Iterated partial sum sequences of regression residuals and tests for changepoints with continuity constraints

Iterated partial sum sequences of regression least squares residuals are defined and large sample properties of sequences of stochastic processes defined by these iterated partial sums are discussed. Also, finite sample properties of the iterated partial sum sequences are obtained. These include a property of least squares residuals of polynomial fits to equispaced data, namely the iterated partial sums sum to 0 provided that the order of iteration is not greater than the order of the polynomial, thus extending the well-known result that residuals sum to 0. Iterated partial sums are shown to play an important role in testing regression parameters for changes at unknown times under the constraint of continuity.