Conditional variance in OLS regression Consider the linear regression model: $$y_{it}=x_{it}\beta+\epsilon_{it}$$
  where $x$
  is single regressor. The conditional mean of any specific observation is:$$E[y_{it}|x_{it}]=x_{it}\beta$$
  under the conditional mean independence assumption. 
The conditional variance, howevever, is where I am confused. On one had, we can take the variance of both sides of the original equation and get:$$var(y_{it})=\beta^{2}var(x_{it})+var(\epsilon_{it})+2cov(x_{it}\beta,\epsilon_{it})=\beta^{2}\sigma_{x}^{2}+\sigma_{\epsilon}^{2}+0$$
On the other hand, think of the following formula for the variance:$$E\left[y_{it}-E\{y_{it}\}\right]^2 =E[\epsilon_{it}^{2}]=\sigma_{\epsilon}^{2}$$
Which one is correct? 
 A: The title of your question "Conditional variance in OLS regression" gives a clue. The first expression 
$$\text{Var}(y_{it})=\beta^{2}\sigma_{x}^{2}+\sigma_{\epsilon}^{2}$$ 
gives the unconditional variance ($x$ is not "conditioned away" and remains in the expression) while the second one 
$$\text{E}\left[y_{it}-\text{E}\{y_{it}\}\right]^2 =\sigma_{\epsilon}^{2}$$ 
gives the conditional variance (conditional on $x$).
A: But your final equation cannot be right if you are not treating the predictors as fixed. For
$$y_i = \beta x_i + \epsilon_i$$
recalling that
$$E\left(Y \right) = E\left[ E \left(Y|X \right) \right]$$
we have 
$$E(y_i) =E\left(  E\left(\beta x_i + \epsilon_i|x_i \right) \right) = \beta \mu_x$$ 
and so
\begin{align} Var(y_i) = E\left(y_i ^2\right) - \left(E(y_i) \right)^2 &= E \left(\beta x_i + \epsilon_i \right)^2 - \beta^2 \mu_x^2 \\ &= \beta^2 E\left(x_i^2 \right) + E(\epsilon^2_i) + 2 E\left( \beta x_i  \epsilon_i \right) - \beta^2 \mu_x^2 \\ & = \beta^2 \left(\sigma_x^2 + \mu^2_x \right) + \sigma^2 -\beta^2 \mu_x^2 \\  & = \beta^2 \sigma^2_x + \sigma^2 \end{align} 
assuming that the $x_i$ are iid. This is similar to what you obtained in your third equation of course.
