Goodness of Fit and Misspecification in Quantile Regressions

The article considers a test of specification for quantile regressions. The test relies on the increase of the objective function and the worsening of the fit when unnecessary constraints are imposed. It compares the objective functions of restricted and unrestricted models and, in its different formulations, it verifies (a) forecast ability, (b) structural breaks, and (c) exclusion restrictions. The quantile-based tests are more informative than their ordinary least squares (OLS) analogues because they allow to analyze the model not only at the center but also in the tails of the conditional distribution. In this example, contrarily to the OLS findings, the quantile-based test uncovers in (a) the forecast weakness of the selected model at the upper quantile; (b) a break occurring in the tails and not in the center of the conditional distribution; and (c) that the excluded variable has a relevant impact at the upper quantile. Monte Carlo experiments analyze the behavior of the different definitions of the test with non-normal errors, comparing least squares and quantile regression results.