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A Residuals-Based Nonparametric Variance Ratio Test for Cointegration

Karsten Reichold

TL;DR

This paper develops a nonparametric variance ratio test for cointegration when applied to regression residuals, extending Br02's variance ratio to a single-equation framework without tuning parameters. It establishes a nuisance-parameter-free limiting null distribution and analyzes local power under near-unit-root alternatives, highlighting dependence on the long-run correlation measure $R^2$ and the deterministic detrending specification. Finite-sample simulations reveal that the variance ratio test typically exhibits smaller size distortions than conventional ADF-type tests, though its local power can be weaker; size-corrected power shows competitive performance across a range of short-run dynamics, with GLS detrending offering limited gains. An empirical illustration on cryptocurrency prices demonstrates the test’s practical applicability and its capacity to yield cointegration evidence consistent with established findings, while also underscoring scenario-dependent interpretation when sample sizes vary. Overall, the variance ratio approach provides a robust, tuning-parameter-free tool to complement traditional residuals-based cointegration tests in applied econometrics.

Abstract

This paper derives asymptotic theory for Breitung's (2002, Journal of Econometrics 108, 343-363) nonparameteric variance ratio unit root test when applied to regression residuals. The test requires neither the specification of the correlation structure in the data nor the choice of tuning parameters. Compared with popular residuals-based no-cointegration tests, the variance ratio test is less prone to size distortions but has smaller local asymptotic power. However, this paper shows that local asymptotic power properties do not serve as a useful indicator for the power of residuals-based no-cointegration tests in finite samples. In terms of size-corrected power, the variance ratio test performs relatively well and, in particular, does not suffer from power reversal problems detected for, e.g., the frequently used augmented Dickey-Fuller type no-cointegration test. An application to daily prices of cryptocurrencies illustrates the usefulness of the variance ratio test in practice.

A Residuals-Based Nonparametric Variance Ratio Test for Cointegration

TL;DR

This paper develops a nonparametric variance ratio test for cointegration when applied to regression residuals, extending Br02's variance ratio to a single-equation framework without tuning parameters. It establishes a nuisance-parameter-free limiting null distribution and analyzes local power under near-unit-root alternatives, highlighting dependence on the long-run correlation measure and the deterministic detrending specification. Finite-sample simulations reveal that the variance ratio test typically exhibits smaller size distortions than conventional ADF-type tests, though its local power can be weaker; size-corrected power shows competitive performance across a range of short-run dynamics, with GLS detrending offering limited gains. An empirical illustration on cryptocurrency prices demonstrates the test’s practical applicability and its capacity to yield cointegration evidence consistent with established findings, while also underscoring scenario-dependent interpretation when sample sizes vary. Overall, the variance ratio approach provides a robust, tuning-parameter-free tool to complement traditional residuals-based cointegration tests in applied econometrics.

Abstract

This paper derives asymptotic theory for Breitung's (2002, Journal of Econometrics 108, 343-363) nonparameteric variance ratio unit root test when applied to regression residuals. The test requires neither the specification of the correlation structure in the data nor the choice of tuning parameters. Compared with popular residuals-based no-cointegration tests, the variance ratio test is less prone to size distortions but has smaller local asymptotic power. However, this paper shows that local asymptotic power properties do not serve as a useful indicator for the power of residuals-based no-cointegration tests in finite samples. In terms of size-corrected power, the variance ratio test performs relatively well and, in particular, does not suffer from power reversal problems detected for, e.g., the frequently used augmented Dickey-Fuller type no-cointegration test. An application to daily prices of cryptocurrencies illustrates the usefulness of the variance ratio test in practice.
Paper Structure (18 sections, 3 theorems, 39 equations, 2 figures, 6 tables)

This paper contains 18 sections, 3 theorems, 39 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The Model and Assumptions
  3. Asymptotic Theory
  4. Preliminary Data Detrending
  5. The Variance Ratio Test
  6. Local Asymptotic Power
  7. Finite Sample Performance
  8. Empirical Size
  9. Size-Corrected Power
  10. A Large Initial Value $u_0$
  11. Empirical Illustration
  12. Summary and Conclusion
  13. Values of $\bar{c}$ and Asymptotic Critical Values
  14. Additional Results
  15. Proofs
  16. ...and 3 more sections

Key Result

Proposition 1

Let $\{x_t\}_{t\in{\mathbb Z}}$, $\{y_t\}_{t\in{\mathbb Z}}$, and $\{u_t\}_{t\in{\mathbb Z}}$ be generated by VR_NoCoint_Test:eq:x, VR_NoCoint_Test:eq:y, and VR_NoCoint_Test:eq:u, respectively, and let $\{w_t\}_{t\in{\mathbb Z}}$ satisfy Assumption VR_NoCoint_Test:ass:FCLTw. Then it holds under the where

Figures (2)

  • Figure 1: Asymptotic power of the tests at the nominal $5\%$ level for $\text{$\text{H}_0$:}\ \rho = 1$ under the local alternative $\rho = \rho_T = 1+c/T$ in cases D0 (first column), D1 (second column), and D2 (third column) for $m=1$. Note: The results are based on $10{,}000$ Monte Carlo replications and standard Brownian motions are approximated by normalized partial sums of $10{,}000$ i.i.d. standard normal random variables.
  • Figure 2: OLS detrended log prices of cryptocurrencies from June 21, 2019 to February 25, 2020.

Theorems & Definitions (8)

  • Remark 1
  • Proposition 1
  • Proposition 2
  • Remark 2
  • Proposition 3
  • Remark 3
  • Remark 4
  • Remark 5