Week 9: Surrogates, Estimation, and Climate Economics

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. Build a surrogate model (neural network or Gaussian process) that replaces an expensive simulation
  2. Use surrogates for simulated method of moments (SMM) estimation
  3. Understand the DICE integrated assessment model for climate economics
  4. Apply deep learning to solve a stochastic climate-economy model
# 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. Surrogate models — fast approximations of slow simulations

Some economic models take hours to solve once. Estimating them (which requires solving thousands of times) is infeasible without a shortcut.

Surrogate models replace the expensive simulation with a fast, differentiable approximation:

  1. Solve the model at \(N\) parameter vectors \(\{\theta_1, \ldots, \theta_N\}\)
  2. Record the moments \(m(\theta_i)\) for each solve
  3. Train a neural network or GP: \(\hat{m}(\theta) \approx m(\theta)\)
  4. Use \(\hat{m}\) in the objective function: \(\min_\theta (\hat{m}(\theta) - m^{\text{data}})^2\)

2. Gaussian processes — uncertainty-aware surrogates

A Gaussian process (GP) provides not just a prediction but an uncertainty estimate. This is valuable for:

  • Active learning: sample where the GP is most uncertain
  • Bayesian optimisation: efficiently search for the best-fitting parameters
  • Credible intervals: know when the surrogate’s prediction is reliable
# Simple Gaussian process regression

from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import RBF, ConstantKernel

# Simulate a model moment as a function of a parameter
np.random.seed(42)
theta_train = np.sort(np.random.uniform(0, 5, 15)).reshape(-1, 1)
# Pretend this is an expensive model solve
moment_train = np.sin(theta_train) + 0.1 * np.random.randn(*theta_train.shape)

# Fit GP
kernel = ConstantKernel() * RBF(length_scale=1.0)
gp = GaussianProcessRegressor(kernel=kernel, alpha=0.01)
gp.fit(theta_train, moment_train)

# Predict
theta_test = np.linspace(0, 5, 200).reshape(-1, 1)
mu, sigma = gp.predict(theta_test, return_std=True)

fig, ax = plt.subplots(figsize=(9, 5))
ax.plot(theta_test, np.sin(theta_test), 'k--', label='True moment function', alpha=0.5)
ax.plot(theta_test, mu, color='#7a2318', linewidth=2, label='GP mean')
ax.fill_between(theta_test.flatten(), mu.flatten() - 2*sigma, mu.flatten() + 2*sigma,
                alpha=0.2, color='#7a2318', label='95% CI')
ax.scatter(theta_train, moment_train, c='black', s=50, zorder=5, label='Training solves')
ax.set_xlabel('Parameter θ', fontsize=11)
ax.set_ylabel('Model moment m(θ)', fontsize=11)
ax.set_title('Gaussian Process Surrogate', fontsize=12)
ax.legend(fontsize=10); ax.grid(True, alpha=0.3)
plt.tight_layout(); plt.show()

3. The DICE model — climate meets economics

The Dynamic Integrated model of Climate and the Economy (Nordhaus, 2017) couples: - An economic growth model (Ramsey-type) - A carbon-cycle model - A climate/temperature model - A damage function linking temperature to GDP

This is a natural application for DEQNs because the state space is high-dimensional (capital, carbon concentrations, temperature layers, technology) and the model is non-stationary.

4. Structural estimation with surrogates

Simulated Method of Moments (SMM):

  1. Choose target moments from data: \(m^{\text{data}}\)
  2. For a given \(\theta\), solve the model and simulate to get \(m(\theta)\)
  3. Minimise \([m(\theta) - m^{\text{data}}]' W [m(\theta) - m^{\text{data}}]\)

With a surrogate, step 2 is instant — enabling gradient-based optimisation over \(\theta\).

Exercises

Exercise 1: Build a GP surrogate for the Brock–Mirman model’s steady-state capital as a function of \(\alpha\) and \(\beta\). How many training points do you need for 1% accuracy?

Exercise 2: Implement a simplified DICE model (2 state variables: capital and carbon concentration). Solve it with a DEQN and plot the optimal carbon tax path.

Next week: Frontiers and Course Synthesis — when to use which method.