面板数据模型的动态随机效应分位数回归

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Introduction
Quantile regression (QR) has attracted considerable attention since its introduction in Koenker and Bassett (1978). While being robust to outliers, QR permits heterogeneity in covariate effects at different parts of the distribution and thus provides more information on the relationship between the outcome variable and the covariates. Koenker (2004) considers QR in fixed effects (FE) panel data models which are capable of controlling for individual heterogeneity, proposing the fixed effect quantile regression (FEQR) estimator and the penalised quantile regression (PQR) estimator. Lamarche (2010) considers PQR in random effects (RE) panel data models, deriving the optimal choice of the penalty coefficient. However, many economic relationships are dynamic and thus cannot be correctly modeled by static panel data models. Galvao (2009) and Galvao and Montes-Rojas (2010) extend QR to the dynamic FE panel data context. This paper considers a dynamic RE panel data model. We propose an L2 penalised quantile regression (L2 PQR) estimator and derive its asymptotic properties. The L2 penalty proposed in this paper is more appropriate for random effects models and can be easily justified in the Bayesian framework of Yu and Moyeed (2001). Moreover, it facilitates an asymptotic analysis while still remaining computationally convenient. Koenker (2004) demonstrates the benefit of imposing a penalty term in QR panel data. The previous literature in QR for panel data uses L1 penalty. Tibshirani (1996) points out and Koenker (2004) notes that the L1 penalty acts more like a model selection device, and thus is more appropriate for FE models in which the individual effects are treated as fixed parameters. In RE models where individual effects are considered as random realisations from a common distribution, using a model selection device on the individual effects is not appropriate. While it is difficult to justify the L1 penalty for RE panels, our L2 penalty naturally results from a Bayesian approach when the individual effects are assumed to be independently and identically distributed as a normal distribution with zero mean. We also show that, in RE dynamic panel data models, an L1 penalty poses a difficulty for the derivation of the asymptotic properties of the resulting estimator (for details, see section 2.1) while the L2 penalty facilitates an asymptotic analysis. The previous literature in QR for panel data emphasises the computational convenience of the L1 penalised QR estimator. Our L2 penalty converts the minimisation of the criterion function into a quadratic programming problem. Many fast algorithms for large-scale quadratic programming are available and large panels can be handled efficiently. The main contribution of this paper is to propose an L2 penalised quantile regression for dynamic RE panel data models and obtain its asymptotic distribution. We also prove a uniform convergence result that is used in previous works (see e.g. Galvao, 2009; Galvao and MontesRojas, 2010) without formal proof. The L2 penalty introduces an asymptotic bias the size of which depends on the variance of the individual specific effect, the coefficient of the lagged dependent variable and the coefficient of the L2 penalty. In order for the penalty term to be 2
JEL Classification: C32, C33 Keywords: Dynamic Quantile Regression, Random Effects, Panel Data Models.
Hale Waihona Puke Address: Faculty of Economics, Sidgwick Avenue, Cambridge, CB3 9DD, United Kingdom. Email address: qf204@. † I would like to thank Alexei Onatski, Vitaliy Oryshchenko, M. Hashem Pesaran and Richard J. Smith for their helpful comments.
L2 Penalised Quantile Regression for Dynamic Random Effects Panel Data Models
QIANG FENG∗† University of Cambridge October, 2011
Abstract This paper considers penalised quantile regression in dynamic random effects panel data models. Previous literature in quantile regression for panel data models advocates the application of penalisation to improve finite sample performance of the quantile regression estimator, focusing on fixed effects and the L1 penalty. In this paper, we consider random effects and propose an L2 penalised quantile regression estimator. We show that the L2 penalty, which arises naturally in a Bayesian framework, is more appropriate for random effects models whereas the L1 penalty, acting more like a model selection device, is more appropriate for fixed effets. Moreover, the L2 penalty facilitates an asymptotic analysis in the presence of lagged dependent variables, whilst the L1 penalty poses difficulties. We obtain the asymptotic properties of the proposed L2 penalised quantile regression estimator, finding that the L2 penalty introduces an asymptotic bias whose size depends on the penalisation coefficient, the variance of the individual effect and the coefficient on the lagged dependent variable. We show that the penalisation coefficient needs to grow at rate T /N , which differs from the rate in fixed effects models. Monte Carlo experimentation suggests that our proposed estimator exhibits encouraging finite sample performance.