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Hi I am a beginner to econometrics!

I have been dealing with bivariate regression. We use the formula $y = \beta_0 + \beta_1 x$.

I am told that if $E(u\mid x) \ne 0$ then the estimate of the slope parameter $\beta_1$ will be biased.

My question is: if $E(u\mid x) = 1$ for all $x$'s, then why would the estimate of the slope parameter be biased? I thought this would mean that only the intercept estimate will be biased.

I thought that the slope parameter will only be biased if the expected value of the errors is different for each $x$. For example if $E(u\mid x_1) = 1$ and the $E(u\mid x_2) = 3$, etc. etc.

Any help will be immensely appreciated! Thank you!

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You are correct that the condition for unbiasedness of the slope coefficient is that

$$E(u \mid X) = const.$$

However, if it is different than zero, we will have bias in the estimation of the constant term.

If you want to go deeper, this thread may be useful, Conditional mean independence implies unbiasedness and consistency of the OLS estimator

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This is my attempt. I think you are confusing $X_s$ to observations $n$. There is no reason for $\beta_1$ to be biased if for all observation the $E(u|X)=1$, as that would mean you simply subtract the constant term, 1 in this case from the error term and the constant, and your equation becomes $y=\beta_0 -1 +\beta_1x + u-1$. So, your $\beta_0$ is biased but not $\beta_1$.

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Let's see what's going on:

$$\begin{align}E[\hat{\beta_1}|x] &=E\left[\frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{S_{xx}}\bigg|\ x\right]=\frac{1}{S_{xx}}\sum E[(x_i-\bar{x})(\beta_1(x_i-\bar{x})+(u_i-\bar{u}))]\\ &=\frac{1}{S_{xx}}\beta_1 S_{xx}+\sum(x_i-\bar{x})E[u_i|x]-E[\bar{u}|x]\sum(x_i-\bar{x}) \\ &= \beta_1+\sum(x_i-\bar{x})E[u_i|x]\end{align}$$ The second term won't go off when $E[u_i|x]\neq c$ (thanks to the correction by @Laconic), and introduce a bias to the expected value. So, it seems you're right.

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    $\begingroup$ Won’t the second term be zero if E[u|x] is constant? $\endgroup$ – The Laconic Mar 18 '19 at 12:01

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