Square terms within exp() can be estimated?

I need some help in understanding the Bayesian estimation in dynare.
My questions are very basic.

First of all, the part of my code is as follows.

log(e_w) = rho_w*log(e_w(-1))+sig_w*eta_w/100;
de_w = e_w - STEADY_STATE(e_w);
de_w2 = de_w^2;

y_ag  = e_ta*(land_g*psi_g)^(pi_l)*((k_ag(-1))^(phi_a)*(h_ag)^(1-phi_a))^(1-pi_l);
psi_g = psi_g(-1)^rho_l * (exp(-alpha_l*de_w2)*v_z)^(1-rho_l);
v_z  = ((z)^(gamma_z))/gamma_z;

Here, I want to estimate rho_l and alpha_l through the Bayesian estimation.
In some literature, it seems that I saw that when estimating it, square terms are removed.
Moreover, square terms are within exp().

So, I’m not sure that we can trust the estimation results of the above code.
If not applicable, how can I correct this problem?

Also, in ‘Residuals of the static equations:’
should residuals be all zero?

I sincerely appreciate your help in advance.

The problem is not estimation per se, but order of approximation. One cannot generally say whether a first order approximation will be sufficient. One thing to check is whether the parameters are identified.