Sanjog Misra

Sanjog Misra

Foundation models, and in particular large language models, can generate highly informative responses, prompting growing interest in using these ''synthetic'' outputs as data in empirical research and decision-making. This paper introduces the idea of a foundation prior, which shows that model-generated outputs are not as real observations, but draws from the foundation prior induced prior predictive distribution. As such synthetic data reflects both the model's learned patterns and the user's subjective priors, expectations, and biases. We model the subjectivity of the generative process by making explicit the dependence of synthetic outputs on the user's anticipated data distribution, the prompt-engineering process, and the trust placed in the foundation model. We derive the foundation prior as an exponential-tilted, generalized Bayesian update of the user's primitive prior, where a trust parameter governs the weight assigned to synthetic data. We then show how synthetic data and the associated foundation prior can be incorporated into standard statistical and econometric workflows, and discuss their use in applications such as refining complex models, informing latent constructs, guiding experimental design, and augmenting random-coefficient and partially linear specifications. By treating generative outputs as structured, explicitly subjective priors rather than as empirical observations, the framework offers a principled way to harness foundation models in empirical work while avoiding the conflation of synthetic ''facts'' with real data.

Max H. Farrell, Tengyuan Liang, Sanjog Misra

This paper integrates deep neural networks (DNNs) into structural economic models to increase flexibility and capture rich heterogeneity while preserving interpretability. Economic structure and machine learning are complements in empirical modeling, not substitutes: DNNs provide the capacity to learn complex, non-linear heterogeneity patterns, while the structural model ensures the estimates remain interpretable and suitable for decision making and policy analysis. We start with a standard parametric structural model and then enrich its parameters into fully flexible functions of observables, which are estimated using a particular DNN architecture whose structure reflects the economic model. We illustrate our framework by studying demand estimation in consumer choice. We show that by enriching a standard demand model we can capture rich heterogeneity, and further, exploit this heterogeneity to create a personalized pricing strategy. This type of optimization is not possible without economic structure, but cannot be heterogeneous without machine learning. Finally, we provide theoretical justification of each step in our proposed methodology. We first establish non-asymptotic bounds and convergence rates of our structural deep learning approach. Next, a novel and quite general influence function calculation allows for feasible inference via double machine learning in a wide variety of contexts. These results may be of interest in many other contexts, as they generalize prior work.

Max H. Farrell, Tengyuan Liang, Sanjog Misra

We study deep neural networks and their use in semiparametric inference. We establish novel rates of convergence for deep feedforward neural nets. Our new rates are sufficiently fast (in some cases minimax optimal) to allow us to establish valid second-step inference after first-step estimation with deep learning, a result also new to the literature. Our estimation rates and semiparametric inference results handle the current standard architecture: fully connected feedforward neural networks (multi-layer perceptrons), with the now-common rectified linear unit activation function and a depth explicitly diverging with the sample size. We discuss other architectures as well, including fixed-width, very deep networks. We establish nonasymptotic bounds for these deep nets for a general class of nonparametric regression-type loss functions, which includes as special cases least squares, logistic regression, and other generalized linear models. We then apply our theory to develop semiparametric inference, focusing on causal parameters for concreteness, such as treatment effects, expected welfare, and decomposition effects. Inference in many other semiparametric contexts can be readily obtained. We demonstrate the effectiveness of deep learning with a Monte Carlo analysis and an empirical application to direct mail marketing.