Lecture 07

• Part 1: Empirical Asset Pricing w/o ML

• Part 2: Return prediction with ML

• Introduction to Empirical Asset Pricing
• Theoretical Foundation: Stochastic Discount Factor (SDF)
• Historical Development within SDF Framework
• GMM and Time-Varying Risk
• Current Developments and Challenges
• Summary and Key Takeaways

• Empirical asset pricing seeks to understand how risk influences asset returns.
• It examines the relationship between expected returns and risk via the risk-return trade-off.
• The Stochastic Discount Factor (SDF) serves as a fundamental concept, harmonizing various asset pricing models.
• Importance: Understanding empirical asset pricing aids investors and asset managers in making informed decisions based on risk assessments.

• The cross-section of returns refers to variations in asset returns based on risk factors.
• Key questions include:
  • Why do certain assets outperform others?
  • How is risk assessed and priced in financial markets?
• Understanding these dynamics is crucial for effective investment strategies.

• Definition: The SDF, denoted , satisfies the equation:
  where is the gross return on asset and is the information set at time .
• The SDF encapsulates the trade-off between risk and expected return, forming the backbone of asset pricing models.

• The SDF represents the marginal utility of consumption, linking asset prices with consumption choices and investor preferences.
• It establishes a framework where risk factors directly impact expected asset returns, crucial for empirical asset pricing insights.

• The Consumption Capital Asset Pricing Model (CCAPM) derives the SDF from intertemporal consumption choices, integrating economic factors.
• Key implication: Asset returns reflect consumers' time preferences and risk aversion, emphasizing connections between macroeconomic influences and asset pricing.

• Capital Asset Pricing Model (CAPM) can be expressed within the SDF approach as:

• This model suggests that expected returns are driven by market risk alone.
• Limitations: While foundational, CAPM fails to address the complexities of asset returns observed in empirical studies.

• CAPM's singular focus on market risk does not adequately capture asset return dynamics.
• The introduction of multifactor models arose to address CAPM's limitations, incorporating additional factors impacting expected returns.
• Empirical findings indicate that multiple factors beyond market risk are significant in explaining asset returns.

• APT presents a framework where the SDF is expressed as a linear combination of multiple risk factors:

• represents systematic risk factors influencing asset returns.
• APT enhances the flexibility and empirical robustness of asset pricing models, allowing for a broader consideration of risks.

• The Fama-French model incorporates three factors: market return, size (SMB), and value (HML):

• It addresses real empirical phenomena such as size and value premiums, providing a more comprehensive view of expected returns.

• The Carhart model expands on Fama-French by introducing a momentum factor.
• This model reflects the empirical observation that past performances influence future returns, enriching the SDF representation and improving predictive power.

• Testing asset pricing models presents challenges, including limitations in available data and common issues related to model misspecification.
• Researchers strive to empirically validate theoretical models to enhance their predictive capabilities in real-world scenarios.

• Generalized Method of Moments (GMM), developed by Hansen, serves as a robust tool for estimating parameters in asset pricing models.
• GMM provides a practical approach to empirically test SDF frameworks against observed data, contributing to model accuracy and relevance.

• Conditional SDF models account for variations in risk over time:

• These models reflect changing market conditions, enhancing asset pricing's adaptability and accuracy in volatile environments.

• The growing number of identified factors presents challenges in effective asset pricing.
• The SDF framework provides context for understanding these factors, connecting them to fundamental economic risks.

• Researchers face challenges in integrating diverse empirical findings into cohesive asset pricing models.
• The adequacy of traditional asset pricing models in contemporary markets is an ongoing debate among scholars and practitioners.

• Recent advances in econometrics and machine learning have the potential to significantly enhance empirical asset pricing.
• Machine learning techniques present promising avenues for improving both model estimation and testing methodologies.

• The SDF framework unifies various asset pricing models under a cohesive theoretical structure.
• It incorporates vital insights from traditional models while allowing for adaptability to evolving market dynamics.
• Grasping the nuances of the SDF is essential for navigating the complexities of asset pricing in today's financial landscape.

• Hansen, L. P. (1982). The Generalized Method of Moments Estimation.
• Fama, E. F., & French, K. R. (1992). The Cross-Section of Expected Stock Returns.
• Carhart, M. M. (1997). On Persistence in Mutual Fund Performance.

• Factor Models: Key for modeling equity returns; provides a parsimonious statistical description of returns’ cross-sectional dependence.

• APT Foundation: Arbitrage Pricing Theory offers a robust economic basis for understanding risk exposures and risk premia.

• Challenges in Estimating Asset Risk Premia:

  • Low signal-to-noise ratio
  • Small sample size
  • Multiple correlated predictors
  • Ambiguity in functional forms

• Machine Learning Applications: Adoption of variable selection and dimensionality reduction has been part of empirical asset pricing, enhancing model robustness.

• Recent Advancements: New methodologies from machine learning facilitate rigorous empirical discoveries, complementing traditional economic theories.

• Objectives of the Paper:

  1. Review recent methodological contributions categorized by purpose.
  2. Discuss asymptotic theory related to data dimensions and compare methodologies.

• Basic Equation:

  • : Excess returns of test assets (e.g., sorted portfolios)
  • : Factor exposure matrix
  • : Factor innovations
  • : Idiosyncratic errors

• Expected Return Decomposition:

  • : Risk premia vector
  • : Pricing errors vector

• No-Arbitrage Condition:

1. Observable Factors:

   • ; if factors are tradable portfolios.

2. Latent Factors and Exposures:

   • Following Connor & Korajczyk (1986), assumes all factors are latent.

3. Observable Exposures but Latent Factors:

   • MSCI Barra model allows for time-varying exposures (Rosenberg, 1974).

• Need for Conditional Models:

  • Static models are inadequate for assets with variable risk exposures and nonlinear payoff structures.

• Conditional Factor Model Specification:

• Model Requirements:

  • : Excess returns vector.
  • : Idiosyncratic errors vector.

Model Frameworks

• Rosenberg (1974):

  • , where is an observable characteristics matrix.

• Instrumented PCA (IPCA):

• Time-Varying Risk Premia:

  • Gagliardini et al. (2016): .

• Nonlinear Extensions:

  • Gu et al. (2021): Introduced a conditional autoencoder model for nonlinear dynamics.

• Challenges:

  • Black-box nature of deep learning models.
  • Need for rigorous theoretical justification.

• Objective:

  • Understand the behavior of expected returns amidst noise from unforeseeable influences.

• Challenges:

  • Expected returns are difficult to measure because of noise in asset prices due to unpredictable news.

• Importance of Improved Measurement:

  • Better measurement aids in refining economic theories to explain expected returns.

• Three Basic Strands:

  1. Cross-sectional regressions (Fama & French, 2008; Lewellen, 2015):

     • Focus on stock-level characteristics affecting expected returns.

  2. Time series regression of portfolio returns on predictors:

     • Challenges include handling high-dimensional predictors (Welch & Goyal, 2007; Koijen & Nieuwerburgh, 2011; Rapach & Zhou, 2013).

  3. Machine Learning Approaches:

     • Increased relevance for empirical asset pricing, leveraging variable selection and dimension reduction.

• Limitations of Traditional Methods:

  • Inability to handle large numbers of predictors effectively.
  • Risk of overfitting and challenges with multiple comparisons.

• Factor Model Variance:

  • Total variance decomposed into systematic risk and idiosyncratic risk.

• TSR (Time Series Regression)

  • Factor exposure estimates:
  • Asset-by-asset time series regressions yield .

• CSR (Cross-Sectional Regression)

  • When exposures are observable:
  • Commonly used for individual stocks, accommodating time-varying characteristics.

• Used when neither factors nor loadings are known:

• SVD provides estimates of latent factors and their loadings.
• PCA allows flexibility in research despite challenges in interpreting factors.

• Extracts latent factors from realized return covariances:

• Provides a framework to generalize risk-premia estimation.

• Addresses flexibility by estimating conditional factors:

• Handles dynamic and unobservable characteristics effectively.

• Introduced by Gu et al. (2021):

  • Proposes a conditional autoencoder structure for modeling risk-return trade-offs.

• Architecture:

  • Structure allows betas to depend on stock characteristics non-linearly.

• Mathematical Representation:

  • Initializing the model and propagating information through layers enhances flexibility in feature extraction.

• Risk Premium of a Factor:

  • Informs about the compensation investors require for holding associated risk.
  • Essential for understanding asset pricing models.

• Estimating Risk Premia:

  • Simple average return for a factor provides a starting point.
  • Models often formulated for non-tradable factors, emphasizing the need for tradable counterparts.

Methodology

1. First-pass Regression (Time Series):

   • Estimates via individual asset regressions.

2. Second-pass Regression (Cross-Section):

   • Estimates risk premia with OLS on estimated .

Generalized Least Squares (GLS) Version

• Replaces OLS in the second pass:

• Building Factor Mimicking Portfolios:
  • Fama & Macbeth (1973) suggest regressing realized returns onto .

• Mimicking Non-Tradable Factors:
  • Goal: Estimate risk premia of non-tradable factors by constructing tradable proxies.

• Approach:
  1. First Pass: SVD of for and .
  2. Second Pass: Run OLS on to obtain risk premia.
  3. Third Pass: Projects onto .

• Identifying Weak Factors:

  • Kan & Zhang (1999): Risk premia becomes distorted when including weak factors.
  • Kleibergen (2009): Proposes valid test statistics across various β values.

• Addressing Weak Factors:

  • Jagedesh et al. (2019): Proposed multivariate variable selection to improve measurement accuracy.

• Importance of Test Assets:
  • Critical for empirical asset pricing analysis.

1. Standard Approach: Use standard portfolios based on characteristics.
2. Expanded Approach: Include broad asset clusters or characteristics-sorted portfolios.
3. Targeted Test Assets: Focus on specific factors of interest.

• Risk Premium and SDF:

  • A factor's risk premium equals its (negative) covariance with the stochastic discount factor (SDF).
  • In the setup of the model, the SDF is expressed as:

SDF Loadings Estimation

• Moment Conditions:

  • Formulate a set of moment conditions:

• Optimization:

• GMM Estimator:

  • Solver defined by:

• PCA for SDF:

  • Strong covariation suggests that SDF can be represented as a function of dominant sources from asset returns.

• SDF Estimation:

  • Can be achieved without relying on knowledge of factor identities.

• Parameterization of SDF:

  • Represented in terms of a small number of linear combinations of factors:

• Estimation Approach:

• Applying DML:

  • Framework proposed to mitigate biases from numerous factors.

• Example Framework:

  • Identify factors of interest and control for expected returns through respective regressions.

• Double Machine Learning (DML) is a framework that aims to estimate causal parameters in the presence of high-dimensional covariates.
• It combines machine learning techniques with traditional econometric methods to control for confounding variables and minimize bias.

• Orthogonalization: DML utilizes two machine learning models to predict the outcomes and the confounders, effectively orthogonalizing the treatment effect from these variables.
• Two-Step Estimation: It employs a two-step procedure where the first step involves predicting the nuisance parameters, and the second step focuses on estimating the causal effect.

• DML is particularly useful in settings with:
  • High-dimensional data: Many predictors compared to the number of observations.
  • Complex relationships: Non-linear and interaction effects among variables.
  • Causal inference: When aiming to derive treatment effects or causal relationships.

1. Model Specification: Specify the model with the treatment variable (e.g., policy intervention) and outcome variable (e.g., economic output).
2. Nuisance Parameter Estimation:
   • Fit machine learning models to predict the outcome and the covariates.
   • Obtain residuals that are orthogonal to the treatment.
3. Estimation of Treatment Effects:
   • Use regression or other econometric methods on the residuals obtained in step 2.
   • Estimate the causal effect of the treatment variable on the outcome variable.

• Optimizing SDF:

• Neural Network Approaches:

  • Use networks to incorporate past performance and characteristics for SDF estimation.

• Purpose:
  • Economic theory aids in identifying the best model but leads to numerous candidate models.
  • Recent notable models with observable portfolios include those by Fama & French (2015), Hou et al. (2015), and others.

• Assessment of Factor Pricing Models:

  • Formalized as statistical hypothesis testing problems.
  • Common focus on zero alpha condition:

• Estimation:

• Quadratic Test Statistic:

• Limitations:

  • Requires ; asymptotic properties may be compromised under certain conditions.

• Comparative Analysis:

  • Testing models is often less informative than comparing them.
  • Classical GRS test relates to optimal Sharpe ratios of portfolios formed from assets.

• Testing Insights:

  • Useful in assessing risk premiums across asset portfolios using factors.

• Model Expansion:

  • Probabilistic methods allow for more robust model comparison.

• Priors:

  • Utilize Spike-and-Slab priors to increase model selection robustness.

• Importance of Testing:
  • Essential for verifying model resilience against various factors and ensuring accurate pricing.
  • Both statistical tests and economic theory must guide model specification to enhance asset pricing predictions.

• Definition of Alphas:

  • Portions of expected returns unexplained by risk factors, often termed anomalies.
  • Significant findings question the effectiveness of traditional asset pricing models.

• Data-Snooping Concerns:

  • The prevalence of multiple testing leads to potential false discoveries in alpha estimation.
  • Examples of early proposals include Lo & MacKinlay (1990) and Sullivan et al. (1999).

Null Hypotheses for Alpha Testing

• Propose single null hypothesis for a collection of alphas:

• Testing Statistics:

  • : test statistic for (often the t-statistic).

Components of Testing

• Let be the number of rejections and the total.
• Both and are random variables, and performance can be modeled to limit relative to .

• Naïve Procedure: Tests individual hypotheses at predetermined level .
• Bonferroni Correction: Adjusts level to to control the rate of false discoveries.

Enhanced Testing Examples

• Giglio et al. (2021a):
  • Develops asymptotic guarantees for valid p-values and t-statistics.
• Bayesian Methods:
  • Joint modeling of factors and improvements in alpha estimation.

• Treats alphas as properties with distributions, focusing on improving power while controlling for false discoveries.
• Addressing changes in alpha estimates can augment statistical power without increasing error rates.

• The interplay of multiple testing in finance necessitates robust methodologies.
• Bayesian approaches offer promising avenues for managing the complexities of alpha testing while mitigating data-snooping biases.

• Key Asymptotic Schemes:
  • Three primary asymptotic schemes are identified in asset pricing:
    1. Fixed N, Large T: Traditional approach emphasizing fixed asset counts and increasing time periods.
    2. Large N, Large T: Both asset counts and periods increase, offering flexibility in model application.
    3. Large N, Fixed T: Focuses on situations where the number of assets grows while the time series remains constant.

• Classical Approach:

  • Developed the central limit theorem for estimating risk premia.
  • Emphasizes the importance of adjusting estimates to account for potential bias in beta estimations.

• Model Implications:

  • The variance of estimators provides insight into the reliability of risk premium estimates.
  • Important adjustments (e.g., Shanken adjustments) are recommended to improve accuracy.

• Comparative Benefits:
  • When both asset count and time increase, it eases the need for complex covariance structure estimations.
  • Simplifies the inference process regarding risk premia.
  • Both OLS and GLS yield similar asymptotic behavior, making model application more flexible.

• New Framework:

  • Offers distinct strategies for modeling ex-post risk premia.
  • Important for applying factor models in practical settings with a fixed timeframe.

• Recommendations:

  • Utilizing this framework can better handle time-varying factors and invisible risk exposures.

• Model Selection:

  • Choosing the appropriate asymptotic scheme is crucial based on the data structure and research goals.
  • Each scheme provides specific benefits and should be considered according to the empirical context.

• Future Directions:

  • Ongoing developments in asymptotic theory will continue to refine investment models and predictions in finance.