Support vector quantile regression for autoregressive data

Abstract In this paper we apply the autoregressive process to the nonlinear quantile regres-sion in order to infer nonlinear quantile regression models for the autocorrelated data.We propose a kernel method for the autoregressive data which estimates the nonlinearquantile regression function by kernel machines. Arti cial and real examples are pro-vided to indicate the usefulness of the proposed method for the estimation of quantileregression function in the presence of autocorrelation between data.Keywords: Autoregressive process, cross validation function, hyper-parameters, kernelfunction, support vector quantile regression. 1. Introduction Since Koenker and Bassett (1978) introduced linear quantile regression, quantile regres-sion has been a popular method for estimating the quantiles of a conditional distributiongiven input variables. Just as classical linear regression methods based on minimizing sumof squared residuals enable us to estimate a wide variety of models for conditional meanfunctions, quantile regression methods o er a mechanism for estimating models for the fullrange of conditional quantile functions, including the conditional median function. By sup-plementing the estimation of conditional mean functions with techniques for estimating anentire family of conditional quantile functions, quantile regression is capable of providing abetter statistical analysis of the stochastic relationships among random variables. An intro-duction to, and look at current research areas of quantile regression can be found in Koenkerand Hallock (2001), Yu et al. (2003), Koenker (2005) and Hwang (2010).Most nonparametric regression methods focus on estimating the regression function forvarious data types (Kim et al., 2008; Shim and Seok, 2008). The estimation of regressionfunction from a data set is usually performed under the assumption that the error terms areiid (Juditsky et al., 1995). This assumption is not satis ed when the correlation is presentin the given data (e.g. time series data), which leads to severe problems on the estimationof a model under the iid assumption.In this paper, we consider the autoregressive model, where y