I want to simulate Wooldridge's (2005) approach to estimate a dynamic non-linear model. I have questions about generating the data.

Assume there is a dynamic probit model
Pr(Yit=1|Xit, Yit-1, ci)=Φ[Xitb+aYit-1+ci],
where Yit is the dependent variable; Yit-1 is the first-order lagged dependent variable; Xit is a vector of other independent variables; ci is the unobserved individual heterogeneity.

There is an initial condition problem when we estimate a dynamic non-linear model. Wooldridge's (2005) approach suggests we can solve this problem using a correlated random-effects model and including the initial value of the dependent variable, Yi0, as an additional regressor. I have two questions when I generate data to simulate this method.

First, how should I generate the initial value Yi0? It does not make sense to generate an exogenous Yi0. Also, the lagged value at t=-1 is not available. Is it correct to generate a Yi0 with Xit, ci and the error term without including a lagged value?

Second, shall I include Yi0 when generating the dependent variable in the following periods (t>1)? Despite that Yi0 is a regressor, I am not sure if I need to use it to generate data or generate data only with Xit, Yit-1, ci and the error term. Thank you for helping me.