Learning non-linear estimation of Constant Elasticity of Substitution (CES) with ADMB

Feb 27, 2020 · 2 min read

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∞(1−exp(−K(a−t0)))\label1 s(a)=L_{\infty} \left(1-exp\left(-K(a-t_0)\right)\right) \label{1}

where the parameters to be estimated are

L∞L_{\infty}

,

KK

, and

t0t_0

. Suppose the observed data are

OiO_i

and

aia_i

, and we want to predict

OiO_i

using

s(ai)s(a_i)

, then ADMB needs to minimize the distance between

OiO_i

and

s(ai)s(a_i)

:

min⁡L∞,K,t0∑i(Oi−s(ai))2\label2 \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=γ(∑i=1nδiXiρ)−υρ\label3 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⁡γ−(υρ)ln⁡(∑i=1nδiXiρ)\label4 \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\ln Y

and

xix_i

. Everything else – the Greek letters – are parameters. I probably need to start with initial guesses. The safest guess would be

ρ=1\rho=1

to get Cobb-Douglas, heh. Also, I think I need to impose the restriction

∑iδi=1\sum_i \delta_i = 1

.

Since I am already tired today, I will continue tomorrow.

Economics
Krisna Gupta
Authors
Krisna Gupta
Lecturer