Assessing the Precision of Turning Point Estimates in Polynomial Regression Functions
研究了评估多项式回归模型中转折点估计精度的三种方法：Delta方法、二次回归精确分布法和马尔可夫链蒙特卡洛方法，发现后两种更可靠，并用环境库兹涅茨曲线数据进行了比较。
Researchers often report point estimates of turning point(s) obtained in polynomial regression models but rarely assess the precision of these estimates. We discuss three methods to assess the precision of such turning point estimates. The first is the delta method that leads to a normal approximation of the distribution of the turning point estimator. The second method uses the exact distribution of the turning point estimator of quadratic regression functions. The third method relies on Markov chain Monte Carlo methods to provide a finite sample approximation of the exact distribution of the turning point estimator. We argue that the delta method may lead to misleading inference and that the other two methods are more reliable. We compare the three methods using two data sets from the environmental Kuznets curve literature, where the presence and location of a turning point in the income-pollution relationship is the focus of much empirical work.