Depth-weighted means of noisy data: An application to estimating the average effect in heterogeneous panels

We study the depth-weighted L-type location estimator of multivariate data when the observations are measured with noise. Under a drifting asymptotic framework, we show that the depth-weighted mean estimators with noisy data are still consistent and asymptotically mean-zero Gaussian under mild conditions. We apply the results to longitudinal data models of heterogeneous agents and develop the depth-weighted mean-group estimator of a vector of random coefficients, which estimates the multivariate average effect in heterogeneous panels or among heterogeneous treatment effects. As an empirical illustration, we examine the relative purchasing power parity.