回归系数

Regression Coefficients

设有两个数据集 $X$ 和 $Y$ ，满足 $Var(X) = 10$ 且 $Var(Y) = 20$ 。我们做如下线性回归：

y \sim \alpha_x + \beta_xx

并得到 $\beta_x = 1$ 。现在再做回归：

x \sim \alpha_y + \beta_yy

那么 $\beta_y$ 的值是多少？

Suppose that we have two datasets $X$ and $Y$ with $Var(X) = 10$ and $Var(Y) = 20$ . We perform the linear regression:

y \sim \alpha_x + \beta_xx

and obtain $\beta_x = 1$ . Suppose now perform the regression:

x \sim \alpha_y + \beta_yy

What is the value of $\beta_y$ ?

解析

设 $r$ 为 $X$ 与 $Y$ 的皮尔逊相关系数。那么

\beta_x = \frac{\sigma_y}{\sigma_x}

\beta_y = r \frac{\sigma_x}{\sigma_y}

因此：

\beta_y = \frac{\sigma^2_x}{\beta_x\sigma^2_y} \to \beta_y = \beta_x\frac{\sigma^2_x}{\sigma^2_y}

把题目给定的常数代入：

\beta_y = 1 \times \frac{10}{20} = \frac{1}{2}

Let $r$ be the Pearson Correlation Coefficient of $X$ and $Y$ . Then

\beta_x = \frac{\sigma_y}{\sigma_x}

\beta_y = r \frac{\sigma_x}{\sigma_y}

Therefore:

\beta_y = \frac{\sigma^2_x}{\beta_x\sigma^2_y} \to \beta_y = \beta_x\frac{\sigma^2_x}{\sigma^2_y}

Plugging in the constants we get:

\beta_y = 1 \times \frac{10}{20} = \frac{1}{2}