Linear regression model when intercept is known

Question

I was given linear model:

$$Y_i = \beta_0 + \beta_1X_i + u_i$$

I know that $\beta_0 = 2$.
Now I should find out the unbiased estimate $\bar{β_1}$.

I know that $\beta_0 = 2$ represents the intercept on this linear model but I am not sure how to find the estimate $\bar{β_1}$ from this.

Could someone please help me to calculate this and give me step by step instructions how to do it? I already tried to study this from the book for two hours but didn't get it.

Answer 1

For simple linear regression you the formula:

\begin{align} \beta_1 &= \frac{\sum_{i=1}^N (x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^N (x_i-\bar{x})^2}\\ \beta_0 &= \bar{y}-\beta_1 \bar{x} \end{align}

So since you already habe $\beta_0$ you obtain $\beta_i$ from

$$ \beta_1 = \frac{\bar{y}-\beta_0}{\bar{x}} $$