This paper deals with the asymptotic distribution of the log likelihood ratio statistic in regression models which have different analytical forms in different regions of the domain of the independent variable. It is shown that under suitable identifiability conditions, the asymptotic chi square results of Wilks and Chernoff are applicable. It is shown by example that if there are actually fewer segments than the number assumed in the model, then the least squares estimates are not asymptotically normal and the log likelihood ratio statistic is not asymptotically $\chi^2$. The asymptotic behavior is then more complicated, and depends on the configuration of the observation points of the independent variable.
"The Log Likelihood Ratio in Segmented Regression." Ann. Statist. 3 (1) 84 - 97, January, 1975. https://doi.org/10.1214/aos/1176343000