Finite-Sample Properties of Model Specification Tests for Multivariate Dynamic Regression Models

Paper: arXiv 2601.21272

Authors: Koichiro Moriya, Akihiko Noda

Abstract

We propose a new model specification test for multiple-equation systems with cross-equation error and dynamic regressor–error dependences. Conventional tests often rely on exogeneity conditions strong enough to ensure consistency of the OLS estimator. These exogeneity conditions are violated when regressors and errors are dynamically dependent, rendering conventional model specification tests invalid. To address these limitations, we clarify the relationship among alternative exogeneity conditions, characterize the consistency of competing multiple-equation estimators, and propose a generalized Durbin estimator for multiple-equation systems with an intercept, cross-equation error and regressor–error dependences. We show that our estimator remains consistent under the weakest exogeneity condition. We then derive its asymptotic distribution and construct Wald tests. Our Monte Carlo experiments confirm that the bootstrap-based Wald test substantially improves finite-sample size control. An application of the bootstrap-based Wald test to the Fama–French multifactor models leaves the null hypothesis unrejected in cases where competing FGLS-based tests reject it.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 8.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous econometric framework extending Durbin’s estimator to multivariate systems under dynamic regressor-error dependence, supported by complete asymptotic proofs. It includes both extensive Monte Carlo simulations for finite-sample performance and a practical empirical application evaluating the Fama-French asset pricing models.

Research Flowchart

  flowchart TD
    A[Goal: Valid Model Specification Tests under Dynamic Regressor-Error & Cross-Equation Dependence] --> B[Clarify Exogeneity Conditions & Estimator Consistency]
    B --> C[Develop Generalized Durbin Estimator for Multiple-Equation Systems]
    D[Input: Monte Carlo Simulations & Fama-French Multifactor Data] --> E[Construct Asymptotic & Bootstrap-Based Wald Tests]
    C --> E
    E --> F[Findings: Bootstrap Wald Test Restores Size Control & Prevents Over-Rejection in Fama-French Models]