Solution to the Segmented Regression Problem

Here's my solution to the "segmented regression" problem that I posed yesterday. Thanks for the comments and suggestions!

You'll recall that what we wanted to do was to end up with a fitted least squares "line" looking like this:

In particular, the "kink" in the line is at a pre-determined point - in this example when x = 30.

Here's how we can achieve this:

The basic regression model is

              y_i = α + β x_i + ε_i    ;     i = , 2, ...., n .                                                              (1)

Suppose that we want the line segments to join when x = x*. Then, define a dummy variable, Di, such that:

             D_i = 0      ;    if xi ≤ x*

             D_i = 1      ;    if xi > x*

The two line segments in the graph above have different intercepts and different slopes, so would probably think of modifying model (1) to become:

             y_i = α + β x_i + γ D_i + δ (x_iD_i) + ε_i    ;     i = , 2, ...., n .                                    (2)

That's a good start, but we still have force the join-point to be at x*.

This requirement amounts to the following restriction on the parameters of the model:

             γ + δ x* = 0,

where x* is just a known number (30 in my example above).

Using this restriction to eliminate γ from equation (2), we get:

             y_i = α + β x_i + δ D_i (x_i - x*) + ε_i    ;     i = 1, 2,..., n .                                     (3)

Here is the EViews output for my estimated regression model:

The EViews workfile is on the code page for this blog, and the data I used are available on the data page.

We can then generate within-sample forecasts, separately, for observations 1 to 30, and observations 30 to 100. If these series are called YFORC1 and YFORC2, this is (part of) what we get:

Notice that YORC1 = YFORC2 at observation 30, as required.

If we then gather X, Y, YFORC1 and YFORC2 into a group, and produce a scatter-plot, here's the result we wanted:

So, it all comes down to the use of a dummy variable and a restriction of the regression coefficients. One without the other won't work.

Ryan commented on the post in question, and suggested that (in my notation) we estimate the model:

                     y_i = α (x_i - 30) + β D_i (x_i -30) + ε_i .

This produces the following results:

Ryan gets the join-point alright, but the fit over the first sub-sample doesn't look very convincing. Sorry, buddy!

© 2013, David E. Giles