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The CES utility function has the form \begin{equation} u(x_1,\dots,x_n)=\left[\sum_{i=1}^n\alpha_ix_i^\rho\right]^{1/\rho}, \end{equation} where $\alpha_i$ is the consumption share parameter and $\sigma=\frac{1}{1-\rho}$ is the elasticity of substitution.

I'm interested in estimating the $\alpha_i$'s and $\sigma$. Since we do not directly observe utility levels, I think we can instead use data to estimate the Marshallian demand functions implied by the utility function: \begin{equation} x_i(p_1,\dots,p_n,M)=\frac{M(\alpha_i/p_i)^\sigma}{\sum_{j=1}^n\alpha_j^\sigma p_j^{1-\sigma}},\quad i=1,\dots,n. \end{equation} Clearly, the function is highly nonlinear in the parameters of interest. So a nonlinear estimation procedure would be required.

Say we have data consumption $x_i$, prices $p_i$ and income $M$. How do we go about estimating the parameters $\alpha_i$'s and $\sigma$?

I can find a lot of papers on estimating the CES production function, but I don't find them helpful since I don't observe the dependent variable ($u$). Given that the CES utility is a common functional form used in the literature, I suspect its estimation must be somewhere in the literature.

I managed to find the nonlinear least squares estimation in Stata, but I don't know how susceptible such a procedure is to misspecification and how much confidence one should put on the statistics (e.g. t-stat or p-value) produced by the program.

Any pointer to the papers/textbook chapters on the topic, as well as statistical packages that implement the estimation procedure will be greatly appreciated.

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