Marilena Furno

Parameter instability in quantile regression

The paper analyzes the behavior of a test for structural break based on quantile regression estimates. It considers the case of an estimated break in conjunction with independent and identically distributed (i.i.d.) and non-i.i.d. errors. It compares the null and the alternative models, where the null imposes stability, while the alternative allows the regression coefficients to change in response to the break. The test relies on the increase of the objective function and the worsening of the fit when unnecessary constraints are imposed. An example with serially correlated real data and a Monte Carlo study taking into account non-normal and non-i.i.d. errors analyze the behavior of the test.

LM TESTS IN THE PRESENCE OF NON-NORMAL ERROR DISTRIBUTIONS

The paper considers different versions of the Lagrange multiplier (LM) tests for autocorrelation and/or for conditional heteroskedasticity. These versions differ in terms of the residuals, and of the functions of the residuals, used to build the tests. In particular, we compare ordinary least squares versus least absolute deviation (LAD) residuals, and we compare squared residuals versus their absolute value. We show that the LM tests based on LAD residuals are asymptotically distributed as a I‡2 and that these tests are robust to nonnormality. The Monte Carlo study provides evidence in favor of the LAD residuals, and of the absolute value of the LAD residuals, to build the LM tests here discussed.

Quantile regression analysis of the Italian school system

We analyze the score on a reading literacy test of 15-year-old Italian students. The data depict a fracture in the Italian school system. By means of quantile regressions (QR) and by repeatedly implementing a QR-based test for structural break, computed in different sub-samples and at various quantiles, we can not only pin down the determinants of the gap but also rank them. We find that the difference in curricula is the main factor in explaining the gap in the students’ scores, and the regional difference is linked to structural and behavioural variables, like poor library facilities and student absenteeism, both mirroring the economic lag of the southern Italian regions. In terms of policy actions, curbing absenteeism in the south could help reduce the regional gap. If instead the target is to enhance excellence, funds could be directed towards academic track and/or north-centre regions.

LAD estimation with random coefficient autocorrelated errors

In this paper we compare the performance of LAD and OLS in the linear regression model with errors which are randomly autocorrelated. This model yields thick-tailed error distributions which make profitable to estimate the model by LAD. The LAD estimator for randomly autocorrelated errors is proved to be asymptotically normal. The Monte Carlo results show that LAD improves upon OLS, unless we revert to a constant autocorrelation model, where the two methods are comparable.

Quantile regression estimates and the analysis of structural breaks

The paper considers a test for structural breaks based on quantile regressions instead of OLS estimates. Besides granting robustness, this allows us to verify the impact of a break in more than one point of the conditional distribution. The quantile regression test is then repeatedly implemented as a diagnostic tool to uncover partial or spurious breaks. The test is also implemented to measure the contribution of each explanatory variable to the instability of the regression coefficients, thus finding which one of the different possible sources of breaks linked to the nature of the explanatory variables is the most effective. A real data example of exchange rates shows the presence of a time-driven break, but only at the lower quartile, while the analysis of the explanatory variable excludes its involvement in the break. Since the asymptotic distribution of the OLS test for structural change depends on i.i.d. normal errors and on the exogeneity of the explanatory variables, a Monte Carlo study analyses the behavior of OLS and quantile regression tests for structural changes with lagged endogenous variables, non-normal errors, spurious or partial breaks, and misspecification.

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.

Misspecification and estimation effect in the Lagrange multiplier tests for heteroskedasticity

The article considers the behaviour of the Lagrange multiplier (LM) tests for conditional and unconditional heteroskedasticity. These tests are based on auxiliary equations having as dependent variable the residuals from the main equation. Some studies have found that the LM tests for conditional heteroskedasticity over-reject the true null, whereas the LM tests for unconditional heteroskedasticity have an under-rejection problem. In order to improve the empirical size of these tests, different ad hoc correcting factors have been presented in the literature. Instead we propose to implement the auxiliary equation with better quality residuals, that is, with residuals from the median regression (LAD) instead of residuals from the mean regression (OLS). We show that the use of LAD residuals does not alter the asymptotic distribution of the LM test. The Monte Carlo study finds that the tests based on LAD residuals improve upon the same test functions implemented using OLS residuals. The improvement granted by LAD has the additional advantage of avoiding the various ad hoc correcting factors defined by different authors to solve the size problems of the analyzed test functions.

Non-linearity tests based on order statistics and quantile regressions

This article presents tests for neglected non-linearity based on order statistics. The tests rely on the estimation of parametric and non-parametric models. The parametric model imposes linearity and is estimated using least squares and, in turn, quantile regressions. The non-parametric model is estimated by implementing nearest neighbor on induced order observations. When we implement the quantile regression estimates, we analyze robust parametric and robust non-parametric results. This avoids faulty inference caused by the lack of robustness of the least squares estimator and allows us to avoid any distributional assumption. The proposed tests are easier to compute than the existing tests based on series expansions. Simulations and examples analyze their behavior and their robustness in the presence of outliers.