Wald inference on varying coefficients
Abhimanyu Gupta, Xi Qu, Sorawoot Srisuma, Jiajun Zhang
TL;DR
The paper develops a unified Wald-type inference framework for nonparametric varying-coefficient models with spatial dependence, enabling easy testing of linear restrictions via a series approximation of the coefficient functions. It systematically extends to baseline SAR, error-dependence (SHAC), misspecification-robust, and varying-spatial-coefficient settings, establishing asymptotic normality for four Wald statistics under broad conditions and finite-sample robustness. The authors demonstrate the approach with two empirical applications—testing CRS in a Chinese production-function context and examining distance-to-employment effects on Boston house prices—finding CRS evidence persists under spatial dependence and distance effects are nonlinear and location-specific. A comprehensive Monte Carlo study validates size and power properties across various designs, including nonparametric weight matrices and varying sieve dimensions, and highlights practical guidance on choosing basis functions and kernels. Overall, the framework provides a practical, extensible toolkit for robust inference on nonparametric varying coefficients in spatially dependent contexts, with broad relevance for regression models featuring cross-sectional and network interactions.
Abstract
We present simple to implement Wald-type statistics that deliver a general nonparametric inference theory for linear restrictions on varying coefficients in a range of regression models allowing for cross-sectional or spatial dependence. We provide a general central limit theorem that covers a broad range of error spatial dependence structures, allows for a degree of misspecification robustness via nonparametric spatial weights and permits inference on both varying regression and spatial dependence parameters. Using our method, we first uncover evidence of constant returns to scale in the Chinese nonmetal mineral industry's production function, and then show that Boston house prices respond nonlinearly to proximity to employment centers. A simulation study confirms that our tests perform very well in finite samples.