Mei Yu Lee

We provide a computational piecewise linear regression (CPLR) method with endogenous segmentation to construct accurate mathematical representations of data. The method models the unemployment–inflation relationship through a deterministic segmentation algorithm, rather than imposing a predetermined functional form to fit all data for one time. The CPLR method procedure sequentially fits local linear regressions on observations ordered by the explanatory variable and detects structural breakpoints according to a specific rule of breakpoint detection property in the coefficient of determination. A minimum segment size is imposed to ensure numerical stability constraint, and the optimal minimum sample size is determined by minimizing the global root mean squared error (RMSE) across candidate segmentations. We establish a theoretical property of the CPLR method algorithm, based on the fact that the breakpoint detection rule follows directly from the minimum least-squares error principle rather than from heuristic segmentation. The example results, using annual data from 1991 to 2022 as an illustration, reveal clear structural breaks and regime-dependent slope heterogeneity across the G7 economies. The number of segments varies substantially across countries, ranging from a stable two-segment structure in the United States to a six-segment structure in France. Positive slopes appear in specific unemployment ranges for several countries, except in the case of the United States. Comparisons with the results of conventional linear and nonlinear regressions show that in several countries, including Japan, Germany, Italy, and France, the CPLR representation provides a closer local fit to the observed data than conventional global linear or nonlinear specifications. The CPLR method, therefore, provides a deterministic and data-driven algorithm for detecting endogenous structural changes and constructing accurate piecewise linear representations without imposing prior theoretical assumptions or relying on statistical inference.

In this paper, we provide a mathematical and statistical methodology using heteroscedastic estimation to achieve the aim of building a more precise mathematical model for complex financial data. Considering a general regression model with explanatory variables (the expected value model form) and the error term (including heteroscedasticity), the optimal expected value and heteroscedastic model forms are investigated by linear, nonlinear, curvilinear, and composition function forms, using the minimum mean-squared error criterion to show the precision of the methodology. After combining the two optimal models, the fitted values of the financial data are more precise than the linear regression model in the literature and also show the fitted model forms in the example of Taiwan stock price index futures that has three cases: (1) before COVID-19, (2) during COVID-19, and (3) the entire observation time period. The fitted mathematical models can apparently show how COVID-19 affects the return rates of Taiwan stock price index futures. Furthermore, the fitted heteroscedastic models also show how COVID-19 influences the fluctuations of the return rates of Taiwan stock price index futures. This methodology will contribute to the probability of building algorithms for computing and predicting financial data based on mathematical model form outcomes and assist model comparisons after adding new data to a database.

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We investigate how firms facing different production modes of software service determine their production mode and pricing decisions in the presence of network effects. We develop a two-stage game model to examine the network effect on the choices of production mode and the strategic behavior of the firms. We find that if all firms adopt cloud service production mode, the resulting software price is higher than if they adopt shrink-wrapped production mode. We also find that network effects lower the level of market price.