Endogenous State Variables Research Articles

Global DSGE models

We introduce our GDSGE framework and a novel global solution method, called simultaneous transition and policy function iterations (STPFIs), for solving dynamic stochastic general equilibrium models. The framework encompasses many well-known incomplete markets models with highly nonlinear dynamics such as models of financial crises and models with rare disasters including the current COVID-19 pandemic. Using consistency equations, our method is most effective at solving models featuring endogenous state variables with implicit laws of motion such as wealth or consumption shares. Finally, we incorporate this method in an automated and publicly available toolbox that solves many important models in the aforementioned topics, and in many cases, more efficiently and/or accurately than their original algorithms.

Dynamic demand for differentiated products with fixed-effects unobserved heterogeneity

Summary This paper studies identification and estimation of a dynamic discrete choice model of demand for differentiated product using consumer-level panel data with few purchase events per consumer (i.e., short panel). Consumers are forward-looking and their preferences incorporate two sources of dynamics: last choice dependence due to habits and switching costs, and duration dependence due to inventory, depreciation, or learning. A key distinguishing feature of the model is that consumer unobserved heterogeneity has a Fixed Effects structure; that is, its probability distribution conditional on the initial values of endogenous state variables is unrestricted. I apply and extend recent results to establish the identification of all the structural parameters as long as the dataset includes four or more purchase events per household. The parameters can be estimated using a sufficient statistic—conditional maximum likelihood (CML) method. An attractive feature of CML in this model is that the sufficient statistic controls for the forward-looking value of the consumer’s decision problem, such that the method does not require solving dynamic programming problems or calculating expected present values.

The Delphic forward guidance puzzle in New Keynesian models

When the central bank has information that can help the private sector better predict the future, should it communicate such information to the public? In purely forward-looking New Keynesian models, such Delphic forward guidance unambiguously reduces ex ante welfare by increasing the variability of inflation and the output gap. We call this phenomenon the Delphic forward guidance puzzle. In more elaborate models with endogenous state variables, a combination of Delphic forward guidance and preemptive policy actions may improve welfare. However, full information revelation is generally not optimal and what information needs to be revealed is highly model-dependent.

ALIENs and Continuous Time Economies

I develop a new computational framework called Actively Learned and Informed Equilibrium Nets (ALIENs) to solve continuous time economic models with endogenous state variables and highly non-linear policy functions. I employ neural networks that are trained to solve supervised learning problems that respect the laws governed by the economic system in the form of general parabolic partial differential equations. The economic information is encoded as regularizers that disciplines the deep neural network in the learning process. The sub-domain of the high dimensional state space that carries the most economic information is learned actively in an iterative loop, enforcing the random training points to be sampled from areas that matter the most to ensure convergence. I utilize a state-of-the art distributed framework to train the network that speeds up computation time significantly. The method is applied to successfully solve a model of macro-finance that is notoriously difficult to handle using traditional finite difference schemes.

Sufficient statistics for unobserved heterogeneity in structural dynamic logit models

We study the identification and estimation of structural parameters in dynamic panel data logit models where decisions are forward-looking and the joint distribution of unobserved heterogeneity and observable state variables is nonparametric, i.e., fixed-effects model. We consider models with two endogenous state variables: the lagged decision variable, and the time duration in the last choice. This class of models includes as particular cases important economic applications such as models of market entry–exit, occupational choice, machine replacement, inventory and investment decisions, or dynamic demand of differentiated products. We prove the identification of the structural parameters using a conditional likelihood approach. The structure of the model implies that there is a sufficient statistic such that the likelihood function conditional on this statistic no longer depends on the unobserved heterogeneity – neither through the current utility nor through the continuation value of the forward-looking decision problem – but still depends on the structural parameters. We apply this estimator to a machine replacement model.

Global DSGE Models

We introduce our GDSGE framework and a novel global solution method, called simultaneous transition and policy function iterations (STPFIs), for solving dynamic stochastic general equilibrium models. The framework encompasses many well-known incomplete markets models with highly nonlinear dynamics such as models of financial crises and models with rare disasters including the current COVID-19 pandemic. Using consistency equations, our method is most effective at solving models featuring endogenous state variables with implicit laws of motion such as wealth or consumption shares. Finally, we incorporate this method in a publicly available toolbox that solves many important models in the aforementioned topics, and in many cases, more efficiently or accurately than their original algorithms.

FiPIt: A simple, fast global method for solving models with two endogenous states & occasionally binding constraints

We propose a simple and fast fixed-point iteration algorithm (FiPIt) to obtain the global, non-linear solution of macro models with two endogenous state variables and occasionally binding constraints. This method uses fixed-point iteration on Euler equations to avoid solving two simultaneous non-linear equations (as with the time iteration method) or creating modified state variables requiring irregular interpolation (as with the endogenous grids method). In the small-open-economy RBC and Sudden Stops models provided as examples, FiPIt is much faster than time iteration and various hybrid methods.

FiPIt: A Simple, Fast Global Method for Solving Models with Two Endogenous States & Occasionally Binding Constraints

We propose a simple and fast fixed-point iteration algorithm FiPIt to obtain the global, non-linear solution of macro models with two endogenous state variables and occasionally binding constraints. This method uses fixed-point iteration on Euler equations to avoid solving two simultaneous nonlinear equations (as with the time iteration method) or creating modified state variables requiring irregular interpolation (as with the endogenous grids method). In the small-open-economy RBC and Sudden Stops models provided as examples, FiPIt is used on the bonds and capital Euler equations to solve for the bonds decision rule and the capital pricing function. In a standard Matlab platform, FiPIt solves both models much faster than time iteration and various hybrid methods. The choice of functions that FiPIt iterates on using the Euler equations can vary across models, and there can be more that one arrangement for the same model. Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

Rational Expectations with Endogenous Information

This paper characterizes a general class of macroeconomic models with incomplete information, when the information process includes endogenous variables. I derive conditions for existence and uniqueness of equilibrium, which apply even when the model contains endogenous state variables, and I introduce an algorithm to solve the general model. As an application I consider a business cycle model with capital where firms must make inferences about aggregate shocks through the movements of endogenous prices. In this model, the central bank's policy rule determines the real effects of nominal shocks, by controlling how informative prices are about the aggregate state. The optimal policy targets acyclical inflation, which makes money neutral. Finally, I demonstrate an advantage of models with endogenous information: the noisy signals are driven by fundamental shocks, rather than ad hoc noise, so data can discipline the information structure. Accordingly, I calibrate the model using US industry-level panel data.

Sufficient Statistics for Unobserved Heterogeneity in Structural Dynamic Logit Models

We study the identification and estimation of structural parameters in dynamic panel data logit models where decisions are forward-looking and the joint distribution of unobserved heterogeneity and observable state variables is nonparametric, i.e., fixed-effects model. We consider models with two endogenous state variables: the lagged decision variable, and the time duration in the last choice. This class of models includes as particular cases important economic applications such as models of market entry-exit, occupational choice, machine replacement, inventory and investment decisions, or dynamic demand of differentiated products. The identification of structural parameters requires a sufficient statistic that controls for unobserved heterogeneity not only in current utility but also in the continuation value of the forward-looking decision problem. We obtain the minimal sufficient statistic and prove identification of some structural parameters using a conditional likelihood approach. We apply this estimator to a machine replacement model.

Solving Multi-Dimensional Dynamic Programming Problems Using Stochastic Grids and Nearest-Neighbor Interpolation

We propose two modifications to the method of endogenous grid points that greatly decreases the computational time for life cycle models with many exogenous state variables. First, we use simulated stochastic grids on the exogenous state variables. Second, when we interpolate to find the continuation value of the model, we split the interpolation step into two: We use nearest-neighbor interpolation over the exogenous state variables, and multilinear interpolation over the endogenous state variables. We evaluate the numerical accuracy and computational efficiency of the algorithm by solving a standard consumption/savings life-cycle model with an arbitrary number of exogenous state variables. The model with eight exogenous state variables is solved in around eight minutes on a standard desktop computer. We then use a more realistic income process estimated by Guvenen et al (2015) to demonstrate the usefulness of the algorithm. We demonstrate that the consumption dynamics differ compared to agents facing a more traditional income process.

Three types of robust Ramsey problems in a linear-quadratic framework

This paper studies robust Ramsey policy problems in a general discrete-time linear-quadratic framework when the Ramsey planner faces three types of ambiguity. This framework includes both exogenous and endogenous state variables. In addition, the equilibrium system from the private sector contains both backward-looking and forward-looking dynamics. We provide recursive characterizations and algorithms to solve for robust policy. We apply our method to a basic New Keynesian model of optimal monetary policy with persistent cost-push shocks. We find that (i) all three types of ambiguity make optimal monetary policy more history-dependent but with different reasons for each type; and (ii) they deliver qualitatively different initial responses of inflation and the output gap following a cost-push shock.

Endogenous Grids in Higher Dimensions: Delaunay Interpolation and Hybrid Methods

This paper investigates extensions of the method of endogenous gridpoints (ENDGM) introduced by Carroll (Econ Lett 91(3):312–320, 2006) to higher dimensions with more than one continuous endogenous state variable. We compare three different categories of algorithms: (i) the conventional method with exogenous grids (EXOGM), (ii) the pure method of endogenous gridpoints (ENDGM) and (iii) a hybrid method (HYBGM). ENDGM comes along with Delaunay interpolation on irregular grids. Comparison of methods is done by evaluating speed and accuracy by using a specific model with two endogenous state variables. We find that HYBGM and ENDGM both dominate EXOGM. In an infinite horizon model, ENDGM also always dominates HYBGM. In a finite horizon model, the choice between HYBGM and ENDGM depends on the number of gridpoints in each dimension. With less than 150 gridpoints in each dimension ENDGM is faster than HYBGM, and vice versa. For a standard choice of 25–50 gridpoints in each dimension, ENDGM is 1.4–1.7 times faster than HYBGM in the finite horizon version and 2.4–2.5 times faster in the infinite horizon version of the model.

Dynamic Portfolio Optimisation with Intermediate Costs: A Least-Squares Monte Carlo Simulation Approach

We propose a dynamic portfolio optimisation strategy that takes liquidity cost into account. Our liquidity cost model is built upon the so-called Marginal Supply-Demand Curve which describes the asset price as a function of the trading volume. We extend the least-squares Monte Carlo algorithm to a stochastic control problem with switching costs and endogenous state variables. This approach is simulation-based, with great flexibility in the choice of investor objective, asset dynamics, portfolio constraints, intermediate consumption and incorporation of different sources of costs, making it easy to implement in practice. We study a portfolio investing in four major sectors in the U.S. market. We benchmark our dynamic strategy against several alternative portfolio strategies in terms of the out-of-sample distribution of terminal wealth and the real market performance. Overall, our dynamic strategy outperforms other standard asset allocation strategies.

Endogenous Grids in Higher Dimensions: Delaunay Interpolation and Hybrid Methods

This paper investigates extensions of the method of endogenous gridpoints (ENDGM) introduced by Carroll (2006) to higher dimensions with more than one continuous endogenous state variable. We compare three different categories of algorithms: (i) the conventional method with exogenous grids (EXOGM), (ii) the pure method of endogenous gridpoints (ENDGM) and (iii) a hybrid method (HYBGM). ENDGM comes along with Delaunay interpolation on irregular grids. Comparison of methods is done by evaluating speed and accuracy by using a specific model with two endogenous state variables. We find that HYBGM and ENDGM both dominate EXOGM. In an infinite horizon model, ENDGM also always dominates HYBGM. In a finite horizon model, the choice between HYBGM and ENDGM depends on the number of gridpoints in each dimension. With less than 150 gridpoints in each dimension ENDGM is faster than HYBGM, and vice versa. For a standard choice of 25 to 50 gridpoints in each dimension, ENDGM is 1:4 to 1:7 times faster than HYBGM in the finite horizon version and 2:4 to 2:5 times faster in the infinite horizon version of the model.

Repeated moral hazard and recursive Lagrangeans

This paper shows how to solve dynamic agency models by extending recursive Lagrangean techniques à laMarcet and Marimon (2011) to problems with hidden actions. The method has many advantages with respect to the promised utilities approach (Abreu et al., 1990): it is a significant improvement in terms of simplicity, tractability and computational speed. Solutions can be easily computed for hidden actions models with several endogenous state variables and several agents, while the promised utilities approach becomes extremely difficult and computationally intensive even with just one state variable or two agents.

A simulation-and-regression approach for stochastic dynamic programs with endogenous state variables

We investigate the optimum control of a stochastic system, in the presence of both exogenous (control-independent) stochastic state variables and endogenous (control-dependent) state variables. Our solution approach relies on simulations and regressions with respect to the state variables, but also grafts the endogenous state variable into the simulation paths. That is, unlike most other simulation approaches found in the literature, no discretization of the endogenous variable is required. The approach is meant to handle several stochastic variables, offers a high level of flexibility in their modeling, and should be at its best in non time-homogenous cases, when the optimal policy structure changes with time. We provide numerical results for a dam-based hydropower application, where the exogenous variable is the stochastic spot price of power, and the endogenous variable is the water level in the reservoir.

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Continuous‐Time Modeling with Spatial Dependence

(Spatial) panel data are routinely modeled in discrete time (DT). However, compelling arguments exist for continuous‐time (CT) modeling of (spatial) panel data. Particularly, most social processes evolve in CT, so that statistical analysis in DT is an oversimplification, gives an incomplete representation of reality, and may lead to misinterpretation of estimation results. The most compelling reason for a CT approach is that, in contrast to DT modeling, it allows adequate modeling of dynamic adjustment processes. This article introduces spatial dependence in a CT modeling framework. We propose a nonlinear structural equation model (SEM) with latent variables for estimation of the exact discrete model (EDM), which links CT model parameters to DT observations. The use of a SEM with latent variables enables a specification that accounts for measurement errors in the variables, leading to a reduction of attenuation bias (i.e., disattenuation). The SEM‐CT model with spatial dependence developed here is the first dynamic SEM with spatial dependence. A simple regional labor market model for Germany, comprising changes in unemployment and population as endogenous state variables, and changes in regional average wages and in the structure of the manufacturing sector as exogenous input variables, illustrates this spatial econometric SEM‐CT framework.El modelamiento de datos panel espaciales se realiza habitualmente utilizando una conceptualización del tiempo discreto (TD). Sin embargo, existen argumentos de peso para conceptualizar el tiempo de manera continua (TC). En concreto, la mayoría de procesos sociales se desarrolla en TC, por lo que el análisis estadístico en DT trae como consecuencia una simplificación excesiva de los procesos, da una representación incompleta de la realidad, y puede conducir a una interpretación errónea de los resultados de la estimación. La razón más convincente para el uso de un enfoque CT es que a diferencia de modelos DT, una conceptualización CT permite el modelado adecuado de los procesos de ajuste dinámico (dynamic adjustment). Este artículo incorpora la dependencia espacial en un marco de modelamiento con CT. Los autores proponen un modelo de ecuaciones estructurales no lineal (nonlinear structural equation model ‐SEM) con variables latentes para la estimación del modelo discreto exacto (exact discrete model‐EDM), que vincula los parámetros del modelo CT a las observaciones de DT. El uso de un SEM con variables latentes permite una especificación que da cuenta de los errores de medición en las variables, dando lugar a una reducción del sesgo de atenuación (es decir, “desatenuacion”). El modelo SEM‐CT con dependencia espacial desarrollado en el presente estudio es el primer SEM dinámico con dependencia espacial.Para ilustrar el marco conceptual SEM‐CT los autores presentan un modelo simple del mercado laboral regional de Alemania. El modelo está compuesto por los cambios en el desempleo y la población como variables endógenas de estado, y los cambios en los salarios regionales promedio y en la estructura del sector manufacturero como variables de entrada exógenas.（空间）面板数据通常基于离散时间（DT）进行建模。然而更令人信服的观点是基于连续时间（CT）进行（空间）面板数据建模。特别是多数社会过程均在连续时间中演化，基于离散时间的统计分析可能过度简化，使得对现实状况的表达不完备，并可能导致对估计结果的错误解释。相比于离散时间（DT）建模，连续时间（CT）建模最具说服力的原因在于在建模过程中允许足够多的动态调整。本文介绍了CT模型框架中的空间依赖性。把CT模型参数链接到DT观察值中，我们提出了用于估计精确离散模型（EDM）的包含潜变量的非线性结构方程模型（SEM）。包含潜在变量的SEM提供了变量测量误差的计算方案，使得衰减偏差（如反衰减性）减小。本文了提出的空间相关SEM‐CT模型是第一个动态空间相关的SEM模型，并以德国一个简单的区域劳动力市场模型为例，以失业和人口构成变化为内生状态变量，以区域平均工资和制造业结构部门变化为外生输入变量，阐述了该空间计量SEM‐ CT模型的框架。

Continuous-Time Modelling with Spatial Dependence

(Spatial) panel data are routinely modeled in discrete time (DT). However, compelling arguments exist for continuous time (CT) modeling of (spatial) panel data. Particularly, most social processes evolve in CT, so that statistical analysis in DT is an oversimplification, gives an incomplete representation of reality, and may lead to misinterpretation of estimation results. The most compelling reason for a CT approach is that, in contrast to DT modeling, it allows adequate modeling of dynamic adjustment processes. This article introduces spatial dependence in a CT modeling framework. We propose a nonlinear Structural Equation Model (SEM) with latent variables for estimation of the Exact Discrete Model (EDM), which links CT model parameters to DT observations. The use of a SEM with latent variables enables a specification that accounts for measurement errors in the variables, leading to a reduction of attenuation bias (i.e., disattenuation). The SEM-CT model with spatial dependence developed here is the first dynamic structural equation model with spatial dependence. A simple regional labor market model for Germany, comprising unemployment change and population change as endogenous state variables, and change in regional average wages and in the structure of the manufacturing sector as exogenous input variables, illustrates this spatial econometric SEM-CT framework.