Actually, from what you describe, I think the law of motion for n should be
n_t =(1-rho_{t})n_{t-1}+m_{t}You inherit employment n_{t-1} from the previous period, some matches dissolve based on rho_t and you get new matches m_t. This is similar to the law of motion for capital. Alternatively, you could shift the whole equation by 1 period:
n_{t+1} =(1-rho_{t})n_{t}+m_{t}and use
predetermined_variables nRegarding the limited information assumption, you need to change the information set of the expectations operator in Dynare. You can use the EXPECTATION(-2) operator for that (see the manual). You can see the change in timing in the working paper version at
ecb.europa.eu/pub/pdf/scpwps/ecbwp304.pdf?0ea454554115cfcd977fa9cc2bd4b493