Change-Point Detection in Time Series Using Mixed Integer Programming*

We use cutting-edge mixed integer optimization (MIO) methods to develop a framework for detection and estimation of structural breaks in time series regression models. The framework is constructed based on the least squares problem subject to a penalty on the number of breakpoints. We restate the l0-penalized regression problem as a quadratic programming problem with integer- and real-valued arguments and show that MIO is capable of finding provably optimal solutions using a well-known optimization solver. Compared to the popular l1-penalized regression (LASSO) and other classical methods, the MIO framework permits simultaneous estimation of the number and location of structural breaks as well as regression coefficients, while accommodating the option of specifying a given or minimal number of breaks. We derive the asymptotic properties of the estimator and demonstrate its effectiveness through extensive numerical experiments, confirming a more accurate estimation of multiple breaks as compared to popular non-MIO alternatives. Two empirical examples demonstrate usefulness of the framework in applications from business and economic statistics.

我们采用前沿的混合整数优化（mixed integer optimization, MIO）方法，构建了面向时间序列回归模型结构断点检测与估计的专用框架。该框架基于对断点数量施加惩罚项的最小二乘问题搭建。我们将L0范数惩罚回归（l0-penalized regression）问题重写为包含整数变量与实值变量的二次规划问题，并证明MIO可通过成熟的优化求解器求得可证最优解。相较于主流的L1范数惩罚回归（LASSO, l1-penalized regression）及其他经典方法，该MIO框架可同时估计结构断点的数量、位置与回归系数，同时支持指定给定断点数量或最小断点数量的可选配置。我们推导了该估计量的渐近性质，并通过大量数值实验验证了其有效性，结果表明相较于主流非MIO类方法，该框架对多重结构断点的估计精度更优。此外，两个实证案例展示了该框架在商业与经济统计应用中的实用价值。