In the lecture we were discussing ways to estimate a model for linear static panel data. For this the Random Effect model was introduced:
$$y_{it} = \alpha + x_{it}'\beta + u_{it},$$
$$\text{where}\;\;\; u_{it} = (\alpha_i - \alpha) + \epsilon_{it}.$$
$$\text{Furthermore},\;\;\; \alpha_i \sim i.i.d (\alpha, \sigma_{\alpha}^2).$$
I have a question regarding the covariance of the error term. Specifically about the alpha part, because it was stated that $$E[(\alpha_i - \alpha)(\alpha_j - \alpha)] = 0$$ if i is not equal to j and equal to the variance of alpha_i when i is equal j. I think I understand the final part (it turns into the variance of alpha_i - alpha, which is the variance of alpha_i, right?), but I'm struggling with the i not equal to j part. The lecturer said that I had to write out the parentheses and then some parts will cancel out, but I fail to find the answer.
Also (unrelated to previous question), the lecturer said that $$E[(\alpha_i - \alpha)\epsilon_{it}]=0$$ because Expectation of alpha_i is alpha. This does not seem correct to me. The answer does, but the explanation not. I would say it is zero, because alpha_i and alpha are both uncorrelated with epsilon?