Tham khảo tài liệu 'handbook of empirical economics and finance _8', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 198 Handbook of Empirical Economics and Finance Kernel Methods with Mixed Data Types So far we have presumed that the categorical variable is of the unordered nominal data type . We shall now distinguish between categorical discrete data types and real-valued continuous data types. Also for categorical data types we could have unordered or ordered ordinal data type variables. For an ordered discrete variable xd we could use Wang and van Ryzin 1981 kernel given by Xd xd X 1 - X 1 - Ầ x Xd-xd if Xf xd if Xf xd. We shall now refer to the unordered kernel defined in Equation as 7 so as to keep each kernel type separate notationally speaking. We shall denote the traditional kernels for continuous data types such as the Epanechnikov of Gaussian kernels by W ộ. A generalized product kernel for one continuous one unordered and one ordered variable would be defined as follows K W0 X X . Using such product kernels we can modify any existing kernel-based method to handle the presence of categorical variables thereby extending the reach of kernel methods. We define Ky Xi x to be this product where y h X is the vector of bandwidths for the continuous and categorical variables. Kernel Estimation of a Joint Density Defined over Categorical and Continuous Data Estimating a joint probability density function defined over mixed data follows naturally using these generalized product kernels. For example for one unordered discrete variable xd and one continuous variable xc our kernel estimator of the PDF would be n f xd xc 1 ĩ Xf xd W nhxc i xc xc This extends naturally to handle a mix of ordered unordered and continuous data . both quantitative and qualitative data . This estimator is particularly well suited to sparse data settings. Li and Racine 2003 demonstrate that s nhp J z f z - h2B _ z XB2 z N 0 V z in distribution where B1 z 1 2 fr V2f z W v v2dv B2 z Ex eD dx x 1 f x y -f x y and V z f z f W2 v dv . Nonparametric Kernel Methods for Qualitative and
