Bootstrap Inference in Cointegrating Regressions: Traditional and Self-Normalized Test Statistics

Abstract

Traditional tests of hypotheses on the cointegrating vector are well known to suffer from severe size distortions in finite samples, especially when the data are characterized by large levels of endogeneity or error serial correlation. To address this issue, we combine a vector autoregressive (VAR) sieve bootstrap to construct critical values with a self-normalization approach that avoids direct estimation of long-run variance parameters when computing test statistics. To asymptotically justify this method, we prove bootstrap consistency for the self-normalized test statistics under mild conditions. In addition, the underlying bootstrap invariance principle allows us to prove bootstrap consistency also for traditional test statistics based on popular modified OLS estimators. Simulation results show that using bootstrap critical values instead of asymptotic critical values reduces size distortions associated with traditional test statistics considerably, but combining the VAR sieve bootstrap with self-normalization can lead to even less size distorted tests at the cost of only small power losses. We illustrate the usefulness of the VAR sieve bootstrap in empirical applications by analyzing the validity of the Fisher effect in 19 OECD countries.

KEYWORDS:

• Bootstrap consistency
• Cointegration
• Self-normalization
• Sieve bootstrap
• Size distortions
• Tuning parameters

Supplementary Materials

The supplementary material contains proofs and additional empirical and simulation results.

Acknowledgments

We are grateful to the editor Atsushi Inoue, an associate editor, and four referees for several insightful and constructive comments that have led to significant changes and improvements of the article. We further thank Katharina Hees, Fabian Knorre, and participants at the Econometrics Colloquium at the University of Konstanz, the IAAE 2021 Annual Conference, the 2021 Asian and North American Summer Meetings of the Econometric Society, and the ${\text{XII}}_t$ Workshop in Time Series Econometrics in Zaragoza for helpful comments. Parts of this research were conducted while Karsten Reichold held a position at the University of Klagenfurt.

Disclosure Statement

The authors have no competing interests to declare.

Data Availability Statement

MATLAB code for empirical applications is available on www.github.com/kreichold/CointSelfNorm