Gauss-Markov assumptions I was trying to determine which of the Gauss-Markov assumptions allow us to see that $b_1$ is an unbiased estimator of $\beta_1$.  I have a feeling it's that $X_{i}$ is not random, but is there anything else that I'm missing?
 A: The Gauss-Markov Theorem is actually telling us that in a regression model, where the expected value of our error terms is zero, $E(\epsilon_{i}) = 0$ and variance of the error terms is constant and finite $\sigma^{2}(\epsilon_{i}) = \sigma^{2} < \infty$ and $\epsilon_{i}$ and $\epsilon_{j}$ are uncorrelated for all i and j the least squares estimator $b_{0}$ and $b_{1}$ are unbiased and have minimum variance among all unbiased linear estimators. Note that there might be biased estimator which have a even lower variance.
Extensive information about the Gauss-Markov Theorem, such as the mathematical proof of the Gauss-Markov Theorem can be found here http://economictheoryblog.com/2015/02/26/markov_theorem/
However, if you want to know which assumption is necessary for $b1$ to be an unbiased estimator for $\beta1$, I guess that assumption 1 to 4 of the following post (http://economictheoryblog.com/2015/04/01/ols_assumptions/) must be fulfilled to have an unbiased estimator. 
A: $$\hat{\beta} = ([inv(X'X)]X')(X\beta + \epsilon)$$
$$\hat{\beta} = \beta + ([inv(X'X)]X')\epsilon$$
$\hat{\beta}$ is an unbiased estimator of $\beta$ under two conditions:


*

*$X$ is non-stochastic $$E(\hat{\beta}) = \beta + E[([inv(X'X)]X')\epsilon]$$ if $X$ is deterministic,  this would reduce to: 
$$E(\hat{\beta}) = \beta + ([inv(X'X)]X') E[\epsilon]$$
The second term on right hand side, $E[\epsilon]$ is zero under one of the Gauss markov assumption. 

*$X$ is stochastic but independent of error ($\epsilon$)
Using this, we can reduce the equation to:
$$E(\hat{\beta}) = \beta + inv(X'X)] E[(X')\epsilon]$$
where $E[(X')\epsilon] = 0$ from an assumption that comes from one of the OLS's properties, $E[X'e] = 0$. 
Reference:
https://web.stanford.edu/~mrosenfe/soc_meth_proj3/matrix_OLS_NYU_notes.pdf
Thanks
Anurag
A: The LS-Estimator is:$$b=\beta + (X'X)^{-1}X'e$$
The estimator is unbiased if $(X'X)^{-1}X'e$ converges to zero, and this is the case, if the designmatrix $X$ is not correlated with the error $e$.
So, the necessary assumption is: $$E[X_{t,k}*e_t]=0$$
