tion m and an idiosyncratic (unobserved) α : To be able to work with various conditional characteristic functions cor- responding assumptions ensuring existence of the conditional distributions need to be made and in what follows we assume that all the conditional density functions and moments exist as generalized functions in S $^{∗}$. In Evdokimov (2011) independence (conditional on the corresponding period X $^{′}$s ) of the regression error from α, and from the X $^{′}$s and error of the other period is assumed: with f$_{·}$$_{|}$$_{·}$ denoting corresponding conditional densities. Conditionally on X$_{i}$$_{2}$ = X$_{i}$$_{1}$ = x the model takes the form 4 with z representing Y$_{1}$, x representing Y$_{2}$, x ∗ standing in for m ( x, α ) , u for U$_{1}$ and u$_{x}$ for U$_{2}$. The convolution equations derived here for 4 or 4a now apply to conditional densities. z = x ∗ + u ; x = x ∗ + u$_{x}$ Y$_{i}$$_{1}$ = m ( X$_{i}$$_{1}$, α$_{i}$ ) + U$_{i}$$_{1}$ ; Y$_{i}$$_{2}$ = m ( X$_{i}$$_{2}$, α$_{i}$ ) + U$_{i}$$_{2}$. f$_{t}$ = f$_{Uit}$ |$_{X}$$_{it}$$_{,α}$$_{i}$$_{,X}$$_{i}$$_{(}$$_{−}$$_{t}$$_{)}$$_{,U}$$_{i}$$_{(}$$_{−}$$_{t}$$_{)}$ ( u$_{t}$ | x, ... ) = f$_{Uit}$ |$_{X}$$_{it}$ ( u$_{t}$ | x ) , t = 1 , 2 27