Abstract
This paper studies estimation and inference for linear quantile regression models with generated regressors. We suggest a practical two-step estimation procedure, where the generated regressors are computed in the first step. The asymptotic properties of the two-step estimator, namely, consistency and asymptotic normality are established. We show that the asymptotic variance-covariance matrix needs to be adjusted to account for the first-step estimation error. We propose a general estimator for the asymptotic variance-covariance, establish its consistency, and develop testing procedures for linear hypotheses in these models. Monte Carlo simulations to evaluate the finite-sample performance of the estimation and inference procedures are provided. Finally, we apply the proposed methods to study Engel curves for various commodities using data from the UK Family Expenditure Survey. We document strong heterogeneity in the estimated Engel curves along the conditional distribution of the budget share of each commodity. The empirical application also emphasizes that correctly estimating confidence intervals for the estimated Engel curves by the proposed estimator is of importance for inference.
Highlights
Since the seminal work of Koenker and Bassett (1978), quantile regression (QR) models have provided a valuable tool in economics, finance, and statistics as a way of capturing heterogeneous effects of covariates on the outcome of interest, exposing a wide variety of forms of conditional heterogeneity under weak distributional assumptions
We propose an estimator for the asymptotic variance-covariance of the QR-generated regressors (GRs) coefficients, and formally establish its consistency
We evaluate the quantile regression with generated regressor (QR-GR) estimator in terms of empirical bias and root mean squared error, and compare its performance with methods that are not designed for dealing with GR
Summary
Since the seminal work of Koenker and Bassett (1978), quantile regression (QR) models have provided a valuable tool in economics, finance, and statistics as a way of capturing heterogeneous effects of covariates on the outcome of interest, exposing a wide variety of forms of conditional heterogeneity under weak distributional assumptions. We show that the asymptotic variance-covariance matrix needs to be adjusted to account for the first-step estimation error. The estimated limiting distribution of the first-step is used to consistently estimate the variance-covariance matrix of the parameters of interest. We propose an estimator for the asymptotic variance-covariance of the QR-GR coefficients, and formally establish its consistency. We establish the asymptotic properties of the QR-GR estimator for non-iid data under weak conditions This is an important generalization for practitioners since it allows for inference in a more general class of models. Researchers can use existing software packages for the first-step estimation and to construct the regressors needed, for example, MLE, OLS, QR or GMM, and apply the QR procedure with our described variance-covariance matrix adjustment.
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