Structural equation modeling mathematical/analytical estimation of latent variables I want to know how latent variables are estimated in structural equation modeling. I do not understand at what point and in what way the standard SEM software comes up with the estimates for latent variable variances/covariances etc. 
To demonstrate, I have simple 3-wave intercept-only panel model based on
$x_{it} = \alpha_{i} + \delta_{it}; \ i = 1,...,n; \ t = 1,2,3$, 
the model is trivial, I just chose it because it is the simplest example I could think of. I am uninterested in the means/intercepts, so assume grand mean centered variables. 
In structural equation modeling, we achive this by specifying a measurement and structural part of the model on wide-format data (this is the basis for fixed- and random-effects models in SEM, as outlined in Bollen & Brand 2010, but here I am using an alternative notation outlined in Bollen 1989, p.395)
$\eta^* = B\eta^* + \zeta^* \\
y^* = \Lambda \eta^*$
where 
$\eta^* = (\alpha, x_1, x_2, x_3)$, $\zeta^* = (\alpha, \delta_1, \delta_2, \delta_3)$, $y^* = (x_1, x_2, x_3)$
$\Lambda = \left[\begin{array}{l}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{array}\right]$, $B = \left[\begin{array}{l}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0\end{array}\right]$, $\Psi = \left[\begin{array}{l}\alpha^{2} & 0 & 0 & 0 \\ 0 & \delta_{1}^{2} & 0 & 0 \\ 0 & 0 & \delta_{2}^{2} & 0 \\ 0 & 0 & 0 & \delta_{3}^{2}\end{array}\right]$.
The model-implied covariance matrix is achieved by putting the equation for $\eta^*$ in reduced-form and working out $y^*y^{*\intercal}$, which (if we respect the assumptions reflected in $\Psi$, i.e., no error covariances etc.) leads to
$\begin{align}\Sigma(\theta) & = \Lambda(I - B)^{-1}\Psi(I - B)^{-1 \intercal}\Lambda^{\intercal} \\
& = \left[\begin{array}{c} \alpha^{2} + \delta_{1}^{2} & & \\ \alpha^{2} & \alpha^{2} + \delta_{2}^{2} &  \\ \alpha^{2} & \alpha^{2} & \alpha^{2} + \delta_{3}^{2} \end{array}\right]
\end{align}$
I understand up to here, and the results are what we should expect: the observed variable at each point in time has just been decomposed into a time-invariant and time-varying error part. 
My questions are: 
1) What are the parameters in $\theta$, given that the factor loadings/regression coefficients in $B$ have been set to 1.0? Aren't they $\alpha^{2}, \delta_{t}^{2}$?
2) How do I solve for $\alpha^{2}, \delta_{t}^{2}$? Do I set the equation equal to the observed covariance matrix $\Sigma(\theta) = S$, work out the partial derivative with respect to each of the unknowns, set to zero and solve? 
 A: Your model isn't clear to me, that's not necessarily bad, but I think you can express it more simply.
You don't need $B$ and $\Psi$ or $\alpha$ in your equations. These aren't necessary for the simple model you specify. You can ignore $\delta$ for this model, if you want to keep things simple and just standardize everything.
If you do this, then the implied correlation matrix is given by:
$\Sigma(\Theta) = \Lambda \Lambda ^T$
The sample correlation matrix is S, and you fit the model by minimizng $F_{ML}$:3
$F_{ML} = log(det(S)) + tr(S \Sigma(\Theta^{-1}) - log(det(\Sigma(\Theta)) - p$
Where det is the determinant, and tr is the trace of the matrix, and p is the number of parameters estimated.
This is the simplest case, other more complex cases extend this, but the basic idea is the same: Find the implied covariance matrix, given the parameters. Minimize F.
I hope this is what you were asking. :)  If not, feel free to ask for clarification.
(I wrote a paper about this once: https://www.researchgate.net/publication/7151927_Confirmatory_factor_analysis_using_Microsoft_Excel) 
