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xiaobaiybz · 2025年02月28日

请老师解释一下这道题,完全没看懂,谢谢

NO.PZ2024120401000073

问题如下:

We regress a stock's returns against a market index according to the following OLS model: Return(i) = β0 + β1*Index(i) + u(i). However, our regression is guilty of omitted variable bias. If our regression indeed suffers from omitted variable bias, which of the following is MOST likely true?

选项:

A.

The OLS assumption that E[u(i)
X(i)] = 0 is incorrect

B.

The OLS assumption that [X(i), Y(i)], i = 1, ..., n are iid. random draws is incorrect

C.

The OLS assumption the large outliers are unlikely is incorrect

D.

The assumption of no perfect multicollinearity is incorrect

解释:

If an omitted variable is a determinant of Y(i), then it is in the error term, and if it is correlated with X(i), then the error term is correlated with X(i).

Because u(i) and X(i) are correlated, the conditional mean of u(i) given X(i) is non-zero. This correlation therefore violates the first least squares assumption, and the consequence is serious: The OLS estimator is biased. This bias does not vanish even in very large samples, and the OLS estimator is inconsistent.

请老师解释一下这道题,完全没看懂,谢谢

1 个答案

李坏_品职助教 · 2025年02月28日

嗨,爱思考的PZer你好:


题目问你,如果你的线性回归模型存在omitted variable bias(就是说模型里面遗漏了变量),会产生什么样的后果?


如果你的模型遗漏了某个重要的解释变量,比如原来应该有三个X可以解释Y,现在你只有两个X,那么遗漏的那个X会被放入残差里面。这样一来会导致两个问题出现:

  1. 残差项就会与其他的X存在相关性。
  2. 那么残差的条件期望就不是0了,因为此时的残差包含了你遗漏的那个X,所以残差的期望不再是0。本题的A选项描述的就是这个问题,所以选A。

从上面的讲义内容可以看出,线性回归的基本假设其中有两条,就被推翻了。

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