Calculating standard error of coefficients in segmented regression

I know this is kinda stupid question to ask here but just to make sure that I am doing this correctly, here it is anyway:

Consider the following equation resulted from segmented regression: $$\hat y_i = \begin{cases} \hat\beta_1 x_i + (\hat\beta_0 - \hat\beta_1 \cdot c) &\text{if } x_i \le c \\ \hat\beta_2x_i + (\hat\beta_0 - \hat\beta_2 \cdot c)&\text{else}\end{cases}$$ where $$c$$ is the break point and $$i = 1,2,\ldots n$$. And according to this, standard error of linear regression coefficient can be calculated as follows: $$\hat y_i = \hat\beta x_i + b,\ i = 1,2,\ldots n\\ \widehat{\text{se}}(\hat{\beta}) = \sqrt{\frac{\frac{1}{n-2} \sum_i \hat{\epsilon}_i^2}{\sum_i (x_i - \bar{x})^2}}$$ Now if I want to calculate the standard error of $$\hat\beta_1$$ from segmented regression, will it be this: $$\widehat{\text{se}}(\hat\beta_1) = \sqrt{\frac{\frac{1}{N-2} \sum_i \hat{\epsilon}_i^2}{\sum_i (x_i - \bar{x})^2}}$$ where $$i$$ is index of $$x_i$$ such that $$x_i \le c$$ and $$N$$ is number of $$x_i$$ such that $$x_i \le c$$? Or do I have to use all $$i$$ (i.e the whole data set) to calculate standard error of $$\hat\beta_1$$?

• The fitted parameter does not apply to the entire data set, so the associated standard error also does not apply to the entire data set. – James Phillips Apr 9 at 15:42