A linearly distributed lag estimator with r-convex coefficients

The purpose of linearly distributed lag models is to estimate, from time series data, values of the dependent variable by incorporating prior information of the independent variable. A least-squares calculation is proposed for estimating the lag coefficients subject to the condition that the rth differences of the coefficients are non-negative, where r is a prescribed positive integer. Such priors do not assume any parameterization of the coefficients, and in several cases they provide such an accurate representation of the prior knowledge, so as to compare favorably to established methods. In particular, the choice of the prior knowledge parameter r gives the lag coefficients interesting special features such as monotonicity, convexity, convexity/concavity, etc. The proposed estimation problem is a strictly convex quadratic programming calculation, where each of the constraint functions depends on r+1 adjacent lag coefficients multiplied by the binomial numbers with alternating signs that arise in the expansion of the rth power of (1-1). The most distinctive feature of this calculation is the Toeplitz structure of the constraint coefficient matrix, which allows the development of a special active set method that is faster than general quadratic programming algorithms. Most of this efficiency is due to reducing the equality constrained minimization calculations, which occur during the quadratic programming iterations, to unconstrained minimization ones that depend on much fewer variables. Some examples with real and simulated data are presented in order to illustrate this approach.