The Stone-Gerry production function generally takes a form of: $$f(x_{i})=\prod_{i=1}^n(x_{i}-a_{i})^{\gamma_i}$$
where $x_i$ represents all inputs in the model, $a_i$ is a constant representing the initial investment which does not contribute to production and $\gamma_{i}$represent output elasticies.
when running a regression and converting it into a production function (which was already discussed on the site) how should our inital regression look when we have constants $a_i$ included?
Stone-Geary production functions attempt to reflect the real-world observation that for production to be feasible, there exist minimum thresholds in the quantities of inputs employed. This is not about initial investment but that one cannot use miniscule levels of inputs and obtain "very little" output -for production to even begin, one needs input amounts above a certain threshold, dictated usually by technology. At the aggregate (market or macro level), perhaps this effect practically vanishes, but for firm-level data, this may be an important feature of actual production.
Assume that we have data for $i=1,...,n$ firms and we postulate the existence of a Stone-Geary production function
$$q_i=f_i(\mathbf x_i;\mathbf a_i) = A\prod_{j=1}^K(x_{ji}-a_{j})^{\gamma_j}$$
The model is non-linear in the parameters, and It will remain non-linear in the parameters even if we take the natural logarithms. So one could in principle consider Non-linear Least Squares (NLS) estimation using the above specification without taking logs, but it is advisable to do take logarithms, because it lends a "degree of linearity" in the model which will only do good to the NLS estimator and the iterative estimation algorithm that it will be used by the software, since no closed form solution exists. So we specify, for $K$ non-constant regressors,
$$\ln q_i =\ln A + \gamma_1\ln(x_{1i}-a_{1})+...+\gamma_K\ln(x_{Ki}-a_{K}) +u_i,\;\;\; i=1,...,n$$
where we want to estimate the gammas and the alphas plus the constant term... Note that the above specification assumes that the "thresholds" (the alphas) are the same for all firms, per regressor (they are not indexed by $i$). So in all we are estimating $2K+1$ parameters with a cross-sectional sample of size $n$ and we must have $2K+1 < n$.
Also, note that $A$ is the level of of output when all inputs exceed their threshold by one unit.
The NLS estimator requires starting values for the coefficients. For the gammas, one could run OLS by setting the alphas equal to zero and use the OLS estimates as starting values. For the alphas, one could start by setting them equal to zero, except if one has some idea about what they could be approximately.
A good thing of such an approach is that the estimated thresholds can be checked for plausibility, using existing knowledge about production processes, which provides a non-statistical test for the validity of the specification.
An open access paper discussing the approach with empirical applications can be found here.
