Nonparametric Regression Using Kernel and Spline Methods

When applying nonparametric regression methods, the researcher is interested in estimating the relationship between one dependent variable, Y , and one or several covariates, X1, . . . , Xq. We discuss here the situation with one covariate, X (the case with multiple covariates is addressed in the references provided below). The relationship between X and Y can be expressed as the conditional expectation E(Y |X = x) = f(x). Unlike in parametric regression, the shape of the function f(·) is not restricted to belong to a specific parametric family such as polynomials. This representation for the mean function is the key difference between parametric and nonparametric regression, and the remaining aspects of the statistical model for (X,Y ) are similar between both regression approaches. In particular, the random variable Y is often assumed to have a constant (conditional) variance, Var(Y |X) = σ, with σ unknown. The constant variance and other common regression model assumptions, such as independence, can be relaxed just as in parametric regression.