# In simple linear regression, is the estimator of an individual response unbiased?

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?

What does $$\hat{\varepsilon^\ast}$$ mean?
Since $$\varepsilon^\ast$$ has mean $$0$$, doesn't it make sense that the best prediction is $$0$$?
• As I wrote in my question, I'm using mean squared error for $\hat{\varepsilon^\ast}$. Feb 7, 2021 at 2:13
• 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 $x*$? Feb 7, 2021 at 17:38