An Algorithm for Distributed Lag Estimation Subject to Piecewise Monotonic Coefficients

Linearly distributed-lag models as a time series tool have very useful applications in many disciplines. In these models, the dependent variable depends on one independent variable and its lags. The specification of the lag coefficients is a crucial question to the efficacy of a model. A new algorithm is proposed for the estimation of lag coefficients subject to the condition that the sequence of the coefficient estimates consists of a certain number of monotonic sections, where the positions of the extrema are also unknowns. The algorithm is iterative, each iteration taking a conjugate gradient step, then forming an estimate of the coefficients and finally adjusting this estimate to satisfy the given constraints. An immediate advantage is that the inversion of an ill-conditioned matrix that frequently occurs in practice is avoided. Moreover, the constraints provide a realistic representation of the prior knowledge and the calculation results in a highly efficient time series estimation. The algorithm and its convergence are described, results from simulation experiments are presented and an application of the algorithm on real annual macroeconomic data concerning the personal consumption expenditures against the GDP for the U.S.A. during 1929 2006 is given.