A residual-based test for autocorrelation in quantile regression models

ABSTRACT

Quantile regression (QR) models have been increasingly employed in many applied areas in economics. At the early stage, applications in the QR literature have usually used cross-sectional data, but the recent development has seen an increase in the use of QR in both time-series and panel data sets. However, testing for possible autocorrelation, especially in the context of time-series models, has received little attention. As a rule of thumb, one might attempt to apply the usual Breusch–Godfrey LM test to the residuals of a baseline QR. In this paper, we demonstrate analytically and by Monte Carlo simulations that such an application of the LM test can result in potentially large size distortions, especially in either low or high quantiles. We then propose a correct test (named the QF test) for autocorrelation in QR models, which does not suffer from size distortion. Monte Carlo simulations demonstrate that the proposed test performs fairly well in finite samples, across either different quantiles or different underlying error distributions.

KEYWORDS:

• Quantile regression
• autocorrelation
• LM test

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 62

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. In the OLS-LM test (i.e. the base model in Equation (4(4) $y_t={\beta }_{0\theta }+{\beta }_{1\theta }{w}_t+{ϵ}_{\theta t},$(4) ) being estimated by OLS), the effect of including $w_t$ in the auxiliary regression is negligible under the conditions as specified in the current simulation setup: (i) $w_t$ is exogenous and (ii) no lag of $y_t$ is included as an explanatory variable. However, when the base model is estimated by quantile regression, it induces non-negligible size distortions either at low or high quantiles as shown in Table 1(a). Hence, one might wonder what makes such a difference. As we will show in the next section, the usual F test does not suffer from any size distortion. In QR, the residuals $e_{\theta t}$ are not orthogonal to $w_t$, which in turn prevents the LM statistic ${\mathrm{LM}}_T$ from being asymptotically equivalent to the usual F statistic. Therefore, we conjecture that such non-equivalence between the LM statistic and the F statistic might cause the non-negligible effect of including $w_t$ in the auxiliary regression.

Additional information

Funding

This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2014S1A5A2A01010546)