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The program outputs parameter estimates from two different parameterizations: the "general changepoint" parameterization (GCP), and the "standard" parameterization (SP) of Kim et al. (2000).
The standard parameterization is (see Kim, et al. 2000, equation 1),
\(\Large E[y|x] = \beta_0 + \beta_1x + \delta_1(x - \tau_1)^+ +\cdots+ \delta_k(x - \tau_k)^+ \Large\) (1)
where (a)+= a if a > 0 and 0 otherwise.
The general changepoint parameterization is,
\(\Large E[y|x] = \sum_{j=1}^{k+1}(\beta_{0,j} + \beta_{1,j}x)^I(\tau_{j-1} \lt x \le \tau_j) \Large\) (2)
where I(A) is the indicator function for {A}, τ0 = min(x), and τk+1 = max(x), and under the constraint that E[y|x] is continuous at τj.
For the relationship between the parameterizations see Table 1 and Appendix A.
| Output Label | Standard | General Changepoint |
|---|---|---|
| Intercept 1 | \(\beta_{0,1}\) \(\Large \beta_{0,1} \Large\) | \(\beta_0\) |
| Intercept j, j ≥ 2 | \(\beta_0 - \sum {}_{h=1}^{j-1} \delta_h \tau_h\) | |
| Slope 1 | \(\beta_{1,1}\) | \(\beta_1\) |
| Slope j, j ≥ 2 | \(\beta_1 + \sum {}_{h=1}^{j-1} \delta_h\) | |
| Slope j - Slope (j-1), j ≥ 2 | \(\beta_{1,j} - \beta_{1, j-1}\) |