Abstract

Many economic theories suggest that the relation between two variables y and x follow a function forming a convex or concave curve. In the classical linear model (CLM) framework, this function is usually modeled using a quadratic regression model, with the interest being to find the extremum value or turning point of this function. In the CLM framework, this point is estimated from the ratio of ordinary least squares (OLS) estimators of coefficients in the quadratic regression model. We derive an analytical formula for the expected value of this estimator, from which formulas for its variance and bias follow easily. It is shown that the estimator is biased without the assumption of normality of the error term, and if the normality assumption is strictly applied, the bias does not exist. A simulation study of the performance of this estimator for small samples show that the bias decreases as the sample size increases.

摘要

许多经济理论表明，两个变量y和x之间的关系遵循形成凸曲线或凹曲线的函数。在经典线性模型 (CLM) 框架中，该函数通常使用二次回归模型建模，目的是找到该函数的极值或转折点。在 CLM 框架中，这一点是根据二次回归模型中系数的普通最小二乘 (OLS) 估计量的比率来估计的。我们推导出该估计量的期望值的分析公式，从中可以轻松得出其方差和偏差的公式。结果表明，在没有误差项正态性假设的情况下，估计量是有偏差的，如果严格应用正态性假设，则不存在偏差。对该估计器对小样本的性能的模拟研究表明，偏差随着样本量的增加而减小。