Deep Learning for Solving and Estimating Dynamic Macro-Finance Models
Many important questions in macro-finance are dynamic. How do firms invest when financing conditions change? How does the financial sector amplify shocks? How do asset prices reflect risks that unfold over time? Economists often answer these questions with continuous-time equilibrium models. These models are powerful, but they are difficult to solve and estimate, especially when the economy has many state variables.
This paper develops a deep-learning method for solving and estimating dynamic macro-finance models. The goal is not to replace economic structure with a black box. The goal is the opposite: use modern machine-learning tools to respect the structure of economic models while making them easier to solve, estimate, and scale.
Traditional numerical methods often work well in low-dimensional settings but become difficult as the number of state variables grows. This is the familiar curse of dimensionality. A model may be conceptually right but computationally hard. Researchers then have to simplify the model, reduce the state space, or separate solution and estimation into steps that may be computationally fragile.
Our approach uses physics-informed neural networks, a class of deep-learning methods designed for differential equations, to approximate the unknown functions inside a continuous-time model. These functions can include value functions, prices, policies, and distributions. The network is trained to satisfy the model's equations, including Hamilton-Jacobi-Bellman equations, Kolmogorov forward equations, and moment conditions. In this sense, the economics disciplines the machine learning: the algorithm is not simply fitting patterns in data; it is learning objects that must obey equilibrium restrictions.
A central advantage is that solution and estimation can be integrated. Many macro-finance models are useful because they connect observed data to hidden economic forces: productivity, financing frictions, risk prices, or constraints. Estimating those objects requires repeatedly solving the model. Deep-learning methods can make that loop more feasible.
We illustrate the method in two applications. One is an industrial-dynamics model with financial frictions, entry, exit, and an equilibrium distribution of banks. The other is a macroeconomic model with a financial sector, binding constraints, nonlinear amplification, and difficult boundary behavior. These examples show that the method can handle canonical models in financial economics and macro-finance, not only toy problems. The associated code package implements forward problems, where the model is solved for given parameters, and inverse problems, where parameters or hidden objects are inferred from model restrictions and data.
The broader implication is that computational methods shape the questions economists can ask. If a model becomes too hard to solve, the researcher often has to make the economy artificially simple. Better computational tools allow richer heterogeneity, more state variables, and tighter links between theory and data. That is especially important in macro-finance, where crises, balance sheet constraints, and asset prices often depend on nonlinear interactions.
The takeaway is that artificial intelligence can be useful in economics when it is embedded inside economic reasoning. The method is not an exercise in asking a neural network to forecast markets. It is a way to solve disciplined models of how markets and the macroeconomy interact, while estimating the parameters that make those models match observed moments. This is a practical use of machine learning: expanding the class of economic models that can be taken seriously to the data.
The paper also reflects a broader methodological shift. As economic questions become more complex, the frontier is not only new data or new theory. It is also the ability to compute the implications of theory in environments that look more like the real economy. Deep learning gives researchers another tool for that task.