How order of stuctural breakpoints is decided? Using 'breakpoints' command in R, I got multiple breakpoints. Yet if I limited the number of breakpoints to 1. It gives just a single breakpoint as desirable. How is this 1st breakpoint is decided and also order of other breakpoints arranged?
 A: The algorithm implemented in breakpoints() (from the strucchange package) estimates the optimal location of $m$ breakpoints for $m = 1, \dots, \mathtt{breaks}$. Optimality is in the sense of minimal residual sum of squares. The reason is that the underlying computations are mostly the same, no matter whether $m$ is 2 or 3 or ... Therefore, from a single fitted breakpoints result, you can extract the optimal location of $m = 1$ breakpoint, $m = 2$ breakpoints etc. Often, the breakpoint locatation for $m = 1$ is one of the two locations for $m = 2$ but this does not have to be the case.
Hence, there is no ordering problem. The practitioner has to answer two questions: (1) What is the number of breakpoints $m$? (2) And given the number $m$, what are the corresponding breakpoint locations?
The breakpoints() function answers this by splitting it up: Question (2) has a unique well-defined answer (in the least squares sense) for every given $m$. Hence, the optimal breakpoint locations for a range of $m$ values are determined first. Subsequently, the best $m$ can be chosen by information criteria or by expert knowledge etc.
Other algorithms such as the PELT algorithm implemented in the changepoints package solve both questions in a single run by only returning the solution for the $m$ that is optimal with respect to a certain information criterion.
