5.3 Broken stick regression

Useful for systems that may suffer a "shock".

Another application of categorical variables.

Model:

$$Av(Y\vert x,z) = \beta_0 + \beta_1 x + \beta_2\times z\times (x - T),$$

where z = 0 if x < T and z = 1 if $x \ge T$, and T is the "breakpoint".

Case 1, x < T, plug in to get

$$Av(Y\vert x,z) = \beta_0 + \beta_1 x + \beta_2\times 0\times (x - T),$$

$$Av(Y\vert x,z) = \beta_0 + \beta_1 x,$$

Case 2, $x \ge T$, plug in to get

$$Av(Y\vert x,z) = \beta_0 + \beta_1 x + \beta_2\times 1\times (x - T),$$

$$Av(Y\vert x,z) = \beta_0 + \beta_1 x + \beta_2 (x - T),$$

$$Av(Y\vert x,z) = \beta_0 - \beta_2 T + (\beta_1 + \beta_2) x,$$

Slope before T is $\beta_1$, slope after T is $\beta_1 + \beta_2$.

Therefore $\beta_2$ measures the change in slope after time T.

Implementation in S-Plus.

1. Create the indicator variable column z.

2. Create a new column by multiplying column z by column x.

3. Run the regression with column x and the "product column".

Technical name: Piecewise linear regression.

Issues: searching for the breakpoint.