Finite-sample generalized confidence distributions and sign-based robust estimators in median regressions with heterogenous dependent errors
We
study the problem of estimating the parameters of a linear median regression
without any assumption on the shape of the error distribution – including no
condition on the existence of moments – allowing for heterogeneity (or
heteroskedasticity) of unknown form, noncontinuous distributions, and very
general serial dependence (linear and nonlinear). This is done through a
reverse inference approach, based on a distribution-free testing theory [Coudin
and Dufour (2009, The Econometrics Journal)], from which confidence sets and
point estimators are subsequently generated. The estimation problem is tackled
in two complementary ways. First, we show how confidence distributions for
model parameters can be applied in such a context. Such distributions – which
can be interpreted as a form of fiducial inference – provide a frequency-based
method for associating probabilities with subsets of the parameter space (like
posterior distributions do in a Bayesian setup) without the introduction of
prior distributions. We consider generalized confidence distributions applicable
to multidimensional parameters, and we suggest the use of a projection
technique for confidence inference on individual model parameters. Second, we
propose point estimators, which have a natural association with confidence
distributions. These estimators are based on maximizing test p-values and
inherit robustness properties from the generating distribution-free tests. Both
finite-sample and large-sample properties of the proposed estimators are
established under weak regularity conditions. We show they are median unbiased
(under symmetry and estimator unicity) and possess equivariance properties.
Consistency and asymptotic normality are established without any moment
existence assumption on the errors, allowing for noncontinuous distributions,
heterogeneity and serial dependence of unknown form. These conditions are
considerably weaker than those used to show corresponding results for LAD
estimators. In a Monte Carlo study of bias and RMSE, we show sign-based
estimators perform better than LAD-type estimators in heteroskedastic settings.
We present two empirical applications, which involve financial and
macroeconomic data, both affected by heavy tails (non-normality) and
heteroskedasticity: a trend model for the S&P index, and an equation used
to study β-convergence of output
levels across U.S. States.
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