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A Residuals-Based Nonparametric Variance Ratio Test for Cointegration

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.


Introduction
Analyzing the relationship between stochastically trending (economic) time series in a single-equation regression framework entails the risk of obtaining misleading spurious regression results (Granger and Newbold, 1974;Phillips, 1986). Practitioners thus rely on statistical tests to assess whether the time series are cointegrated. In this context, it is popular in applications to employ so-called no-cointegration tests for the null hypothesis of no cointegration against the alternative of cointegration based on regression residuals estimated by ordinary least squares (OLS).
Alternative approaches include, e. g., single-equation cointegration tests based on conditional error correction models (e. g., Kremers et al., 1992;Zivot, 2000) and system-based tests (e. g., Phillips and Ouliaris, 1990;Johansen, 1991;Shintani, 2001;Breitung, 2002;Harris and Poskitt, 2004;Cai and Shintani, 2006). Systembased approaches have the advantage that they do not require the specification of a left-hand side variable and may also allow to test for the number of (linearly independent) cointegrating relations in the system. If, however, there exist reasons for a specific choice of the left-hand side variable, it is convenient and intuitively appealing to analyze the relationship between the variables in a single-equation framework.
Among the most popular residuals-based no-cointegration tests are the parametric augmented Dickey-Fuller (ADF, Dickey and Fuller, 1979;Said and Dickey, 1984) type test, proposed in Engle and Granger (1987) and asymptotically justified in Phillips and Ouliaris (1990), the semiparametric Z α test (Phillips, 1987;Phillips and Ouliaris, 1990), and the parametric MSB test (Perron and Ng, 1996;Pesavento, 2007). 1 The three tests share the common feature that they require the choice of tuning parameters (e. g., the number of lags in an auxiliary regression and/or kernel and bandwidth choices to estimate a long-run variance parameter) to accommodate the correlation structure in the data. Although these tuning parameter choices allow for asymptotically valid inference, they are likely to have adverse effects on the performance of the tests in finite samples.
In contrast, this paper proposes a nonparametric no-cointegration test, which requires neither the specification of the correlation structure in the data nor the choice of tuning parameters. The test is an extension of Breitung's (2002) nonparametric variance ratio unit root test (originally applied to observed univariate time series) to regression residuals. 2 The test statistic is easy to compute as it is defined as a (re-scaled) ratio between the sample variances of the regression residuals and their partial sums. Under the null hypothesis, the sample variances converge to random variables whose distributions are scale dependent on the same long-run variance parameter. This makes the limiting null distribution of the test statistic nuisance parameter free without estimating the long-run variance parameter directly. Under the alternative of cointegration, the test statistic converges to zero at rate equal to sample size, which makes the test consistent. In the following, we refer to the test as the (nonparametric) variance ratio (no-cointegration) test. 3 The paper derives asymptotic theory for the variance ratio no-cointegration test in a setting that allows for the presence of deterministic time trends both in the regression equation and in the regressors. In the presence of deterministic components, we derive the asymptotic properties of the variance ratio nocointegration test under both OLS detrending and general least squares (GLS) detrending. Moreover, the paper compares the variance ratio test in terms of local asymptotic power, empirical size and size-corrected power with the ADF, Z α and MSB tests in a detailed simulation study.
We follow, e. g., Pesavento (2007) and Perron and Rodríguez (2016) and impose a directional restriction on the model, which excludes cointegration between the right-hand side variables. In this case, local asymptotic power of the variance ratio test and its competitors is a function of a single nuisance parameter, R 2 , which measures the long-run correlation between the regression errors and the regressors (cf. Pesavento, 2004;2007). In addition, we construct a simulation setting that allows to analyze the effects of different short-run dynamics in the data generating process (DGP) on the performance of the tests in finite samples while controlling for effects of R 2 . This justifies a comparison between local asymptotic power of the tests and their power in finite samples. The results reveal that local asymptotic power properties do not serve as a useful indicator for the performance of residualsbased no-cointegration tests in finite samples and we explain why Pesavento (2007, p. 127) and Perron and Rodríguez (2016, p. 99) come to opposing conclusions.
Finally, an empirical illustration applies the variance ratio test and its competi-2 Breitung's (2002) test, in turn, is a generalization of the unit root test of Shin and Schmidt (1992).
3 Please note that Westerlund (2005) has extended Breitung's (2002) unit root test to test for cointegration in panel data. However, little is known about the asymptotic and finite sample properties of Breitung's (2002) unit root test when applied to regression residuals in a pure time series setting. In particular, the nonparametric variance ratio no-cointegration test is not considered in the two insightful contributions of Pesavento (2007) and Perron and Rodríguez (2016). tors to daily prices of the four cryptocurrencies with highest market capitalization. Test decisions are heterogeneous across tests, but the variance ratio test, the ADF test, and the ADF test based on a modified information criterion provide reliable evidence for the presence of cointegration between the four cryptocurrencies with the highest market capitalization. The results are in line with those in related literature pointing towards the presence of cointegrating relationships in the cryptocurrency market (cf., e. g. Keilbar and Zhang, 2021;Bykhovskaya and Gorin, 2022).
The paper proceeds as follows: Section 2 introduces the model and its underlying assumptions. Section 3 defines the variance ratio no-cointegration test under OLS-and GLS detrending and derives its asymptotic properties. Section 4 analyzes the performance of the variance ratio test in finite samples and Section 5 contains the empirical illustration. Section 6 summarizes and concludes. All proofs are relegated to the Appendix, which also contains additional asymptotic and finite sample results. Supplementary Material, available on the author's homepage, provides further finite sample results.
Throughout, x denotes the integer part of a real number x, 1 m denotes the m-dimensional vector of ones and I m denotes the (m × m)-dimensional identity matrix. The symbols p −→ and w −→ signify convergence in probability and weak convergence, respectively, as the sample size T → ∞.

The Model and Assumptions
We consider the model denotes the long-run covariance matrix of {w t } t∈Z .
The asymptotic results depend on the specification of the deterministic components in (2.1) and (2.2). This paper considers three different cases: no deterministics (D0), intercept only (D1) and intercept and linear trend (D2). In case D0, we set x 0 = µ = 0 and remove d t τ from (2.2). In case D1, we allow for x 0 = O P (1) and set µ = 0 and d t = 1. Finally, in case D2, we allow for x 0 = O P (1) and a deterministic trend in x t and set d t = [1, t] . 5 5 Perron and Rodríguez (2016) also consider the case with a deterministic trend in the regressors but only a constant (rather than a constant and a linear trend) in the regression. Following Pesavento (2007), we abstain from considering this case as it implies that the limiting null distributions of the test statistics depend on the drift parameter µ being exactly zero or not -and thus leaves the opportunity of choosing wrong critical values in applications. Considering more general deterministic components, e. g., polynomial time trends, in (2.2) and/or (2.1) is also possible but beyond the scope of this paper.

Preliminary Data Detrending
If deterministic components are included in (2.2), it is standard practice in applications to first detrend z t := [y t , x t ] according to the choice of d t using ordinary least squares (OLS). In this case, the OLS detrended time series are defined as (3.1) In case D1 and D2 the definition reduces to z t = z t − T −1 T s=1 z s and respectively. For notational brevity, we define z t = [ y t , x t ] := z t in case D0. With these definitions in place, it follows from (2.2) that where u t is defined analogously to z t . The OLS residuals in (3.3) are given by where β denotes the OLS estimator of β in (3.3).
Remark 1. Perron and Rodríguez (2016) suggest to use GLS detrended data rather than OLS detrended data in cases D1 and D2 to increase local asymptotic power of residuals-based no-cointegration tests. Forρ := 1 +c/T , with some constantc ≤ 0, define zρ t := z t −ρz t−1 and dρ t := d t −ρd t−1 , t = 2, . . . , T . The GLS detrended variables are constructed as y In what follows, the main text focuses in cases D1 and D2 on OLS detrended data, whereas a series of remarks is dedicated to the use of GLS detrended data.

The Variance Ratio Test
Typically, residuals-based no-cointegration tests require tuning parameter choices (e. g., the number of lags in an auxiliary regression and/or kernel and bandwidth choices to estimate a long-run variance parameter) to accommodate the correlation structure in the data. Various simulation results indicate that these tuning parameter choices can affect the finite sample performance of no-cointegration tests considerably. This is particularly unfavorable in situations where different tuning parameter choices lead to different test decisions.
To obtain a tuning parameter free no-cointegration test, we apply the nonparametric variance ratio test of Breitung (2002), originally proposed to test for a unit root in an observed univariate time series, to the OLS residuals u t defined in (3.4), i. e., based on OLS detrended data. The variance ratio test statistic is thus defined as In contrast to, e. g., the MSB and Z α test statistics, the variance ratio test statistic does not depend on a consistent estimator of the long-run variance parameter Ω ξ·v . Instead, it depends on a random variable η T , whose limiting null distribution is, as we shall see in the proof of Proposition 1, scale dependent on Ω ξ·v . In this sense, the variance ratio test statistic may be interpreted as a selfnormalized version of the MSB test statistic. 7 Scale dependence of the limiting null distribution of η T on Ω ξ·v is key to obtain a nuisance parameter free limiting distribution of the variance ratio test statistic under the null hypothesis of no cointegration. Breitung and Taylor (2003).
7 This self-normalizing feature of the variance ratio test fits well to the (bootstrap-assisted) self-normalized testing approach of Reichold and Jentsch (2022) for hypotheses on the cointegrating vector.
The limiting null distribution of the variance ratio test statistic is nonstandard but free of any nuisance parameters. It only depends on the dimension m of x t , through the dimension of W v (r), and on the deterministic component in (2.2), as the detrended processes appear in the limit. 8 We proceed with analyzing the behavior of the variance ratio test statistic under the alternative of cointegration. Proposition 2 shows that in the presence of cointegration, the variance ratio test statistic converges to zero at rate equal to sample size. Hence, the variance ratio test is a left-tailed test, rejecting the null hypothesis of no cointegration in favor of the alternative of cointegration for small (i. e., close to zero) realizations of VR. Table A.2 in Appendix A provides corresponding asymptotic critical values in cases D0, D1, and D2 for m = 1, . . . , 5.

Remark 2. The variance ratio test applied to the residuals from the regression with GLS detrended data is defined as VR
as defined in Remark 1. Remarks 3 and 4 in Section 3.3 derive the limiting distribution of VR (GLS) both under the null hypothesis of no cointegration and under local alternatives and also provide some guidance on the choice ofc.

Local Asymptotic Power
To analyzes the performance of the variance ratio test statistic under local alternatives, we set ρ = ρ T = 1 + c/T , with c ≤ 0, in (2.3). For c = 0, the regression errors {u t } t∈Z are integrated of order one, i. e., there is no cointegration between {y t } t∈Z and {x t } t∈Z . Under the local alternative c < 0, the error process {u t } t∈Z is a near unit root process such that T −1/2 u rT is an Ornstein-Uhlenbeck process, with R 2 := Ω −1 ξξ Ω ξv Ω −1 vv Ω vξ = 1 − Ω ξ·v /Ω ξξ and W v (r) := m −1/2 1 m W v (r) (L ?)emma 5.1]PeRo16cf., e. g.,. The coefficient R 2 lies between zero and one and measures the squared long-run correlation between the regression errors {u t } t∈Z and the regressors {x t } t∈Z . Pesavento (2004;2007) shows analytically that R 2 is the only nuisance parameter affecting local asymptotic power of the ADF, MSB, and Z α tests. The following propositions shows that R 2 is also the only nuisance parameter affecting local asymptotic power of the variance ratio test. where Proposition 3 shows that the limiting distribution of the variance ratio test statistic under local alternatives depends on the location parameter c, the number of regressors m and on the nuisance parameter R 2 . Local asymptotic power of the variance ratio test at the nominal level α is given by the probability that G VR,c is smaller than the α-quantile of the limiting null distribution of VR. For c = 0, G VR,c coincides with the limiting null distribution given in Proposition 1, i. e., local asymptotic power at c = 0 is equal to α. To ease comparisons between the limiting distribution of the variance ratio test and the limiting distribu-tions of the tests considered in Pesavento (2007) under local alternatives, rewrite Remark 3. In case of GLS detrending, it follows from Perron and Rodríguez (2016, Lemma 5.3) and similar arguments as used in Perron and Rodríguez (2016, Proof of Theorem 5.2) and in the proof of Proposition 1 that the variance ratio test statistic converges under the local alternative

VR,c (c) does not depend onc and coincides with G VR,c as defined in Proposition 3 in case D0. The limiting null distribution of the variance ratio test based on GLS detrending is given by G (GLS)
VR,0 (c).

Remark 4.
To provide some guidance on the choice ofc, we follow Perron and Rodríguez (2016, p. 93) and choosec such that P G  Figure 1 illustrates the local asymptotic power curve of the variance ratio test at the nominal 5% level (in cases D1 and D2 based on OLS detrended and GLS detrended data) for m = 1 and R 2 ∈ {0, 0.4, 0.8}. In case of GLS detrending,c is 9 In this notation, the limiting distributions of the ADF, Z α and MSB statistics under local alternatives are given by c chosen as suggested in Table A.1 in Appendix A -with results being qualitatively similar for other choices ofc. The figure also displays the local asymptotic power curves of the ADF test, the Z α test, and the MSB test based on OLS detrended data derived in Pesavento (2007) and the local asymptotic power curve of the ADF and MSB tests based on GLS detrended data derived in Perron and Rodríguez (2016), with the value ofc as suggested in Perron and Rodríguez (2016 , Table 1). 11 As expected, local asymptotic power of all tests decreases when R 2 increases, with the power loss being least pronounced in case D0 and most pronounced in case D2. Figure 1 further reveals that local asymptotic power of the ADF test (in the presence of deterministics based on OLS detrended data), the Z α test, and the MSB test is very similar throughout and considerably larger than local asymptotic power of the variance ratio test, irrespective of whether the variance ratio test is based on OLS detrended data or GLS detrended data. 12 In cases D1 and D2, GLS detrending increases local asymptotic power of the variance ratio test for R 2 = 0.8, but this power improvement seems to be negligible when compared with local asymptotic power of the other tests considered here. On the contrary, for R 2 = 0 local asymptotic power of the variance ratio test based on GLS detrended data is much smaller than local asymptotic power of its OLS-based counterpart. For R 2 = 0.4, GLS detrending seems to lead to minor power improvements for small values of c, but as c moves further away from zero GLS detrending becomes disadvantageous. We conclude that in terms of local asymptotic power, the variance ratio test does not benefit from GLS detrending, which is in line with the findings in Breitung and Taylor (2003) and Nielsen (2009) for Breitung's (2002) unit root test applied to observed univariate time series. In contrast, GLS detrending improves local asymptotic power of the ADF and MSB tests considerably throughout, which is in line with the findings in Perron and Rodríguez (2016). Finally, note that results for other values of m are qualitatively similar, with local asymptotic power being lower overall for larger values of m.
The results clearly demonstrate that the ADF, Z α , and MSB tests outperform 11 Perron and Rodríguez (2016) derive, among others, the limiting distributions of the ADF, Z α , and MSB tests based on GLS detrended data under local alternatives. However, the analytical expressions of the limiting distributions of the ADF test and the Z α test (the Zρ test in their notation) in Theorem 5.2 contain typos. The square root should be removed from the numerator in the second term of the limiting distribution of the ADF test and from both the numerator and the denominator in the second term of the limiting distribution of the Z α test (T ?)heorem 1]Pe07cf.. The results in Figure 1 are based on the corrected version of the limiting distribution of the ADF test. 12 Hosseinkouchack (2014) analyzes local asymptotic power of Breitung's (2002) unit root test when applied to an observed (potentially OLS detrended) univariate time series and finds similar power losses. the variance ratio test in terms of local asymptotic power. These power losses might not come as a surprise, given the self-normalizing property of the variance ratio test and the fact that self-normalized tests are well known for having smaller local asymptotic power than their (semi-)parametric counterparts (cf., e. g., Shao, 2015). However, local asymptotic power analyses are unable to reveal finite sample effects of both different short-run dynamics in the DGP and tuning parameter choices. The next section complements the local asymptotic power analysis with a careful assessment of the performance of the tests in finite samples.

Finite Sample Performance
We generate data according to (2.1) and (2.2) with m = 1 regressor, i. e., for t = 1, . . . , T . The regression errors are generated as u t = ρu t−1 + ξ t using the following five different short-run dynamics: To ensure stationarity of ξ t , the parameter values are restricted to |φ|, |θ| < 1 and a 1 +a 2 < 1, with a 1 , a 2 ≥ 0. In the IID and GARCH cases, the long-run covariance matrix of [ξ t , v t ] is given by Ω IID = Ω GARCH = Σ, whereas in the ARMA case it is given by In the AR and MA cases, the long-run covariance matrices Ω AR and Ω MA can be directly deduced from Ω ARMA by setting θ and φ, respectively, equal to zero. Although the long-run covariance matrix of [ξ t , v t ] differs across types of short-run dynamics, the only nuisance parameter affecting local asymptotic power of the residuals-based no-cointegration tests, i. e., R 2 (compare the discussion in Section 3.3), is equal to σ 2 εv in each case. The DGP thus allows to analyze the effects of different short-run dynamics in ξ t on the performance of residuals-based no-cointegration tests in finite samples while controlling for effects of R 2 . 14 In contrast, the local asymptotic power analysis in Section 3.3 only allows to assess the effect of R 2 .
Remark 5. Pesavento (2007, p. 127) and Perron and Rodríguez (2016, p. 99) state that local asymptotic power results serve as a useful indicator for the performance of residuals-based no-cointegration tests in finite samples. However, Pesavento (2007) only considers a DGP similar to our IID case, while Perron and Rodríguez (2016) allow the errors to be serially correlated -using a vector autoregression model of order one to generate [ξ t , v t ] -but restrict the diagonal and off-diagonal elements of the long-run covariance matrix of [ξ t , v t ] to one and a constant r, respectively, such that R 2 = r 2 . Perron and Rodríguez (2016, p. 99) correctly point out that holding the long-run covariance matrix fixed implies that changes in the autoregressive parameters are compensated by changes in the covariance matrix of [ξ t , v t ] . Consequently, changes in the autoregressive parameters do not affect the performance of the tests in their DGP. In contrast, we shall see below that the DGP considered in this paper is able to detect severe effects of different short-run dynamics on the performance of residuals-based no-cointegration tests. The finite sample results in this section thus differ considerably from the local asymptotic power results in Section 3.3.
Parameters are chosen as follows: In all cases, we set β = 1. Moreover, we set We compare the variance ratio test with the ADF test, the MSB test, and the Z α test in terms of empirical size and size-corrected power. In cases D1 and D2, the tests are based on OLS detrended data. In addition, we also consider the variance ratio test and the ADF test based on GLS detrended data, indicated by the superscript "(GLS)", wherec is chosen as suggested in Table A.1 in Appendix A and in Table 1 of Perron and Rodríguez (2016), respectively. In contrast to the variance ratio test, the ADF, MSB, and Z α tests require tuning parameter choices. The number of lags for the ADF test and the MSB test is selected using AIC. For the ADF test and its GLS-version, we also analyze the results based on a modified AIC (MAIC) criterion proposed in Ng and Perron (2001) for unit root testing in an observed univariate time series, taking into account further modifications suggested in Perron and Qu (2007). 15 We also analyze the performance of the GLS-version of the MSB test in combination with the MAIC. The use of MAIC is indicated by the superscript " * ". For both information criteria, the minimal number of lags is zero and the maximal number of lags is restricted to not exceed 12(T /100) 1/4 , the upper bound suggested in Perron and Qu (2007). 16 The Z α test is based on a kernel estimator of a long-run variance parameter. We present results for the quadratic spectral (QS) kernel together with the corresponding data-dependent bandwidth-selection rule of Andrews (1991). 17 For convenience, Appendix D describes the construction of the ADF, MSB, and Z α tests in more detail. 18 Table 1 presents empirical sizes of the tests under the null hypothesis of no cointegration (ρ = 1) in cases D1 and D2 for T = 100. The results directly reveal that the performance of the tests depends heavily on the short-run dynamics in ξ t . In particular, size distortions in the MA and ARMA cases can be more severe than in the IID, AR, and GARCH cases. Moreover, relative to the IID case with R 2 = 0, changing the short-run dynamics in ξ t can have much larger adverse effects on the performance of the tests than increasing R 2 to 0.8.

Empirical Size
Focusing on the performance of the different tests in detail reveals that the variance ratio test is less size-distorted than the ADF and MSB tests. The variance ratio test often also outperforms the Z α test, especially in the MA case with θ < 0.9 and in the ARMA(0.3,0.6) case. GLS detrending worsens the performance of the variance ratio test, especially in case D2, and of the ADF test, but the GLS detrended version of the ADF test in combination with the MAIC performs rela-15 Perron and Qu (2007) suggest to construct the ADF statistic based on GLS detrended data and to determine the number of lags in the auxiliary regression based on OLS detrended data. 16 The Supplementary Material provides results based on BIC and MBIC, with MBIC defined analogously to MAIC (compare Section D.2 in the Appendix). The tests based on MBIC perform similar to those based on MAIC. In contrast, the tests based on BIC are more prone to severe size distortions than those based on AIC, but often reveal some power advantages compared to their AIC based counterparts. These differences in size-corrected power, however, do not alter the overall picture emerging from the discussion in Section 4.2. In particular, the tests based on both BIC and MBIC suffer from similar power reversal problems as observed for the tests based on AIC and MAIC in Section 4.2.
17 Using the Bartlett kernel instead of the QS kernel often leads to slightly larger size distortions.
18 When the ADF and MSB tests are used in applications, it seems to be more popular to perform OLS detrending rather than GLS detrending and employing the AIC or BIC rather than the MAIC or MBIC. The following results thus also allow to assess whether practitioners should stick to these "default" choices or not. tively well. Similarly, the GLS detrended version of the MSB test in combination with the MAIC has much smaller size distortions than the MSB test, but it tends to be very conservative.
Comparing the ADF test with the ADF * test reveals that using the MAIC criterion is also advantageous under OLS detrending. In most cases, the variance ratio test and the ADF * test perform similarly, but the ADF * test has some performance advantages over the variance ratio test in the MA case with θ > 0.3 and in the ARMA(0.3,0.6) case. These performance advantages are the more pronounced the larger R 2 .
Increasing the sample size reduces the size distortions of the tests, especially for VR (GLS) in case D2, and makes the MSB (GLS) * less conservative, compare Table B.4 in Appendix B, which presents the results for T = 250. Beyond that, the results are qualitatively similar to those for T = 100. Overall, the findings also hold in case D0, although the variance ratio test seems to be rather conservative in this case even for T = 250, compare Table B.3 in Appendix B.

Size-Corrected Power
To analyze the finite sample properties of the tests under deviations from the null hypothesis, we generate data under the alternatives ρ = ρ T = 1 + c/T using an equidistant grid of 21 points over the interval [0,60] for the values of −c. 19 To account for the large performance differences between the tests in terms of empirical sizes under the null hypothesis (c = 0), the analysis focuses on sizecorrected (empirical) power. To this end, test decisions are based on case-specific empirical critical values obtained from simulations under the null hypothesis rather than on asymptotic critical values. All size-corrected power curves thus start at the nominal 5% level.
Size-corrected power of the tests decreases when R 2 becomes larger. However, for all three deterministic specifications different short-run dynamics in ξ t can have more adverse effects on the performance of the tests than increasing R 2 from zero to 0.8. Figure 2 presents the size-corrected power curves of the tests for the different short-run dynamics in case D2 with T = 100 and R 2 = 0.4. 20 For most short-run dynamics, the Z α test performs best, while the MSB test performs 19 Note that the alternatives move closer to the null hypothesis as the sample size increases. Hence, for (very) large sample sizes, the size-corrected power results should match the local asymptotic power results in Figure 1 worse. The ranking of the remaining tests is highly case dependent.
In general, the variance ratio test performs relatively well and much better than suggested by the local asymptotic power results. In particular, it outperforms the MSB (GLS) * test and both the OLS and GLS versions of the ADF * test -thus its biggest competitors under the null hypothesis -in the MA cases and in the ARMA(0.3,0.6). In the MA case with θ = 0.9, the variance ratio test is even the most powerful test, but this result should be interpreted with caution, as all tests are heavily size-distorted in this case. In other cases, the variance ratio test is less powerful than its competitors for small deviations from the null hypothesis but outperforms the MSB (GLS) * test and both the OLS and GLS versions of the ADF * test for larger deviations from the null hypothesis. 21 Importantly, for some short-run dynamics, the power curves of the MSB (GLS) * test and both the OLS and GLS versions of the ADF * test reveal power reversal problems. This phenomenon is well known in the unit root literature for the MSB and ADF tests in combination with the AIC criterion in the MA case with a large θ. However, the MAIC criterion has been introduced as a possible way to prevent this degeneracy (cf. the discussion in Perron and Qu, 2007). Comparing the power curves of the ADF * and ADF (GLS) * tests with those of the ADF and ADF (GLS) tests reveals that using the MAIC criterion rather than the AIC criterion indeed prevents power reversal problems associated with the ADF test in the MA case with θ = 0.9. 22 In other cases, however, the tests based on MAIC still suffer from power reversal problems (e. g., in the AR case with φ = 0.6 and in the ARMA case with φ = 0.6 and θ = 0.3) and perform worse than those based on AIC (e. g., in the IID and GARCH cases).
With respect to preliminary data detrending, the results suggest that the GLS versions of the ADF and ADF * tests perform similarly to the OLS versions of the test, whereas the GLS version of the variance ratio test is often (considerably) less powerful than the OLS version. 23 On the other hand, the MSB (GLS) * test is in most cases clearly more powerful than the MSB test.
Finally, increasing the sample size is beneficial for the power of the tests, but results for T = 250 are qualitatively similar, see, e. g., Figure B.2 in Appendix B in case D2 with R 2 = 0.4. It is noteworthy that short-run dynamics in ξ t even have adverse effects -including power reversal problems for the ADF tests and the MSB (GLS) * test -on the power of the tests for larger sample sizes, especially in case D2 with R 2 = 0.8. However, as T increases, short-run dynamics in ξ t become less pervasive such that the size-corrected power results approach the local asymptotic power results analyzed in Section 3.3, see, e. g., the results for T = 1,000 in case D2 with R 2 = 0.4 in Figure B.3 in Appendix B. 24 In general, we notice that for small to medium sample sizes, the finite sample performance of the tests are dominated by the short-run dynamics in ξ t such that size-corrected power results deviate considerably from local asymptotic power results. Pesavento (2007) and Perron and Rodríguez (2016) do not detect these discrepancies between the finite sample performance of the tests and the local asymptotic power results because their DGPs only allow for very mild effects of short-run dynamics (compare the discussion in Remark 5). As the sample size increases, the short-run dynamics in ξ t become less important and the performance of the tests is then dominated by R 2 . In this case, size-corrected power results are more similar to local asymptotic power results. The sample size required for a sufficient degree of similarity between finite sample results and local asymptotic power results generally increases with the order of the deterministic component and R 2 , i. e., the required sample size is much smaller in case D0 with R 2 = 0 than in case D2 with R 2 = 0.8.
To provide an overall assessment, it is important to note that residuals-based no-cointegration tests are used to avoid analyzing the relationship between stochastically trending (economic) variables in a spurious regression framework. For practitioners, small upward size distortions may thus have a higher weight in the overall assessment of the general performance of the tests than high power under the alternative. We thus conclude that both the variance ratio test and the ADF * test perform best among the tests considered in this simulation study and thus may be deemed useful for testing for cointegration in applications. However, practitioners should be aware of the individual specific shortcomings of the tests. 25 To make the analysis more robust against individual specific shortcomings of the tests, practitioners could use a Fisher-type combination test or a "union-of-rejections" decision rule (cf. Bayer and Hanck, 2013).

A Large Initial Value u 0
Several contributions in related literature highlight strong effects of a large initial value u 0 on the power of unit root (a ?)nd references therein]HLT09see, e. g., and residuals-based no-cointegration (see, e. g., Perron and Rodríguez, 2016) tests. To comment briefly on the effect of a large (in a well-defined sense) initial value u 0 on the power of the variance ratio test, we use the same set-up as before but generate initial values u 0 of order T 1/2 . 26 To this end, we follow Harvey et al. (2009) and Perron and Rodríguez (2016) and generate u 0 as u 0 = λ u /(1 − ρ 2 T ) 1/2 , where ρ T = 1 + c/T , c < 0, and λ u is a fixed constant. 27 We compare the performance of the variance ratio test with the performance of the tests already analyzed in the previous subsections. To simplify the comparison with the size-corrected power curves discussed in Section 4.2, we again consider size-corrected power results of the tests, using the same empirical critical values as in the previous subsection. Figure 3 presents the results for c = −20, T = 100 and R 2 = 0.4 in case D2. Again, the performance of the tests highly depends on the short-run dynamics in ξ t . In the MA case with θ = 0.9 and in the ARMA(0.3,0.6) case, size-corrected power of all tests decreases as λ u moves away from zero, whereas in the AR case with φ = 0.9, size-corrected power of the tests is rather unaffected by changes in λ u . In the majority of cases, however, size-corrected power of the ADF and of the ADF * tests increases as λ u moves away from zero, with clear performance advantages of ADF * over ADF, whereas power of the other tests decreases for larger initial values u 0 . 28 These adverse effects are more pronounced for the GLSversions of the tests than for their OLS counterparts, which fits well to the findings in Perron and Rodríguez (2016). Noticeably, the VR test is, except for the MA and ARMA(0.3,0.6) cases, rather unaffected by initial values u 0 corresponding to small to medium values of λ u . Results for T = 250 and/or other choices of c and R 2 are qualitatively similar and we observe comparable effects also in case D1. 26 It is clear from the proof of Proposition 1 that u 0 does not affect the distribution of the VR statistic under the null hypothesis in cases D1 and D2, neither in the limit nor in finite samples. In case D0, however, an initial value u 0 of order T 1/2 does affect the distribution of the VR statistic under the null hypothesis both in finite samples and in the limit. Since in applications an intercept is typically included in (2.2), we do not comment upon this issue any further.
27 Effects in case u 0 is drawn from a normal distribution with mean zero and variance λ 2 u /(1 − ρ 2 T ) are less pronounced. 28 Replacing (M)AIC with (M)BIC in the construction of the ADF and MSB tests yields similar results.

Empirical Illustration
This illustration considers daily closing prices (in U.S. dollar) of cryptocurrencies. In particular, we test for cointegration between the four cryptocurrencies with the highest market capitalization (as of February 25, 2020, excluding stable coins), namely Bitcoin (BTC), Ethereum (ETH), XRP and Bitcoin Cash (BCH). We focus on the latest T = 100, T = 200 and T = 250 time points of the data set analyzed in detail in Keilbar and Zhang (2021), which ends in February 25, 2020. 29 Figure 4 shows the OLS detrended logarithms of the T = 250 daily prices of the four cryptocurrencies between June 21, 2019 and February 25, 2020. Keilbar and Zhang (2021) find evidence that the four series are integrated of order one. In addition, maybe apart from the beginning of the period, Figure 4 indicates a strong co-movement of the four series.
We choose Bitcoin (i. e., the cryptocurrency with the highest market capitalization) as the left-hand side variable and calculate the nine test statistics considered in the previous section allowing for a deterministic time trend (D2). Table 2 provides the results for the three periods. For each period, test decisions at the 5% significance level are heterogeneous across tests. The ADF and ADF * tests reject the null hypothesis of no cointegration in all three periods, whereas the ADF (GLS) * test and the MSB (GLS) * test never reject. For the T = 100 and T = 200 peri- ods, the test decisions of the VR test are in line with the those of the ADF and ADF * tests, but for the T = 250 period the VR test leads to an opposing result.
Comparing the values of the test statistics between the T = 200 and the T = 250 periods reveals that the ADF and ADF * test statistics decrease (i. e., become more significant) as the sample size increases, whereas the remaining seven test statistics -including the VR statistic -increase (i. e., become less significant). Thus, it seems that a more detailed analysis is needed to decide whether the series are cointegrated in the T = 250 period or not. On the other hand, in light of the simulation results in Section 4, the results in the T = 100 and T = 200 periods seem to provide reliable evidence for the presence of cointegration. 30 For investors, the presence of a cointegrating relationship between the four cryptocurrencies with the highest market capitalization clearly complicates diversification of their cryptocurrency portfolios. On the other hand, it allows them to use a cointegration-based trading strategy to increase profits (cf, e. g., Leung and Nguyen, 2019;Keilbar and Zhang, 2021).

Summary and Conclusion
This paper derives asymptotic theory for Breitung's (2002) nonparametric variance ratio unit root test when applied to regression residuals and analyzes its asymptotic and finite sample properties in this case. The results reveal that the variance ratio test has smaller local asymptotic power than its competitors. However, in finite samples, short-run dynamics in the errors can have severe effects on the performance of the tests both under the null hypothesis and under the alternative. In terms of empirical size, the variance ratio test and the ADF test based on a modified AIC (BIC) perform best and, in particular, outperform the ADF test based on the (unmodified) AIC (BIC). As both tests also perform relatively well in terms of size-corrected power, they may be deemed useful for testing for cointegration in applications. In particular, both test behave nicely in the important case where the regression errors have a moving average component with a small negative coefficient. However, practitioners should be aware that under some short-run dynamics the variance ratio test can be less powerful than its competitors under small deviations from the null hypothesis (but then quickly catches up), whereas the ADF test is prone to power reversal problems for larger deviations from the null hypothesis -whether or not the AIC (BIC) is modified. Finally, an empirical illustration testing for cointegration between daily prices of cryptocurrencies shows the usefulness of the variance ratio test in practice.
Future research could strive to make the ADF test (and the MSB test) more robust against power reversal problems. Moreover, it might be possible to increase local asymptotic power of the variance ratio test by extending Nielsen's (2009) family of unit root tests -which contains Breitung's (2002) test as a special case -to cointegration testing. However, this comes at the cost of introducing an index parameter to be chosen by the practitioner. A detailed simulation study then needs to assess whether increasing local asymptotic power of the variance ratio test is also beneficial for its finite sample performance both under the null hypothesis and under the alternative.

Declaration of Interest
The author has no conflicts of interest to declare. Phillips, P.C.B., Ouliaris, S. (1990). Asymptotic Properties of Residual Based Tests for Cointegration. Econometrica 58, 165-193.

C Proofs
The proofs frequently use the fact that in cases D1 and D2 the deterministic component d t fulfills By construction, it holds that B ξ (r) = Ω 1/2 ξ·v W ξ·v (r) + Ω ξv (Ω −1/2 vv ) W v (r), which implies B ξ (r) = Ω 1/2 ξ·v W ξ·v (r)+Ω ξv (Ω −1/2 vv ) W v (r). Analogously, it follows from (2.4) and the fact that OLS detrending annihilates x 0 (and µt) in x t that where it follows from B v (r) = Ω 1/2 vv W v (r) that B v (r) = Ω 1/2 vv W v (r). In case D0, it holds that, using the notation P (r) = P (r), Since the vector of numerator and denominator of the variance ratio test statistic can be expressed as a continuous functional of T −1/2 u rT up to an error of o P (1), the weak convergence results in (C.8) and (C.10) hold jointly (P ?)roof of Lemma 1]Ph87cf., e. g.,. The limiting null distribution of the variance ratio test statistic stated in the proposition thus follows from the continuous mapping theorem and the fact that the scalar long-run variance parameter Ω ξ·v > 0 in (C.8) and (C.10) cancels out.
It follows from Markov's inequality, stationarity of u o