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How Measurement Error Affects OLS

this is another data science question I saw on twitter:

with some corrections:

basically we're trying to answer how $\alpha$, $\beta$, etc changes if errors are added to the $y$ values.

note that ryxcommar's notation for normal distribution is $N(\mu,\sigma)$ instead of $N(\mu,\sigma^2)$. we'll be using this for the answer.

so $y^* = \alpha + \beta x + \epsilon + u$ where $u$ is measurement error distributed $N(2,2)$

since $x$ isn't affected by the measurement error, $\beta$ is also unaffected.

which means $u$ only affects $\alpha$ & $\epsilon$.

OLS residuals always have mean $0$, because the constant is carried over to the $\alpha$ term. so $\alpha^* = \alpha + 2$. and $\epsilon^* \sim N(0,1) + N(0,2) \sim N(0,\sqrt{5}) + cov(e,u)$. this is why the second correction is nice to have, so $\epsilon \sim N(0,\sqrt{5})$

in conclusion:

• $\alpha^* = \alpha + 2$

• $\beta^* = \beta$

• $\mu^* = \mu = 0$

• $\sigma^* = \sqrt{\sigma + 2^2}$