[Paper Review] Time-Invariance Coefficients Tests with the Adaptive Multi-Factor Model

Liao Zhu, Robert A. Jarrow+1 more|arXiv (Cornell University)|Nov 9, 2020

Financial Markets and Investment Strategies24 references6 citations

TL;DR

This paper tests time-invariant beta coefficients in the Adaptive Multi-Factor (AMF) model, which is derived from the Generalized Arbitrage Pricing Theory (GAPT) and estimated using a Groupwise Interpretable Basis Selection (GIBS) algorithm on ETF factors. The AMF model shows time-invariant betas for all periods under 6 years, outperforming the Fama-French 5-factor model, which exhibits unstable betas even in short horizons, especially during crises.

ABSTRACT

The purpose of this paper is to test the time-invariance of the beta coefficients estimated by the Adaptive Multi-Factor (AMF) model. The AMF model is implied by the generalized arbitrage pricing theory (GAPT), which implies constant beta coefficients. The AMF model utilizes a Groupwise Interpretable Basis Selection (GIBS) algorithm to identify the relevant factors from among all traded ETFs. We compare the AMF model with the Fama-French 5-factor (FF5) model. We show that for nearly all time periods with length less than 6 years, the beta coefficients are time-invariant for the AMF model, but not for the FF5 model. This implies that the AMF model with a rolling window (such as 5 years) is more consistent with realized asset returns than is the FF5 model.

Motivation & Objective

• To test the time-invariance of beta coefficients in multi-factor models under the Generalized Arbitrage Pricing Theory (GAPT).
• To evaluate whether the Adaptive Multi-Factor (AMF) model with Groupwise Interpretable Basis Selection (GIBS) produces more stable betas than the Fama-French 5-factor (FF5) model.
• To validate the AMF model using no-arbitrage and goodness-of-fit tests, ensuring its empirical reliability.
• To compare the performance of AMF and FF5 in in-sample and out-of-sample settings across varying time periods.
• To investigate whether the constant beta assumption in AMF is empirically justified over different investment horizons, especially during financial stress.

Proposed method

• The AMF model uses LASSO regression on a high-dimensional set of ETFs to estimate factor exposures via the GIBS algorithm, selecting relevant risk factors for each stock.
• The model is estimated using price differences rather than returns, preserving time-invariant beta coefficients in the price process as required by GAPT.
• No-arbitrage tests confirm the validity of the GAPT framework by checking for non-zero intercepts (Jensen's alpha) in the factor model.
• Time-invariance of betas is tested using a linear model (constant betas) versus a generalized additive model (GAM) with time-varying splines, comparing model fit via ANOVA.
• False Discovery Rate (FDR) correction (BHY method) is applied to p-values from the ANOVA tests to control for multiple testing across stocks.
• Rolling window analysis is applied to assess stability of betas over different time periods, with window lengths up to 6 years.

Experimental results

Research questions

• RQ1 Are the beta coefficients in the AMF model time-invariant across different investment horizons, particularly under 6 years?
• RQ2 How does the stability of beta coefficients in the AMF model compare to that of the Fama-French 5-factor model across various time periods?
• RQ3 Does the AMF model with GIBS-selected factors provide better in-sample and out-of-sample fit than the FF5 model?
• RQ4 Is the constant beta assumption in the AMF model empirically valid, especially during financial crises?
• RQ5 To what extent does the GAM-based time-invariance test detect time-varying betas, and how does overfitting affect the results in short time windows?

Key findings

• The AMF model exhibits time-invariant beta coefficients for all time periods with duration less than 6 years, regardless of the starting year.
• The Fama-French 5-factor model shows significant time variation in betas even for periods under 5 years, particularly during the financial crisis.
• The AMF model achieves higher in-sample adjusted R² and better out-of-sample R² than the FF5 model, indicating superior fit and robustness.
• In the GAM-based time-invariance test, the AMF model still outperforms the FF5 model, with 100% of time periods showing lower percentages of stocks with time-varying betas.
• The percentage of stocks with time-varying betas is consistently lower in the AMF model than in the FF5 model across all time periods, as shown in the heatmap comparison (Figure 9) and difference map (Figure 10).
• The GAM test results are exploratory due to overfitting risks in short windows (e.g., n=156 observations with p>50 parameters), suggesting caution in interpreting time-varying signals.