Estimating systemic importance with missing data in input-output graphs

TL;DR

This paper develops bounds on the error in estimating the importance of firms in a production network when data is incomplete, using influence vectors derived from input-output matrices, and compares these to PageRank methods.

Contribution

It provides sharp bounds on influence vector errors under incomplete data and explores extensions to other economic models and missing data distributions.

Findings

Derived bounds on influence vector errors with incomplete data

Analyzed impact of data missingness on influence measures

Compared influence vector estimates to PageRank algorithms

Abstract

In the context of the Cobb-Douglas productivity model we consider the $N \times N$ input-output linkage matrix $W$ for a network of $N$ firms $f_1, f_2, \cdots, f_N$. The associated influence vector $v_w$ of $W$ is defined in terms of the Leontief inverse $L_W$ of $W$ as $v_W = \frac{\alpha}{N} L_W \vec{\mathbf{1}}$ where $L_W = (I - (1-\alpha) W')^{-1}$, $W'$ denotes the transpose of $W$ and $I$ is the identity matrix. Here $\vec{\mathbf{1}}$ is the $N \times 1$ vector whose entries are all one. The influence vector is a metric of the importance for the firms in the production network. Under the realistic assumption that the data to compute the influence vector is incomplete, we prove bounds on the worst-case error for the influence vector that are sharp up to a constant factor. We also consider the situation where the missing data is binomially distributed and contextualize the bound on the influence vector accordingly. We also investigate how far off the influence vector can be when we only have data on nodes and connections that are within distance $k$ of some source node. A comparison of our results is juxtaposed against PageRank analogues. We close with a discussion on a possible extension beyond Cobb-Douglas to the Constant Elasticity of Substitution model, as well as the possibility of considering other probability distributions for missing data.