Ex. 6.6

Suppose that all you have is software for fitting local regression, but you can specify exactly which monomials are included in the fit. How could you use this software to fit a varying-coefficient model in some of the variables?

Soln. 6.6

This exercise relates to Section 6.4.2 Structured Regression Functions in the text.

Consider a varying-coefficient model formulated as follows. We divide \(p\) predictors in \(X\) into a set \((X_1, X_2,...,X_q)\) with \(q < p\), and the remainder of the variables are denoted by a single vector \(Z\). Assume the linear model

\[\begin{equation} \label{eq:6-6a} f(X) = \alpha(Z) + \beta_1(Z)X_1 + \cdots + \beta_1(Z)X_q.\non \end{equation}\]

Note that the coefficients \(\beta\) can vary with \(Z\).

Assume that all but \(j\)-th term \(\beta_j(Z)\) is known, we can estimate \(\beta_j(Z)\) by a local regression

\[\begin{equation} \min_{\beta_j(z_0)}\sum_{i=1}^NK_\lambda(z_0, z_i)(y_i-\alpha(z_0) - x_{i1}\beta_1(z_0) - \cdots - x_{qi}\beta_q(z_0))^2.\non \end{equation}\]

This is done for each \(\beta_j\) in turn, repeatedly, until convergence. At each step, only a one-dimensional local regression is needed.