On Distributed-Lag Modeling Algorithms by r-Convexity and Piecewise Monotonicity

Linearly distributed-lag models as a time series tool have useful applications in many disciplines. In these models, the dependent variable is given approximately by a weighted sum of an independent variable and a prescribed number of its lags. The specification of the lag coefficients depends usually on a structural assumption about the form of the underlying relation, which is relevant to the efficacy of a model. Two least squares algorithms for distributed-lag modeling subject to restrictions on the signs of finite differences of the lag coefficients are considered. The algorithms include a quadratic programming calculation that estimates the lag coefficients subject to the nonnegativity of the rth consecutive differences of the coefficients and an iterative procedure that provides piecewise monotonic lag coefficients by a combination of the conjugate gradient procedure and piecewise monotonic data approximation. The important feature of these methods is the ability to obtain properties such as monotonicity, piecewise monotonicity, convexity and r-convexity that occur in a wide range of underlying relations for a realistic representation of the lag coefficients. An application to real quarterly macroeconomic data derived from the International Monetary Fund for the period 1959:Q2–2013:Q2 is presented in order to illustrate some modeling aspects of our methods. Dependent variable is the Annual Rate of Change of the GDP in United States and independent variable is the Annual Rate of Change of the Money Supply for United States.