Nonparametric Estimation and Inference in Psychological and Economic Experiments

The goal of this paper is to provide some statistical tools for nonparametric estimation and inference in psychological and economic experiments. We consider a framework in which a quantity of interest depends on some primitives through an unknown function $f$. An estimator of this unknown function can be obtained from a controlled experiment in which $n$ subjects are gathered, and a vector of stimuli is administered to each subject who provides a set of $T$ responses. We propose to estimate $f$ nonparametrically using the method of sieves. We provide conditions for consistency of this estimator when either $n$ or $T$ or both diverge to infinity, and when the answers of each subject are correlated and this correlation differs across subjects. We further demonstrate that the rate of convergence depends upon the covariance structure of the error term taken across individuals. A convergence rate is also obtained for derivatives. These results allow us to derive the optimal divergence rate of the dimension of the sieve basis with both $n$ and $T$ and thus provide guidance about the optimal balance between the number of subjects and the number of questions in a laboratory experiment. We argue that in general a large value of $n$ is better than a large value of $T$. Conditions for asymptotic normality of linear and nonlinear functionals of the estimated function of interest are derived. These results are further applied to obtain the asymptotic distribution of the Wald test when the number of constraints under the null is finite and when it diverges to infinity along with other asymptotic parameters. Lastly, we investigate the properties of the previous test when the conditional covariance matrix is replaced by a consistent estimator.