Adaptive and unbiased predictors in a change point regression model

Consider the problem of prediction in a change point regression model. That is, assume a simple linear regression model holds for all x (independent variable) less than γ_k, and a different simple linear regression model holds for x #62; γ_k. However, γ_k, the change point, is unknown but can be one of m possible values (γ₁ ≤ γ₂ ≤ ⋯ ≤ γ_m). For x = x₀ #62; γ_m we want to predict a future value of the dependent variable. Adaptive predictors (estimators) and unbiased predictors (estimators) are studied. The adaptive estimator is one which estimates by least squares for the kth model, provided the residual sum of squares for the kth model is smallest. Some theoretical justification for the adaptive estimator is given through a decision theory formulation. A small simulation study comparing the adaptive estimator and unbiased estimator is also offered.