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I am using linear regression.

$Y_i = \beta_0 + \beta_1 x_i + \varepsilon_i$

$\varepsilon_i \overset{iid}{\sim} Normal(0, \sigma^2)$

At $X = x^\ast$, let's define the mean response as

$\mu^\ast = \beta_0 + \beta_1 x^\ast$.

The estimator for the mean response is

$\hat{\mu^\ast} = \hat{\beta_0} + \hat{\beta_1} x^\ast$.

Since $E(\hat{\mu^\ast}) = E(\hat{\beta_0} + \hat{\beta_1} x^\ast) = \beta_0 + \beta_1 x^\ast = \mu^\ast$, $\hat{\mu^\ast}$ is unbiased.


Now, let's define $Y^\ast$ as an individual response at $x^\ast$. This individual response deviates from the mean response by $\varepsilon^\ast$. Thus,

$Y^\ast = y^\ast + \varepsilon^\ast$.

Now, I think that the estimator is this:

$\hat{Y^\ast} = \hat{y^\ast} + \hat{\varepsilon^\ast}$.

Is this estimator unbiased? I'm not sure. If we use the MSE as $\hat{\varepsilon^\ast}$, then

$E(\hat{Y^\ast}) = E(\hat{\beta_0} + \hat{\beta_1} x^\ast + \hat{\varepsilon^\ast}) = \beta_0 + \beta_1 x^\ast = \mu^\ast$,

which is CLEARLY not $Y^\ast$.

Thus, is this estimator truly biased, or did I make a mistake with my math?

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What does $ \hat{\varepsilon^\ast}$ mean?
Since $ \varepsilon^\ast$ has mean $0$, doesn't it make sense that the best prediction is $0$?

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    $\begingroup$ As I wrote in my question, I'm using mean squared error for $\hat{\varepsilon^\ast}$. $\endgroup$ Feb 7, 2021 at 2:13
  • $\begingroup$ In the last calculation you use the fact that its expected value is 0. But, I still don’t know what it is. Do you have a formula for it? Does it depend on any of the observed data or depend on $x*$? $\endgroup$
    – John L
    Feb 7, 2021 at 17:38

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