Firm Dynamics and Lead-Lag Effects: Statistical Analysis of Growth Rates

This research is dedicated to the analysis of firm growth rate correlations, with the primary focus on deciphering signals embedded within the time series data while effectively discerning them from random noise fluctuations. To achieve this, we employ methodologies derived from Random Matrix Theory and non-linear dynamics tools. The study focuses on a meticulous examination of eigenvectors across diverse frequency ranges, aiming to reveal patterns and dependencies within the dataset. Additionally, it employs an unconventional approach known as Hilbert Complex Principal Component Analysis (HPCA) to explore potential lead-lag relationships among these growth rates. The thesis begins by providing a comprehensive literature review, including an overview of Random Matrix Theory and its analytical tools, such as the Marcenko-Pastur distribution. Subsequently, the research focuses into the statistical analysis of log-returns and their correlations. This analysis extends to studying mode signals in both the time and frequency domains, including testing non-linearities using surrogate data methods. The thesis further explores whether sales returns and price returns originate from the same generative process, initially examining structures of prices' mode across various sampling frequencies and finally comparing them to synthetic data. Eventually, the research focuses on lead-lag effects, introducing a lagged correlation matrix and proposing methods to identify time lags that maximize lead-lag relationships. At this point we introduce the non-conventional approach of Hilbert Principal Component Analysis, drawing inspiration from climate science, and applies it to the dataset to uncover potential lead-lag relationships between variables.