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?

  • 6
    $\begingroup$ How can $\hat{\beta_1}$ be unbiased for $\beta_1+1$ when it is already unbiased for $\beta_1$? $\endgroup$ Commented Sep 23, 2021 at 10:06

1 Answer 1



$$\mathbb E[\hat\beta_1]=\beta_1\ne \beta_1+1$$

But $\hat\beta_1+1$ will be unbiased for $\beta_1+1$.

$$ \mathbb E[\hat\beta_1 + 1] = \mathbb E[\hat\beta_1] + \mathbb E[1] = \beta_1 + 1 $$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.