Suppose y is a function of x but the slope of the relationship changes at x=k₁. A regression line for such a function is achieved by defining a new variable x such that

x₁ = 0 if x<k₁
and
x₁ = x − k₁ for k₁≤x

The variable x₁ can be defined more succinctly as x₁=u(x-k₁), where u(z) is the function such that u(z)=0 if z<0 and u(z)=z for z≥0.

The regression of y on x and x₁ gives an equation such as

y = d₀ + c₀x + c₁x₁

The coefficient c₁ gives the change in the slope of the relationship at k₁.

Thus the slopes of the relationship are:

dy/dx = c₀ for x<k₁
and
dy/dx = c₀+c₁ for x≥k₁

For more than one bendpoint the procedure is analogous and the slope of the relationship is the sum of the coefficients up to the different levels of x.

The Constrained Case

This covers the case in which the slopes of the relationship over different intervals are required to be the same. For example, suppose y is a function of time such that there are bend points at k₁ and k₂. Furthermore suppose the slope of the relationship in the third interval (from k₂<x has to be the same as the slope in the first interval, from 0<x<k₁.

Now consider the case in which a regression of y on variable x, z and w has to be of the form

y = a + bx + bz + cw

This form is the same as

y = a + b(x+z) + cw

That is to say, y must be regressed on (x+z) and w. Adding two variables together forces the regression coefficients to be the same.

Likewise if the regression has to be of the form

y = a + bx −bz + cw
then
y = a + b(x-z) +cw

Now consider again the previous example in which the slope in the third interval of a trend line is required to be the sames as the slope in the first interval. First two additional variables x₁ and x₂ need to be defined as

x₁ = u(x-k₁)
and
x₂ = u(x-k₂)

An unconstrained regression would yield an equation of the form

y = d₀ + c₀x + c₁x₁ + c₂₂

The slope of the relationship in the first interval is c₀ and in the third interval it is c₀+c₁+c₂. For the slopes in the first and third interval to be equal requires that

c₀ = c₀+c₁+c₂
which reduces to
c₂ = −c₁

Therefore the regression equation is of the form₂

y = d₀ + c₀x + c₁x₁ − c₁x₂
which reduces to
y = d₀ + c₀x + c₁(x₁−x₂)

Thus y must be regressed on x and (x₁−x₂).

This method can be generalized.

Suppose the relationship cyclic relationship is of the form

Then the regression would use a generated variable of the form

The other variable in the regression would just be the trend variable.