arXiv subjects

Takeshi Fukasawa

Publications and source records attributed to Takeshi Fukasawa.

7 recordsLinked to original sources

arXiv2026

Lagrange multipliers in Maximum likelihood estimations and Least squares problems with Constraints

This study investigates a statistical property of Lagrange multipliers in constrained Maximum Likelihood Estimation (MLE) and Least Squares (LS) problems from the perspective of numerical optimization. Building on large-sample theory, we show that the associated Lagrange multipliers converge to zero as the sample size increases, provided the distribution is correctly specified in MLE or the residuals are normally distributed in LS. Although this asymptotic behavior has long been recognized in statistics, it has received little explicit attention in numerical optimization and has rarely been exploited in algorithmic design. Importantly, the insight extends beyond classical low-dimensional settings: even in modern high-dimensional applications, such as deep learning, where the number of parameters may exceed the sample size, the same reasoning applies provided the generalization performance is good. This observation has two main implications. First, many constrained optimization algorithms, including the Augmented Lagrangian Method, Sequential Quadratic Programming, and Interior Point methods, require initial values for the multipliers, and choosing zero is statistically justified. Numerical experiments for constrained regressions and dynamic discrete choice model estimations support this implication by showing that initializing multipliers at zero usually lead to stable and efficient performance. Second, penalty-based approaches that convert constrained problems into unconstrained ones can perform well when the true multipliers are small. This helps explain why penalty-based methods often perform well in practice.

econ.EM↗

arXiv2026

Sequential algorithm for structural estimations with equilibrium constraints

This study examines sequential algorithms with the Zero Jacobian Property (ZJP) for estimating structural models subject to equilibrium constraints. For the Maximum Likelihood Estimation (MLE) and the Generalized Method of Moments (GMM), the current study shows that these algorithms attains fast (near-quadratic) local convergence in large samples to the solution of the constrained optimization problem. If consistent initial estimates of the parameters are available, the algorithms yield an asymptotically efficient estimator even after one iteration. It then proposes a novel algorithm called Sequential Linearly Constrained (SLC) algorithm, which is applicable to a broader class of structural models than existing methods. A key advantage of the SLC algorithm is that it can be implemented without explicitly computing the Jacobian of the equilibrium constraints and can be multiple times faster than the Nested Fixed Point (NFXP) approach. The current study illustrates its performance through two numerical experiments: a dynamic discrete game with time-varying unobserved heterogeneity and a dynamic demand model.

econ.EM↗

arXiv2025

The Use of Symmetry for Models with Variable-size Variables

This paper presents a universal representation of symmetric (permutation-invariant) functions with multidimensional variable-size variables. These representations help justify approximation methods that aggregate information from each variable using moments. It further discusses how these findings provide insights into game-theoretic applications, including two-step policy function estimation, Moment-based Markov Equilibrium (MME), and aggregative games. Regarding policy function estimation, under certain conditions, estimating a common policy function as a function of a firm's own state and the sum of polynomial terms (moments) of competitors' states is justified, regardless of the number of firms in a market, provided a sufficient number of moments are included. For MME, this study demonstrates that MME is equivalent to Markov Perfect Equilibrium if the number of moments reaches a certain level and regularity conditions are satisfied. Regarding aggregative games, the paper establishes that any game satisfying symmetry and continuity conditions in payoff functions can be represented as a multidimensional generalized aggregative game. This extends previous research on generalized (fully) aggregative games by introducing multidimensional aggregates.

econ.GN↗

arXiv2025

When do firms sell high durability products? The case of light bulb industry

This study empirically investigates firms' incentives on the choice of product durability, and its social optimality, by developing a dynamic structural model of durable goods with forward-looking consumers and oligopolistic multi-product firms. Based on the observations of the light bulb market, it specifies a model where firms produce multiple products with different durability levels and set product prices based on dynamic incentives. It proposes and applies novel estimation algorithms that alleviate the computational burden and data requirement for estimating demand and marginal cost parameters of dynamic demand models. Using light bulb market data in Japan, structural parameters are estimated. This study obtains the following results. First, large firms have incentives to collude to eliminate high durability incandescent lamps, though it is profitable to sell them for each firm. In contrast, when they can collude on prices, they don't have incentives to eliminate high durability bulbs. Second, eliminating high durability incandescent lamps leads to larger producer and total surplus, though it leads to lower consumer surplus.

econ.GN↗

arXiv2025

Fast and simple inner-loop algorithms of static / dynamic BLP estimations

This study investigates computationally efficient inner-loop algorithms for estimating static/dynamic BLP models. It provides the following ideas for reducing the number of inner-loop iterations: (1). Add a term relating to the outside option share in the BLP contraction mapping; (2). Analytically represent the mean product utilities as a function of value functions and solve for value functions (for dynamic BLP); (3). Combine an acceleration method of fixed-point iterations, especially the Anderson acceleration. They are independent and easy to implement. This study shows the good performance of these methods using numerical experiments.

econ.EM↗

arXiv2025

Computationally Efficient Methods for Solving Discrete-time Dynamic models with Continuous Actions

This study investigates computationally efficient algorithms for solving discrete-time infinite-horizon single-agent/multi-agent dynamic models with continuous actions. It shows that we can easily reduce the computational costs by slightly changing basic algorithms using value functions, such as the Value Function Iteration (VFI) and the Policy Iteration (PI). The PI method with a Krylov iterative method (GMRES), which can be easily implemented using built-in packages, works much better than VFI-based algorithms even when considering continuous state models. Concerning the VFI algorithm, we can largely speed up the convergence by introducing acceleration methods of fixed-point iterations. The current study also proposes the VF-PGI-Spectral (Value Function-Policy Gradient Iteration Spectral) algorithm, which is a slight modification of the VFI. It shows numerical results where the VF-PGI-Spectral performs much better than the VFI- and PI-based algorithms especially in multi-agent dynamic games. Finally, it shows that using relative value functions further reduces the computational cost of these methods.

econ.GN↗

arXiv2024

Jacobian-free Efficient Pseudo-Likelihood (EPL) Algorithm

This study proposes a simple procedure to compute Efficient Pseudo Likelihood (EPL) estimator proposed by Dearing and Blevins (2024) for estimating dynamic discrete games, without computing Jacobians of equilibrium constraints. EPL estimator is efficient, convergent, and computationally fast. However, the original algorithm requires deriving and coding the Jacobians, which are cumbersome and prone to coding mistakes especially when considering complicated models. The current study proposes to avoid the computation of Jacobians by combining the ideas of numerical derivatives (for computing Jacobian-vector products) and the Krylov method (for solving linear equations). It shows good computational performance of the proposed method by numerical experiments.

econ.EM↗