Econometrics proofs for 3rd year economics coursework

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EconometricAli
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Hey everyone,
I'm doing an 3rd year economics course in university and I'm already running into difficulty trying to show proofs of the following:
Typeset coursework questions: Question 3 asks to prove E[X̄] = β and var(X̄) = σ²/n for independent, identically distributed X1 to Xn with mean β and variance σ²; Question 4 asks to show least squares residuals sum to zero.


Now I certainly don't wish to ask for spoon feeding just some direction or source to where I can look up and read for the solution. My knowledge on these are still weak and the deadline is by the end of this Friday. I'm going to try the textbook tomorrow to see alternatives because I couldn't understand the lecture as well as I had hoped.

Thank you
 

Discussion

3a is easy: [math]E\left(\sum X_I\right)= \sum E(X_I)= \sum \beta= n\beta[/math] so the expected value of [math]\frac{1}{n}\sum X_i[/math] is [math]\beta[/math].

Pretty much the same thing for 3b.

For 4, what does "linear model" imply?