CAF - Estimating choice models with piecewise smooth objective functions. Application to joint retirement

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Estimating Choice Models with Piecewise

Smooth Objective Functions: Application to

Joint Retirement Tatiana Rosá1 | Siqi Wei 2 1CAF Development Bank. trosa@caf.com 2IE University:.siqi.wei@ie.edu We study choice models with piecewise smooth objective functions and provide conditions under which introducing latent variables derived from regional components yields a censoredmodel–like representation. These latent variables can then be treated as potential outcomes, enabling tractable estimation via the stochastic EM algorithm. We illustrate the framework using two examples: responses to taxes and the (S, s) model. We further estimate a joint retirement decision model for European couples using an interdependent duration framework with our methods. The results provide empirical evidence of complementarities between spouses in the retirement process in Europe. K E Y W O R D S Choice models, Piecewise smooth, Censored models, Latent variables, Joint retirement Small sections of text that are less than two paragraphs may be quoted without explicit permission as long as this document is acknowledged. Findings, interpretations and conclusions expressed in this publication are the sole responsibility of its author(s) and cannot be, in any way, attributed to CAF, its Executive Directors or the countries they represent. CAF does not guarantee the accuracy of the data included in this publication and is not, in any way, responsible for any consequences resulting from its use. ©2025 Corporación Andina de FomentoC A F - D O C U M E NT O D E T R AB A J O # 2 0 2 5 / 1 6

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Estimación de modelos de elección con funciones objetivo suavemente definidas por tramos: aplicación a la jubilación conjunta Tatiana Rosá1 | Siqi Wei 2 1CAF Development Bank. trosa@caf.com 2IE University:.siqi.wei@ie.edu Estudiamos modelos de elección con funciones objetivo suavemente definidas por tramos y presentamos condiciones bajo las cuales la introducción de variables latentes derivadas de componentes regionales produce una representación análoga a la de un modelo censurado. Estas variables latentes pueden tratarse como resultados potenciales, lo que permite una estimación tratable mediante el algoritmo EM estocástico. Ilustramos el marco con dos ejemplos: respuestas a impuestos y el modelo (S, s). Además, estimamos un modelo de decisión conjunta de jubilación para parejas europeas utilizando un marco de duraciones interdependientes con nuestros métodos. Los resultados aportan evidencia empírica de complementariedades entre cónyuges en el proceso de jubilación en Europa. K E Y W O R D S Modelos de elección, Suavidad por tramos, Modelos censurados, Variables latentes, Jubilación conjunta Pequeñas secciones del texto, menores a dos párrafos, pueden ser citadas sin autorización explícita siempre que se cite el presente documento. Los resultados, interpretaciones y conclusiones expresados en esta publicación son de exclusiva responsabilidad de su(s) autor(es), y de ninguna manera pueden ser atribuidos a CAF, a los miembros de su Directorio Ejecutivo o a los países que ellos representan. CAF no garantiza la exactitud de

los datos incluidos en esta publicación y no se hace responsable en ningún aspecto de las consecuencias que resulten de su utilización. ©2025 Corporación Andina de FomentoLANE ET AL. 2 1|INTRODUCTION Many structural models in applied economics are defined by optimization problems whose objective functions are piecewise smooth (Pudney, 1989). Such piecewise smooth features naturally arise from nonlinear budget constraints (such as kinked budget constraints) or nonsmooth operators embedded in the objective (such as max and min operators or fixed adjustment costs). These piecewise features are central in a broad set of applications– including taxation and bunching, durable (S, s) adjustment, and household or firm problems with nonconvexities–yet they complicate both solution and estimation. The difficulty is not only numerical: nondifferentiability makes the model challenging to solve. When the optimum occurs at a corner or at a boundary where multiple regions meet, the mapping from observed choices to unobservables can become one-to-many: the same observed choice may be consistent with a range of latent shocks. In that sense, these problems resemble censored or incomplete-data models, even when the original model is written as a standard choice problem. This feature complicates likelihood-based estimation, since the likelihood contribution of boundary observations typically requires integrating over the set of latent shocks compatible with the observed choice. A common practice is to solve the model region by region–obtaining local optima in each piece and selecting the global optimum–paired with simulation-based estimation methods such as indirect inference (Smith Jr, 1993; Gouriéroux and Monfort, 1997). However, this approach requires specifying an auxiliary model, and identification tends to be less transparent, with statistical and computational efficiency depending on the choice of auxiliary specification and optimization method. This paper develops an alternative route. We provide conditions under which a broad

class of optimization-based choice models with piecewise smooth objective functions can be re-expressed in a reduced form that resembles an artificial censored model. This censored representation brings the problem into the domain of incomplete-data models, allowing us to apply estimation tools such as the stochastic EM (SEM) algorithm. The key step is to introduce a finite set of well-defined latent variables that link the optimal choice to observables and unobservables. Concretely, these latent variables are constructed as (i) unconstrained global maximizers of the auxiliary functions (piecewise regional components) over theentiredomain and (ii) maximizers at the boundaries between regions. Under our conditions–most importantly, global unimodality of each smooth component over an appropriate open extension of its region–any optimal choice for the original piecewise objective must coincide with one element of this finite set. Moreover, the observed choice data allow the researcher to infer which element of the latent set is selected. Consequently, these introduced latent variables can be interpreted as "practical" potential options offered to each unit and they select the optimal one from the finite list. This transformation effectively reframes the original optimization problem into a potential outcome framework. On the one hand, from an optimization perspective, the transformation decomposes the global piecewise maximization into a finite comparison across a set of well-defined global and boundary solutions—rather than regional subproblems—providing an alternative representation of the optimization problem. More importantly, the potential outcome framework facilitates alternative estimation methods. We propose using the stochastic EM (SEM) algorithm (Dempster et al., 1977; Celeux et al., 1996), an established method for models with incomplete or censored data. Leveraging the potential outcome framework, SEM alternates between an E-step, where we draw values for the introduced latent variables from the posterior distribution given the observables, and an M-step, where we estimate the potential outcome framework withLANE ET AL. 3

the E-step draws, until the parameters converge to a stationary distribution. The algorithm replaces integration with tractable iterative updates and has the potential to achieve fast convergence when combined with acceleration techniques (Wei, 2024). Two illustrative examples are discussed: 1) responses to taxes and transfers under progressive income taxation (Saez, 2010), and 2) the (S, s) model for infrequent adjustment (Attanasio, 2000; Bover, 2010). This paper characterizes a class of optimization-based choice models with piecewise smooth objective functions that can be transformed into a reduced form resembling censored models. We develop an alternative solution method and propose the stochastic Expectation– Maximization (SEM) algorithm for their estimation. Piecewise smooth objective functions in choice models can arise from various factors, such as bounded choice sets (e.g., non-negativity restrictions), nonlinear constraints (e.g., progressive tax systems), or objective functions containing nonlinear expressions of the choice variable (e.g., max and min operators) (Pudney, 1989). Estimating such models, which often requires solving them, can be challenging due to the frequent occurrence of nonlinearity and non-differentiability, which not only complicate the solution method but also can lead to data censorship. A potential approach involves sequentially obtaining local optima in each region and then selecting the global optimum, paired with simulation-based estimation methods such as indirect inference (Smith Jr, 1993; Gouriéroux and Monfort, 1997). However, this approach requires specifying an auxiliary model, and identification tends to be less transparent, with statistical and computational efficiency depending on the choice of auxiliary specification and optimization method. In this paper, we provide conditions under which choice models with piecewise smooth objective functions can be transformed into a reduced form resembling an artificial censored model: while some choices are observed directly, there are censored outcomes consistent

with a range of error terms, creating a one-to-many relationship between observables and unobservables. This transformation not only provides an alternative solution method to the choice model but also facilitates the application of the SEM algorithm, which is particularly useful for handling incomplete data. The key to the transformation is the introduction of a set of well-defined latent variables that link the optimal choice to the relevant observables and unobservables. Specifically, the latent variables that we propose to introduce are the unconstrained global maximizers of the auxiliary functions (piecewise regional components) over theentiredomain, as well as their maximizers at the boundaries between regions. The definition of these latent variables depends on the auxiliary functions (e.g., through first-order conditions) and thus depends on other observables, unobservables, and parameters of interest. We show that when the auxiliary functions have a unique global maximizer with no local maxima over the entire domain, the optimal choice that maximizes the piecewise smooth objective function must correspond to one of the introduced latent variables. Moreover, researchers can infer which latent variable corresponds to the optimal choice based on the observed choice data. Consequently, these introduced latent variables can be interpreted as "practical" potential options offered to each unit and they select the optimal one from the finite list. This transformation effectively reframes the original optimization problem into a potential outcome framework. On the one hand, from an optimization perspective, the transformation decomposes the global piecewise maximization into a finite comparison across a set of well-defined global and boundary solutions—rather than regional subproblems—providing an alternative representation of the optimization problem. More importantly, the potential outcome framework facilitates alternative estimation methods. While the transformation applies generally, it is particularly useful in cases whereLANE ET AL. 4 some observed choices are consistent with a range of unobservables—typically arising

at corner or boundary solutions—creating a one-to-many mapping between choices and unobservables and thereby resembling a censored model. In such settings, direct estimation methods such as maximum likelihood require integrating over the entire feasible range of unobservables for censored observations, which is often computationally demanding or infeasible. In such cases, we propose using the stochastic Expectation–Maximization (SEM) algorithm (Dempster et al., 1977; Celeux et al., 1996), an established method for models with incomplete or censored data. Leveraging the potential outcome framework, SEM alternates between an E-step, where we draw values for the introduced latent variables from the posterior distribution given the observables, and an M-step, where we estimate the potential outcome framework with the E-step draws, until the parameters converge to a stationary distribution. The algorithm replaces integration with tractable iterative updates and has the potential to achieve fast convergence when combined with acceleration techniques (Wei, 2024). Two illustrative examples are discussed: 1) responses to taxes and transfers under progressive income taxation (Saez, 2010), and 2) the (S, s) model for infrequent adjustment (Attanasio, 2000; Bover, 2010). Finally, as an empirical application, we estimate the interdependent duration model of Honoré and de Paula (2018) with our method to study the joint retirement decisions among European couples. In this model, the retirement timing decision is determined through within-household Nash bargaining, considering potential complementarities in retirement decisions. The objective function of the bargaining problem has the piecewise feature, as the complementarity depends on the timing of the last person retiring, introducing a max operator into the objective function. This results in data "censoring" for jointly retiring couples, complicating both solving and estimating the model.

Applying the method, we introduce three pairs of well-defined latent variables as potential outcomes. The final choice of each household, observed by researchers, must correspond to one of these three options. This effectively transforms the original Nash bargaining problem into a potential outcome framework, which we leverage to develop the SEM estimation method. The proposed algorithm demonstrates numerical stability, satisfactory finite-sample performance, and computational efficiency. The model is estimated using a sample of 1969 couples from 19 countries in the Survey of Health, Ageing, and Retirement in Europe (SHARE) data. Results provide empirical evidence of the existence of complementarities in the retirement process among European couples. A simulation exercise shows that complementarity reduces the median retirement age by approximately 5 months for wives and 2 months for husbands. Literature. This paper broadly relates to two strands of literature. The first is the literature on choice-model estimation methods, including the textbook by Pudney (1989), the recent survey by Blomquist et al. (2023), and simulation-based approaches such as Smith Jr (1993), Gouriéroux and Monfort (1997), Celeux et al. (1996), and Dempster et al. (1977). Our main contribution is to connect a broad class of optimization-based choice models with kinks, corners, or nonconvexities—arising either from nonlinear budget constraints or from nonsmooth operators in the objective function—to censored latent-variable models. The equivalence clarifies how these models’ one-to-many mapping from observed choices to unobservables parallels the structure of censoring, thereby enabling estimation through incomplete-data methods such as the stochastic EM algorithm.1 The paper also contributes empirically to the retirement literature, particularly studies considering the role of spousal interactions in retirement decisions, by providing empirical

1Bertanha et al. (2023) discuss and use the connection between bunching behavior and a censored model for estimation; we generalize and formalize this connection in a broader setup.LANE ET AL. 5 evidence of spousal complementarities in retirement decisions among European couples (Blundell et al., 2016; García-Miralles and Leganza, 2024; Honoré and de Paula, 2014, 2018; Hospido and Zamarro, 2014; Johnsen et al., 2022; Lalive and Parrotta, 2017; Michaud and Vermeulen, 2011; Michaud et al., 2020).2 The rest of the paper proceeds as follows. Section 2 characterizes a class of choice models with piecewise-smooth objective functions, develops the transformation of the original optimization problem into a potential-outcome framework, and introduces the SEM estimation method. Section 3 presents two illustrative examples. Section 4 applies the method to the interdependent duration model to empirically analyze joint retirement behavior among European couples. Finally, Section 5 concludes. 2|CHOICE MODELS WITH CENSORED REDUCED FORM In this section, we characterize a class of optimization-based choice models with piecewise smooth objective functions, which have a reduced form resembling a censored model, providing an alternative solution approach and facilitating the use of the stochastic ExpectationMaximization (SEM) estimation method for these choice models. Choice models. We start with choice models that characterize units’ behavior. For each unit i= 1, ...,N, acontinuousdecision yi is made by maximizing an objective function V(y;x i,u i,θ): yi =arg max y∈Sy V(y;x i,u i,θ). (1) For example, V(y) may represent an individual’s utility function or a firm’s profit function.

While xi and yi are observable to researchers, ui is unobservable and is assumed to follow a distributionf u(u;γ). The unknown parameter vectorκ≡[θ;γ]is of interest. 3 The estimation of such models can be straightforward when the objective function V(y) is well behaved such that policy functions characterizing the optimal choice yi can be derived. In such cases, one could pursue likelihood-based estimation, since it is fully specified, or exploit the moment restrictions implied by the model. However, we focus on a specific class of models in whichV(y) exhibits non-differentiability due to its piecewise feature, making the solution to the choice model problem and thus the estimation procedure more complex. Specifically, V(y;x i,u i,θ) = KX k=1 1(y∈S k)Vk(y;x i,u i,θ), (2) where {Sk}K k=1 are mutually exclusive and collectively exhaustive subsets of Sy, meaningSK k=1 Sk =S y and Sk ∩S j =∅ for k̸=j . The model is piecewise in the sense that the functional form of the objective depends on the region in which the actionylies. This piecewise structure can arise from many sources in different applications, such as nonlinear budget constraints (e.g., progressive taxation) and nonlinear components in the objective function (e.g., max and min operators), as discussed later in Sections 3–4. 2See also Hurd (1990), Blau (1998), Gustman and Steinmeier (2000), Coile (2003), Gustman and Steinmeier (2004), Blau and Gilleskie (2006), Banks et al. (2007), Van der Klaauw and Wolpin (2008), Pozzoli and Ranzani (2009).

3The framework can be extended to incorporate unknown unit-specific heterogeneity with panel data, and the following discussion still applies.LANE ET AL. 6 This structure introduces two main challenges: (i) non-differentiability at the boundaries between regions Sk, which complicates the model solution; and (ii) solutions yi at these boundaries often lead to censoring, meaning that multiple values of ui are consistent with the observedy i. A common approach to estimating such models is to sequentially obtain local maximizers in each region Sk and then select the global maximizer, paired with simulation-based estimation methods such as indirect inference (Smith Jr, 1993; Gouriéroux and Monfort, 1997). This approach, however, requires additional care to account for possible corner solutions within each region, involves specifying an auxiliary model, and is generally less statistically efficient than MLE as well as less transparent in terms of identification. From a computational perspective, the simulated moments may be non-smooth, and the method can be computationally demanding in large parameter spaces.4 In the remainder of this section, we propose a new approach to estimating piecewise choice models. We show that under certain assumptions, the choice model can be transformed into a potential outcome framework, which we then estimate using the SEM procedure. This method avoids the need for region-by-region optimization, operates directly on the likelihood or moment conditions implied by the original model, and offers potential gains in computational efficiency, thus presenting a transparent and practically effective alternative. 2.1|Potential Outcome Framework Now we characterize a class of choice models whose optimization problem can be reformulated as a latent "potential outcome" framework, where the potential outcomes correspond to the maximizers of functions Vk(y), both global and boundary-local. This framework will be further leveraged for estimation in the following subsection.

We impose the following assumptions. Assumption 1 is a regularity condition ensuring well-defined solutions, while Assumptions 2–3 are structural conditions on the objective function and its domain. Assumption 1 (Existence and uniqueness of the solution) For any θ∈Θ , the objective functionV(y)attains a unique maximizer a.s. with respect to(x i,u i)over the domainS y. Assumption 2 (Global unimodality of auxiliary functions) For all (xi, ui) and θ∈Θ , and for any region Sk, k= 1, . . ., K, that contains nonempty interior, there exists an open set ˜Sk ⊇S k such that Vk(y) is continuous on ˜Sk, admits no local maxima other than a unique global maximum, and the maximizery ∗ k ∈ ˜Sk is characterized by the conditiong k(y∗ k) =0. Assumption 2 imposes assumptions on each auxiliary function Vk(y): while the overall objective function V(y) may exhibit irregular features globally, each component Vk(y) is assumed to be well-behaved, possessing a unique global maximizer on an open set ˜Sk ⊇S k and no other local maxima. The domains ˜Sk need not be identical across k, nor must they coincide with the global feasible set Sy. Moreover, when Vk(y) is differentiable, the condition gk(y∗ k) = 0 corresponds to the standard first-order condition. Assumption 2 ensures that the global maximizers of the auxiliary functions are well-defined. Assumptions 1–2 allow for a certain degree of discontinuity. A special case is when V(y) exhibits jumps at region boundaries. For example, Section 3 presents an example with

removable jumps due to fixed adjustment cost. 4While approaches such as Chernozhukov and Hong (2003) reformulate this problem as a quasi-posterior sampling task, they can require long sampling runs, especially when the parameter dimension is large.LANE ET AL. 7 Assumption 3 (Existence of boundary maximizers) For all (xi, ui) and θ∈Θ , and for any k̸=j , let Skj ≡∂S k ∩∂S j denote the common boundary between regions Sk and Sj, and let Sk0 ≡∂S k ∩∂S y denote the portion of the boundary of Sk that coincides with the boundary of the feasible set Sy. For any nonempty set Skj with0 ⩽j < k⩽K , the boundary maximizer y∗ kj ≡arg max y∈Skj V(y)exists and is characterized by an equationg kj(y∗ kj) =0. Assumption 3 ensures that we can characterize local maximizers of the objective function along all boundaries. Together with the global maximizers y∗ k, the boundary-local maximizers y∗ kj will serve as auxiliary latent variables, forming the basis of the potential outcome framework discussed in the remainder of this section. The following result formalizes how these auxiliary variables characterize the global maximizer of the objective function. Theorem 1 (Latent potential outcome representation) Under Assumptions 1-3, the global maximizer y of the objective function V(y) over the feasible set Sy must belong to the following finite candidate set: y∈  [ {k:So k̸=∅} {y∗ k}  ∪  [ 0⩽j<k⩽K

{y∗ kj}  . Moreover, the mapping fromyto the elements of this set satisfies: Ify∈S o k,theny ∗ k =y,for1⩽k⩽K; Ify∈S kj,theny ∗ kj =y,for0⩽j < k⩽K, whereS o k denotes the interior of regionS k. Proof By Assumption 1, a global maximizer y exists over the feasible set Sy. Since the sets {Sk}K k=1 are mutually exclusive and collectively exhaustive, y must lie either in the interior of some regionS ◦ k fork=1, ...,K, or on a boundaryS kj for 0⩽j < k⩽K. If y∈S ◦ k for k= 1, ...,K: On S◦ k, V(y) =V k(y) by construction, so y is also a local maximizer of Vk. Since Vk has no other local maxima except the unique global maximum y∗ k (by Assumption 2), it follows thaty=y ∗ k. If y∈S kj for 0 ⩽j < k⩽K: Since y maximizes V(y) globally and lies in Skj, it must coincide with the boundary-local optimizery ∗ kj, by Assumption 3. Therefore, the global maximizer must lie in the finite candidate set: y∈ K[ k=1 {y∗ k} ! ∪   [ 0⩽j<k⩽K {y∗ kj}  . First, Theorem 1 offers an alternative perspective on solving the optimization problem

compared to the region-based optimization approach. By reframing the problem in this way, it avoids the need to discuss corner solutions that typically arise when searching for a local optimizer. More importantly, by introducing the well-defined auxiliary latent variables y∗ k for k= 1, ...,K and y∗ kj for k̸=j , Theorem 1 transforms the original optimization problem intoLANE ET AL. 8 apotential outcomeframework: gk(y∗ k;x,u,θ) =0, for 1⩽k⩽K;g kj(y∗ kj;x,u,θ) =0, for 0⩽j < k⩽K, (3) y= KX k=1 1(y∈S o k)y∗ k + X 0⩽j<k⩽K 1(y∈S kj)y∗ kj. (4) whereu∼f u(u;γ), and parameter vectorκ≡[θ;γ]is of interest. Specifically, the auxiliary variables y∗ k and y∗ kj, defined by the functions g(·), can be interpreted as latent potential options available to unit i, as in Equation (3). Unit i selects the option that maximizes the objective function V(y), and only this chosen outcome is observed by the researcher, as formalized in Equation (4). It is important to emphasize that unit i optimizes over the entire feasible domain Sy; the finite set of alternatives in Equations (3)–(4), comprising y∗ k and y∗ kj, merely provides an equivalent representation that captures all possible optima and serves as a practical device for estimation. Censored models. Equations (3)–(4) constitute a valid framework, equivalent to the choice model in the

likelihood as long as Assumptions 1–3 are satisfied. In simpler cases, where there is a one-to-one mapping between each observed choice y and the unobservable u through the functions g(·), direct estimation based on Equations (3)–(4) is often straightforward. Our main interest, however, lies in cases where some observed choicesy are consistent with a range of unobservable values, creating a one-to-many mapping between y and u. In this case, the framework resembles a censored model, and its estimation— such as via Maximum Likelihood—often requires integrating over the entire feasible range of unobservables for each given choicey, making computation more demanding. In the following subsection, we propose a simulation-based estimation method, the SEM method, to estimate the choice model leveraging the potential outcome framework.5 Our assumptions are that the choice model is identified under the regularity conditions following Newey and McFadden (1994), ensuring that the targeted estimator (either MLE or GMM) is consistent and asymptotically normal. The focus on this paper is on the estimation procedure of the model. 2.2|SEM Estimation Method The stochastic EM (SEM) algorithm (Celeux et al., 1996) is a simulated version of the classical Expectation-Maximization (EM) algorithm (Dempster et al., 1977). As an iterative method, SEM exchanges between an E-step, where latent variables are sampled from their posterior distribution conditional on the observables, and an M-step, where we estimate the model using data and latent draws, until the parameters converge to the stationary distribution. Based on the potential outcome framework in Equations (3)–(4), we propose the following general steps of conducting SEM. Define the latent variable vectory∗ = [y∗ 1 , ...,y∗ K, y∗

1,2, ...,y∗ K−1,K]. Starting from an initial guess ˆκ(0), we iterate over the following two steps for s= 1, 2, ....,S until convergence to a stationary distribution: E-step: Given ˆκ (s), drawy ∗ i from the posterior distributionf(y ∗ i |yi,x i; ˆκ(s)). 5Alternatively, the framework can also be readily utilized for Bayesian estimation methods, especially given its similarity to SEM in the sampling of latent variables. However, we will not pursue this further.LANE ET AL. 9 M-step: Estimate the model using the E-step draws : ˆκ(s+1) =arg minκ H(κ;y ∗,x) T ×W×H(κ;y ∗,x), where the known functionH(·) can be either the score functions or the moment restrictions based on functionsg(·), andWis a weighting matrix. The final estimator is the average of the lastS 0 iterations, bκ= PS s=S−S0+1 ˆκ(s)/S0. The E-step requires drawing values for y∗ k and y∗ kj given the observables at parameter value ˆκ(s). In practice, for each observation i, if Equations (3)–(4) imply a unique value of ui consistent with yi, we recover ui and compute y∗ k and y∗ kj. If these equations imply a range of values of ui consistent with yi, we sample ui from the truncated distribution of fu(u; ˆγ(s))restricted to that range and then computey ∗ k andy ∗ kj. The M-step estimates the model in Equations (3)–(4) using pseudo-complete data. It

can be either likelihood-based, where H(·) denotes the score function, or moment-based, where H(·) denotes moment conditions implied by the functions g(·). While a likelihoodbased M-step yields an estimator asymptotically equivalent to the MLE under the regularity conditions in Nielsen (2000), a moment-based M-step yields an estimator defined by the specified moment conditions, often offering easier implementation and lower computational burden (Arcidiacono and Jones, 2003; Arellano and Bonhomme, 2016). The advantage of SEM is that it avoids complex optimization involving integration over latent variables and instead consists of a sequence of much simpler M-step estimations under pseudo-complete data. While its baseline convergence spe