NBER美国国民经济研究局-NBER-Measuring-_Dark-Matter_-in-Asset-Pricing-Models_83页_1mb

Summary of "Measuring 'Dark Matter' in Asset Pricing Models"

Core Content

This working paper introduces a novel information-based fragility measure for Generalized Method of Moments (GMM) models, particularly in the context of asset pricing. The measure is designed to detect model fragility and overfitting tendencies by analyzing the informativeness of cross-equation restrictions in GMM models. The term "dark matter" is used metaphorically to refer to model components or parameters that are difficult to estimate directly from the data but have significant effects on model performance.

Main Points

• Definition of Dark Matter Measure: The measure evaluates the fragility of GMM models by comparing the asymptotic variances of two estimators: one based on the full set of moment restrictions (full GMM) and another based on a subset (baseline GMM). It identifies the largest discrepancy in all linear directions, indicating how much the model relies on cross-equation restrictions rather than observable data.

• Connection to Model Properties: The dark matter measure is linked to two critical properties:

  • Refutability: As the dark matter measure increases, the power of specification tests decreases, meaning the model becomes harder to reject.
  • Overfitting: Models with higher dark matter measures tend to overfit the data, leading to poor out-of-sample performance, especially under local instability in the data-generating process (DGP).

• Sensitivity Analysis: The measure is closely related to sensitivity analysis, which assesses how model parameters respond to small changes in the DGP. The authors propose using a baseline model to define "reasonable" perturbations, thereby improving the robustness of fragility assessments.

• Empirical Applications: The paper applies the dark matter measure to two models:

  • A rare-disaster risk model, where parameters like the likelihood and magnitude of disasters are hard to estimate directly.
  • A long-run risk model with a nine-dimensional parameter space, demonstrating that models with similar in-sample fits can have dramatically different fragility properties.

• Robust Estimation: The paper advocates for the use of recursive (two-stage) GMM estimation in the presence of high model fragility, despite its worse in-sample fit. This is because recursive GMM can provide better out-of-sample performance when the model is misspecified.

Key Information

• Theoretical Framework: The dark matter measure is derived from asymptotic variances and eigenvalue decomposition, making it computationally feasible even for complex dynamic models.

• Sample-Size Interpretation: The measure can be interpreted as the amount of additional data required for the baseline estimator to match the precision of the full estimator, which is influenced by the informativeness of cross-equation restrictions.

• Statistical Foundations: The authors build on the semiparametric local minimax efficiency bounds and extend them to Markov processes with local instability. This provides a formal basis for interpreting the dark matter measure as an informational metric.

• Literature Connection: The measure relates to existing work on:

  • Model fragility and local instability in time series.
  • Structural estimation using cross-equation restrictions.
  • Bayesian learning and robustness considerations in economic models.

• Practical Implications: The measure helps diagnose model fragility in asset pricing applications, particularly in cases where the peso problem (under-representation of rare events in the data) is present. It also highlights the importance of incorporating parameter uncertainty and agent robustness in model evaluation.

Conclusion

The dark matter measure provides a quantitative tool for assessing the internal refutability and external validity of GMM models. It helps economists identify models that are overly reliant on untestable restrictions, which can lead to poor out-of-sample performance and lack of robustness. The measure is especially useful in the presence of local instability or misspecification, offering a new perspective on model evaluation and selection in asset pricing and macroeconomic modeling.