# Estimating elasticity of substitution in nested CES functions

I have aggregate data on $L_t, K_t$ and $X_t$, and want to estimate elasticity of substitution parameters, $\gamma$ and $\sigma$ for these factors. Assuming the production function takes the following form: $$Y_t=(A_lL^{\gamma}+[A_kK_t^{\sigma} +A_xX_t^{\sigma}]^\frac{\gamma}{\sigma})^{\frac{1}{\gamma}}$$

Technology parameters, $A$s are not observable and hence need to be controlled for in an econometric specification. I am thinking of first estimating the parameter, $\sigma$, on the inner CES function combining $K$ and $X$. Then I should be able to estimate the outer CES parameter, $\gamma$. In a way, the two parameters are not jointly estimated. Would this method be valid, I mean, in statistical sense? Will I get consistent and unbiased estimates? I've read papers on non-linear regression methods, but was wondering if this simple approach is feasible. Thanks.

• This is related to the question I asked. The linked reference therein estimates a normalized nested CES production function. Might be of interest. – Graeme Walsh May 1 '16 at 16:24
• Possible duplicate of CES production function estimation – luchonacho Feb 25 '17 at 15:20
• Doesn't look like a duplicate. That linked question is a two input production function and the above is a three input production function that is nested. – BKay Feb 27 '17 at 13:08

Neglecting technology parameters and assuming constant returns to scale, the parameters $$\sigma$$ and $$\gamma$$ are jointly estimable via dynamic (least squares) programming. See this paper.