Overview of Sections and Core Contributions
The following sections provide a comprehensive guide to the aggregation of micro volatility, ensuring that the reader is well-accompanied through the formal steps. The aim is to explain the dynamics of populations of agents with the necessary rigor to avoid incorrect results.
- Section Overview:
- Literature Review: In Section \ref{sec:literature}, we review related strands of literature.
- Data and Methods: Section \ref{sec:data1} introduces the data and methods used.
- Empirical Exploration: In Section \ref{sec:empirical}, we explore the size distribution and the distribution of growth rates in our data.
- Mathematical Definitions: Section \ref{sec:log_scale} contains useful mathematical definitions and properties.
- Firm-Level Data Nonlinearities: Section \ref{sec:firms_sectors} discusses the necessity of acknowledging nonlinearities in firm-level data to avoid incorrect outcomes.
- Diversification Argument: Section \ref{sec:lucas} provides a concise formal review of the diversification argument as in Lucas (1977) and the contribution of Gabaix (2011). It clarifies the framework for studying the decay of variance with population size.
Once the sectoral to aggregate (linear) relations are clarified, we proceed to study groups of agents with the goal of completing the nonlinear part of the aggregation in Section \ref{sec:agg_group}. The core contributions of this paper are formally explored in this section. Understanding the variance of groups of agents allows us to aggregate them through linear equations to arrive at aggregate volatility, thereby connecting micro parameters to the macro volatility observed.
- Robustness and Margins:
- Size Distribution Control: Section \ref{sec:size_dist_control} controls for results robustness when changing agents' size distribution.
- Extensive Margins: Section \ref{sec:bme} shows how extensive margins can be accounted for and discusses estimations of cross covariance elements.
Finally, Section \ref{sec:conclusion1} summarizes how all the mechanisms found combine in our empirical benchmark system to let the aggregate show its observed variance.
The developments in the Appendix are crucial, although not included in the main body for brevity. They cover:
- Estimations of uncertainty introduced by off-diagonal covariances.
- Clarification of accounts in the frequency domain.
- Derivation of moments of log-normal and log-Laplace distributions.
- Introduction of the codes defining the tests and estimation procedures used in the paper.