the importance of flexible modeling. Our methods are readily motivated by the econometric challenges of studying U.S. residential rental rates and their dependence on unemployment and mortgage policies following the Global Financial Crisis. Our investigation deals with key features of the data, including considerable zip-code-level heterogeneity and skewness in the distribution of rents. The separation of modeling features into those that are practically relevant from those that are not warranted in this context is handled by model comparison.
Koenker-Basset-1978 introduced quantile regression as a minimization problem involving an asymmetrically weighted linear loss function, but subsequent work has noted the duality between that approach and modeling through a likelihood function built on the asymmetric Laplace (AL) distribution Koenker-Machado-1999, Yu-Moyeed-2001. The latter approach becomes very potent when the AL distribution is expressed as a mixture of normal and exponential distributions Kozumi-Kobayashi-2011. The mixture formulation permits estimation by simple, yet powerful, MCMC algorithms, and has enabled extensions of the quantile methodology to a variety of other settings including censored data Kozumi-Kobayashi-2011, binary data Benoit-Poel-2010, Ojha-Rahman-2021, ordinal outcomes Rahman-2016, Alhamzawi-2016, Maheshwari-Rahman-2023, linear mixed models Luo-Lian-Tian-2012, and panels of binary Rahman-Vossmeyer-2019, Bresson-etal-2021, ordinal Alhamzawi-Ali-Longitudinal2018, or dynamic censored data Kobayashi-Kozumi-2012. Recent work by Goncalves-etal-2020 considered extensions to the case of dynamic quantile linear models. While the application of the AL distribution has unlocked a plethora of new research opportunities, the AL distribution itself is not without its limitations. For instance, the skewness is completely determined once a quantile is chosen and the mode of the distribution is always fixed at the value of the location parameter. These limitation can be circumvented by introducing a shape parameter into the mean of the normal kernel in the AL mixture representation leading to the generalized asymmetric Laplace (GAL) distribution Yan-Kottas-2017, Rahman-Karnawat-2019.
We extend GAL modeling to the random effects panel setting by proposing an efficient MCMC sampler which offers a variety of algorithmic improvements through suitable transformations of the mixture variables, block sampling of scale and shape parameters, and block sampling of the individual-specific and common effect parameters Nascimento-Goncalves-2021. These changes eliminate the problem of high autocorrelation in the MCMC draws, but are also applicable to the MCMC estimation models based on the simpler AL distribution Luo-Lian-Tian-2012 while also allowing for correlated random effects. For both the GAL and AL panel data models, we adapt the methods of Chib-1995 and Chib-Jeliazkov-2001 to enable model comparison through marginal likelihoods, which, with few exceptions Kobayashi-Kozumi-2012, Maheshwari-Rahman-2023 is broadly lacking in the quantile literature. Several simulation studies carefully examine the properties and practical appeal of the proposed techniques.
The empirical contribution of the paper involves the study of U.S. residential rental rates during the recovery period following the Global Financial Crisis. We construct a novel data set that includes median rental rates in $14,533$ zip codes from 2010 to 2016, as well as zip-code level economic, demographic, mortgage, and tax policy controls. Our methodology is particularly appealing in this setting because housing prices and rents are heavily skewed and heterogeneous across regions. For instance, from 2010 to 2016, the Cleveland MSA region's change in “All Transaction House Price Index” was about 8.42, whereas the San Francisco MSA region's change was about 156.\footnote{Based on data from the FRED database at the Federal Reserve Bank of St.\! Louis.} Skewness, along with heterogeneity in economic outcomes, has been identified as an important driver of public policy and political economy considerations Benhabib-Bisin-2018.
The data reveal a positive impact of unemployment on residential rental rates as uncertain job prospects reduce the willingness and ability of households to commit to homeownership and instead shift demand towards rental units. We also find that home mortgage deductions decrease rental prices by making homeownership more attractive. This finding is particularly relevant in the context of the Tax Cuts and Jobs Act, which was passed in 2017. The law lowered mortgage deductions, suggesting that one consequence of the policy change is an expected rise in rents. Lastly, model comparisons across many quantiles and samples reveal that the data overwhelmingly support the more flexible GAL modeling framework.
The remainder of the paper is organized as follows. In Section (ref), we present the proposed modeling, estimation and model comparison framework. This section also presents improved algorithms for the simpler AL-based model. Section (ref) illustrates the proposed algorithms in multiple simulation studies. Section (ref) describes the data, presents our rental rates application and discusses the results, while Section (ref) concludes.
This section introduces our proposed model, discusses the challenges associated with its estimation, and presents the MCMC estimation algorithm. The section also offers an improved algorithm for estimating AL-based models. Besides, this section describes the computation of marginal likelihood for the proposed framework and the AL-based model.
We focus on a panel data model which takes the form
where $y_{it}$ denotes the $t$-th response on the $i$-th unit, $x_{it}$ is a $k$ vector of covariates, $\beta_{p_{0}}$ is a $k$ vector of common parameters at the $p_{0}$-th quantile (henceforth, simply $\beta$), $z_{it}$ is an $l$ vector of variables with $z_{it} \subseteq x_{it}$, and $\alpha_{i}$ is an $l$ vector of subject-specific random effects that induces dependence between observations on the same individual.\footnote{An unfortunate rift in terminology has persisted between statistics and econometrics in the panel (longitudinal) context. In statistics, $\beta$ and $\{ \alpha_i \}$ are called fixed and random effects, respectively, because the former do not vary with $i$, whereas the latter are subject-specific. In econometrics these terms are used to distinguish between alternative ways of dealing with $\{ \alpha_i \}$ -- fixed effects estimators remove the heterogeneity (if possible) by data transformations such as mean- or first-differencing, whereas random effects estimators model the $\{ \alpha_i \}$ explicitly through a distribution.}
Although not immediately obvious from the notation, the setup is rather general and can capture dynamics, unknown covariate functions, and correlated random effects, depending on what is included in $x_{it}$ (and potentially also in $z_{it} \subseteq x_{it}$). In particular, dynamic modeling can be pursued by including lags of $y_{it}$ in $x_{it}$ and ensuring that the lag coefficients satisfy stationarity. Flexible functional modeling for some covariate $s$ can be implemented through a set of basis functions $\mathcal{B} = \{ b_1, \ldots, b_m \}$, e.g., B-splines, natural splines, truncated power series, wavelets, etc., Ruppert-etal-2003 so that $f(s) = \sum_{j=1}^{m} b_j(s) \delta_j$, in which case $x_{it}$ includes $(b_1 (s_i), \ldots, b_m (s_i) )'$, while $(\delta_1, \ldots, \delta_m )'$ becomes part of the regression parameter vector $\beta$. In addition, correlated random effect models where the heterogeneity can be correlated with certain observed covariates is handled by interacting those covariates with $z_{it}$ and including the result in $x_{it}$ Chamberlain-1984, Mundlak-1978, Chib-Jeliazkov-2006. Random effect models are also indispensable in settings with multivariate heterogeneity or time-invariant covariates because the data transformations (i.e., mean- or first-differencing) underlying “fixed effects” estimators in econometrics (i) do not remove slope heterogeneity and (ii) wipe out any time-invariant covariates.
We parameterize the model in Equation (ref) by letting $\varepsilon_{it} \stackrel{iid}{\sim} \textrm{GAL}(0, \sigma, p_{0}, \gamma)$, using the quantile-fixed GAL distribution Yan-Kottas-2017, Rahman-Karnawat-2019 -- a generalization stemming from the mixture representation of the AL distribution -- to decouple the modeling of skewness from the quantile parameter. A variable $s$ is said to follow a quantile-fixed GAL distribution, i.e., $s \sim \textrm{GAL}(\mu, \sigma, p_{0}, \gamma)$, where $\mu$, $\sigma$, $p_{0}$, and $\gamma$ represent the location, scale, quantile, and skewness parameters, respectively, if it has density
where $s^{\ast} = \frac{s - \mu}{\sigma}$, $p\equiv p(\gamma,p_{0})=I(\gamma<0) + [p_{0} - I(\gamma<0)]/g(\gamma)$, $p_{\gamma_{+}} = p - I(\gamma > 0)$ and $p_{\gamma_{-}} = p - I(\gamma < 0)$, $g(\gamma) = 2 \Phi(-|\gamma|) \exp(\gamma^{2}/2)$ and $\gamma \in (L,U)$, where $L$ is the negative square root of $g(\gamma)=1-p_{0}$ and $U$ is the positive square root of $g(\gamma)=p_{0}$ (see Section 2 in Rahman-Karnawat-2019 for more details). The term “quantile-fixed” refers to the fact that the density in Equation (ref) satisfies $\int_{-\infty}^{\mu} f_{GAL}( \varepsilon_{it} | \mu, \sigma, p_{0}, \gamma )d \varepsilon_{it} = p_{0}$. The special case of AL density results when $\gamma=0$, this is more clearly seen from the mixture representation presented in Equation (ref).
Figure (ref) offers a visualization of the differences between the quantile-fixed GAL and AL densities. The figure shows three different quantiles when $\sigma = 1$, the standard case. We observe that the GAL distribution, unlike the AL distribution, allows the mode to vary rather than being fixed at $\mu=0$ at all quantiles. Additionally, at the median $p_{0}=0.50$, the GAL distribution can be positively ($\gamma<0$) or negatively ($\gamma>0$) skewed and can have tails that are heavier or narrower than the AL distribution. These characteristics make GAL significantly more flexible than AL, but the value of the extra flexibility is an application-specific empirical question.
Because of the additional flexibility that the GAL distribution offers over the AL distribution, we refer to the model based on the former as the Flexible Random Effects Quantile (FREQ) model, and the model based on the latter as the Random Effects Quantile (REQ) model.
The distributional assumption on the error term, $\varepsilon_{it} \stackrel{iid}{\sim} \textrm{GAL}(0,\sigma,p_{0},\gamma)$ implies that $y_{it}|\alpha_{i} \stackrel{ind}{\sim} \textrm{GAL}(x'_{it} \beta + z'_{it} \alpha_{i},\sigma,p_{0},\gamma)$ for $i = 1, \ldots, n$ and $t = 1, \ldots, T_{i}$. Assuming a density $f(\alpha|\Omega)$ for the random effects, the complete data likelihood can be expressed as
where $\alpha = (\alpha_{1}, \ldots, \alpha_{n})$, $y = (y_{1}, \ldots, y_{n})$ with each $y_{i} = (y_{i1}, \ldots, y_{iT_{i}})'$ for $i = 1,\ldots, n$. The density $f(\alpha_{i}|\Omega)$ can be any suitable distribution, but is typically assumed normal Luo-Lian-Tian-2012, e.g., here we let $\alpha_{i}|\Omega \stackrel{iid}{\sim} N(0_{l}, \Omega)$ for $i = 1, \ldots, n$. The complete data likelihood in Equation (ref) can be combined with priors on the parameters to obtain the joint posterior distribution, but this posterior does not yield known conditional posteriors suitable for a tractable MCMC algorithm. Hence, we utilize the mixture representation of the GAL distribution obtained by introducing a shape parameter into the mean of the normal kernel in the normal-exponential mixture of the AL distribution and mixing with respect to a half-normal distribution Yan-Kottas-2017, Rahman-Karnawat-2019.
For $\varepsilon_{it} \sim \textrm{GAL}(0,\sigma,p_{0},\gamma)$, the mixture representation can be expressed as,
where $s_{it} \sim N^{+}(0,1)$, $\omega_{it} \sim \mathcal{E}(1)$, $u_{it} \sim N(0,1)$, $A\equiv A(p)=\frac{1-2p}{p(1-p)}$, $B\equiv B(p)=\frac{2}{p(1-p)}$, $C=[I(\gamma>0)-p]^{-1}$, and $p$ is as defined earlier. Here, $N^{+}, \mathcal{E}, N$ denote half-normal, exponential, and normal distributions, respectively. Note that the GAL mixture distribution reduces to an AL mixture distribution when $\gamma$ is set to 0, as mentioned earlier. Substituting the mixture representation given by Equation (ref) into Equation (ref), the model can be written as $y_{it} = x'_{it}\beta + z'_{it}\alpha_{i} + \sigma A\omega_{it} + \sigma C |\gamma|s_{it} + \sigma (B \omega_{it})^{\frac{1}{2}}u_{it}. $ In this formulation, the scale parameter appears in the conditional mean which is not suitable for estimation Kozumi-Kobayashi-2011. Therefore, we make the following transformation $h_{it} = \sigma s_{it}$, $\nu_{it} = \sigma \omega_{it}$ and rewrite the model as,
Stacking the model given by Equation (ref) for each individual $i$, we get
where $y_{i} = (y_{i1}, \cdots, y_{iT_{i}})'$, $ X_{i} = ( x'_{i1}, \cdots, x'_{i T_{i}} )'$ is the design matrix of size $ T_{i} \times k $ for each individual $i$, $Z_{i} = (z'_{i1}, \cdots, z'_{iT_{i}})'$ is $T_{i} \times l$ matrix of covariates associated with the random effects, $\nu_{i} = (\nu_{i1}, \cdots, \nu_{i T_{i} } )'$, $ h_{i} = (h_{i1}, \cdots, h_{iT_{i}})'$, $u_{i} = (u_{i1}, \cdots, u_{i T_{i}})'$ and the diagonal matrix
The model in Equation (ref) implies that $y_{i}|\beta,\alpha_{i}, \nu_{i}, h_{i}, \sigma, \gamma \sim N_{T_{i} } \left( X_{i}\beta + Z_{i}\alpha_{i} + A\nu_{i} + C|\gamma|h_{i}, \hspace{.05in} \Lambda_{i} \right)$, which is combined with the priors
to obtain the joint posterior distribution. In the notation above, $IG$ and $IW$ denote an inverse-Gamma and inverse-Wishart distribution, respectively. Let $\Theta=(\beta,\alpha,\nu,h,\sigma, \gamma, \Omega)$, then the posterior distribution can be expressed as follows,
where $f(y|\Theta)$ denotes the density, conditional on $\alpha$, resulting from the stacked FREQ model given by Equation (ref).
The FREQ model has several appealing properties: (i) it can accommodate both common and random effect parameters, (ii) random effects can be associated with multiple variables, in addition to the constant, allowing for both slope and intercept heterogeneity, (iii) quantile regression gives us the ability to explore the entire distribution of the outcome variable $y$, and (iv) flexibility in the skewness parameter allows better fit for various settings. Economic data that are skewed, exhibit power laws, or present odd asymmetries would benefit from the model offered in this paper. Such data include distributions of income, bank assets, social networks, and house or rental prices, the latter of which is explored in this paper.
The FREQ model can be estimated by sampling the objects of interest, $(\beta,\alpha,\nu,h,\sigma,\gamma,\Omega)$ from their respective conditional posterior densities as in Algorithm (ref). We first sample the parameters $(\beta,\alpha)$ in a block conditional on remaining parameters where $\beta$ is sampled marginally of $\alpha$, and then $\alpha$ is sampled conditional on $\beta$. Both the conditional posteriors follow a normal distribution with hyperparameters updated as shown in Algorithm (ref). We utilize block sampling for two reasons: (i) to account for possible correlation between the two parameters, and (ii) to reduce the inefficiency factors in the MCMC draws Chib-Carlin-1999,Greenberg-2012.
The random effects covariance matrix, $\Omega$, is sampled from an inverse-Wishart distribution with updated hyperparameters. The scale and shape parameters $(\sigma,\gamma)$ are jointly sampled marginally of $(\nu, h)$ using a random-walk Metropolis-Hastings (MH) algorithm Chib-Greenberg-1995. Here, the target density is the product of the GAL likelihood and the prior distributions, given by Equation (ref) and Equation (ref), respectively; while the proposal values are drawn from a bivariate truncated normal distribution. We note that joint sampling of $(\sigma,\gamma)$ is critical in reducing the autocorrelation of MCMC draws and hence to the efficiency of the algorithm. The mixture variable, $\nu$, is sampled from a generalized inverse Gaussian (GIG) distribution, draws from which are generated using the technique in Devroye-2014. Lastly, the mixture variable, $h$ conditional on remaining parameters, is sampled from a half-normal distribution with updated hyperparameters. The derivations of the conditional posterior distributions and further details of Algorithm (ref) are presented in (ref)
To exemplify the practical utility of the FREQ model, we also estimate the more established REQ model using the sampler presented in Algorithm (ref). This sampling algorithm has two important improvements from the sampler proposed in Luo-Lian-Tian-2012. First, $(\beta, \alpha)$ are sampled in a single block which significantly lowers the autocorrelation in the MCMC draws and improves the mixing of the Markov chain. Because we can achieve lower inefficiency factors in Algorithm (ref), the number of MCMC draws can typically be reduced, thereby decreasing computational burdens and run times. Second, we correct the updating for $\sigma$ by including the terms involving the exponential variable in the updated hyperparameters. The resulting MCMC algorithm is fast, efficient, and maintains the tractability of the sampling distributions.
Before ending this section, we note that the FREQ model presented in Section (ref) and the estimation procedure explained above (and that of REQ model) considers multivariate heterogeneity i.e., random effects in the intercept and slope parameters. This leads to the distribution, $\alpha_{i} \sim N(0_{l}, \Omega)$, where $\Omega$ is an $l \times l$ covariance matrix that is assigned an inverse-Wishart prior distribution. In the special case of intercept heterogeneity only, we have $\alpha_{i} \sim N(0, \varphi^{2})$ for $i=1,2, \cdots,n$. Since $\varphi^{2}$ is a scalar, it is assigned an $IG$ prior distribution i.e., $\varphi^{2} \sim IG(c_{1}/2, d_{1}/2)$. This leads to changes in the conditional posterior density of $(\beta, \alpha)$ and the conditional posterior of $\Omega$ is replaced by that of $\varphi^{2}$. Specifically, $\beta$ is sampled as in Step 1(a) of Algorithm (ref) except that $V_{i} = \varphi^{2} Z_{i}Z'_{i} + \Lambda_{i}$; whereas for Step 1(b), $\alpha_{i}$ is sampled from $N( \tilde{\alpha_{i}}, \tilde{A_{i}} )$, where $\tilde{A_{i}}^{-1} = \left( Z'_{i}\Lambda_{i}^{-1}Z_{i} + (\varphi^{2})^{-1} I_{l} \right)$ and the expression for $\tilde{a}_{i}$ is unaltered. The parameter $\varphi^2$ is sampled from an updated $IG (\tilde{c}_{1}/2,\tilde{d}_{1}/2)$, where, $\tilde{c}_{1} = nl+ c_{1}$ and $\tilde{d}_{1} = \sum_{i=1}^{n} \alpha'_{i}\alpha_{i} + d_{1} $. Step (3) of Algorithm (ref) remains largely unaltered, except that in the full likelihood given by Equation (ref), the density $f(\alpha_{i}|\Omega)$ is replaced by $f(\alpha_{i}|\varphi^{2})$. Finally, Step (4) and Step (5) of Algorithm (ref) remains unchanged, except that $z_{it} = 1$ and $\alpha_{i}$ is a scalar. This modified MCMC algorithm is utilized to estimate the residential rental rates model in Section (ref), where intercept heterogeneity is of considerable importance.
Similarly, estimating the REQ model with intercept heterogeneity requires some modification to Algorithm (ref). In particular, the sampling of $\beta$ in Step 1(a) now requires $V_{i} = \varphi^{2} Z_{i}Z'_{i} + \Lambda_{i}$; while sampling $\alpha_{i}$ in Step 1(b) needs $\tilde{A_{i}}^{-1} = \left( Z'_{i}\Lambda_{i}^{-1}Z_{i} + (\varphi^{2})^{-1} I_{l} \right)$. All other expression in the conditional posteriors remain unchanged. The parameter $\varphi^2|\alpha,y$ is sampled from an $IG(\tilde{c}_{1}/2, \tilde{d}_{1}/2)$, where $ \tilde{c}_{1} = nl+ c_{1}$ and $\tilde{d}_{1} = \sum_{i=1}^{n} \left( \alpha'_{i}\alpha_{i} \right) + d_{1}$. Finally, Step (3) and Step (4) of Algorithm (ref) remains unchanged, except note that $z_{it} = 1$ and $\alpha_{i}$ is a scalar. This modified algorithm for the REQ model is also used to estimate the rental rates application in Section (ref) for comparison.
To properly address model uncertainty, Bayesian model comparison proceeds by representing the posterior model probability of model $\mathcal{M}_{s}$ given the data $y$ as, \[ \Pr(\mathcal{M}_{s}| y ) \propto \Pr(\mathcal{M}_{s}) m(y|\mathcal{M}_{s}), \] where $\Pr(\mathcal{M}_{s})$ is the prior model probability and $m(y|\mathcal{M}_{s})$ is the marginal likelihood. Given the sampling density $f(y|\mathcal{M}_{s}, \Theta_{s})$ and prior distribution $\pi(\Theta_{s}|\mathcal{M}_{s})$ under model $\mathcal{M}_{s}$, the marginal likelihood is defined as the integral \[ m(y|\mathcal{M}_{s}) = \int f(y|\mathcal{M}_{s},\Theta_{s}) \pi(\Theta_{s}|\mathcal{M}_{s}) \, d\Theta_{s}, \] which can also be expressed, using Bayes' theorem, as
where the numerator is the product of the likelihood function and prior density, and the denominator is the joint posterior density Chib-1995, Chib-Jeliazkov-2001. Equation (ref) is known as the basic marginal likelihood identity since it holds for all values in the parameter space. However, marginal likelihood estimate is typically computed at a high-density point (such as the mean or mode), denoted $\Theta^{\ast}_{s}$, to minimize estimation variability. The numerator quantities in Equation (ref) are generally directly available, and therefore, the problem of marginal likelihood estimation is reduced to finding an estimate of the posterior ordinate in the denominator of Equation (ref).
Well-known properties of Bayesian model comparisons based on marginal likelihoods and their ratios, or Bayes factors, are that they lead to finite-sample model probabilities, do not require competing models to be nested and have appealing asymptotic properties that give rise to information criteria Greenberg-2012. Another important, yet underappreciated, point is that marginal likelihoods provide a measure of sequential out of sample predictive fit, which can be seen by writing
Therefore, the adequacy of the model as captured by the marginal likelihood corresponds to the cumulative out-of-sample predictive record where the fit of $y_i$ is measured with respect to the posterior density using data $\{ y_j \}_{j<i}$ up to the $i$th data point. This is in sharp contrast to in-sample measures of fit that condition on the entire data set $y$. Also, the marginal likelihood is invariant to permutations in the indices of the data, so that the same $m(y|\mathcal{M}_{s})$ will be obtained if the data are rearranged.
We next consider the computation of the marginal likelihood for the FREQ and REQ models.
The marginal likelihood for the FREQ model is derived following Chib-Jeliazkov-2001 since the conditional posterior for ($\sigma, \gamma$) does not have a tractable form and is sampled using an MH algorithm (see Algorithm (ref)). Let $\Theta = (\beta, \Omega, \Theta_{1})$ where $\Theta_{1} = (\sigma,\gamma)$, then the joint posterior density for the FREQ model (marginally of $\alpha$, $\nu$, and $h$) can be expressed as,
where $(\beta^{\ast},\Omega^{\ast}, \Theta_{1}^{\ast})$ denotes a high density point of $(\beta, \Omega, \Theta_{1})$. The latent variables $(\alpha, \nu, h)$ are marginalized to reduce the computational burden since computing high dimensional ordinates is costly and leads to inefficient estimates. Moreover, in the decomposition presented in Equation (ref), we have intentionally placed the intractable posterior ordinate $\pi(\Theta_{1}^{\ast}|y)$ first so as to avoid the MH step in the reduced MCMC run -- the process of running an MCMC sampler with one or more parameters fixed at some value Greenberg-2012. We first estimate $\pi(\Theta_{1}^{\ast}|y)$, followed by $\pi(\beta^{\ast}|y,\Theta_{1}^{\ast})$, and lastly, $\pi(\Omega^{\ast}|y,\beta^{\ast},\Theta_{1}^{\ast})$.
To get an estimate of $\pi(\Theta_{1}^{\ast}|y)$, we first need to express the ordinate in a computationally convenient formulation. We know $\Theta_{1}$ is sampled using an MH step, which requires a proposal density and a transition kernel. Define the transition kernel from $\Theta_{1}$ to $\Theta_{1}^{\ast}$ as,
where $q(\Theta_{1}, \Theta_{1}^{\ast}|y,\beta,\Omega,\alpha)$ denotes the proposal density for the transition from $\Theta_{1}$ to $\Theta_{1}^{\ast}$, and
denotes the probability of making the move from $\Theta_{1}$ to $\Theta_{1}^{\ast}$. Note that the conditioning of the proposal density on $y$ and the remaining parameters is only for the sake of generality, and a particular proposal density may be independent of both $y$ and $(\beta, \Omega, \alpha)$. Since the transition kernel, (i.e., Equation (ref)) satisfies the reversibility condition, we exploit this property and, through suitable modifications following Chib-Jeliazkov-2001, arrive at the following expression,
where $E_{1}$ represents expectation with respect to the posterior distribution $\pi(\Theta_{1},\beta,\Omega,\alpha|y)$ and $E_{2}$ represents expectation with respect to the distribution $\pi(\beta,\Omega,\alpha|y,\Theta_{1}^{\ast}) \times q(\Theta_{1}^{\ast},\Theta_{1}|y)$. In this formulation, the numerator in Equation (ref) can be estimated by using draws $\{ \Theta_{1}^{(m)},\beta^{(m)},\Omega^{(m)},\alpha^{(m)} \}_{m=1}^{M}$ from the complete MCMC run and taking an average of $\alpha_{MH}(\Theta_{1}, \Theta_{1}^{\ast}|y,\beta,\Omega,\alpha) \, q(\Theta_{1}, \Theta_{1}^{\ast}) | y,\beta,\Omega,\alpha)$, where $\alpha_{MH}(\Theta_{1}, \Theta_{1}^{\ast}|y,\beta,\Omega,\alpha)$ is given by Equation (ref) and $q(\Theta_{1}, \Theta_{1}^{\ast})|y,\beta,\Omega,\alpha)$ is bivariate truncated normal distribution described in Algorithm (ref).
To compute the denominator in Equation (ref), we note that the distribution $\pi(\beta,\Omega,\alpha|y,\Theta_{1}^{\ast})$ is conditioned on $\Theta_{1}^{\ast}$. Therefore, we conduct a reduced run of Algorithm (ref), i.e., sample $\beta$, $\alpha$, $\Omega$, $v$, and $h$ with $\Theta_{1} = (\sigma, \gamma)$ fixed at $\Theta_{1}^{\ast}=(\sigma^{\ast}, \gamma^{\ast})$. Additionally, at each iteration of the reduced run, we generate, $\Theta_{1}^{(m)} \sim q(\Theta_{1}^{\ast}, \Theta_{1}|y,\beta^{(m)},\Omega^{(m)}, \alpha^{(m)})$. The draws $\{ \beta^{(m)}, \Omega^{(m)}, \alpha^{(m)}, \Theta_{1}^{(m)} \}$ obtained from such a procedure are draws from $\pi(\beta,\Omega,\alpha|y,\Theta_{1}^{\ast}) \times q(\Theta_{1}^{\ast},\Theta_{1}|y)$ which can be utilized to compute the denominator. Therefore, an estimate of the posterior ordinate, $\pi(\Theta_{1}^{\ast}|y)$, can be obtained as,
where $M$ and $M_{1}$ denote the number of MCMC draws from the complete and (first) reduced MCMC runs.
Next, we need to estimate $\pi(\beta^{\ast}|y,\Theta_{1}^{\ast})$ and $\pi(\Omega^{\ast}|y,\beta^{\ast},\Theta_{1}^{\ast})$. We already have the sequence of draws $\{ \beta^{(m)}, \Omega^{(m)}, \alpha^{(m)}, \nu^{(m)}, h^{(m)} \}_{m=1}^{M_{1}}$ from the reduced MCMC run conditioned on $\Theta_{1}^{\ast}$. These draws are utilized to estimate $\pi(\beta^{\ast}|y,\Theta_{1}^{\ast}) = \int \pi(\beta^{\ast} |y, \Theta_{1}^{\ast}, \alpha, \Omega, \nu, h) \, \pi( \alpha, \Omega, \nu, h|y,\Theta_{1}^{\ast}) \, d\alpha \, d\Omega \, d\nu \, dh, $ as follows,
To estimate $\pi(\Omega^{\ast}|y,\beta^{\ast},\Theta_{1}^{\ast})$, we conduct a second reduced MCMC run, i.e., run Algorithm (ref) for $M_{2}$ iterations with $(\beta, \Theta_{1})$ fixed at $(\beta^{\ast}, \Theta_{1}^{\ast})$. The resulting draws $\{ \alpha^{(m)}, \Omega^{(m)}, \nu^{(m)}, h^{(m)} \}_{m=1}^{M_{2}}$ are utilized to estimate $\pi(\Omega^{\ast}|y, \beta^{\ast}, \Theta_{1}^{\ast}) = \int \pi(\Omega^{\ast}| y, \beta^{\ast}, \Theta_{1}^{\ast}, \alpha, \nu, h) \, \pi( \alpha, \nu, h|y,\beta^{\ast}, \Theta_{1}^{\ast})\, d\alpha \, d\nu \, dh,$ as given by,
Substituting the expression from Equations (ref)-(ref) in Equation (ref), we have an estimate of the joint posterior ordinate $\pi(\beta^{\ast}, \Omega^{\ast}, \Theta_{1}^{\ast}|y)$.
The other quantities in the marginal likelihood (see Equation (ref)) are prior ordinates and the likelihood of the FREQ model. Both quantities require straightforward evaluations. All the prior distributions are completely known (see Equation (ref)), so prior ordinates can be easily evaluated at a chosen high-density point $\Theta^{\ast} = (\beta^{\ast}, \Omega^{\ast}, \sigma^{\ast}, \gamma^{\ast})$. The likelihood also requires evaluation at $\Theta^{\ast}$, but we first need to express it marginally of $(\alpha, \nu, h)$ since we marginalized them while computing the joint posterior ordinate. The required FREQ model likelihood can be written as,
where $f_{GAL}$ denotes the density of GAL distribution. The likelihood can be computed at $\Theta^{\ast} = (\beta^{\ast}, \Omega^{\ast}, \sigma^{\ast}, \gamma^{\ast})$ using Monte Carlo integration as follows,
where $\{\alpha_{i}^{(j)}\}$ are draws from $f(\alpha_{i}|\Omega^{\ast})$ for $i=1,\cdots,n$, and $J$ is some large number. Additionally, $(\nu,h)$ are automatically marginalized since $f_{GAL}(\cdot)$ is the GAL density that does not involve any mixture variables Rahman-Karnawat-2019.
The derivation of the marginal likelihood for REQ model follows Chib-1995 since all the conditional posteriors have a known form (see Algorithm (ref)). Let $\Theta = (\beta, \sigma, \Omega)$, then the joint posterior (marginally of $\alpha$ and $\nu$) can be expressed as,
where the $\ast$ on the parameters denotes a high-density point. Each expression on the right-hand side of Equation (ref) can be written in terms of the conditional posteriors (see Algorithm (ref)) and an estimate is obtained by taking the ergodic average of the conditional posterior density with MCMC draws either from the complete or reduced runs.
The posterior density $\pi(\beta^{\ast}|y)$ is expressed as $\pi(\beta^{\ast}|y) = \int \pi(\beta^{\ast}|y,\nu, \sigma, \Omega) \, \pi(\nu,\sigma,\Omega|y) \, d\nu \, d\sigma \, d\Omega$ and its estimate is computed as $ \hat{\pi}(\beta^{\ast}|y) = G^{-1} \sum_{g=1}^{G} \pi(\beta^{\ast}|y,\nu^{(g)},\sigma^{(g)}, \Omega^{(g)})$, where the G draws are from the complete MCMC run. The remaining two terms are reduced conditional density ordinates and require MCMC draws from two separate reduced runs. To obtain an estimate of $\pi(\Omega^{\ast}|y,\beta^{\ast}) = \int \pi(\Omega^{\ast}|y,\beta^{\ast},\sigma,\alpha, \nu) \, \pi(\sigma,\alpha,\nu|y,\beta^{\ast}) \, d\sigma \, d\alpha \, d\nu$, we conduct a (first) reduced run, i.e., run Algorithm (ref) for $G_{1}$ iterations with $\beta$ fixed at $\beta^{\ast}$. We then compute an estimate of the ordinate as $ \hat{\pi}(\Omega^{\ast}|y,\beta^{\ast}) = G_{1}^{-1} \sum_{g=1}^{G_{1}} \pi(\Omega^{\ast}|y,\beta^{\ast},\sigma^{(g)},\alpha^{(g)},\nu^{(g)})$. Finally, an estimate of the third term $\pi(\sigma^{\ast}|y, \beta^{\ast}, \Omega^{\ast}) = \int \pi(\sigma^{\ast}|y, \beta^{\ast}, \Omega^{\ast}, \alpha, \nu) \, \pi(\alpha,\nu|y,\beta^{\ast},\Omega^{\ast}) \, d\alpha \, d\nu$, is obtained as $ \hat{\pi}(\sigma^{\ast}|y,\beta^{\ast}, \Omega^{\ast}) = G_{2}^{-1} \sum_{g=1}^{G_{2}} \pi(\sigma^{\ast}|y,\beta^{\ast},\Omega^{\ast}, \alpha^{(g)},\nu^{(g)})$, where the $G_{2}$ Gibbs draws are from the second reduced run of Algorithm (ref) with $(\beta, \Omega)$ fixed at $(\beta^{\ast}, \Omega^{\ast})$.
With an estimate of the joint posterior ordinate now available, we need to compute the prior ordinates and the likelihood to estimate the marginal likelihood for REQ model. The prior ordinates are readily available since the prior distributions for ($\beta, \sigma, \Omega$) have tractable forms. The likelihood is calculated marginally of ($\alpha, \nu$) since we marginalized them while computing the joint posterior ordinate. The required likelihood can be written as,
where $f_{AL}$ denotes the density of the AL distribution. The above expression can be computed using Monte Carlo integration at $\Theta^{\ast}=(\beta^{\ast},\sigma^{\ast},\Omega^{\ast})$ as follows,
where $\{\alpha_{i}^{(j)}\}$ are draws from $f(\alpha_{i}|\Omega^{\ast})$ for $i=1,\cdots,n$, with $J$ being a large number. Note that $\nu$ is automatically marginalized by virtue of not using the mixture representation of AL distribution in the likelihood.
In this section, we conduct multiple simulation studies to illustrate the performance of the proposed MCMC algorithm for estimating the FREQ model, estimated using Algorithm (ref), and compare the results to those from the REQ model, estimated using Algorithm (ref). Moreover, we compute the marginal likelihood for both the models to examine the benefits (i.e., better model fitting), if any, of the FREQ model vis-a-vis the REQ model. The data for the simulation studies are generated from the following panel data model,
where $\alpha = (\alpha_{1}, \alpha_{2})' \sim N_{2}( [0,0]', [1, 0; 0, 1])$, $\beta = (\beta_{1}, \beta_{2}, \beta_{3})' = (10, 5, 2 )'$, $z_{2it} \sim \mathrm{Unif}(1,3)$, $x_{2} \sim N(0,0.25)$, $x_{3} \sim N(2,0.25)$, and the errors $\varepsilon$ were generated from a standard logistic distribution $\mathcal{L}(0,1)$. We generate 9 different data samples with $T_{i}= (5, 10, 15)$ and $n=(100,250,500)$, where $T_{i}$ denotes the number of repeated observations for each individual $i$ and $n$ represents the number of individuals.
In each simulation study, the posterior estimates of the parameters in the FREQ model are obtained based on the simulated data and the following prior distributions: $\beta \sim N(0_{k}, 100 I_{k})$, $\Omega \sim IW(\omega_{0}, O_{0})$, $\sigma \sim IG(5/2,8/2)$, and $\gamma \sim \mathrm{Unif}(L,U)$. Here, $\omega_{0}=5+l$, $O_{0}= (\omega_{0} - l-1)\ast I_{l}$, and $(L,U)$ are obtained as mentioned in Section (ref). The same prior distributions are employed for the REQ model. Table (ref) reports, for each simulated dataset, the MCMC results at five different quantiles obtained from 10,000 iterations after a burn-in of 2,500 iterations. Inefficiency factors are calculated using the formula, $ 1 + 2 \sum_{t=1}^{T} \rho_{k}(t) \bigg( \frac{T-t}{T} \bigg), $ where $\rho_{k}(t)$ denotes the autocorrelation for the $k$th parameter at lag $t$, and $T$ is the value at which the autocorrelations taper off (typically, 0.05 or 0.10). In the MH sampling of $(\sigma, \gamma)$, the tuning factor $\iota$ is adjusted to obtain an acceptance rate of approximately 30 percent. Convergence of the MCMC draws is quick, as demonstrated in the trace plots for the 25th quantile from Simulation Study 1 in Figure (ref). The trace plots for the remaining quantiles in Simulation Study 1 and all quantiles in the other 8 simulation studies are similar and reflect quick convergence to the joint posterior distribution.
The results, presented in Table (ref), show that the posterior estimates of the regression coefficients $\beta$ are close to the true values $(10,5,2)$ with small standard deviations, and hence the algorithm is successful in recovering the true values of the parameters. Inefficiency factors are low, which indicate that the MCMC draws mix well and the proposed algorithm is efficient. This pattern is observed at all quantiles and across simulation studies. For the scale parameter $\sigma$, the posterior estimates vary with quantiles, have small standard deviations, and low inefficiency factors. The posterior estimates of the components of covariance matrix (i.e., $\Omega_{11}, \Omega_{22}$, and $\Omega_{12}$) are close to the true values used to generate the data. However, inefficiency factors for these parameters tend to be higher as compared to other model parameters indicating slower mixing. But, this is not unusual and have earlier been reported in the literature Chib-Jeliazkov-2006. Such slow mixing occurs because $\Omega$ is at second level of modeling hierarchy and depends on the data only via $\{ \alpha_{i}\}$, the random effects parameter.
The posterior estimates of the shape parameter $\gamma$ suggest that the GAL distribution allows considerable flexibility in skewness while modeling the data relative to the AL distribution. For example, in Simulation Study 9, the posterior mean of $\gamma$ at the 25th (75th) quantile is 1.20 ($-1.28$) which corresponds to a skewness of $-0.06$ ($0.13$). This is in contrast to the fixed skewness of $1.64$ and $-1.64$ in the REQ model. Additionally, the posterior mean at the 50th quantile is $-0.00$ with a high standard deviation ($0.02$), which implies zero skewness. In summary, all estimates of $\gamma$ suggest a skewness that is much closer to the skewness of underlying data generating process.
The above discussion clearly establishes the flexibility of FREQ model vis-a-vis the REQ model. However, a more pertinent question is what do we gain from this flexibility? In particular, does the flexibility translate to a better model fit that may justify the additional effort required to estimate the FREQ model? To answer this question, we also estimate the REQ model using Algorithm (ref), compute the log-marginal likelihoods for the two models, and report them in Table (ref). Looking at the results from Simulation Study 9, we see that the log-marginal likelihood is higher for the FREQ model at the 10th, 25th, 75th, and 90th quantiles. This lends evidence that the FREQ model is better supported by the data. Moreover, the difference between log-marginal likelihoods of the FREQ and REQ models is higher at the 10th or 90th quantiles as compared to 25th or 75th quantiles, which suggests that the gains from flexibility increase as we move towards the tail of the distribution. At 50th quantile, the log-marginal likelihood is higher for the REQ framework, but here a direct comparison of marginal likelihoods may be misleading. This is because the posterior mean of $\gamma$ is statistically equivalent to zero, thus pointing to a zero skewness framework, i.e., the REQ model. Therefore, both frameworks are rather equivalent at the 50th quantile. Across the simulation studies, we observe that the FREQ framework better fits the data at non-50th quantiles. Whereas, for the 50th quantile, the two models are equivalent. In conclusion, the FREQ model tends to be a better model than its REQ counterpart at quantiles away from the median. The appropriate model (AL vs. GAL) will be application- and data-specific. However, in order for a researcher to have a more thorough understanding of the data distribution, both approaches should be considered and compared in a model comparison analysis. We conduct this exercise in the next section where we study residential rental rates in the US.
The US housing sector has received considerable attention as a result of the Global Financial Crisis (GFC) when mortgage delinquencies and foreclosures increased. Homeownership rates fell from 69% in 2004 to 62% in 2016. Because of the nexus between house prices and residential rents (see Loewenstein-Willen for a recent study), a drop in homeownership is likely to increase demand for rental units. However, the GFC also featured a drop in household income and an increase in unemployment.\footnote{Aggregate figures for mortgage delinquencies, unemployment, household income, and homeownership are available on FRED.} The concurrence of these events, theoretically, leads to ambiguous effects on rental rates. In this application, we take an empirical approach and explore changes to residential rental rates in the post-GFC United States. In particular, we examine how median rental prices in 14,533 zip codes are influenced by unemployment rates and mortgage policies.
In studying rental rates, two concerns must be addressed: (1) heterogeneity because regions of the US vary greatly by housing supply, income, and other factors that influence prices, and (2) skewness in the distribution of prices because that distribution has a large right tail and significant outliers. Exhibiting these concerns, Figure (ref) presents box plots of median rental rates in 5 states and the entire US for 2010 and 2016. Apparent from the figure are the dramatic differences across the states, where California's lower quartile is above most other states' upper quartile. Additionally, all states display a strong upper (right) skew. Thus, we employ our FREQ model to accommodate heterogeneity and allow for skewness flexibility in the error term. The performance of the FREQ model is tested relative to the REQ model using our novel marginal likelihood approach. This exercise provides important insights for understanding how the data support the different specifications across various quantiles.
There is a vast literature on house prices and rental rates both before and after the Global Financial Crisis. Studies have examined various price determinants including zoning, regulation, and housing supply Glaeser-etal-2005, Jackson2018, income differences Quigley-Raphael-2004, and tax policy Chatterjee-Eyignungor-2015. Many of these studies highlight how heterogeneity in city-specific features, such as average income and land availability, can lead to discrepancies in the effect of the boom and bust on prices. Additionally, a few international papers have focused on the distribution or quantiles of rental prices Thomschke-2015, Marz-etal-2016, Waltl2018. We contribute to this literature by implementing a novel quantile regression approach to study rental rates in the United States. Moving beyond mean regression and exploiting large differences in population, income, and economic activity provides a deeper understanding of the determinants of rental markets. Further, our new methodology and model comparison approaches allow us to uncover potential biases that may result from ignored heterogeneity and erroneous distributional assumptions.
We construct a novel zip-code-level data set where our outcome variable of interest, $y_{it}$, is the median monthly rental price of zip code $i$ at year $t$. The sample includes $n=14,533$ zip codes in the United States from 2010-2016 ($T=7$). The residential rental price data come from the Zillow Rental Index (ZRENT). Our covariates include annual controls for each zip code's population, demographics, socioeconomic status, agriculture, property ownership, mortgage characteristics, and unemployment. The covariates are constructed from the Statistics of Income (SOI) Tax Stats, Individual Income Tax Statistics, provided by the Internal Revenue Service (IRS). Table (ref) presents descriptions and summary statistics of our variables. We proxy for “population” using the total number of tax returns filed in the zip code and the remaining variables are generally a function of that measure.
We model the data using the FREQ and REQ models, where $y_{it}= LnRent_{it}$, $x_{it}$ includes the remaining variables in Table (ref), and $z_{it}$ is a constant to control for zip-code-level heterogeneity. Heterogeneity is an important concern. While a researcher can control for a host of demographic, socioeconomic, and location characteristics, much is left unobserved. City-level policies, nearby neighborhood spillovers, and commuting effects may enter the error term, which heavily influence rental prices. Thus, our specifications for both models include zip code random effects. Additionally, we include time dummies to capture aggregate changes to prices.
Prior distributions play an important role in Bayesian inference, particularly in model comparison where marginal likelihoods and hence their ratios, the Bayes factors, become arbitrary with improper priors, or sensitive to the prior with formally proper, but increasingly diffuse priors. Therefore, we employ a training sample approach where we take 10% of our data as a training sample and retain the remainder as a comparison sample. The data in the training sample are used to construct a first-stage posterior distribution which is used as a proper informative training sample prior when analyzing the comparison sample. Information from the training sample is not lost, as it is now part of the prior density used in evaluating the marginal likelihood over the remaining 90% of the data.\footnote{When estimating the model on the training sample, we used the following relatively uninformative priors: $\beta \sim N(0_{k}, 25 I_{k})$, $\varphi^{2} \sim IG(12/2, 10/2)$, $\sigma \sim IG(10/2,8/2)$ and $\gamma \sim \mathrm{Unif}(L,U)$, where $(L,U)$ are obtained as mentioned in Section (ref).}
Before getting to the parameter estimates, we bring attention to the marginal likelihood results. We compare the performance of the FREQ model, relative to the REQ model, for the full sample of US zip codes, as well as several state-specific models (Arizona, California, and Illinois). We consider these smaller samples of states to empirically explore the model fit when $n$ is smaller and when there are varying degrees of heterogeneity and skewness in the sample. Table (ref) presents the log-marginal likelihood estimates for the four samples across five quantiles. We find that the FREQ model has a higher marginal likelihood than the REQ model in all samples at the 10th, 25th, 75th, and 90th quantiles. The differences, particularly further in the tails, are quite dramatic, giving the FREQ model a posterior model probability of $\approx 1$ over the REQ model. At the 50th quantile, we find that the REQ is the favored model in all samples except California. In fact, at the 50th quantile, $\gamma$ is statistically equivalent to 0 in the FREQ model (for all states except California), implying the AL parameterization is more appropriate. Overall, these results demonstrate strong support from the data for the additional flexibility of the FREQ model. Researchers especially interested in the tail of their distribution of interest should employ this more flexible approach in their applied work to improve model fit.
Turning attention to the parameter estimates, Table (ref) presents the results for the FREQ model and Table (ref) presents the results for the REQ model. We find that unemployment is positively associated with residential rental rates. All else equal, a 1 percentage point increase in the fraction of the population that receives unemployment compensation is associated with an increase in monthly residential rental rates of 0.33% (at the 50th quantile). In a 2012 speech to the National Association of Home Builders, Ben Bernanke, then Chairman of the Federal Reserve, stated that “High unemployment and uncertain job prospects may have reduced the willingness of some households to commit to homeownership.” By not committing to homeownership, individuals shift their preferences toward renting. The increase in demand for rental units puts upward pressure on prices, explaining the positive result. We find that the effect gets incrementally larger at higher quantiles, i.e., regions of the US that are more expensive. As an economy recovers from a crisis, attention should be paid to the price of rental units. Policymakers may want to focus on limiting the upward pressure on these prices as individuals and families may have a more difficult time recovering from the economic downturn if rents increase.
We also find negative effects from our home mortgage variable (HMrate). Thus, the fraction of the population taking home mortgage tax deductions is negatively associated with rental prices. The ability to deduct mortgage interest on individual income taxes makes homeownership more attractive than renting. This result is important in light of the Tax Cuts and Jobs Act (TCJA), which was signed into law in the United States in 2017. The Act lowered the mortgage deduction limit and put a limit on how much an individual can subtract from their taxable income. Our model results suggest that this decrease in home mortgage deductions puts upward pressure on rental prices, a costly unintended consequence. Our results are in line with HembreDantas2022, who find that reductions in homeownership subsidies increase rental payments.
The other results in Tables (ref) and (ref) largely align with intuition. Specifically, we find that income is positively associated with rental rates. Average adjusted gross income has a positive effect across the quantiles and the fraction of the population paying alternative minimums (i.e., high-income taxpayers) is also positively associated with rental rates. Whereas, the fraction of the population claiming earned income tax credits (EITC), which represents low-income working individuals, is negatively associated with rental rates. Additionally, the year indicators, which are relative to 2010, are positive and get incrementally larger, capturing aggregate increases in prices.
In this section, we present the FREQ and REQ results when the sample is restricted to zip codes in Illinois. The model specifications remain the same as before. We chose Illinois (IL) for two reasons: (1) we wish to explore empirical parameter estimates in a smaller sample setting and (2) IL provides extensive variation in land value from expensive metropolitan regions (e.g., Chicago) to rural, farming areas. Table (ref) presents the FREQ results and Table (ref) presents the REQ results. Importantly, recall from Table (ref) that the data support the FREQ model over the REQ model at all quantiles except the 50th.
In looking at the results for the 10th quantile, a major discrepancy between the FREQ and REQ is apparent. The REQ model results suggest that unemployment compensation is negatively associated with residential rental prices. Meaning, in regions of IL that are inexpensive (10th quantile), an increase in the fraction of the population receiving unemployment compensation should decrease rental rates. However, the FREQ model results suggest that the effect of unemployment is not statistically different from zero (i.e., unemployment has no effect on rental prices). In considering the marginal likelihood results (Table (ref)), we know that the posterior model probability of the FREQ model is approximately 1, relative to the REQ model, demonstrating that the data overwhelmingly support the specification with the flexibility in the skewness of the error provided by the GAL distribution. Thus, the results of the FREQ model are validated, whereas those of the REQ model are negated.
This example demonstrates the dangers of ignoring skewness in the error distribution. Had a researcher or policymaker solely considered a model with the AL distributional assumption (which is commonly done), they would have arrived at an erroneous conclusion about the relationship between unemployment and rental rates. We caution against this approach and instead motivate researchers to use model comparison to uncover the best model and to especially consider the GAL approach at higher and lower quantiles, where the benefits are most dramatic.
This article has considered the Bayesian analysis of a random effects quantile regression model for panel data under the generalized asymmetric Laplace distribution, which eliminates the dependence of distributional skewness on the quantile parameter. New computationally efficient MCMC sampling algorithms have been developed for parameter estimation, as well as model comparison, in both the FREQ and REQ versions of the model. Key to the improved properties of our posterior simulator is the idea of carefully designed parameter blocking. Various features of the proposed modeling framework and estimation methodology have been studied in simulation studies.
The paper has also devoted considerable attention to studying the behavior of U.S. residential rental rates following the Global Financial Crisis. Our methodology fits this purpose very well due to the strong right skew of rental rates and the extensive heterogeneity at the zip-code level across different regions. Our results reveal that unemployment has positive effects on rental rates and mortgage deductions have negative effects. Regions of the U.S. characterized by high unemployment also exhibited declines in homeownership, leading to an increase in demand for rental units and putting upward pressure on prices. The negative effect of mortgage deductions sheds light on the unintended consequences of the Tax Cuts and Jobs Act (TCJA) as a potential contributor to the large increases in rental prices since 2017.
Based on our model comparisons, we find that the data overwhelmingly support the FREQ model in various subsamples and at all quantiles, especially away from the median, suggesting that researchers interested in the tails of the distribution could find the more flexible GAL modeling framework decidedly more useful.