Auto-selection Of Varying Coefficient In Quantile Regression

In statistical studies, data and models are essential two parts, they have equal importance. It is a very critical issue that whether a model can correctly describe the data information. So on their research has attracted wide attention. Statistical research summarized data information to solve practical problems through data analysis, the usual practice is to introduce data into statistical models to study. Notably, the statistical model of the complex relationship between itself is only an approximate description of the variables, the starting point is the intuitive concept of some people are able to accept, or sometimes the relationship between the number itself is based on certain assumptions made by the test of researchers. But the crucial point is that whether the established by researchers could reflect the implied data relationship, whether the statistical model could solve the problem of statistical research want. If some errors appeared during data acquisition due to measurement error or other irresistible factors that appear factors, the inference of statistical model will be impacted so much. How to deal with this situation is worthy work.30years ago, Koenker and Bassett (1978) proposed quantile regression model. Since that time, this model is applied to almost every field of the society. Compared to the least squares regression, quantile regression has more lenient application conditions, the more strong ability to mine information from the data. It make the regression analysis based on the condition quantile of the dependent variable with the predict variable, this could get the entire quantile regression model. Therefore, compared to ordinary least squares regression, quantile regression is much better when the concern problem is special.It is easy to interpret the characteristics and reduce the difficulty of estimation when parametric quantile regression is used. However, non-parametric quantile regression model and semi-parametric quantile regression model have attracted much more attentions. The reason is that, not only in theory but also in the practice, the two models are more flexible than the parametric regression model, In modern statistics, regression analysis of high-dimensional data is a hot topic, but there always exists "curse of dimensionality" problem, this problem makes it difficult to implement non-parametric quantile regression model in practice. Semi-parametric quantile regression model contains both a wealth of non-parametric model with flexible properties, but also has the ability to capture the linear relationship between the variables, thus easier to interpret this part of the model. Semi-parametric quantile regression model has attracted unprecedented attention in the statistical study. And this is a top hot research field. This paper proposes a partially linear varying coefficient quantile models.This study is mainly motivated by the problem how those data only numbers without a deep insight prior knowledge of the background data. In the situation of data without a deep insight prior knowledge, the proposed new partially linear coefficient models with some of the coefficients of the independent variables may be a constant, some of them may be a non-parametric function. So that the complex relationship between the variables can be observed in this model, and also it capture the dynamics feature. So by adding the penalty function in the objective function, the data is automatically direct select the independent variable is a constant or non-parametric function coefficient. It could avoid invalidation that may be caused due to a misspecified model. In the varying coefficient model framework, the penalty function unlike other penalties as a direct penalty function of the amount of the coefficient to be taken into account, instead the norm penalty of the first derivative of the coefficients. And the use of adaptive Lasso penalty idea that the small norm of the first derivative of coefficients is penalized to be a constant, and those larger norm first derivative coefficients are reserved as a varying coefficient that depend on the an indicator variable. This is an automatic choice for varying coefficient quantile regression model. This method is based on data derived from the model structure without any a priori assumption, and that derived structures can make a good recommendation to assume further follow-up model. This model combines the advantages of semi-parametric and quantile regression models. In addition, especially for those just given numeric data but without deep insight background information, we are able to establish the dependent variable and the independent variables linear or dynamic relationship to provide advice and assistance to the subsequent modeling and decision-making.Estimation method proposed in this paper is automatically selected and estimated varying coefficient quantile regression model. Assume that all the coefficients is nonlinear, the first step gets a consistent estimate of the coefficients, and then calculate the weight of each independent variable coefficient the first derivative norm penalty in objective function, if the argument is a constant factor, then the norm of its first derivative is small, the penalty is estimated to get out of this coefficient is constant. And vice versa. To demonstrate this method for practical economic problems have good explanatory, this paper selected Boston housing prices data as this method application example. Using this data analysis to get a low level of education in the areas of high population ratio of people are reluctant to buy a house, so naturally willing to pay lower prices, house prices in the region will be low. The low level of education of the population in low rate areas, this region will be more secure, environment is also good, and people are willing to live in such a good place. The factor of low level education is real an indicator variable that some impacts of the independent variables on the price vary with the indicator variable.