[Paper Review] The Incidental Parameters Problem in Testing for Remaining Cross-section Correlation

Artūras Juodis, Simon Reese|arXiv (Cornell University)|Oct 8, 2018

Spatial and Panel Data Analysis4 citations

TL;DR

This paper identifies a critical flaw in the CD test for cross-section correlation when applied to panel data models with many incidental parameters, showing that the test statistic diverges as the time dimension grows—even under correct modeling of cross-section dependence. To resolve this, the authors propose a weighted CD test statistic that restores standard normal inference under the null, addressing the incidental parameters problem in empirical applications.

ABSTRACT

In this paper we consider the properties of the Pesaran (2004, 2015a) CD test for cross-section correlation when applied to residuals obtained from panel data models with many estimated parameters. We show that the presence of period-specific parameters leads the CD test statistic to diverge as length of the time dimension of the sample grows. This result holds even if cross-section dependence is correctly accounted for and hence constitutes an example of the Incidental Parameters Problem. The relevance of this problem is investigated both for the classical Time Fixed Effects estimator as well as the Common Correlated Effects estimator of Pesaran (2006). We suggest a weighted CD test statistic which re-establishes standard normal inference under the null hypothesis. Given the widespread use of the CD test statistic to test for remaining cross-section correlation, our results have far reaching implications for empirical researchers.

Motivation & Objective

• To investigate the behavior of the CD test statistic when applied to residuals from panel models with many estimated parameters.
• To demonstrate that the CD test statistic diverges as the time dimension increases, even when cross-section dependence is correctly modeled.
• To identify this divergence as an instance of the incidental parameters problem in panel data econometrics.
• To propose a modified, weighted CD test statistic that restores asymptotic standard normality under the null hypothesis.
• To ensure reliable inference for remaining cross-section correlation in widely used panel models such as two-way fixed effects and common correlated effects estimators.

Proposed method

• Theoretical analysis of the CD test statistic's asymptotic distribution under the null hypothesis of no remaining cross-section correlation.
• Derivation of the asymptotic behavior of the CD test when period-specific parameters are present, showing divergence as T → ∞.
• Construction of a weighted CD test statistic that adjusts for the influence of incidental parameters by reweighting the test components.
• Proof that the weighted CD test statistic converges in distribution to a standard normal under the null, even with many incidental parameters.
• Application of the weighted test to two prominent panel models: the two-way fixed effects model and the common correlated effects estimator (CCEP).
• Use of U-statistic theory and asymptotic distribution theory to establish the validity of the proposed test under general conditions.

Experimental results

Research questions

• RQ1 Does the CD test statistic remain asymptotically valid when applied to residuals from panel models with many incidental parameters?
• RQ2 What is the asymptotic behavior of the CD test statistic when period-specific effects are included in the model?
• RQ3 Can a modified test statistic restore standard normal inference under the null hypothesis in the presence of incidental parameters?
• RQ4 How does the incidental parameters problem affect the reliability of the CD test in common panel data models like two-way fixed effects and CCEP?
• RQ5 Is the proposed weighted CD test robust to the presence of unobserved common factors and time-specific effects?

Key findings

• The standard CD test statistic diverges in probability as the time dimension T increases, even when cross-section dependence is correctly modeled.
• This divergence constitutes a clear instance of the incidental parameters problem in panel data models with many individual and time-specific effects.
• The divergence occurs under both the two-way fixed effects model and the common correlated effects estimator (CCEP), undermining standard inference.
• The proposed weighted CD test statistic achieves asymptotic standard normal distribution under the null hypothesis, restoring valid inference.
• The weighted test is robust to the presence of unobserved common factors and time-specific effects, ensuring correct size in finite samples.
• Empirical applications relying on the standard CD test may produce misleading conclusions about remaining cross-section correlation due to this asymptotic bias.