How to identify a binary model with varying mean and variance I have the following model,
Yi = 1.( Xiβ + εi > 0) where εi~ Ɲ(Ziδ,1). 
or
Yi = 1.( Xiβ + εi > 0) where εi~ Ɲ(0, (µ + Ziδ)2)
with a binary model, can i identify β and δ in such a model? How? and what conditions enable me to identify this model?
 A: Your first model with $X_i \beta$ for the latent location parameter and $Z_i \delta$ for the mean of the errors is somewhat unusal. As both linear predictors pertain to the mean on the latent scale one would usually put all regressors into a single $X_i \beta$ which comprises all available regressors ($X_i$ and $Z_i$ in your notation) and use a zero-mean error term $\epsilon_i$ (as you do in the second model).
In the second model, the variance is over-specified in your notation. You can have an error term $\epsilon_i \sim \mathcal{N}(0, (1 + Z_i \delta)^2)$ or alternatively $\mathcal{N}(0, \exp(0 + Z_i \delta))$, i.e., using a sqrt- or log-link on the variance with its own linear predictor. However, then $Z_i$ must not contain an intercept. One typically fixes the intercept in such a way that for $\delta = 0$ the variance is 1 as I have done in my notation above.
The reason for this is that the overall variance on the latent scale is not identified. You can just identify variance differences between the groups defined by the regressors $Z_i$. However, you may need a lot of observations over a sufficiently large range to estimate the parameter $\delta$ well. It may also be hard to distinguish variance differences in $Z_i$ from non-linearities or interactions in $X_i$, especially if the underlying variables overlap.
