Goodness of Fit and Misspecification in Quantile Regressions
Marilena Furno*
July 2009
Abstract
The paper 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, (c) exclusion restrictions. The quantile based tests are more informative than their OLS analogues since they allow to analyze the model not only at the center but also in the tails of the conditional distribution. In our example, contrarily to the OLS findings, the quantile based test uncovers in (a) the forecast weakness of the selected model at the upper quantile; in (b) a break occurring in the tails and not in the center of the conditional distribution; in (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 nonnormal errors, comparing least squares and quantile regression results.
Keywords: quantile regression, test, structural break, forecasts, exclusion restriction.
* Department of Economics, Università di Cassino. Address for correspondence: Via Orazio 27/D, 80122, Napoli, Italy e-mail:
[email protected].
1. Introduction Inference in the quantile regression framework is an on-going topic of research. Quantile regression is a robust procedure and ensures estimation and inference not excessively sensitive to small departures from the assumed model.1 The value added of quantile regression with respect to other procedures is to allow a detailed analysis of the regression, not only at the mean but also in the tails of the conditional distribution, at different quantiles. This is a relevant feature not only for estimation but for testing purposes as well. Estimation and inference based on global hypotheses may average out important features, for instance effects that take place in the tails and that on average cancel out. The widespread use of OLS is linked to its ease of interpretation even under misspecification, since OLS provides the minimum mean squared error linear approximation to the conditional expectation. An analogous result is now available for the quantile regression. Angrist et al. (2006) show that, under misspecification, quantile regression provides the best linear predictor to the conditional quantile under a weighted mean squared error loss function. The paper discusses a test on the correct specification of a quantile regression. The test compares the objective functions of the same model estimated both under the null and the alternative, where the two hypotheses define different specifications of the same equation. The increase of the objective function in case of incorrect model specification allows to select the appropriate formulation. A Monte Carlo study considers the behavior of this test. The case study here discussed, the score of the test on mathematical skill of the PISA (Program for International Student Assessment) survey for the year 2000 in Italy, shows the validity of the quantile regression analysis, which is extremely useful to improve the understanding of the model. Indeed with the quantile based test we find evidence in favor of a break, as in the OLS analogue, but in addition we spot the levels, i.e. the quantiles, where the break is more pronounced. We find that the break is very effective in the tails while it is quite irrelevant around the median. Then we uncover a weak point in the forecast ability of the model, which forecasts well at most quantiles but not at the higher one, which can be relevant if the policy target is to support and enhance high quality. Finally we find that the excluded regressor is an appropriate explanatory variable only for the top scoring students, at the 95th quantile, while at the remaining quantiles its contribution is not statistically different from zero. Once again this is important if the target is to improve excellence. There are rather few works on tests of specification for parametric quantile regressions. The seminal paper by Koenker and Machado (1999) defines a likelihood ratio test for exclusion restrictions in the quantile regression and provides the guideline for the discussion of the test here 1
Quantile regression is robust with respect to outliers in the error distribution. 2
presented. Kim and White (2002) define a test of specification based on the quantile version of the information matrix equality. Furno (2007) builds on the Koenker and Machado (1999) likelihood ratio test to define an F test and Furno (2009 b) discusses a Lagrange multiplier test, where both these test are designed to verify the presence of structural break in quantile regression. On this same issue Qu (2008) presents a cusum and a Wald type test. Related work on robust tests of specification can be found in Huskova and Picek (2005). They present a test for structural change based on bootstrapping M-estimators. Their approach extends to quantile regression, but they do not analyze this issue. In a paper by Gagliardini et al. (2005) robust GMM tests for structural break are proposed, which are asymptotically equivalent to Wald, Lagrange multiplier and likelihood ratiotype statistics, but once again this work does not focus on quantile regression. Finally, He and Zhu (2003) look at a cusum process of the gradient vector of a quantile estimator in order to avoid any smoothing procedure involved in nonparametric estimation. This paper considers the comparison of parametric models estimated, under both the null and the alternative, with quantile regression. A generalization of this approach, left to future research, is to compute non-parametrically the model under the alternative. This would allow to define an omnibus test, which would not require a specific functional form under the alternative. Section 2 briefly reviews estimation, goodness of fit and inference in the quantile regression. Section 3 discusses the test and Section 4 its implementations. Section 5 considers an application with real data while sections 6 and 7 present a Monte Carlo study. The last section concludes the paper.
2. Goodness of Fit and Inference in Quantile Regressions Consider the standard linear regression model yt = xt β + εt, where yt is the dependent variable, xt is the k-row vector of a single observation for the k explanatory variables, εt is the i.i.d. error term having continuous and strictly positive density f(.) at the selected quantile, in a sample of size n. The objective function of the quantile regression estimator, for the chosen quantile θ, is given by V(b(θ)) = ∑ y ≥ x bθ yt − xt b + ∑ y t < xt b (1 − θ ) yt − xt b = Σρ (ε t ) with ρ(εt)=εt(θ-I(εt
