Week 10: Frontiers and Course Synthesis

Deep Learning for Macroeconomics — Honours, The University of Edinburgh
Instructor: Juan Zurita · juan.zurita@ed.ac.uk

Learning objectives

By the end of this notebook you will be able to:

  1. Describe sequence-space methods and how they relate to DEQNs
  2. Understand the basics of continuous-time heterogeneous-agent models
  3. Choose the right computational method for a given economic model
  4. Identify open research questions at the frontier of computational macro
# Setup
import numpy as np
import matplotlib.pyplot as plt
import torch
import torch.nn as nn

torch.manual_seed(42)
np.random.seed(42)
device = 'cuda' if torch.cuda.is_available() else 'cpu'
print(f"PyTorch {torch.__version__} on {device}")

1. Sequence-space methods

An alternative to recursive methods: solve for the entire sequence of equilibrium objects \(\{c_t, k_t\}_{t=0}^T\) simultaneously.

Auclert et al. (2021) showed that many HA models can be solved efficiently in sequence space using Jacobians — and neural networks can represent these Jacobians.

DEQNs in sequence space

Instead of parameterising \(c(k, z; \theta)\) recursively, parameterise the history-dependent policy: \[c_t = \text{NN}(k_0, z_0, z_1, \ldots, z_t; \theta)\]

This is what Scheidegger calls Sequence-Space DEQNs (Lecture 10).

2. Continuous-time heterogeneous agents

The HJB-KFE system for continuous-time HA models:

\[\rho V(a, z) = \max_c \big\{ u(c) + V_a(a,z) [ra + wz - c] \big\} + \text{(shock transition terms)}\]

\[0 = -\frac{\partial}{\partial a}[s(a,z) g(a,z)] + \text{(shock transition terms)}\]

where \(g(a,z)\) is the stationary distribution. PINNs solve both equations simultaneously — a powerful approach pioneered by Fernández-Villaverde, Hurtado & Nuño (2023).

3. Decision guide: which method when?

START
  │
  ├─ Dimensions ≤ 3? ──── YES ──→ VFI or projection (PNM methods)
  │                                 Fast, well-understood, guaranteed convergence
  │
  ├─ Continuous time? ──── YES ──→ PINNs (Week 7)
  │                                 Natural for HJB/KFE systems
  │
  ├─ Occasionally binding constraints? ── YES ──→ DEQNs + Fischer-Burmeister (Week 6)
  │                                               Handles kinks naturally
  │
  ├─ Heterogeneous agents? ── YES ──→ DEQNs + Young's method (Week 8)
  │                                    Scales beyond Krusell-Smith moments
  │
  ├─ Need estimation? ── YES ──→ Surrogates + SMM (Week 9)
  │                               Fast gradient-based optimisation
  │
  └─ High-dimensional (10+ states)? ── YES ──→ DEQNs (Weeks 4-5)
                                                Only feasible approach

4. Open questions and research frontiers

  1. Convergence guarantees for DEQNs — gradient descent can get stuck in local minima. When can we guarantee global convergence?

  2. Transfer learning across models — can a network trained on one calibration be fine-tuned for another? (Huge speed-up for estimation)

  3. Real-time model solving — can we solve DSGE models fast enough for nowcasting and real-time policy analysis?

  4. Interpretability — can we extract economic insights from trained networks? (Feature importance, attention maps)

  5. Multi-agent deep learning — can we solve game-theoretic models (multiple interacting agents) with neural networks?

5. Course synthesis

Week Topic Key method Key model
1 Neural networks as approximators MLP, PyTorch Brock–Mirman (supervised)
2 Deep learning fundamentals Backprop, Adam Function approximation
3 Automatic differentiation Autograd Economic derivatives
4 DEQNs I Euler residual loss Deterministic growth
5 DEQNs II Quadrature for expectations Stochastic RBC
6 Constraints Fischer–Burmeister Consumption–savings
7 PINNs PDE residual loss Cake-eating HJB
8 Heterogeneous agents Young’s method + DEQN Aiyagari / Krusell–Smith
9 Surrogates & estimation GP, SMM DICE climate model
10 Frontiers Sequence-space, continuous-time HA Research directions

The unifying idea

Every method in this course follows the same pattern:

  1. Parameterise the unknown function (policy, value, distribution) as a neural network
  2. Define a loss from the model’s own equations (Euler, HJB, complementarity, PDE residual)
  3. Minimise the loss using gradient descent + automatic differentiation
  4. Verify using Euler-equation errors, simulation, or comparison to known solutions

Final exercise: choose your own model

Pick a dynamic economic model from your macroeconomics courses (or from research papers) and:

  1. Write down the equilibrium conditions
  2. Identify which method from this course is most appropriate
  3. Implement a DEQN (or PINN) solution
  4. Verify your solution and compute Euler-equation errors

This could be the start of a dissertation chapter or an Honours thesis project.

Course references

  • Azinovic, M., Gaegauf, L., & Scheidegger, S. (2022). “Deep Equilibrium Nets.” International Economic Review, 63(4).
  • Scheidegger, S. (2025). Deep Learning for Solving and Estimating Dynamic Economic Models.
  • Fernández-Villaverde, J., Hurtado, S., & Nuño, G. (2023). “Financial Frictions and the Wealth Distribution.” Econometrica, 91(4).
  • Auclert, A., Bardóczy, B., Rognlie, M., & Straub, L. (2021). “Using the Sequence-Space Jacobian to Solve and Estimate Heterogeneous-Agent Models.” Econometrica, 89(5).
  • Nordhaus, W. (2017). “Revisiting the Social Cost of Carbon.” PNAS, 114(7).

Thank you for taking this course. Good luck with your research!