# Nonparametric Estimation and Inference in Economic and Psychological Experiments

## TL;DR

This paper develops nonparametric estimation tools for analyzing responses in economic and psychological experiments, providing conditions for consistency, optimal convergence rates, and inference procedures considering complex covariance structures.

## Contribution

It introduces a sieve-based nonparametric estimator for experimental data with multiple responses, offering new insights into optimal experiment design and inference under diverse covariance conditions.

## Key findings

- Optimal divergence rate for sieve basis dimension identified
- Guidance on balancing subjects and questions in experiments provided
- Asymptotic normality of estimators established under covariance estimation

## Abstract

The goal of this paper is to provide some tools for nonparametric estimation and inference in psychological and economic experiments. We consider an experimental framework in which each of $n$subjects provides $T$ responses to a vector of $T$ stimuli. We propose to estimate the unknown function $f$ linking stimuli to responses through a nonparametric sieve estimator. We give conditions for consistency when either $n$ or $T$ or both diverge. The rate of convergence depends upon the error covariance structure, that is allowed to differ across subjects. With these results we derive the optimal divergence rate of the dimension of the sieve basis with both $n$ and $T$. We provide guidance about the optimal balance between the number of subjects and questions in a laboratory experiment and argue that a large $n$is often better than a large $T$. We derive conditions for asymptotic normality of functionals of the estimator of $T$ and apply them to obtain the asymptotic distribution of the Wald test when the number of constraints under the null is finite and when it diverges along with other asymptotic parameters. Lastly, we investigate the previous properties when the conditional covariance matrix is replaced by an estimator.