Q:

OLS: $E[\epsilon_{it}^T\epsilon_{it}] \not= 0$ in 1st equation biases standard errors in 2nd equation?

Suppose ${X_{it}},{Y_{it}}$ are time series with $X_{it}\sim N(0.1,1)$, ($\sigma^2(Y_{it}) = 1$ and $mean(Y_{it})$ is similar to that for $X_{it}$, but changes when the dummy = 1). and $t \in \{1,2,...,200\}$, $i \in \{1,2,...,N\}$. In a real world setting this will be periodic stock market returns over $N$ firms (but you can ignore this). There is a dummy, $D_t$ that's equal to unity over $t \in \{150,151,...,200\}$ and equal to zero otherwise. The time series model to be estimated with OLS $\forall i$ is: 
$(1) Y_{it} = \alpha_i + \beta_i X_{it} + \gamma_i D_{t} + \epsilon_{it}$
This model generally adheres to the Gauss-Markov assumptions for each $i$. However, we have $E[\epsilon_{it}^T \epsilon_{jt}] \not= 0$ for all $i$ and $j$.
The next step is to construct a vector of gammas using the $N$ estimates of model $(1)$. Call this vector $\bf{\hat{\gamma}}$. We then use this in the cross-sectional model:
$(2) \hat{\gamma}_i = a + b Z_i + u_i$
where $Z_i$ is some cross-sectional variable that doesn't cause any violations in OLS assumptions and is relevant for explaining $\hat{\gamma}_i$. 

The claim in the applied econometrics literature is that $E[\epsilon_{it}^T \epsilon_{jt}] \not= 0$ in model $(1)$ leads to (i) No problem for the OLS coefficient estimates in $(2)$, but (ii) Biased standard errors in $(2)$. 

Can someone please post ideas about why this is the case? 
I don't understand what $\epsilon_{it}^T$ is in the expression $E[\epsilon_{it}^T \epsilon_{jt}] \not= 0$. Of course $\epsilon_{it}$ is a scalar and you can't transpose a scalar. This is seen HERE, where they apply this methodology.

A:

In order to be sure you need to go into the details, this implies comparing the true variance covariance matrix with the one you get in the second ols stage. 
The true one:
This can be obtained by replacing eq.2 into eq.1, pooled OLS follows, and from it, the true $\hat a , \hat b$ variance covariance matrix:
$Y_{it} = \alpha_i + \beta_i X_{it} + aD_t + bD_tZ_{i} +D_t u_{i} + \epsilon_{it}$ 
Using matrix notation to split the  equation in $\gamma$ parameters and others leads to:
$Y = X\theta + Z\gamma + \varepsilon$ 
where we are interested in $V(\hat \gamma)$ , $\gamma=[a \; b] $, Z is a two column vector $Z=[D_t \; D_tZ_i]_{[i=1,..,N;t=1,...,T]}$ ( a similar structure defines X but this is not of interest) and where $V(\varepsilon) =\Sigma$ has a full structure of between firms covariances that's why it's not diagonal ($\sigma^2I_{NT}$)
like in the GAUSS-MARKOV assumptions. By Frish-Waugh we can express $\gamma$ ols as :
$\hat \gamma = (Z'M_{X}Z)^{-1}Z'M_{X}Y$ where $M_X= I-X(X'X)^{-1}X' $
which implies the following true variance:
$V(\hat \gamma) = H\Sigma H'$ where $H = (Z'M_{X}Z)^{-1}Z'M_{X}$
The other one
Under the assumption of non correlated firms (and time periods but this is not the issue), $\Sigma$ has a simplier diagonal structure $\Delta$. This means that $\Delta$ triangular terms are 0. Under an even simpler specification,( the one that is estimated by default by econometric and statistical software for OLS) $\Sigma$ follows GAUSS-Markov assumptions meaning that even the diagonal terms are equal thus $\Sigma$ is downgraded to $\sigma^2I$
This implies that not considering between firms correlation would lead to $V(\hat\gamma) $as:
$V(\hat \gamma) = H\Delta H'$ or $V(\hat \gamma) = H\sigma^2I H' \equiv \sigma^2(Z'M_xZ)^{-1}$
which, as it can be seen, are not equal to the true one. 

