Regression Estimation and Time Series Analysis

Abstract

Let (X_i,Y_i)_N be a sequence of random pairs valued in ℝ^d ×ℝ such that the regression function of Y on X, defined for x ∈ ℝ^d by

$$ {\text{r(x)}} = {\text{E}}\left( {{\text{ }}{{\text{Y}}_{\text{i}}}\mid {X_{\text{i}}} = {\text{x}}} \right){\text{ for any i}} = 1.2.....$$

((3.1.1))

exists. Such a condition is in particular satisfied when the process (X_i, Y_i) is stationary, but this assumption will not be necessary for a lot of results presented here. Here r has to be understood as an arbitrary element of the equivalence class of functions defined by (3.1.1). The knowledge of r is helpful in constructing estimates of future values of Y given X)x and it is useful in understanding the relation between the variables X and Y.