Using Moment Constraints in GME Estimation

Abstract

In this contribution, we explore the sensitivity of parameter estimates derived through the generalized maximum entropy (GME) approach under alternative specifications of the width of the error term supports. Although many recommend a “three-sigma” rule for setting the width of this term, there can be noticeable differences in the results if it is expanded beyond that, as others in the literature have suggested. We use a Monte Carlo analysis to see how imposing a moment-based condition into the GME problem, as an additional constraint, affects the results. We find that it removes the sensitivity of the parameter estimates to the width of the supports for the error term and that this remains robust even when the data is ill-conditioned. Based on this, we recommend that researchers impose this condition when doing GME-based estimation, to improve the performance of the estimator.

Appendix

General Derivation for Basic GME Problem

We can write the GME problem as

$$\begin{aligned} & \hbox{max} - \sum\limits_{i = 1}^{2I + 1} {p_{i}^{b} \ln p_{i}^{b} } - \sum\limits_{j = 1}^{N} {\sum\limits_{k = 1}^{2K + 1} {p_{jk}^{e} \ln p_{jk}^{e} } } \hfill \\ &{\text{s.t.}}\quad \quad y_{j} = x_{j} \left( {\sum\limits_{i = 1}^{2I + 1} {p_{i}^{b} z_{i}^{b} } } \right) + \sum\limits_{k = 1}^{2K + 1} {p_{jk}^{e} z_{k}^{e} } \quad \forall j \hfill \\ & {\text{where}}\;\;z_{k}^{e} \in \left\{ {z_{1}^{e} , \ldots z_{K + 1}^{e} , \ldots ,z_{2K + 1}^{e} } \right\},z_{K + 1}^{e} = 0\;\quad {\text{and}}\quad - z_{k}^{e} = z_{2(K + 1) - k}^{e} \quad \forall k \ne K + 1 \hfill \\ \end{aligned}$$

which can be rewritten as

$$\begin{aligned} & \hbox{max} - \left[ {\sum\limits_{i = 1}^{I} {p_{i}^{b} \ln p_{i}^{b} } + (1 - \sum\limits_{i \ne I + 1} {p_{i}^{b} } )\ln (1 - \sum\limits_{i \ne I + 1} {p_{i}^{b} } ) + \sum\limits_{i = I + 2}^{2I + 1} {p_{i}^{b} \ln p_{i}^{b} } } \right] \\ & \quad - \sum\limits_{j = 1}^{N} {\left[ {\sum\limits_{k = 1}^{K} {p_{jk}^{e} \ln p_{jk}^{e} } + (1 - \sum\limits_{k \ne K + 1} {p_{jk}^{e} } )\ln (1 - \sum\limits_{k \ne K + 1} {p_{jk}^{e} } ) + \sum\limits_{k = K + 2}^{2K + 1} {p_{jk}^{e} \ln p_{jk}^{e} } } \right]} \\ & \quad {\text{s}} . {\text{t}} .\\ & \quad y_{j} = x_{j} \left( {\sum\limits_{i = 1}^{I} {p_{i}^{b} z_{i}^{b} } + (1 - \sum\limits_{i \ne I + 1} {p_{i}^{b} } )z_{I + 1}^{b} + \sum\limits_{i = I + 2}^{2I + 1} {p_{i}^{b} z_{i}^{b} } } \right) \\ &\quad \quad \quad+ \left( {\sum\limits_{k = 1}^{K} {p_{jk}^{e} z_{k}^{e} } + (1 - \sum\limits_{k \ne K + 1} {p_{jk}^{e} } )z_{K + 1}^{e} + \sum\limits_{k = K + 2}^{2K + 1} {p_{jk}^{e} z_{k}^{e} } } \right)\quad \forall j \\ \end{aligned}$$

Which give the first-order conditions

$$\begin{aligned} \frac{\partial L}{{\partial p_{i}^{b} }} & = - \left( {1 + \ln p_{i}^{b} } \right) + \ln (1 - \sum\limits_{{i^{\prime} \ne I + 1}} {p_{{i^{\prime}}}^{b} } ) + 1 - \sum\limits_{j = 1}^{N} {\lambda_{j} x_{j} \left( {z_{i}^{b} - z_{I + 1}^{b} } \right)} = 0\quad \forall i \ne I + 1 \\ \frac{\partial L}{{\partial p_{jk}^{e} }} & = - \left( {1 + \ln p_{jk}^{e} } \right) + \ln (1 - \sum\limits_{{k^{\prime} \ne K + 1}} {p_{{jk^{\prime}}}^{e} } ) + 1 - \lambda_{j} \left( {z_{k}^{e} - z_{K + 1}^{e} } \right) = 0\quad \forall j,k \ne K + 1 \\ \end{aligned}$$

taking the k = 1 and k = 2 K + 1 cases, we can derive the result $\lambda_{j} = \frac{1}{{z_{1}^{e} - z_{2K + 1}^{e} }}\ln \left( {\frac{{p_{j,2K + 1}^{e} }}{{p_{j,1}^{e} }}} \right)$ and substitute it into the following expression derived from the FOCs for $p_{i}^{b}$

$$\begin{aligned} \ln p_{{i^{\prime } }}^{b} - \ln p_{i}^{b} & = \sum\limits_{{j = 1}}^{N} {\lambda _{j} x_{j} \left( {z_{i}^{b} - z_{{i^{\prime } }}^{b} } \right)\quad } \forall i,i^{\prime } \in {\mathbf{I}} = \left\{ {1, \ldots ,I,I + 2, \ldots ,2I + 1} \right\}, \\ & \qquad\qquad\qquad \qquad \qquad \qquad \qquad \qquad I + 1 \notin {\mathbf{I}} \\ \end{aligned}$$

And we obtain $\frac{{p_{{i^{\prime}}}^{b} }}{{p_{i}^{b} }} = \left[ {\prod\limits_{j = 1}^{N} {\left( {\frac{{p_{j,2K + 1}^{e} }}{{p_{j,1}^{e} }}} \right)^{{x_{j} }} } } \right]^{{\frac{{z_{i}^{b} - z_{{i^{\prime}}}^{b} }}{{z_{1}^{e} - z_{2K + 1}^{e} }}}} \quad \forall i,i^{\prime} \in {\mathbf{I}}\;\quad {\text{and}}\quad I + 1 \notin {\mathbf{I}}$.