Nonparametric Decomposition of Time Series Data with Inputs

The backfitting algorithm commonly used in estimating additive models is used to decompose the component shares explained by a set of predictors on a dependent variable in the presence of linear dependencies (multicollinearity) among the predictors. Multicollinearity of independent variables affects the consistency and efficiency of ordinary least squares estimates of the parameters. We propose an estimation procedure that address this problem by estimating shares of each predictor one at a time, the sequence depends on the initial guess of the relative importance of the variables in the model. Simulated data show that backfitting the ordinary least squares procedure and additive smoothing splines are comparable and superior over the ordinary least squares in estimating share of the contribution of the different predictors to the dependent variable. The superiority of the predictive ability of the method is also more apparent as multicollinearity worsens. The method is used in modeling sales data with predictors including marketing activation indicators, competition measures, distribution indicators, economic and weather indicators. The sales data (time series) that is characterized by severe multicollinearity and inadequate linear fit illustrates the advantage of additive model estimated through backfitting with spline smoother over ordinary least squares and the backfitted linear smoother in terms of predictive ability (MAPE) and interpretability of the estimated shares of the predictors to the dependent variable.