联合模型评估

Joint Model Evaluation

专题: Statistics / 统计
难度: L2
来源: OpenQuant

题目详情

使用 OLS，将 $y$ 对 $X_1$ 回归后，模型的 $R^2$ 为 0.45。又将 $y$ 对 $X_2$ 回归后，模型的 $R^2$ 为 0.3。设把 $y$ 对 $X_1, X_2$ 一起回归得到的新模型的 $R^2$ 的下界和上界记为 $[\min, \max]$ 。求新模型的 $\min$ 和 $\max$ $R^2$ 。

Using OLS, we regress $y$ onto $X_1$ and find that the model has an $R^2$ of 0.45. We also regress $y$ onto $X_2$ and find that the model has an $R^2$ of 0.3. Let $[\min, \max]$ denote the lower and upper bound of $R^2$ of a model which regresses $y$ onto $X_1, X_2$ . Find both the $\min$ and $\max$ $R^2$ values for the new model.

解析

回忆一下， $R^2$ 衡量的是模型的解释力，即因变量 $y$ 的变动中有多少可以被自变量 $X_1, X_2$ 解释。当我们向回归模型中加入新变量时， $R^2$ 不会下降；它只会增加或保持不变。若两个变量完全共线，则 $R^2$ 会保持不变；若其中一个变量能解释另一个变量解释不了的那部分 $y$ 的变动，则 $R^2$ 会上升。

据此，联合回归的最小 $R^2$ 为 0.45，最大 $R^2$ 为 0.75。

Original Explanation

Recall that $R^2$ quantifies the explanatory power of the model - the variation in the dependent variable ( $y$ ) that can be explained by the independent variables ( $X_1, X_2$ ). When we add a new variable to a regression model, the $R^2$ value can never decrease; rather, it can either increase or stay the same. It will stay the same in the case that both variables are perfectly collinear and increase if there is some variation in $y$ that is explained by one variable and not the other.

Based on this, the minimum $R^2$ of the combined regression will be 0.45 and the maximum $R^2$ of the combined regression will be 0.75.