I have a simple question:
Assume I have an AR(1) process b = rho_bb(-1) + e_b* and another process which depends on this process, a = rho_aa(-1) + psi_bb + e_a. Now assume I can observe b and estimate rho_b and the variance of the shock e_b. Then, I observe additionally the variable a and estimate both processes at the same time in one mod-file. That is, I estimate the parameters rho_a, rho_b and psi_b and the variances of the shocks e_a and e_b. It turns out, that the estimate for rho_b is slightly different for both cases, 0.4398 in the first case and 0.4423 in the second one (I have attached the mod-file and data file).
Just to be sure, this is a numerical problem? Theoretically, we should get exactly the same estimate for the AR(1) process of b, right? I am asking because in a bigger model, the difference is much higher (0.5 vs. 0.8) for rho_b. What would be the best way to estimate both processes? First estimate the AR(1) process of b only, then calibrate rho_b and the variance of e_b to the estimated values and then estimate the process for a (with b being still an observable)?