Essentials of Empirical Finance
Essentials of Empirical Finance
This page collects a set of materials on empirical finance, centered around my Notes, Essentials of Empirical Finance, which can be downloaded* here on the right.
The Notes provide a graduate-level introduction to the main statistical and econometric tools used in empirical finance, combining theoretical foundations with applications to financial data.
The first part of the Notes provides a rigorous probabilistic and statistical analysis of univariate financial returns and prices.
The second part covers the fundamentals of cross-sectional financial econometrics, with a continuous focus on multivariate financial data and returns.
Three appendices on Probability, Statistics, and Linear Algebra provide the mathematical and statistical foundations required throughout the Notes, making the material largely self-contained.
To complement the theoretical material, a collection of links to shared Google Colab notebooks is provided below. The material is organized by chapter, with a brief summary of the topics covered in each chapter followed, in square brackets, by links to the corresponding notebooks.
The notebooks implement in Python most of the methods and examples discussed in the Notes, allowing readers to reproduce the results, work directly with financial data, and develop a more practical understanding of the techniques detailed in the Notes.
Chapter 1, Prices and Returns, introduces financial prices, linear and logarithmic returns, portfolio returns, and single- and multi-period investment strategies, together with some key empirical facts about financial markets. [Link to Notebook 1]
Chapter 2, The Log Random Walk Model, develops the basic probabilistic model for the evolution of log prices and returns, including its Gaussian specification and its implications over long investment horizons. [Link to Notebook 2]
Chapter 3, Estimation of the Log Random Walk Model, discusses the estimation of expected returns and volatility, including sample estimators and the RiskMetrics approach, with particular attention to estimation uncertainty. [Link to Notebook 3]
Chapter 4, On the (non-) Normality of Returns, investigates the empirical distribution of financial returns, examining higher moments, normality tests, empirical distribution functions and density estimation, as well as alternatives to the normal distribution. [Link to Notebook 4]
Chapter 5, Multivariate Problems in Probability and Statistics, extends the analysis to multiple financial assets, introducing random vectors, variance-covariance matrices and multivariate optimization, with applications to portfolio allocation and to improved estimation of expected returns and covariance matrices. [Link to the Notebook 5]
Chapter 6, The Linear Regression Model, provides a comprehensive treatment of linear regression, Ordinary and Generalized Least Squares, goodness of fit, statistical inference, forecasting and the interpretation of regression coefficients, concluding with an application to Style Analysis.
Chapter 7, Factor Models, introduces factor representations of asset returns, their estimation and their use for obtaining efficient representations of large variance-covariance matrices, while also discussing the economic and statistical nature of factors.
Chapter 8, Principal Component Analysis, develops the spectral decomposition of variance-covariance matrices and Principal Component Analysis, with applications to covariance-matrix denoising and the construction of implicit statistical factors.
* © 2026 Francesco Rotondi. All rights reserved.
These Notes are made freely available for personal and educational use only. They may be downloaded and cited with appropriate attribution. Reproduction, redistribution, modification, or commercial use of the Notes, in whole or in substantial part is strictly forbidden.