Abstract

A spline regression is a regression that allows for discontinuities at points along the regression line. For example, connecting two straight line segments at a point, called a spline knot or join point, is the most basic form of a spline regression. In contrast, dummy variable regression allows for abrupt breaks or “jumps” in a regression line. For example, if a savings account is earning 1% interest, then the value of the savings can be represented by a straight line with a rising slope. If the interest rate then increases to 2%, there is no sudden change (no “jump”) in the savings. Instead, the line rotates at that point (known as a spline knot) and rises more steeply to represent the higher interest rate. Spline regression requires that the function itself be continuous, even if its slope changes at specified points. In other words, a straight line can be separated into pieces with different slopes as long as those pieces touch at the transition points. This is known as piecewise-linear regression. Curved lines can also be connected in a similar manner. Quadratic and cubic splines are popular, but any set of polynomial pieces can be connected at spline knots. Unlike polynomial splines, fractional spline regressions are not limited to integer-valued exponential powers but can have decimal places such as 2.1357. Sometimes the spline knot locations or the degree of the polynomial splines are not known in advance and must be determined by a search procedure. This can be accomplished using stepwise regression when the spline knot adjustment variables are limited to a set of integer-valued polynomials. This entry first introduces and summarizes spline regression techniques and then provides more details regarding the use of these techniques.