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\@startsection{section}{1} \z@{1.0\linespacing\@plus\linespacing}{.8\linespacing}{Introduction} The control function approach is commonly used in empirical economic studies to cope with endogenous explanatory variables heckman1985control,wooldridge2015control. This approach typically involves two stages: the first stage projects the endogenous explanatory variables to a set of instruments and other exogenous variables and obtains the residuals; these residuals serve as extra explanatory variables, also known as control functions when estimating the structural equations in the second stage. In this paper, we propose a quasi-Bayesian method that combines a frequentist-type first-stage estimation with some Bayesian approach in the second stage. The study is motivated by structural discrete choice models that apply the control function method to correct for endogeneity bias. In such scenarios, second-stage estimation remains challenging for frequentist methods but can be more conveniently addressed by Bayesian procedures, as the latter transform the difficult integration and/or optimization problem into a sampling problem of posterior distributions. Therefore, the proposed quasi-Bayesian approach offers researchers great flexibility in combining nonparametric frequentist methods with computationally convenient Bayesian approaches. With a clear understanding of the theoretical foundation regarding the asymptotic behavior of the posterior obtained from the second stage--referred to as the quasi-posterior, since it uses frequentist estimates from the first stage--this new procedure can be particularly useful in econometric models characterized by complex likelihood functions and endogeneity issues.
To formalize the idea, we consider independent and identically distributed ($i.i.d.$) observations ${\bm Y}^n=(Y_i,i=1,\cdots,n)^\top$. The component $\zeta_0$ represents a vector of unknown functions that one needs to estimate from the first-stage in order to form the control functions. Throughout the paper, we assume that the true $\zeta_0$ is identifiable and can be estimated from the subvector ${\bm Y}^n_1=(Y_{1i},i=1,\cdots,n)^\top$ that consists of the observables from the first stage. We denote the corresponding estimator by $\widehat{\zeta}_n$. The second stage postulates a probability density function, $p_{\theta}^n(\cdot;\zeta)$ for the subvector ${\bm Y}^n_2=(Y_{2i},i=1,\cdots,n)^\top$ of observables with the unknown parameter $\theta\in\Theta\subset \mathbb{R}^p$, for some integer $p\geq 1$. The unique truth is denoted by $\theta_0$. The Bayesian procedure in the second stage posits a prior distribution $\Pi(\theta)$ and then updates it to the quasi-posterior distribution given the limited-information likelihood\footnote{The current approach is based on the “limited information" strategy in the sense that the information contained in both stages (or equations) are not simultaneously considered.} in the second stage. Using Bayes' rule pragmatically, the quasi-posterior of the finite-dimensional parameter $\theta$ is
for any given measurable set $\mathcal{A}\subset \Theta$. We refer to the estimation/inferential procedure based on the above quasi-posterior as quasi-Bayesian, because the term $p^n_{\theta}({\bm Y}_2^n;\widehat{\zeta}_n)$ is not a genuine likelihood and it depends on some plug-in estimator $\widehat{\zeta}_n$. Two key questions arise: (1). Is the quasi-posterior distribution of $\theta$ asymptotically Gaussian? (2). What are the center and dispersion of this quasi-posterior distribution? To our best knowledge, such problems have not been investigated in the literature even when the estimator $\widehat{\zeta}_n$ is estimated parametrically. This paper fills this void and allows for general first-stage nonparametric estimation.
We investigate the asymptotic properties of this quasi-Bayesian procedure from a frequentist perspective. The main challenge lies in the fact that one encounters a nonstandard posterior distribution that depends on the first-stage estimation. We show that the credible set formed by the quantiles of the quasi-posterior does not achieve correct coverage in large samples, as it fails to account for the estimation uncertainty from the first stage. Although one can not directly rely on the credible set for inference, the quasi-posterior still contains useful information. In particular, we prove that the quasi-Bayesian point estimator, such as the quasi-posterior mean, is asymptotically equivalent to a frequentist two-stage estimator; that is, it is root-$n$ asymptotically normal with the same sandwich-type covariance matrix as its frequentist counterpart. In essence, the quasi-posterior distribution is correctly centered but has incorrect dispersion. We also show that a bootstrap method, when appropriately applied to the quasi-Bayesian approach, yields confidence intervals with the desired coverage probability and is therefore valid for inference.
To illustrate our theory, we focus on an endogenous multinomial Probit (MNP) model from petrin2010control, which we refer to as the Petrin--Train model in the sequel. The framework is popular in modeling consumer choices over different products, say choosing from $j\in\{0,\cdots, J\}$, with the $0$-th alternative being the null category (non-purchase option). The endogenous variables are usually the prices in the context. To address the endogeneity problem, researchers often construct the Hausman-type price instruments hausman1994diff. The first stage involves the estimation of a vector of $(\zeta_j)_{j=1}^J$, as the conditional mean functions of the endogenous variables given the exogenous covariates and instrument variables. Given the estimated $\widehat{\zeta}_n=(\widehat{\zeta}_{n,j})_{j=1}^J$, we can extract the residuals as the control functions for the second-stage MNP model. In this scenario, the likelihood function $p^n_{\theta}({\bm Y}_2^n;\widehat{\zeta}_n)$ is the product of the conditional choice probabilities that involve complicated multiple integrals. As reviewed in train2009discrete, the Bayesian approach enjoys practical advantages over frequentist methods in estimating the MNP-type models. This is also confirmed in our context both with respect to the Monte Carlo simulation as well as the empirical study.
Going beyond the Petrin-Train model, our study reflects two generic features of modern econometric models. On one hand, many models used to describe observed data may be so complex that the likelihoods associated with these models are computationally intractable. These analytical difficulties can be alleviated by simulation-based procedures, such as the Bayesian method, which has proven successful in many areas. On the other hand, the control function approach provides an effective solution to deal with the endogeneity problem. This approach treats endogeneity as an omitted variable bias problem, in which the inclusion of estimates of the first-stage errors as additional covariates corrects the inconsistency of the second stage. The idea of combining a frequentist first stage and a Bayesian second stage has been practiced in empirical research on estimating the distribution of students' preferences for public schools agarwal2018demand, though lacking a formal theoretical investigation for such a “hybrid” or quasi-Bayesian procedure. agarwal2018demand also suggest a bootstrap method for inference, which inspires our current proposal. Our study sheds theoretical light on the asymptotic normality of the two-stage quasi-Bayesian estimator and the bootstrap validity therein.
There is a clear advantage of our proposal over the frequentist two-step method or the Bayesian full information paradigm in the current context, because it separates two tasks: one can correct for the endogeneity bias via robust nonparametric first-stage estimation, while tackling the complicated likelihood in the second stage by a Bayesian approach. This separation of tasks allows for considerable algorithmic flexibility. Regarding the frequentist two-step approach, the second stage requires the simulated method of moments mcfadden1989sme,hajivassiliou1998scores, which calls for Monte Carlo simulation, in addition to solving hard optimization problems for finding the frequentist point estimators. Given the complicated form of the influence function, it might be a daunting task to compute the correction term or the standard error analytically. When it comes to inference, one has to rely on bootstrap for the frequentist approach too. Referring to the full information Bayesian approach, one would focus on the posterior that is proportional to $\tilde{p}^n_{\theta,\zeta}({\bm Y}^n)d\Pi(\theta)dQ(\zeta)$, where $ \tilde{p}^n_{\theta,\zeta}({\bm Y}^n)$ is the density function for the joint distribution of $\bm{Y}^n_1$ and $\bm{Y}^n_2$ combining two stages together, and $Q(\cdot)$ is a suitable prior on the functional component $\zeta$. The joint likelihood inevitably requires a full model encompassing both stages, for which the applied researcher may be reluctant to hypothesize such a structure murphy2002two. If the researcher does not want to assume a parametric model for $\zeta$, the full-information Bayes must put a nonparametric prior $Q(\zeta)$ on it. Although the non-parametric Bayesian inference is one of the most vibrant research areas in econometric theory, verification of the related Bernstein--von Mises (BvM) theorem either relies on the Gaussian processes priors or explores particular model structures; see ghosal2017bayesian. In addition, designing a feasible algorithm for the posteriors remains a skillful task when the joint likelihood functions from both stages are combined.
\@startsection{subsection}{2} \z@{.8\linespacing\@plus.7\linespacing}{.7\linespacing}{Related Literature} Studying asymptotic properties of quasi-Bayesian procedures forms an important line of work in econometrics. Regarding the discrepancy between quasi-Bayesian credible sets and frequentist confidence sets, we highlight a recurring theme when the generalized information identity fails chernozhukov2003mcmc. In this case, the local asymptotic normality (LAN) expansion van1998asymptotic generates a centering term whose asymptotic variance is of the sandwich form that does not match the (minus) second derivative matrix of the (quasi) log-likelihood function. These phenomena appear in studies of the generalized method of moments (GMM) or the limited information approach kim2002limited,chernozhukov2003mcmc,chib2018moment, as well as in misspecified models kleijn2012mis,muller2013risk,kim2014quasi. Given the natural link between GMM and two-step estimation newey1994large, it is tempting to think that our framework is nested by the works mentioned above. This is not the case. We emphasize that one distinct feature of our problem is that the first-stage estimation is based on a frequentist method and then plugged into the posterior of the second stage. One can view this as approximating the posterior of $\zeta$ using the Dirac delta function. Such feature violates the standard assumptions in proving the BvM theorem.
Another line of research closely related to this paper is the profile sampler lee2005profile,cheng2008profile. This literature recommends sampling from the profile likelihood $pl_n(\theta)\equiv \sup_{\zeta}p^n_{\theta}({\bm Y}^n;\zeta)$ for semiparametric models indexed by $(\theta,\zeta)$. The infinite-dimensional parameter $\zeta$ therein must be estimated via the profile maximum likelihood so that there is no additional adjustment term. See related discussions on the profiling strategy to eliminate nuisance parameters/functions in the last paragraph on page 1357 of newey1994var. Such procedures are not required in our setup. The crux of our analysis is to characterize the first-stage estimation effect on the quasi-posterior. On the technical ground, this paper also refines the proof of lee2005profile and cheng2008profile: we show the asymptotic negligibility of the posterior outside a shrinking neighborhood of the truth, where the radius depends on the convergence rate of first-stage estimation. Our proof relaxes the convergence rate requirement to match the well-known order of $o(n^{-1/4})$ in general semiparametric models newey1994var\footnote{In comparison, the result of Lemma A.1 of lee2005profile is stated for some neighborhood with a fixed radius, yet later on, this lemma is cited to cover the case with a shrinking radius of the order $o(n^{-1/3})$.}. The proof of our BvM theorem also differs from existing literature from two aspects. First, we need to take into account the first-stage nonparametric estimation when we carefully partition the integral range into the dominating one and negligible parts. Second, despite the semiparametric nature of the problem, we can avoid checking the prior stability condition in the semiparametric BvM literature ghosal2017bayesian, because we simply estimate the nonparametric element rather than assigning certain nonparametric prior to it.
\@startsection{subsection}{2} \z@{.8\linespacing\@plus.7\linespacing}{.7\linespacing}{Organization} The rest of our paper is organized as follows. Section (ref) contains the main theoretical results. This relatively long section is divided into three subsections, in which we introduce necessary assumptions, investigate the asymptotic behavior of the quasi-posterior, and show the validity of bootstrap methods for inference. Section (ref) applies the general theory to the Petrin--Train model. We present low-level regularity conditions and state the asymptotic results. Section (ref) conducts Monte Carlo simulations and provides an empirical illustration. The last section concludes. Proofs of the main results are presented in Appendix A. The supplementary material contains several Appendices. Technical lemmas and their proofs are collected in Appendix B. The formulation of the BvM theorem given parametric first-stage estimation is presented in Appendix C. In Appendix D, we provide details of the influence function for a special case of the Petrin--Train model. Appendix E offers additional simulation evidence comparing our bootstrap proposal with another method.
\@startsection{subsection}{2} \z@{.8\linespacing\@plus.7\linespacing}{.7\linespacing}{Notations} For a function $f(\cdot)$ of a random vector $Y$ that follows distribution $P$, we use the standard empirical process notations: $P_0f=\int f(y)dP_0(y),\mathbb{P}_n f=n^{-1}\sum_{i=1}^{n}f(Y_i)$, and $\mathbb{G}_n f=n^{1/2}\left(\mathbb{P}_n-P_0\right)f$. We also write $\mathbb{E}_0$ instead of $P_0$ in bounding center stochastic terms. Definitions of the $P_0$-Glivenko-Cantelli class and the $P_0$-Donsker class follow those in van1996empirical and van1998asymptotic. We denote the bootstrap empirical measure as $\mathbb{P}_n^*$ and the bootstrap empirical process as $\mathbb{G}^*_n=\sqrt{n}(\mathbb{P}_n^*-\mathbb{P}_n)$. For a sequence of random variables $Z^*_n$, we write $Z^*_n=o_{P^*}(1)$ if the law of $Z_n^*$ is governed by the bootstrap law $\mathbb{P}^*$ and if $\mathbb{P}^*(|Z_n^*|>\epsilon)=o_{P_0}(1)$ for any $\epsilon>0$.
For any $p$-dimensional vector $\nu$, we write $\nu^{\otimes 2}\equiv \nu\nu^\top$. For any $\alpha\in \mathbb{R}$, let $\lfloor \tau\rfloor$ be its integer part. Considering a multi-index $k=(k_1,\cdots,k_d)$, define $k_{.}=k_1+k_2+\cdots+k_d$. For any $L>0,\tau\geq 0$ and non-negative function $L$ on $\mathbb{R}^d$, define the $\tau$-H\"older class with envelope $L$, denoted by $\mathcal{C}^{\tau,L}(\mathbb{R}^d)$ (Section 3 of ichimura2010semi) to be the set of all functions $f:\mathbb{R}^d\mapsto\mathbb{R}$ that have finite mixed partial derivatives $D^kf$ of all orders up to $k_{.}\leq \lfloor \tau\rfloor$,
For any real vector $b$, let $\Vert b\Vert_E$ denote the Euclidean norm of $b$. For any $J$-dimensional vector of functions $\bm{h}(x)$ with its $j$-th component equal to $h_j(x)$, we let
where $\Vert h_j\Vert_{\sup}\equiv \sup_{x\in\mathcal X} |h_j(x)|$, and $\mathcal{X}$ denotes its support.
\@startsection{section}{1} \z@{1.0\linespacing\@plus\linespacing}{.8\linespacing}{Main Theoretical Results} Bayesian methodology is attractive in its own right. Nonetheless, our problem is fundamentally non-Bayesian, as researchers aim to estimate unknown fixed parameters using the control function approach to correct the endogeneity bias. This motivates us to investigate the large sample behavior of the resulting quasi-posterior under a fixed true probability model $P_0=P_{\theta_0,\zeta_0}$ that generates the observed data. A thorough understanding will provide solid justification for the quasi-Bayesian methods, which can be attractive to non-Bayesian practitioners because of their convenience. Our main use of the quasi-posterior is to interpret it as a frequentist's device, similar to that of a sampling distribution. This is crucial to justify whether the resulting credible set can be used as a confidence set asymptotically, in view of the first stage estimation uncertainty.
\@startsection{subsection}{2} \z@{.8\linespacing\@plus.7\linespacing}{.7\linespacing}{Assumptions and Discussions} The (limited-information) log-likelihood function is of fundamental importance in the second-stage Bayesian formulation. Throughout the paper, we focus on the case with i.i.d. data, so the likelihood function becomes $p_{\theta}^n(\bm{Y}_2^n;\zeta)=\prod_{i=1}^n p_{\theta}(Y_{2i};\zeta)$ and the log-likelihood is denoted by $\ell_n(\theta;\zeta)\equiv\log p_{\theta}^n(\bm{Y}_2^n;\zeta)=\sum_{i=1}^n\log p_{\theta}(Y_{2i};\zeta)$. We define the following first-order derivative with respect to (w.r.t.) the finite-dimensional parameter $\theta$:
where $\theta$ is partitioned into the parameter of interest $\beta$ and the nuisance parameter $\eta$. The metric associated with the parameter $\theta$ is $d_{\Theta}(\cdot,\cdot)$. The second-order derivative is denoted by $\ddot{l}_{\theta}\left( Y_2,\theta;\zeta\right)$ in the same vein. Thereafter, the score function is $S_{\theta}\left( \theta,\zeta\right) =P_0\dot{l}_{\theta}\left( Y_2,\theta;\zeta\right)$, with the corresponding empirical version as $S_{\theta,n}\left( \theta,\zeta\right) =\mathbb{P}_n\dot{l}_{\theta}\left( Y_2,\theta;\zeta\right)$. The negative Hessian matrix $V_0=-P_0\ddot{l}_{\theta}\left( Y_2,\theta_0;\zeta_0\right)$ is the information matrix for the second-stage likelihood, when $\zeta_0$ is known. The unknown function in the first stage may contain $J$ components, so we write $\zeta_0=(\zeta_{0j})_{j=1}^J$. Consider a map defined by $\Psi(\theta,\zeta)\equiv S_{\theta}(\theta,\zeta)$. The following pathwise derivative of $\Psi$ plays a key role in determining the effect of the first-stage estimation newey1994var:
where each $\dot{\Psi}_{\zeta_j}$ is a bounded linear functional that maps $\zeta_j-\zeta_{0j}$ to the real line. The proper functional space for $\zeta$ is denoted by $\mathcal{G}$ with its pseudo metric $d_G(\cdot,\cdot)$.
We list all regularity conditions followed by some heuristic discussions in order. They are not necessarily the weakest possible assumptions.
Assumptions (ref) to (ref) are generic, concerning the identifiability of the model parameters, the asymptotics of the first stage nonparametric estimation, as well as the existence of a frequentist two-stage estimator. Considering the convergence rate requirement in Assumption (ref), we explicitly work with nonparametric first stage estimation. Given this convergence rate, the set $\mathcal{G}_n$ allows us to localize the analysis around the truth related to the $P_0$-Donsker properties in Assumption (ref). In addition, one may also impose the shape constraint or perform proper trimming to further regularize the problem. The parametric case is easier and we formulate the necessary changes in the supplementary material. The frequentist two-stage estimator $\widehat{\theta}_n$ in Assumption (ref) merely serves as a theoretical device, as it will become the centering point in the local asymptotic normal (LAN) expansion ghosh2002bayesian,chernozhukov2003mcmc,lee2005profile. In our context of analyzing structural discrete choice models, $\widehat{\theta}_n$ is difficult to compute, which motivates the quasi-Bayesian procedure in the first place. Assumption (ref) on the prior density is also standard ghosh2002bayesian,chernozhukov2003mcmc.
Compared with the standard conditions for the BvM theorem van1998asymptotic,ghosal2017bayesian, our setting involves two complications due to the presence of the estimated control function. The first task is to ensure that the posterior concentrates in any small neighborhood of the true parameter uniformly over a set $\mathcal{G}_n$ to which the estimated $\widehat{\zeta}_n$ belongs. The second distinction is that one needs an adjustment term in the LAN expansion, which accounts for the first-stage estimation error. Assumption (ref) (i) is needed to ensure that the likelihood ratio $\prod_{i=1}^n[p_{\theta}(Y_{2i};\widehat{\zeta}_n)/p_{\widehat{\theta}_n}(Y_{2i};\widehat{\zeta}_n)]$ decays exponentially fast for $\{\theta:d_{\Theta}(\theta,\theta_0)>\delta\}$ with a fixed radius $\delta>0$ as in Lemma S1. Meanwhile, Assumptions (ref) and (ref) (iii) are used to strengthen the exponential type decay for $\{\theta:Cr_n\leq d_{\Theta}(\theta,\theta_0)\leq\delta\}$, with the rate $r_n$ stated in Assumption (ref). The differentiability condition in Assumption (ref) with respect to $\theta$ and $\zeta$ are needed to account for the two stages in our quasi-Bayesian method. Finally, Assumption (ref) is used to establish the quadratic expansion of the log-likelihood over the range $\{\theta: d_{\Theta}(\theta,\theta_0)\leq Cr_n\}$ in Lemma S3 of the supplementary material. Therein we also require Assumption (ref) (ii) to kill various smaller order terms via the stochastic equicontinuity. We also need Assumption (ref), when studying the frequentist coverage property of the quasi-Bayesian credible set. \@startsection{subsection}{2} \z@{.8\linespacing\@plus.7\linespacing}{.7\linespacing}{Large Sample Behaviors of Quasi-Posteriors} We develop the asymptotic theory by drawing on two themes. The first is the frequentist two-step estimation and inference newey1994var,chen2003estimation,ichimura2010semi. The crux therein is how to characterize the influence of the first-step nonparametric estimation on the remaining parametric components and how to make proper adjustments for inference. This analysis is also central to our study, and we formally show how the quasi-posterior ignores the effect from the first-stage estimation. The second theme is the asymptotic analysis of Bayesian methods for semiparametric models. Our theoretical findings have not been reported before; they shed new light on the BvM theorem van1998asymptotic,ghosal2017bayesian. For standard semiparametric models using full information Bayesian methods, the BvM theorem states that the marginal posterior for the finite-dimensional parameter is approximately a normal distribution centered at the semiparametric efficient estimator. Hence, the point estimators and credible sets are produced by one stroke therein. However, estimation and inference have to be dealt with separately for the quasi-posterior distribution in our setting.
Let $\tilde{\pi}$ be the density function of $h\equiv \sqrt{n}(\theta-\widehat{\theta}_n)$ conditional on the data $\bm{Y}_2^n$ and the first stage estimate $\widehat{\zeta}_n$:
Furthermore, the limiting normal density is
In order to metrize the weak convergence, we consider the following “total variation of moments” norm chernozhukov2003mcmc for a real-valued function $f$ on $\mathcal{S}$ as
for the choice of $a\geq 0$ in Assumption (ref). Let $V_0^{1/2}$ denote the square-root of the positive definite matrix $V_0$.
Theorem (ref) may look similar to the standard parametric BvM theorem at first glance. However, the departures are the presence of the first-stage estimation $\widehat{\zeta}_n$ and the centering point $\widehat{\theta}_n$. The latter one is a two-stage frequentist estimator whose asymptotic covariance matrix takes the sandwich form $V_0^{-1}\Omega_0V_0^{-1}$, with $\Omega_0$ corresponding to the variance of $\Gamma_2(Y_{2i})+\Gamma_1(Y_{1i})$ in Assumption (ref). This has important implication about the frequentist coverage property of the credible set.
Let $\mathcal{C}_n(\alpha)$ be the quasi-Bayesian credible set constructed from quantiles of the quasi-posterior such that $\Pi(\theta\in \mathcal{C}_n(\alpha)|{\bm Y}^n_2;\widehat{\zeta}_n)=1-\alpha$ for a given nominal level $\alpha\in(0,1)$. An important consequence of Theorem (ref) is that $\mathcal{C}_n(\alpha)$ does not have the correct asymptotic coverage probability in general, as stated in Corollary (ref). Recall the definition of $\Delta_{n,0}$ in Assumption (ref).
Referring to Corollary (ref), the variance of the quasi-posterior of $\theta$ is governed by the information matrix $V_0$, which only captures the variance of $\Gamma_2(Y_{2i})$. If the nuisance function is known, or we plug in $\zeta_0$ rather than $\widehat{\zeta}_n$, then the Bayesian approach for the second stage would give us
The centering point is the infeasible MLE based on $\zeta_0$ such that:
Given equations ((ref)) and ((ref)), one can easily show the frequentist validity if $\zeta_0$ is known exactly. For the quasi-Bayesian approach that replaces $\zeta_0$ with $\widehat{\zeta}_n$, its centering point in ((ref)) is the frequentist two-stage estimator such that:
Referring to the statement in ((ref)), we can see that the discrepancy between $\Omega_0$ and $V_0$ prevents the right hand side of ((ref)) from converging to the desired coverage probability $1-\alpha$ in general.
With the additional restriction $V_0=\Omega_0$, which corresponds to the case where the generalized information equality holds chernozhukov2003mcmc, $\mathcal{C}_n(\alpha)$ will have the correct coverage probability coincidentally, as ((ref)) shows. In our analysis of the endogenous discrete choices model in Section (ref), this occurs in the absence of endogeneity.\footnote{Appendix D calculates the influence from the first-stage estimation for the Petrin--Train model with three alternatives, see equations (S.1) to (S.3).} Intuitively, this holds as the likelihood function of the second stage is free from the first stage control variables without the endogeneity. We believe the scenario is rather special, where one does not need the control function approach from the beginning. The general coverage failure of the quasi-Bayesian credible set is similar to what kleijn2012mis found when they studied the Bayesian procedure for misspecified models. Considering the formal decision theory of the interval estimation problem, muller2013risk showed that the asymptotic risk of a vanilla posterior is worse than that of a modified posterior using the sandwich covariance matrix. Also, see kim2014quasi for the asymptotic theory of the posterior odds ratio for testing hypotheses with the limited-formation Bayesian approach.
We are interested in whether the mean of this quasi-posterior is a legitimate point estimator for $\theta_0$. For this purpose, we define the quasi-posterior mean and covariance matrix as follows:
Below, we show that the quasi-posterior mean is asymptotically equivalent to the frequentist two-stage estimator $\widehat{\theta}_n$. However, the quasi-posterior variance is not the same as the asymptotic variance of the frequentist two-stage estimator.
\@startsection{subsection}{2} \z@{.8\linespacing\@plus.7\linespacing}{.7\linespacing}{Validity of Bootstrap Inference} This section applies nonparametric bootstrap to our quasi-Bayesian method. The resulting inferential procedure yields asymptotically valid confidence intervals and avoids analytical correction for the first stage estimation error needed to be carried out case-by-case. We show that the bootstrap point estimator mimics the asymptotic behavior of the quasi-Bayesian point estimator. The essence of this analysis lies in the bootstrap likelihood with multinomial weights. The technical challenge is to deal with the infinite-dimensional $\zeta$ from the first stage. Our theory also connects the bootstrap procedure suggested by agarwal2018demand to a more general context with two-stage semiparametric estimation. In principle, the theory works for other exchangeable weights van1996empirical. However, the most convenient approach is Efron's nonparametric bootstrap efron1979bootstrap, because it only requires generating a sequence of new data sets by sampling the rows of the original data with replacement and then updating the posterior with the same prior. It is straightforward to compute all bootstrap calculations in parallel.
Let the bootstrap weights $\bm{M}_{n}=\left( M_{n1},...,M_{nn}\right)^{\top}$ follow the multinomial distribution $Mult\left( n,\left(n^{-1},...,n^{-1}\right) \right)$ efron1979bootstrap. The resulting marginal posterior distribution is given by the Bayes formula:
where $p^{n*}_{\theta}({\bm Y}_2^n;\zeta)\equiv\prod_{i=1}^n p_{\theta}^{M_{ni}}(Y_{2i};\zeta) $ stands for the bootstrap likelihood function and $\widehat{\zeta}^*_n$ stands for the first-stage bootstrap estimator. The bootstrap quasi-Bayesian point estimator is
Also, denote by $\widehat{\theta}_n^*$ the bootstrap counterpart of the frequentist two-stage estimator in Assumption (ref). Algorithm (ref) below provides the implementation details.
Figure (ref) provides a graphical illustration of Algorithm (ref) and compares it to the quasi-Bayesian credible set $\mathcal{C}_n(\alpha)$ described in Section (ref). The histogram and density plots in the figure are drawn based on one simulated sample in Section (ref). We emphasize that it is crucial to bootstrap the Bayesian point estimator (which is taken to be the quasi-posterior mean) in each bootstrap replication in order to ensure the frequentist validity of the inference procedure. In the supplementary material, we compare our proposal with an alternative procedure that only makes a single draw from the quasi-posterior in each bootstrap sample.
Let $\theta_{0,j}$ be the $j$th coordinate of the parameter $\theta_0$. The $1-\alpha$ bootstrap percentile interval for $\theta_{0,j}$ can be formed as
where $Q^*_{n,j}(a)$ denotes the $a$-th quantile of the bootstrap distribution $\left\{\widetilde{\theta}^*_{n,b,j} :b=1,\ldots,B\right\}$. Besides the bootstrap percentile interval, we also consider a percentile-$t$ interval by bootstrapping a $t$-ratio. For the parameter $\theta_j$, calculate the standard error $\widetilde s_{n,j}$ as the square root of the $j$-th diagonal element of the quasi-posterior variance matrix $n^{-1}\widetilde{\mathrm{Var}}_n(\theta)$. Corollary (ref) implies that $\widetilde s_{n,j}\to_{P_0}\sqrt{(V_0^{-1})_{jj}}$. We introduce the following bootstrap percentile-$t$ interval:
where $q_{n,j}^{\ast}(a)$ denotes the $a$-th quantile of the bootstrap distribution of the $t$-ratio: \[ \left\{ (\widetilde{\theta}_{n,b,j}^*-\widehat{\theta}_{n,j})/\widetilde{s}^*_{n,b,j} :b=1,\cdots,B\right\}. \] Although the $t$-ratio studentized by the quasi-posterior standard error would not make its limiting distribution equal to the standard normal, the resulting bootstrap percentile-$t$ interval in ((ref)) is asymptotic distribution, see Section 10.19 in hansen2022econometrics for a related discussion for bootstrapping frequentist two-stage estimators.\footnote{We thank one anonymous referee for making this suggestion.}
\@startsection{section}{1} \z@{1.0\linespacing\@plus\linespacing}{.8\linespacing}{The Petrin--Train Model} We illustrate our theory in Section (ref) by applying it to a class of endogenous discrete choice models originally proposed by petrin2010control. We revisit Example 1 in petrin2010control with normal latent errors, which does not enforce the independence of irrelevant alternatives (IIA) assumption in Logit models. The joint normality of the latent errors in both stages facilitates application of the control function approach. We relax the first-stage specification in petrin2010control to allow nonparametric estimation. We henceforth refer to this model as the Petrin-Train model throughout the paper.
As demonstrated below, Petrin-Train model has two features: (I) the conditional choice probability and thus the likelihood function contain complicated multi-dimensional integrals; (II) it involves the pilot estimation of (functional) nuisance parameters. Many econometric models share similar structures. For instance, agarwal2018demand estimate the distribution of students' preferences for public schools by first estimating the believed assignment probabilities at various schools using frequentist methods and then conducting Bayesian estimation in the second stage. Their second-stage likelihood involves a multiple integral similar to the MNP, albeit with fairly complicated decision rules in their context. Our analysis of the Petrin-Train model can serve as a template for studying the more complex choices models.
Let $U_{ij}$ for $j=0,1,\cdots,J$ denote the latent utility of alternative $j$ for individual $i$, with the following linear specification:
The individual $i$'s choice $C_i\in\{0,1,\cdots,J\}$ is determined by maximizing the utility across $(J+1)$ alternatives. We treat the choice $0$ as the baseline and work with utility differences as follows:
In the sequel, we denote
with $U^{\dagger}_{ij}=U_{ij}-U_{i0}$, $X^{\dagger}_{ij}=X_{ij}-X_{i0}$ and $\varepsilon^{\dagger}_{ij}=\varepsilon_{ij}-\varepsilon_{i0}$. Hence, the utility maximization leads to the following decision rule:
Among the choice-specific covariates $X^{\dagger}_{ij}$, some components denoted by $X_{ij}^{\dagger e} $ may be correlated with the error term $\varepsilon^{\dagger}_{ij}$. The endogenous regressors $X^{\dagger e}_{ij}$ admit the following representation\footnote{For notation simplicity, we work with a scalar endogenous variable $X_{ij}^{\dagger e}$ so that the range of each function $\zeta_j(\cdot)$ is the real line.}:
where $Z_{ij}$ collects the instrumental variables and other exogenous regressors, and $v^{\dagger}_{ij}$ stands for the unobservable error that is correlated with $\varepsilon^{\dagger}_{ij}$ The functions $(\zeta_j)_{j=1}^J$ are completely unspecified, except for some standard smoothness restrictions as detailed later. Following Example 1 of petrin2010control, the joint normality of latent errors $\varepsilon^{\dagger}_{ij}$ and $v^{\dagger}_{ij}$ leads to the decomposition of $\varepsilon^{\dagger}_{ij}$:
where $\lambda_{0}v^{\dagger}_{ij}$ is the control function\footnote{In principle, we can allow for different $\lambda_{0,j}$ over $j=1,\cdots,J$. In this case, the model can be written in the form of $U^{\dagger}_{ij}=(X^{\dagger}_{ij})^\top\beta_0+ (\tilde{v}_{ij}^\dagger)^{\top}\lambda_0+\epsilon^{\dagger}_{ij}$, by generating proper longer vectors of $\tilde{v}_{ij}^\dagger$ and $\lambda^\top_0=(\lambda_{0,j})_{j=1,\cdots,J}$. See Chapter 6 of lee2010micro for the general formulation. For ease of exposition, we do not consider such complications.}, and the the reminder errors $(\epsilon^{\dagger}_{ij})_{j=1}^J$ are also jointly normal. Using ((ref)), the latent utility of the second stage can be written as:
The conditional choice probability for the $j$-th alternative in the second stage involves a multivariate integral:
The function $G$ is a $J$-dimensional normal distribution with mean zero and covariance matrix $\Sigma_0$. For identification purpose, we assume that $\Sigma_0$ can be parameterized as $\Sigma_0=\Gamma(\eta_0) $ which depends on some given transformation $\Gamma(\cdot)$ and unknown parameter $\eta_0$. For example, one can take $(\Sigma_0)_{jk}=\eta^{|j-k|}_0$ for the $(j,k)$-th element in $\Sigma_0$. We collect the regression coefficients as $\beta^\top_0=(\tilde\beta^\top_0,\lambda_0)$ as the parameter of interest and treat $\eta_0$ as nuisance parameter.
Our quasi-Bayesian method applied to the Petrin--Train model consists of two stages. The first stage estimates the functions $\zeta_j$ nonparametrically (e.g., kernel regression) from equation ((ref)) and then obtains the residuals $(\hat v^{\dagger}_{ij})_{j=1}^J$. The second stage corresponds to the MNP model ((ref)) with $(v^{\dagger}_{ij})_{j=1}^J$ replaced by the first stage residual estimates. As conditional choice probability $P_0\left(C_i=j\mid X_i,Z_i\right)$ is analytically intractable, Bayesian approach becomes more appealing as it explores the conditional conjugate structure induced by MNP. The posterior of $\beta$ can be drawn using Markov Chain Monte Carlo (MCMC) algorithms coupled with data augmentation techniques albert1993bayesian,mcculloch1994exact,nobile1998hybrid,imai2005bayesian. Chapter 5 of train2009discrete presents a comprehensive discussion about the advantages of Bayesian methods for MNP-type models. We also refer interested readers to anceschi2023conjugacy for a recent review on the contemporary development of various fast and scalable MCMC or deterministic algorithms. We conduct the bootstrap inference following Algorithm (ref). Note that the second-stage Bayesian procedure does not require maximization of any function. This reduces the computational burden compared to bootstrapping a frequentist two-stage estimator, whose second stage involves maximization of the simulated likelihood function associated with the MNP model. The computational advantage of the Bayesian second-stage over a frequentist second-stage is amplified when bootstrap resampling is called for valid inference. Our simulation exercise in Section (ref) illustrates that the quasi-Bayesian method can save the computation time to a large extent.
In the following, we present low level conditions that are sufficient to specialize Theorems (ref) and (ref) to the Petrin--Train model. We introduce notation: denote the collection of vectors across non-baseline choices by ${\bm U}^{\dagger}_i= [U^{\dagger}_{i1},..., U^{\dagger}_{iJ}]^\top$. We write
where $W_{ij}^\top=((X^{\dagger}_{ij})^\top,v^{\dagger}_{ij})$. Let $R(C_i)$ specify the truncation region associated with the choice variable. If $C_i=0$, $R(C_i)$ consists of the region such that each component of the latent utility is non-positive. For $C_i=j\neq 0$, $R(C_i)$ restricts $U^{\dagger}_{ij}$ to be positive and greater than all other $U^{\dagger}_{ik}, k\neq j$. Referring to the expression ((ref)) with $\theta=(\beta^\top, \eta^\top)^\top$, the score functions w.r.t. $\beta$ and $\eta$ take the following forms:
cf. equations (4) and (5) of hajivassiliou1996simulation. Clearly, the score functions also involve multivariate integrals, which makes the analytical correction for the first-stage estimation error difficult. In Assumption (ref) below, $\mathbb{W}_{J}$ denotes the Wishart distribution for $J\times J$ positive-definite random matrices.
The following proposition specializes Theorems (ref) and (ref) to the Petrin--Train Model. It establishes the asymptotic normality of the quasi-Bayesian point estimator and the asymptotic validity of the bootstrap confidence intervals.
We provide heuristic discussions about how Assumptions (ref) to (ref) imply the high-level assumptions in Section (ref). We separate the discussion for the first and the second stages.
Regarding the first stage, we need to properly control the complexity of the underlying functional class which contains the $\zeta_0$. Herein, we work with the H\"older class. The desired Glivenko-Cantelli or Donsker properties are satisfied with sufficient smoothness restrictions (relative to the dimensionality of first-stage regressors) in Assumption (ref). For nonparametric estimation in the first stage, one may use kernel smoothing estimators, including the local constant or local polynomial estimators in chen2003estimation,ichimura2010semi, or sieve estimators in chen2007sieve. For example, if we estimate $\zeta_{0j}$ by local constant kernel estimator with the kernel function $K(\cdot)$ and bandwidth $h^{d_z}$, the linear representation in Assumption (ref) is
The control of bias $b_{n,j}$ or remainder terms $R_{n,j}$ are also standard; see Equation (3.14) and the discussion in ichimura2010semi.
Referring to the second-stage likelihood structure, the high-level conditions mainly rely on the differentiability conditions, which are indeed satisfied for the MNP. Assumption (ref) provides more primitive conditions for showing the Glivenko-Cantelli and Donsker properties in Assumption (ref). In Section S3 of the supplementary material, we provide detailed expressions of $g_j(\cdot,\theta)$ in Assumption (ref) in the case of three alternatives ($J+1=3$), which capture the influence from first-stage estimation. Indeed, the required moment or smoothness condition is easily satisfied under mild regularity conditions, such as the compact support of covariates. Nonetheless, the expression itself is lengthy, which again advocates bootstrap for inference. Given the expression of $g_j(\cdot,\theta)$, the term $\Gamma_1(\cdot)$ appearing in the influence function in Assumption (ref) can be written as $\Gamma_1(\cdot)=\sum_{j=1}^J v^{\dagger}_{ij}\mathbb{E}_0[\partial g_j(\cdot,\theta_0)/\partial \theta|Z_{ij}]$.\footnote{This is a special case of the expression (3.15) from ichimura2010semi, because the conditional mean regression in the first stage does not depend on $\theta$ from the second stage.} Regarding the prior choice in Assumption (ref), one can also impose the restriction on the eigenvalue rather than the first diagonal element as in imai2005bayesian. Examples of other proper normalizations for the unrestricted or restricted covariance can be found in train2009discrete.
Next, we draws readers' attention to several important restrictions of our current exposition. Namely, our asymptotic results are built on the point identification, smooth criterion functions in the second stage, and the i.i.d. data structure.
\@startsection{section}{1} \z@{1.0\linespacing\@plus\linespacing}{.8\linespacing}{Monte Carlo Simulation} We conduct Monte Carlo simulations for the Petrin--Train Model. Simulation results confirm that the quasi-Bayesian credible set does not have desirable coverage probabilities, and this problem can be solved by using bootstrapping the quasi-posterior mean or $t$-ratio. Our simulation design is a multinomial choice model with three or four choices and one endogenous regressor for each choice. For each alternative, the latent utility level takes the following form:
The true value of $\tilde\beta$ is $1$. The endogenous regressor $X^e_{ij}$ depends on $\xi_{ij}$ that is excluded from equation ((ref)):
Observables $\xi_{i0},.., \xi_{iJ} $ and error terms $v_{i0},..., v_{iJ}$ in the first stage ((ref)) are independent standard normal. The error term $\varepsilon_{ij}$ in the latent utility ((ref)) is generated by $\varepsilon_{ij}=\lambda v_{ij}+\epsilon_{ij}$ for $j=0,1,\cdots,J$, where $\epsilon_{ij}$ are jointly normal with means equal to zero, and variances $\sigma_{0}^2=1$, $\sigma_{j}^2= \sigma^2= 0.75$ for $j=1,\cdots J$. In the case of three choices ($J=2$), the correlation coefficient for any pair of $\epsilon_{ij}$ is $corr(\epsilon_{ij},\epsilon_{is})=\rho$ for $j\neq s$. In the case of four choices ($J=3$), $corr(\epsilon_{i0},\epsilon_{i1})=corr(\epsilon_{i2},\epsilon_{i3})=\rho$, and $corr(\epsilon_{ij},\epsilon_{is})=0$ for other $(j,s)$ pairs. We set $\rho=\sigma/2$ and $\lambda=0.6$. The latent utility relative to the null category becomes
and the endogenous regressors can be written as
with the instrument $Z_{ij}=(\xi_{ij},\xi_{i0})$ and the conditional mean function $\zeta_j(Z_{ij})=\tau(\xi_{ij})-\tau(\xi_{i0})$. Substituting the control function $v_{ij}$ into the (differenced) utility equation ((ref)) leads to
where $\epsilon^{\dagger}_{ij}= \epsilon_{ij}-\epsilon_{i0}$ is independent of $X^{\dagger e}_{ij}$ and $v_{ij}^{\dagger}$.
We consider two designs for the functional form for $\tau(\cdot)$ in ((ref)): Design I with $\tau(z)= 0.9z + 0.9z^2 + \ln(z+1)^2$ and Design II with $\tau(z)=0.9z+0.9z^2+\exp(0.9z)$. Our first stage estimates the model ((ref)) using a kernel regression\footnote{Alternatively, if one proceeds with the differenced endogenous variable in equation ((ref)) directly, one can estimate a nonparametric additive model with regressors $(\xi_{ij},\xi_{i0})$.} with the bandwidth chosen by leave-one-out cross-validation and obtains the residual $\widehat v_{ij}$. This step is implemented by the R package $\mathtt{np}$ hayfield2008np. Our second stage estimates the coefficient $\beta=(\tilde\beta,\lambda)^\top$ in the MNP model ((ref)), after substituting $ v_{ij}^{\dagger}$ by its estimator $\widehat v^{\dagger}_{ij}$ obtained in the first stage. The Bayesian second stage draws the quasi-posterior of $\beta$ using a Gibbs sampler with data augmentation imai2005bayesian, which is implemented by the R package $\mathtt{MNP}$ imai2005mnp.
Table (ref) presents the empirical coverage probabilities and the average lengths of the confidence intervals for the scalar parameter $\tilde\beta$ produced by different inference procedures: QB stands for the quasi-Bayesian credible set $\mathcal{C}_n(\alpha)$ discussed in Corollary 2. QBB and QBB-$t$ represent the quasi-Bayesian bootstrap percentile and percentile-$t$ intervals given by Algorithm (ref), (2.12) and (2.13). For comparison, we also consider a frequentist two-stage method that has the same first stage as our quasi-Bayesian method, but employs simulated maximum likelihood estimation for the second stage ((ref)), which is implemented by the R package $\mathtt{mlogit}$ croissant2020estimation.\footnote{Setting the argument “$\mathtt{probit = TRUE}$” when calling the function $\mathtt{mlogit}$ estimates a multinomial probit model.} The resulting frequentist bootstrap percentile (or percentile-$t$) interval is denoted as F2B (or F2B-$t$). The sample size is $1000$. The number of bootstrap replications is $500$, and the number of simulation replications is $1000$. Table (ref) reveals that the quasi-Bayesian credible interval (QB) systematically under-covers the true parameter $\tilde\beta$ in all cases. Bootstrap quasi-Bayesian intervals (QBB and QBB-$t$) significantly improve the coverage performance and yield coverage probabilities close to nominal levels. Since QBB-$t$ is studentized using the posterior standard error that does not account for the first stage uncertainty, it is not expected to have better coverage performance than QBB -- a fact also confirmed by simulation evidence. We also observe that bootstrap confidence intervals QBB and QBB-$t$ are longer than the credible interval QB, as bootstrap resampling incorporates estimation uncertainty from the first stage.
As Table (ref) shows, the frequentist bootstrap confidence intervals F2B and F2B-$t$--especially the former--also have good coverage performance in finite samples. The second stage of the frequentist approach uses a simulated maximum likelihood estimator and is computationally much slower than the Bayesian second stage. For example, to construct a confidence interval in the case of three alternatives ($J+1=3$) and Design I, quasi-Bayesian bootstrap intervals QBB and QBB-$t$ on average take about $130$ seconds per simulation replication, while the frequentist counterparts F2B and F2B-$t$ take more than $4400$ seconds. When it comes to four alternatives ($J+1=4$) and Design I, QBB and QBB-$t$ take about $170$ seconds to compute whereas F2B and F2B-$t$ take about $8700$ seconds. This confirms the computational advantage of using a Bayesian second stage to handle complicated likelihood functions derived from structural models, especially when the bootstrap is used for inference. All codes are run using R version 4.4.1 (2024-06-14) on a desktop machine (3.8GHz 8-Core Intel Core i7, 40GB Memory) with macOS system (x86 64-apple-darwin20). When simulating the posterior, the initial $2000$ Gibbs draws are discarded, and the following $2000$ draws are stored.
The quasi-Bayesian method in Table (ref) imposes a non-informative prior distribution on the parameter $\tilde\beta$. Figure (ref) illustrates how the coverage performance is affected by the variance of the prior distribution. We consider the prior variance for $\tilde\beta$ equal to $100$, $1,000$, and $+\infty$ ($+\infty$ corresponds to the non-informative prior used in Table (ref)). The prior mean for $\tilde\beta$ is set as zero in all cases. Figure (ref) shows that the under-coverage of QB and the improvement achieved by QBB are robust to the variance of the prior.
\@startsection{section}{1} \z@{1.0\linespacing\@plus\linespacing}{.8\linespacing}{Empirical Application} We apply the quasi-Bayesian approach to investigate U.S. firms' incorporation decisions, initially studied by eldar2020regulatory. This research concerns about how the corporate law at the state level affects firms' choice of locations to incorporate their business, in particular, whether firms prefer to incorporate in states where the corporate laws are more friendly to share-holders or more protective of managers. In the U.S, two states dominate the incorporation market: in year 2013, $63.9\%$ of firms were incorporated in Delaware, and $8.5\%$ in Nevada. Interestingly, these two dominating states have quite different legal characteristics: Delaware's laws are more market-oriented and facilitate takeover activities, while Nevada's laws provide more protection for managers' interests. In the dataset, each state's legal characteristics are measured by three indices: ATS counts the number of anti-takeover statutes, resulting in a score from $0$ to $5$; DIR (or OFF) measures the extent to which state laws protect directors (or officers) from liability by providing them exemption and indemnification. The DIR index takes a value from $0.5$ to $6$ and the OFF index ranges from $0.5$ to $5$. We use the data in year 2013, which contains 2,922 firms. Let $i$ denote the firm and $j$ denote the incorporation location, we consider an MNP model with three alternatives: Delaware ($j=1$), Nevada ($j=2$), and other states ($j=0$). A firm's latent utility is expressed as:
where we consider three specifications for the regressor $Law_j$ for the location alternative $j$: $Law_j=ATS_j$, $DIR_j,$ or $OFF_j$, respectively. Other regressors are firm-level characteristics and are the same for all specifications: $Inst.~Owner_i$ represents the share of institutional ownership of firm $i$, and $Small_i$ is a dummy variable for small-sized firms. The concern about endogeneity arises from the variable $Inst.Owner$. Following eldar2020regulatory, we use a dummy instrumental variable $S\&P500$ which indicates whether the firm is included in the S&P 500 index. Inclusion in the S&P 500 index is positively correlated with a firm's institutional ownership. On the other hand, the S&P 500 is mainly decided by the market's view about the firm's representativeness and thus does not depend on the firm's decision. The first-stage regression model is
where $\tau$ is an unknown function. The control function takes the form $\widehat{v}_{ij}=ATS_j\times \hat{\nu}_i$. Our first stage estimates the function $\tau(\cdot)$ in ((ref)) by a local-constant kernel regression where the bandwidths are selected by the leave-one-out cross-validation. Including $\widehat{v}_{ij}$ as an additional regressor, the Bayesian second stage draws the quasi-posteriors of $\tilde\beta_1$, $\tilde\beta_2$ and $\tilde\beta_3$, upon which the point estimates and $90\%$ bootstrap confidence intervals QBB and QBB-$t$ are constructed in the same way as the simulation exercise in Section (ref). Results are presented in Table (ref) with the percentile and percentile-$t$ intervals reported in brackets and parentheses, respectively. For comparison, Table (ref) also includes the frequentist two-stage estimator and its bootstrap intervals F2B and F2B-$t$.
The negative estimates of $\tilde\beta_1$ in Table (ref) suggest that firms on average prefer less legal restrictions on the takeover and less protections on directors/officers, which explains why a majority of firms choose to incorporate in Delaware which has market oriented laws friendly to shareholders. On the other hand, a positive estimate of $\tilde\beta_3$ suggests that a firm's preference about corporate laws varies with respect to its size. Small firms are more likely to welcome the protectionist laws that restrict takeover activities and protects directors/offices from liability. This explains why Nevada, with $ATS=5$, occupies a notable market share for incorporation. Our results align with the findings of eldar2020regulatory, who estimated a multinomial Logit model. In Table (ref), the quasi-Bayesian and frequentist two-stage approaches generate very similar point estimates and confidence intervals, but the former is about 50 times faster. For one specification of the law variable, QBB and QBB-$t$ cost about 20 minutes while F2B and F2B-$t$ cost 16 hours.
\@startsection{section}{1} \z@{1.0\linespacing\@plus\linespacing}{.8\linespacing}{Conclusion} Our research highlights two key aspects of modern econometric models. First, the complexity of these models leads to analytically intractable likelihoods, which can be more conveniently dealt with by the Bayesian method. Second, the control function approach effectively handles the endogeneity problem by including the first-stage residuals as additional covariates, rectifying the second stage's inconsistency. Our study extensively investigates this novel quasi-Bayesian method. Given the widespread use of the Bayesian approach in contemporary econometrics and machine learning, our method provides practitioners with enhanced flexibility to integrate cutting-edge algorithms from various estimation stages. This methodology can be beneficial for other structural models when the underlying asymptotic theory is well understood.
\@startsection{section}{1} \z@{1.0\linespacing\@plus\linespacing}{.8\linespacing}{Appendix A: Proofs of Main Results} This appendix collects proofs for Theorems (ref), (ref), Corollary (ref), (ref), and Proposition (ref). Our proof of ((ref)) in Theorems (ref) is patterned in line with generic arguments in the proof of Theorem 1.4.2 from ghosh2002bayesian or Theorem 1 from chernozhukov2003mcmc. The crux is to address the first-stage nonparametric estimator $\widehat{\zeta}_n$. This is reflected in a different choice of splitting ranges of the integration and a more delicate expansion of the (quasi-)log-likelihood ratio in our proof.
Next, we prove the coverage property of the resulting quasi-Bayesian credible set, given the results in ((ref)) and ((ref)). We denote the $p$-dimensional standard normal measure of a generic set $A$ by
Recall the definition of $\Delta_{n,0}$ in Assumption (ref).