Separate AR(1) process estimated differently


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)?

I am thankful for any help or comment,
all the best,
data_AR_1.xls (20.5 KB)
AR_1.mod (681 Bytes)

It is not a numerical problem.

In theory the two estimators (just b, vs a&b) of rho_b are equal asymptotically. But there is no reason they should be identical in finite samples, in fact you would expect them to never be equal in finite samples (it would occur with probability zero).

The reason is that the two estimators are using different methods to identify b. In the first case you are finding the rho_b that best fits the finite sample b data. In the second case you are finding the rho_b that best fits the finite sample b and a data jointly.