Learning non-linear estimation of Constant Elasticity of Substitution (CES) with ADMB
I recently discovered a free software called ADMB. Its function is to estimate non-linear regressions. I need non-linear regression estimation to get an elasticity parameter that is not equal to 1.
The example that ADMB uses for Robust Linear Regression is the (von Bertalanffy) growth curve model:
$$ s(a)=L_{\infty} \left(1-exp\left(-K(a-t_0)\right)\right) \label{1} $$where the parameters to be estimated are
$$L_{\infty}$$,
$$K$$, and
$$t_0$$. Suppose the observed data are
$$O_i$$and
$$a_i$$, and we want to predict
$$O_i$$using
$$s(a_i)$$, then ADMB needs to minimize the distance between
$$O_i$$and
$$s(a_i)$$:
$$ \min_{L_{\infty}, K, t_0} \sum_{i} (O_i-s(a_i))^{2} \label{2} $$What I need to do is replace model \ref{1} with the Constant Elasticity of Substitution function:
$$ Y = \gamma \left(\sum_{i=1}^n \delta_i X_i^{\rho}\right)^{-\frac{\upsilon}{\rho}} \label{3} $$but of course the natural-log version:
$$ \ln Y = \ln \gamma - \left(\frac{\upsilon}{\rho}\right) \ln \left(\sum_{i=1}^n \delta_i X_i^{\rho}\right) \label{4} $$So I need to alter equation \ref{1} in the ADMB example to the equation I want, namely \ref{4}. In \ref{4}, the observables are
$$\ln Y$$and
$$x_i$$. Everything else – the Greek letters – are parameters. I probably need to start with initial guesses. The safest guess would be
$$\rho=1$$to get Cobb-Douglas, heh. Also, I think I need to impose the restriction
$$\sum_i \delta_i = 1$$.
Since I am already tired today, I will continue tomorrow.