On Segmented Multivariate Regression

This paper concerns segmented multivariate regression models, models which have different linear forms in different subdomains of the domain of an independent variable. Without knowing that number and their boundaries, we first estimate the number of these subdomains using a modified Schwarz criterion. The estimated number of regions proves to be weakly consistent under fairly general conditions. We then estimate the subdomain boundaries (“thresholds”) and the regression coefficients within subdomains by minimizing the sum of squares of the residuals. We show that the threshold estimates converge (at rates, 1/n and n−1/2, respectively at the model’s threshold points of discontinuity and continuity) and that the regression coefficients as well as the residual variances are asymptotically normal. The basic condition on the error distribution required for the veracity of our asymptotic results is satisfied by any distribution with zero mean and a moment generating function (having bounded second derivative around zero). As an illustration, a segmented bivariate regression model is fitted to real data and the relevance of the asymptotic results is examined via simulations.