# Structural Estimation, Simulations, and Initial Values

I want to estimate model parameters and fear about the impact of initial values of simulations.

Short model overview

Consider a firm producing a homogeneous output good whose output price, $$P_t$$, is $$dP_t=mP_tdt+s P_t dW_t.$$

Denoting the production output by $$Q_t$$, the firm's profits, $$\Pi_t$$, are $$\Pi_t=P_tQ_t-C(Q_t),$$ where $$C(Q_t)$$ is a cost function.

Suppose $$C(Q_t)=c_0+c_1Q_t+c_2Q_t^2$$ with $$c_0,c_1,c_2\geq0$$.

The model includes further state variables and choice variables (investment rates, equity issuance, etc.).

Estimation by moment matching

To estimate the model, I simulate many firms (arising from different shocks $$dW_t$$ and find their optimal production/financing/... policies) and seek parameter values which make the simulated firms behave like firms in the real data (e.g., match average capacity utilization and investment rates, etc.). The estimator is thus the method of simulated moments (SMM).

My Question

The simulated panel somewhat depends on the initial values, $$P_0$$. If $$P_0$$ is high, then so is $$P_t$$ and $$Q_t$$. In particular fixed costs like $$c_0$$ would fail to correct for these changes: If $$P_0$$ is very large, then $$P_tQ_t$$ is large and $$c_0$$ can be large. But if $$P_0$$ is small, then so is $$P_tQ_t$$ and $$c_0$$ can't be large (otherwise the firm never produces anything).

However, estimation results should be independent of initial values? As output price is a geometric Brownian motion, there is no stable, long-run, stationary distribution from which I could draw initial values.

Q: How do I avoid that the simulated panel (and thus my estimated parameters) depends on the initial values of my simulations?