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UBC Theses and Dissertations

Asymptotic inference for segmented regression models Wu, Shiying


This thesis deals with the estimation of segmented multivariate regression models. A segmented regression model is a regression model which has different analytical forms in different regions of the domain of the independent variables. Without knowing the number of these regions and their boundaries, we first estimate the number of these regions by using a modified Schwarz' criterion. Under fairly general conditions, the estimated number of regions is shown to be weakly consistent. We then estimate the change points or "thresholds" where the boundaries lie and the regression coefficients given the (estimated) number of regions by minimizing the sum of squares of the residuals. It is shown that the estimates of the thresholds converge at the rate of (Op(ln²n/n), if the model is discontinuous at the thresholds, and Op{n-¹/2) if the model is continuous. In both cases, the estimated regression coefficients and residual variances are shown to be asymptotically normal. It is worth noting that the condition required of the error distribution is local exponential boundedness which is satisfied by any distribution with zero mean and a moment generating function provided its second derivative is bounded near zero. As an illustration, a segmented bivariate regression model is fitted to real data and the relevance of the asymptotic results is examined through simulation studies. The identifiability of the segmentation variable is also discussed. Under different conditions, two consistent estimation procedures of the segmentation variable are given. The results are then generalized to the case where the noises are heteroscedastic and autocorrelated. The noises are modeled as moving averages of an infinite number of independently, identically distributed random variables multiplied by different constants in different regions. It is shown that with a slight modification of our assumptions, the estimated number of regions is still consistent. And the threshold estimates retain the convergence rate of Op(ln² n/n) when the segmented regression model is discontinuous at the thresholds. The estimation procedures also give consistent estimates of the residual variances for each region. These estimates and the estimates of the regression coefficients are shown to be asymptotically normal. The consistent estimate of the segmentation variable is also given. Simulations are carried out for different model specifications to examine the performance of the procedures for different sample sizes.

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