Testing Shape Restrictions with Continuous Treatment: A Transformation Model Approach

TL;DR

This paper introduces a new statistical testing method for shape restrictions like convexity or concavity in transformation models, useful for verifying treatment effects without estimating transformations or error distributions.

Contribution

It develops a bootstrap-based testing procedure for shape restrictions in semiparametric models that does not require estimating the transformation or error distribution.

Findings

The tests effectively verify convexity or concavity in empirical data.

Bootstrap critical values provide accurate inference.

Application demonstrates testing of loan demand convexity.

Abstract

We propose tests for the convexity/linearity/concavity of a transformation of the dependent variable in a semiparametric transformation model. These tests can be used to verify monotonicity of the treatment effect, or, equivalently, concavity/convexity of the outcome with respect to the treatment, in (quasi-)experimental settings. Our procedure does not require estimation of the transformation or the distribution of the error terms. The statistic takes the form of a U statistic or a localised U statistic, and we show that critical values can be obtained by bootstrapping. In our application we test the convexity of loan demand with respect to the interest rate using experimental data from South Africa.