# Why is $E(\hat{\beta}_1) = \beta_1 + 1$ also an unbiased estimator of true $\beta_1$

let regression model be $$Y_i = \beta_0 + \beta_1X_i +u_i$$

If the ordinary least square estimator of $$\beta_1$$, $$\hat{\beta}1 = \frac{\sum^n_{i=1}(x_i -\bar{X})(y_i-\bar{Y})}{\sum^n_{i=1}(x_i -\bar{X})^2}$$ is an unbiased estimator for true $$\beta_1$$. Is $$\hat{\beta_1}$$ also an unbiased estimator of the true $$\beta_1 + 1$$? In other words, is $$E(\hat{\beta}_1) = \beta_1 + 1?$$

Question: Why would $$E(\hat{\beta_1}) = \beta_1 + 1$$ still be an unbiased estimator?

• How can $\hat{\beta_1}$ be unbiased for $\beta_1+1$ when it is already unbiased for $\beta_1$? Sep 23 at 10:06

$$\mathbb E[\hat\beta_1]=\beta_1\ne \beta_1+1$$
But $$\hat\beta_1+1$$ will be unbiased for $$\beta_1+1$$.
$$\mathbb E[\hat\beta_1 + 1] = \mathbb E[\hat\beta_1] + \mathbb E[1] = \beta_1 + 1$$