You have complete data but can’t optimize because functions are discontinuous
In their new paper, Xiaohong Chen, Wayne Yuan Gao and Likang Wen suggest:
Don’t optimize discontinuous functions → Reformulate with ReLU
ReLU, which stands for stands for Rectified Linear Unit, transforms a 40-year-old problem into something you can solve with standard machine learning tools (PyTorch, TensorFlow) on a laptop. ReLU is a simple mathematical function:
ReLU(t) = max(t, 0)
In plain English: If the input is positive, keep it but if the input is negative, replace it with zero.
Economic Applications
- Binary Choice Models (Core application)
Examples: Labor force participation (work/don’t work), product purchase decisions (buy/don’t buy), transportation mode choice (drive/bus/bike)
What it estimates: Parameters showing how income, education, prices, etc. affect these yes/no decisions
Why ReLU helps: Traditional methods took forever to estimate; ReLU makes it routine - Conditional Quantile Models
Example: How does income affect not just average consumption, but consumption across the distribution (poor vs. rich)?
ReLU allows flexible heterogeneity - Stochastic Volatility
Example: How does economic uncertainty affect financial market volatility?
ReLU captures time-varying risk
ReLU-Based and DNN-Based Generalized Maximum Score Estimators
Authors: Xiaohong Chen, Wayne Yuan Gao, Likang Wen
From: Yale University, University of Pennsylvania, University of Pennsylvania
When you can’t even get all the data you need
In their new paper, Yicong Jiang, Lucas Janson suggest:
Don’t estimate entire distributions → Just means and variances using data fusion with machine learning
The Problem: You want to study relationships between variables that are never measured together. For example, housing wealth and consumption spending are never observed jointly because they are in separate datasets, yet both datasets share common demographics (age, education, income). Using the Consumer Expenditure Survey (CEX) for consumption data and the Survey of Consumer Finances (SCF) for wealth data to estimate, in the traditional way, the correlation between wealth and consumption would have yielded a coefficient between -0.5 and +0.8, rather useless bounds. Estimating exact bounds required estimating entire probability distributions = statistically hard, computationally expensive, operationally difficult.
With data fusion it is possible to combine separate datasets using shared variables to study relationships between unmeasured pairs.
How It Works
Instead of estimating full distributions, use standard machine learning to estimate:
E[Wealth | Demographics] using one dataset
E[Consumption | Demographics] using the other dataset
Var[Wealth | Demographics] and Var[Consumption | Demographics]
Then use the Cauchy-Schwarz inequality to combine them into tight bounds on your answer. With data fusion and machine learning the correlation between consumption and wealth is between 0.31 and 1.10 with a 95% confidence. As a result, a positive relationship is confirmed despite never observing wealth and consumption together.
Other economic applications
- Loan discrimination: Study race + loan approval when regulations prohibit measuring both on same person
- Privacy-preserving research: Combine tax records (wealth) + health surveys (health) without revealing individual identities
Semiparametric Inference for Partially Identifiable Data Fusion Estimands via Double Machine Learning
Authors: Yicong Jiang, Lucas Janson
From: Harvard University
