A Comparison of Semiparametric Tests for Fractional Cointegration

There are various competing procedures to determine whether fractional cointegration is present in a multivariate time series, but no standard approach has emerged. We provide a synthesis of this literature and conduct a detailed comparative Monte Carlo study to guide empirical researchers in their choice of appropriate methodologies. Special attention is paid on empirically relevant issues such as assumptions about the form of the underlying process and the ability of the procedures to distinguish between short-run correlation and long-run equilibria. It is found that several approaches are severely oversized in presence of correlated short-run components and that the methods show diﬀerent performance in terms of power when applied to common-component models instead of triangular systems.


Introduction
The concept of cointegration derives its popularity from the fact that it allows to model equilibrium relationships between non-stationary time series. In practice, however, standard cointegration analysis can often not be applied, since the I(1)/I(0) framework is too restrictive. For example, the series of interest may be persistent but not have a unit root, or the deviations from the equilibrium may be more persistent than the I(0) model allows.
Fractional cointegration overcomes these shortcomings, by allowing for non-integer integration orders of the variables in the system and any (possibly non-zero) memory order in the cointegrating residuals as long as it is reduced compared to the original system. Consequently, fractional cointegration promises to facilitate the modeling of a larger number of equilibrium relationships compared to standard cointegration.
This has led to the development of various testing and rank estimation procedures to determine whether fractional cointegration is present in a multivariate time series.
Semiparametric approaches, on the other hand, have the advantage that they allow the researcher to focus on the long-run relationship between the series and do not require the specification of shortrun dynamics. This literature encompasses the spectral-based rank estimation procedure of Robinson & Yajima (2002) and its extension by Nielsen & Shimotsu (2007), a Hausmann-type test based on the multivariate local Whittle estimator introduced by Robinson (2008a), a number of residual-based tests for the null hypothesis of no fractional cointegration developed by Marmol & Velasco (2004), Chen & Hurvich (2006), Hualde & Velasco (2008), and Wang et al. (2015), a variance-ratio test proposed by Nielsen (2010), a test based on a GPH-type estimate of the cointegration strength introduced by Souza et al. (2018) and a rank estimation procedure based on an eigenanalysis of the autocovariance function from Zhang et al. (2018).
Unfortunately, the domain of applicability of most of these procedures is much more restrictive than the definition of fractional cointegration. Some are only applicable in stationary systems -some only in non-stationary systems. Some procedures require the reduction in memory to be more than 1/2some only require the memory of the cointegrating residuals to be less than 1/2. Furthermore, there are different assumptions about the form of the fractionally cointegrated system. Some approaches assume that one of the observed series itself is an observation of the common underlying trend. Other approaches assume an unobserved common underlying trend. We refer to these models as the triangular system and the common-components model. Which of these assumptions is more suitable in practice depends on the specific application. On the one hand, it may be appropriate to think of the risk-free interest rate as an observed common component that is perturbed by risk premia if the yields of risky bonds are realized so that a triangular model can be used. For cointegrated pairs of stocks, on the other hand, it is unclear why the price of one stock should be interpreted as a perturbed version of another stock price so that a common-components model is more appropriate. Finally, even though the development of each of these procedures to determine whether fractional cointegration is present is a major theoretical contribution, relatively little effort has been devoted to analyze how they perform compared to each other.
Here, we try to address these issues by providing a survey of all the rank estimation and testing procedures discussed above. To study the relative performance of the competing approaches, we conduct an extensive Monte Carlo analysis of their size and power properties. It is found that several proceduresnamely those of Nielsen & Shimotsu (2007) or Robinson & Yajima (2002), Marmol & Velasco (2004), and Hualde & Velasco (2008) show severe finite sample size distortions in multivariate systems with correlated short-run components. The relative performance in terms of power depends on the form of the system under considerations. For triangular systems and non-stationary common-components models the test of Souza et al. (2018) performs best overall, whereas the test of Chen & Hurvich (2006) is preferable for stationary common-components models.
The rest of the paper is structured as follows. The next Section gives the definition and model of fractional cointegration we adopt and briefly reviews the basic estimation methods required by the tests.
Section 3 is divided into two subsections describing two types of tests, 3.1 containing the tests based on a spectral matrix and 3.2 summarizing the tests based on cointegrating residuals, Section 4 presents finite sample results, and Section 5 concludes.

Fractional Cointegration -Models and Definitions
A p-dimensional vector-valued time series X t has long memory if its spectral density fulfills where G is a real, symmetric and non-negative definite matrix, Λ j (d) = diag λ −d1 e iπd1/2 , ..., λ −dp e iπdp/2 is a p × p diagonal matrix, Λ j (d) is its complex conjugate transpose and '∼' implies that for each element the ratio of real and imaginary parts on the left-and right-hand side tends to one.
The element in the a-th row and b-th columns of the spectral matrix f X (λ) is denoted by f ab (λ) ∼ g ab λ −2d for a, b ∈ {1, ..., p} where g ab denotes the respective element of G. The periodogram of X t at the Fourier frequencies is given by with w X (λ) = 1 √ 2πT T t=1 X t e iλt , and λ j = 2πj/T , for j = 1, ..., T /2 , where · denotes the greatest integer smaller than the argument.
There is a number of different definitions of fractional cointegration in the literature. The most common one goes back directly to Engle & Granger (1987). According to this definition the p-dimensional vector-valued time series X t is cointegrated of rank r, if all components of X t are integrated of order d (denoted by I(d)), and there exists a non-singular matrix β so that the r linear combinations v t = β X t are I(d − b a ) = I(d va ) with d > b a > 0 for all a = 1, ..., r. The matrix β is called the cointegrating matrix and each of its columns is a cointegrating vector. The elements of the vector v t are the cointegrating residuals. Other definitions are given by Johansen (1995), Flôres Jr & Szafarz (1996), , and Robinson & Yajima (2002) who also provide a discussion of the implications of the different definitions.
Standard cointegration is a special case of the definition above where d = 1 and d va = 0 for all a. In this setup the system is non-stationary, whereas the cointegrating residuals are stationary. In contrast to that, fractional cointegration allows for a more flexible model so that several cases can be distinguished: or non-stationary cointegration (0.5 < d v < d).
In general, (fractional) cointegration is an equilibrium concept where the persistence of the cointegrating residual d v determines the speed of adjustment towards the cointegration equilibrium β X t , and shocks have no permanent influence on the equilibrium as long as d v < 1 holds.
As an example, consider the fractionally (co-)integrated bivariate model with X t = (X 1t , X 2t ) , where and Here u t = (u 1t , u 2t ) is a weakly-dependent zero-mean process with constant covariance matrix Ω u and spectral density matrix f u (λ), e t (with variance σ 2 e and spectral density f e (λ)) is a univariate weaklydependent zero-mean process that is allowed to be correlated with u t , and L denotes the lag-operator so , and it is of the same dimension as the process that it is applied to. Furthermore, 1(·) denotes the indicator function that takes the value one if its argument is true and is zero, otherwise. Finally, it is assumed The truncated processes ∆ −(d−ba) u at 1(t > 0) are fractionally-integrated processes of type-II, which means they are only asymptotically stationary for d < 1/2, but in contrast to type-I processes they are still defined for d > 1/2. For a detailed discussion cf. Marinucci & Robinson (1999). In this bivariate model there can be at most one cointegrating relationship. In this case r = 1 and β itself is a cointegrating vector. Obviously, if the linear combination β X t = v t has reduced memory, the same is true for every scalar multiple of it. To identify the cointegrating vector, it is therefore customary to apply some kind of normalization such as setting the first element of the vector to unity. In Equations (3) to (5), fractional cointegration arises if ξ 1 , ξ 2 = 0, and b 1 , b 2 > 0. In this case the normalized cointegrating vector is β = 1, − ξ1 ξ2 = 1, −β and the cointegrating residual v t is Note that this model is a common-components model, but it also nests a triangular system. This is obtained as a special case if Ω u,22 = 0 so that X 2t is a direct (rescaled) observation of the underlying common trend and only X 1t is perturbed with a cointegration error so that b = b 1 .
Standard cointegration in the I(1)/I(0) framework is obtained as a special case if d = 1 and b 1 = b 2 = 1. It is also possible to have ξ 1 , ξ 2 = 0, so that both X 1t and X 2t contain the common component Y t , but they are not cointegrated. This is the case if b 1 = b 2 = 0.

Tests for no fractional cointegration
In the following, we provide a comprehensive review of semiparametric tests and estimation procedures that can be used to determine the order of fractional cointegration in a p-dimensional vector-valued time series X t . According to the definition discussed above, this requires that the components of X t are integrated of the same order.
In practice, this can either be assumed based on domain specific knowledge, or it can be tested with tests for the equality of memory parameters that allow for cointegration introduced by, for example, Robinson & Yajima (2002), Nielsen & Shimotsu (2007), Hualde (2013), and Wang & Chan (2016). In particular Robinson & Yajima (2002) discuss in detail how to partition a vector-valued time series into subvectors with equal memory parameters. These can then be used for further cointegration analysis.
In the following, it will be assumed that all components of X t are I(d), which means we abstract from these pre-testing issues to focus on the actual tests for the null of no fractional cointegration. For all tests the hypotheses are defined by In contrast to standard I(1)/I(0) cointegration, the memory parameter d is unknown in fractionally cointegrated systems and has to be estimated. Since we are in a setting that potentially entails cointegration, multivariate memory estimation might not be feasible so that the memory parameters are estimated univariately. If not stated otherwise, the estimates involved in the tests are the means of the univariate memory estimates for the components of the system.
The tests presented in this Section apply the most common estimators: the log-periodogram estimator d GP H of Geweke & Porter-Hudak (1983) and Robinson (1995b), the local Whittle estimator d LW of Künsch (1987) and Robinson (1995a), or the exact local Whittle estimator d ELW of Shimotsu & Phillips (2005) and Shimotsu (2010). All of these estimators are periodogram based and employ the first m Fourier frequencies. The general requirement is that m < T /2 tends to infinity more slowly than T , so that 1 m + m T → 0 as T → ∞ and even the largest frequency 2πm/T is asymptotically local to the zero frequency.
To estimate the cointegrating relationship β X t = v t when r = 1, the vector is partitioned such that X t = (y t , x t ), where y t is a scalar and x t is (p − 1) × 1. By doing so, the focus is on one possible cointegrating relation y t =βx t + v t whereβ is (p − 1)-dimensional.
As in standard cointgration analysis the vectorβ can be estimated with ordinary least squares (OLS) as long as d > 1/2 so that the series remains non-stationary. In stationary long-memory time series, OLS is inconsistent in presence of correlation between the stationary regressors and the innovation term v t (cf. Robinson (1994)). Robinson (1994) and  introduce an alternative estimator of the cointegrating vector that is based on the periodogram local to the zero frequency. In contrast to OLS, Estimating the linear cointegrating relationship with NBLS requires calculating the averaged crossperiodogram of x t with itself and y t by I av xx (λ j ) = 2π T m j=1 ω x (λ j )ω x (λ j ) and I av xy (λ j ) = 2π T m j=1 ω x (λ j ) ω y (λ j ). The NBLS estimate ofβ is then defined by The bandwidth m has to fulfill the usual local-to-zero condition as T → ∞. If not specified otherwise, this is the estimator we employ to estimate the cointegrating vector. Other estimators suggested in the literature include estimation based on the eigenvectors of a version of I av X (λ j ) (cf. Chen & Hurvich (2003)) and joint estimation with the memory parameters in multivariate local Whittle approaches such as those of Robinson (2008b) and Shimotsu (2012).
The following review is divided into tests based on the spectral density local to the origin (Section 3.1) and tests based on estimates of the cointegrating residuals (Section 3.2). Of course, this distinction is not clear cut, since some of the residual-based approaches also use the spectral properties of the potential cointegrating residuals and for example the test of Nielsen (2010) is presented as a variance-ratio test.
Many different categorizations would be possible. Here, we refer to those approaches as "spectral-based" that rely on the properties of the spectrum of the observed series X t itself, and those that rely on the spectrum of the cointegrating residual are called "residual-based".

Tests based on the spectral matrix
A number of procedures to determine the fractional cointegrating rank of the p-dimensional time series X t are based on properties of the rescaled spectral matrix local to the zero frequency. This is denoted by G in Equation (1) and has reduced rank if and only if X t is fractionally cointegrated. If fractional cointegration is present, the number of eigenvalues that are equal to zero corresponds to the cointegrating rank r and therefore to the number of cointegrating relationships.
Based on this property Robinson & Yajima (2002) introduce an information criterion to determine the fractional cointegration rank that is extended to non-stationary processes by Nielsen & Shimotsu (2007).
To obtain an estimate G of G, the first step of the procedure consists in applying the univariate exact local Whittle estimator of Shimotsu & Phillips (2005) and Shimotsu (2010) to each component of X t separately, using bandwidth m. In contrast to a multivariate local Whittle estimate that has the inverse of G in its objective function and is therefore not consistent under fractional cointegration, each of the univariate estimates is consistent for the memory order d of X t . The memory order d is therefore estimated by the pooled estimator d ELW that is the arithmetic mean of the univariate estimates. The The bandwidths have to fulfill m1 m → 0 in order to ensure faster convergence of d ELW than of G( d ELW ). 1 Denote the empirical eigenvalues calculated from G( d ELW ) and sorted in descending order by δ a,G for a = 1, ..., p. The cointegrating rank can then be estimated using a model selection criterion that is based on the partial sum of the sorted eigenvalues r N S = arg min k=0,...,p−1 where n(T ) is a function which fulfills n(T ) + 1 √ m1 n(T ) → 0 as T → ∞ so that n(T ) goes to zero but more slowly than the estimation error in the eigenvalues that is of order O P m −1/2 1 . Asymptotically, the expression is therefore minimal if only estimates of non-zero eigenvalues are included in the sum.
To deal with situations in which the scales of the components in X t are different, Nielsen & Shimotsu (2007) suggest to base the procedure on the correlation matrix P .., g pp ) contains the diagonal elements of . This is admissible since the rank of P is the same as that of G in the limit. Nielsen & Shimotsu (2007)  A similar rank estimation procedure based on the average of finitely many tapered periodogram ordinates local to the origin was also proposed by Chen & Hurvich (2003).
The aforementioned inconsistency of the multivariate local Whittle estimator under fractional cointegration is the basis for a test procedure originally proposed by  This idea is formalized by Robinson (2008a). The test statistic is based on the objective function of the multivariate local Whittle estimator (cf. Lobato (1999), Shimotsu (2007)) with Similar to the previous procedure, the memory parameter d is estimated by pooling the univariate estimates obtained by applying the local Whittle estimator to each of the component series. The equally weighted average is denoted by d LW .
To obtain a test statistic, the derivative s * (d) from (8) is evaluated at this averaged univariate estimate: where g * aa , a = 1, ..., p, are the diagonal elements of G * ( d LW ). The scaled derivative m 1/2 s * ( d LW ) is asymptotically normal so that the test follows a χ 2 1 -distribution if appropriately standardized by the term in the denominator.
The test generates power because G(d) is singular under the alternative of fractional cointegration so that the inverse G * ( d LW ) −1 of the estimate and consequently the trace s * d LW become large. This is a score-type test that avoids the calculation of the multivariate local Whittle estimator that can be numerically expansive. Since the efficiency of the multivariate estimate is obtained with a single Newton step from the univariate estimate in direction of the multivariate one, s * d LW is directly proportionate to the difference between the efficient and the inefficient estimate.
This test allows series of dimensions larger than two, but it is restricted to processes with d ∈ (−1/2, 1/2) and focuses on the empirically relevant range d ∈ (0, 1/2). Hence, non-stationary processes are not allowed. An extension based on a trimmed version of the local Whittle estimator is proposed, but the size properties of this test in simulations appear to depend heavily on the sample size. 2 An alternative way to allow for non-stationary processes would be to base the test on the objective function of the multivariate exact local Whittle estimator (as in Shimotsu (2012), but without allowing for fractional cointegration) and univariate ELW estimates. Since the exact local Whittle estimates have the same asymptotic properties as the local Whittle estimate for d ∈ (−1/2, 1/2), the test would have the same limiting distribution.
For a bivariate process with known d ∈ (0, 1], Souza et al. (2018) propose a test based on an estimate of b obtained from the determinant of the trimmed and truncated spectral matrix of the fractionally differenced process via a log-periodogram regression.
Denote the fractionally differenced process by These results are available from the authors upon request. and can be approximated by whereg is a constant and finite scalar.
Under cointegration, f ∆ d (λ) does not have full rank near the origin (like G in (1)) so that its determinant D ∆ d (λ) approaches zero as λ → 0 + . The memory reduction b can be estimated from the logged version of Equation (10) using a log-periodogram type regression, In order to make the estimation of b feasible, the empirical determinant D ∆ d (λ) has to be calculated from an estimate f ∆ d (λ) of the spectral density at the Fourier frequencies with order numbers j = The latter is obtained from the locally averaged periodogram is thus a local average of the periodogram at frequency j and the l + 1 frequencies to its left and right and the λ j are spaced so that the local averages are non-overlapping.
The resulting estimator for the cointegrating strength b is given by , andZ * is the mean of the Z * j . Under the null hypothesis of no fractional cointegration we have b = 0. Under this condition, and assuming that l and m fulfill the condition l+1 m + m T + 1 m + log m m → 0 as T → ∞, the estimate b GP H is consistent and asymptotic normal with variance is the polygamma function of order 1 and Γ(·) denotes the gamma function.
The null hypothesis of no fractional cointegration can thus be tested using a simple t-test: The method has no restrictions regarding the range of d and b but is only applicable to bivariate processes.
For practical purposes, d is usually unknown and has to be estimated, but as shown in our simulation study in Section 4 this has no severe implications for the quality of the test. However, a thorough theoretical examination of this aspect would be interesting for further research.

Tests based on cointegrating residuals
By the definition of fractional cointegration the memory d v of the linear combination v t = β X t is lower than that of X t itself. Under the null hypothesis of no fractional cointegration one can still write v t = β X t = y t −βx t , since y t can still depend on the values of the other components of X t . The difference to the cointegrated case is only that d v = d. It is therefore natural to test for fractional cointegration by Under weak non-stationary fractional cointegration so that d > d v > 1/2, Marmol & Velasco (2004) suggest a Hausman (1978)-type F-test that compares the OLS estimate β OLS of the cointegrating vector with an alternative estimate β N B with opposite consistency characteristics.
The OLS estimator β OLS is consistent forβ under the alternative (as long as d > 1/2) but inconsistent under the null hypothesis. Marmol & Velasco (2004) propose an alternative estimator β N B that is consistent for the vectorβ under the null hypothesis but inconsistent under the alternative. The estimator is given by and where I xx (λ j ) and I xy (λ j ) are the respective elements of the periodogram I ∆X∆X (λ j ) of the differenced process ∆X t and m 2 is subject to the usual bandwidth conditions.
The estimator is closely related to the narrow band least squares estimator β m from (6) Since the process is non-stationary, the memory is estimated by local Whittle from the differenced process. ALternatively, d could be estimated using a tapered local Whittle estimator, or by the exact or fully extended local Whittle estimator.
The test statistic compares both estimates ofβ where the normalizing variance V M V is estimated from the periodogram of the OLS residuals v OLS t and that of x t so that This leads to the test statistic The choices of m and m 2 are not linked, but both have to satisfy the condition (m d−2 + m γ−1 log T ) The asymptotic distribution is non-standard and depends on the memory parameter d. It is given by Critical values are tabulated in Marmol & Velasco (2004) for dimensions up to p = 5 and different forms of detrending that affect the type of the fractional Brownian bridges. The test statistic W M V diverges under the alternative since both β N B and V −1 M V diverge under fractional cointegration. Although the consistency of the test is derived assuming d > d v > 0.5, Marmol & Velasco (2004) state that the test remains consistent if the stationarity border is crossed by the cointegrating residuals, i.e. d > 0.5 > d v . Our simulations in Section 4 confirm this.
A direct residual based test is proposed by Chen & Hurvich (2006) who estimate the possible cointegrating subspaces using eigenvectors of the averaged periodogram local to the zero frequency. The process X t is assumed to be stationary after taking (q − 1) integer differences which allows d ∈ (q − 1.5, q − 0.5). In order to account for possible over-differentiation the complex-valued taper h t = 0.5(1 − e i2πt/T ) of Hurvich & Chen (2000) is applied to the data. The tapered discrete Fourier transform and periodogram of X t are defined by Next, define the averaged periodogram matrix of X t by where m 3 is a fixed positive integer fulfilling m 3 > p + 3. The eigenvalues of I av X (λ j ) sorted in descending order are denoted by δ a,I av X and the corresponding eigenvectors are given by χ a,I av X , for a = 1, ..., p. Under the alternative hypothesis, if there are r > 0 cointegrating relationships, the matrix consisting of the first r eigenvectors provides a consistent estimate of the cointegrating subspace.
To construct a test for the null hypothesis of no fractional cointegration the potential cointegrating residuals v t are estimated by multiplying X t with the eigenvectors χ a,I av X so that v av at = χ a,I av X X t , for a = 1, ..., p.
The memory of the p residual processes is estimated with the local Whittle estimator using bandwidth m but calculated using shifted Fourier frequencies λj withj = j + (q − 1)/2 to account for the tapering of order q. These estimates are denoted by d va, LW , and they remain consistent and asymptotic normal.
Since there can be at most p − 1 cointegrating relationships in a p-dimensional time series, the first residual corresponding to the largest eigenvalue cannot be a cointegrating residual. Its memory must therefore equal the common memory d of X t . The last residual v av pt corresponding to the smallest eigenvalue, on the other hand, is most likely to be a cointegrating residual if there is cointegration so that its memory is reduced by b under cointegration.
The test idea of Chen & Hurvich (2006) is therefore to compare the estimated memory orders from the residual series v av 1t and v av pt . Hence, the test compares the estimated memory parameters d (first residual) and d v (last residual). Chen & Hurvich (2006) A conservative test statistic is therefore given by The tests rejects if W CH is larger than the standard normal quantile z 1−α/2 . It is very versatile, since it does not impose restrictions on the cointegration strengh b and can be applied to stationary as well as non-stationary long memory processes, but it requires a priori knowledge about the location of d in the parameter space to determine the order of differencing.
Hualde & Velasco (2008) propose another testing strategy in a residual-based regression framework.
As before, the series X t is partitioned such that X t = (y t , x t ) and they consider the single-equation The test idea is based on the observation that the fractionally differenced residual ∆ dx v t is unrelated to the long-run level of x t under the null hypothesis. This is because ∆ dx v t is I(0) and x t is I(d). The cross-spectrum of x t and ∆ dx v t should therefore be zero at frequencies local to zero. Possible dependence between the short-run components u t and e t in (3) would manifest itself in form of a non-zero crossspectrum at higher frequencies.
The test statistic of Hualde & Velasco (2008) is therefore based on the quantity τ m defined as The projection vector ζ(λ j ) estimates the discrete Fourier transform (DFT) of the residual process v t from w ∆ dv ,d X (λ j ) -the DFT of the fractionally differenced process ∆ dv,d X t . As usual for these semiparametric approaches, it is assumed that m ≤ T /2 and m/T → 0, as T → ∞.

This leads to the test statistic
where the weights are defined by a j = 1 if j ∈ {0, T /2} and a j = 2 otherwise. Under the null hypothesis this test statistic follows an asymptotic χ 2 p−1 -distribution. Under the alternative the test develops power, since d v is estimated from the NBLS estimate of the cointegrating residuals. Since these have reduced memory under the alternative, the first component of ∆ dv,d X t (y t ) is I(b) instead of I(0) and the cross spectrum of the underdifferenced estimate of v t and x t in τ m becomes non-zero. As before, the memory orders are estimated using consistent estimators that account for the (possible) non-stationarity of the data -for example the exact local Whittle estimator of Shimotsu & Phillips (2005).
A modified test with more power in bivariate systems Here, the respective elements of the spectral matrices are denoted by f ab (λ j ) andf ab (λ j ) with a, b ∈ {1, 2}. This is the same as τ m but with f 12 (λ j ) replaced by f 12 (λ j ) that is constructed using d v so that it also diverges under the alternative and constitutes an additional source of power. The asymptotic χ 2 p−1 -distribution is unaffected by this modification. It is not necessary to impose any restrictions on the range of d and d v except for those implied by fractional cointegration, and processes of dimensions higher than two are allowed. The asymptotic χ 2 p−1 distribution depends only on the dimension of the process. Furthermore, the memory parameters are allowed to differ as long as two components of X t share the same memory parameter and the vector is sorted so that the component with the highest memory comes first.
Nielsen (2010) introduces a sequential testing approach in order to test the null hypothesis of no fractional cointegration and to determine the cointegrating rank. The method is based on a variance-ratio statistic and imposes the assumption that the process X t is non-stationary and the potential cointegrating residual process is stationary with d v < 0.5 < d. Denote the demeaned process by Z t = X t − X t , where X t is the vector of arithmetic means of the component series. The fractionally integrated version of Z t is denoted by Z t = ∆ − Z t . Then the variance ratio is given by Taking the ratio has the advantage of eliminating the processes' variance from the asymptotic distribution. The eigenvalues of K T ( ) sorted in ascending order are denoted by δ a,K with a = 1, ..., p.
Similar to the spectral matrix G, the rank of K T ( ) is reduced to p − r under fractional cointegration.
This leads to a non-parametric trace statistic whose structure is similar to the trace statistic of Johansen & Juselius (1990) in the parametric context where r is the number of cointegrating relations under the null hypothesis. Using (15) the cointegrating rank can be determined by a sequence of tests of the null hypothesis H 0 : r = r 0 vs. H 1 : r > r 0 .
The limiting distribution is given by is a n−r dimensional vector of mutually independent standard fractional Brownian motions of type II, and the Brownian motions driving the fractional Brownian motions B n−r d and B n−r d+ are identical. This asymptotic distribution is non-standard and depends on the dimension p, the cointegrating rank r, the order of fractional integration and d. In practice d can be estimated consistently, and the other parameters are known. Critical values for d = 1, = 0.1, and p − r = 1, 2, ..., 8 are given by Nielsen (2010), who recommends to use = 0.1 to integrate the process because it leads to higher power than larger values whereas smaller values improve power slightly but lead to size distortions at the same time.
For more details confer Nielsen (2009). Note that choosing a different order of fractional summation changes the limiting distribution which implies that the test performance is free from user-chosen tuning parameters.
To see why this test can be considered to be residual-based, note that where η a denotes the eigenvector corresponding to the ath eigenvalue. Since the first r eigenvectors are consistent estimates of the cointegrating space (cf. Theorem 3 in Nielsen (2010)), the first r eigenvalues are thus given by the ratio of the sum of the squared cointegrating residuals and the sum of squares of their times integrated version v t .
Here the squares are estimators of the respective process variances and it is assumed that d > 1/2 > d v . Therefore, under the null hypothesis of no fractional cointegration the enumerator grows with rate O P (T 2d ) and the more persistent denominator grows with rate O P (T 2(d+ ) ), so that the eigenvalue has rate O P (T −2 ).
Under the alternative of fractional cointegration with d v < 1/2, the process v t is stationary so that the process variance is finite and the enumerator grows with rate O P (T ). The denominator that may or may not be stationary due to the integration with is O P (T max{1/2,d−b+ } ). Consequently, the eigenvalue is O P (T min{0,1−2(d−b+ )} ), so that it goes to zero more slowly than under the null hypothesis.
The test is restrictive in that it requires non-stationary processes and, preferably, stationary residual processes, but as shown by his Monte Carlo simulation the test still exhibits power if d v > 0.5 and b > 0.
Furthermore, it is applicable to multivariate systems and is able to estimate the number of cointegrating relations. Wang et al. (2015) propose a simple residual-based test in a bivariate setting, where X t = (X 1t , X 2t ) .
The test statistic is based on the partial sum of ∆ dv Z 2t , which is the demeaned second component series fractionally differenced with the memory order of the potential cointegrating residual v t . It is given by where f 22 is the spectral density of either u 2t or e t in (3), depending on whether a triangular model or a common-components model is assumed.
Under the null hypothesis d v = d so that ∆ dv Z 2t is I(0) and the appropriately rescaled sum is asymptotically standard normal. Under the alternative ∆ dv Z 2t is I(b), so that the test statistic diverges with rate O P (T b ).
To make this test statistic feasible the spectral density f 22 can be estimated from the periodogram of the fractionally differenced process ∆ d Z 2t following the approach of Hualde (2013): While Wang et al. (2015) are agnostic about the method that is used for the estimation of the memory parameters d and d v , they assume that d > 1/2 so that the cointegrating vector can be estimated using ordinary least squares. The memory orders can then be estimated from v OLS t and Z 2t using any of the common semiparametric estimates such as ELW with bandwidth m as in f 22 that fulfills the usual bandwidth conditions. The method does not impose any restrictions on the fractional cointegrating strength b. As the Monte Carlo simulations below show, the non-stationarity requirement (d > 1/2) can be circumvented if the cointegrating residual v t is based on the NBLS estimate of the cointegrating vector instead of the OLS estimate. Zhang et al. (2018) propose an alternative estimator of the cointegrating space that is based on the eigenvectors of the non-negative matrix where Ω Z (j) = 1 T T −j t=1 Z t+j Z t is the autocovariance matrix at lag j and j 0 is a fixed integer. The matrix M is thus the sum of the outer products of the first j 0 autocovariance matrices with themselves.
The outer product is used instead of the covariance matrices Ω Z (j) to ensure that there is no information cancellation over different lags in M . It is assumed that d > 0.5 and d v < 0.5. If r is unknown, the p potential cointegrating residuals are estimated using the eigenvectors so that v M at = χ a,M X t . By the same argument as in the procedure of Chen & Hurvich (2006), the residual corresponding to the smallest eigenvalue is most likely a cointegrating residual with reduced memory of d v = d − b and the residual corresponding to the largest eigenvalue is I(d).
The cointegrating rank can be estimated using a simple criterion based on the summed autocorrelations of the potential cointegrating residuals. Define where v M at is the mean of v M at . The cointegrating rank estimator counts the instances when the averaged autocorrelation is smaller than a threshold c 0 ∈ (0, 1): If the residual v M at is stationary (d v < 1/2), the rescaled sum of autocorrelations Q a (k 0 )/k 0 converges to zero asymptotically for k 0 → ∞, since the autocorrelations are asymptotically proportionate to k 2dv−1 .
Under certain regularity conditions this estimate is consistent. Even though the consistency is only proven for r ≥ 1 in Theorem 4.2 of Zhang et al. (2018), our simulations below show that it also works well in discriminating between r = 0 and r = 1.
It should be noted that the authors define r = p if all components of X t are I(0). This leads to some abuse of notation and r cannot be interpreted as the cointegrating rank in a narrow sense. Based on their simulations Zhang et al. (2018) recommend to use j 0 = 5, k 0 = 20 and c 0 = 0.3. The estimator is easy to implement and applicable to higher dimensional processes. However, the requirement of d > 0.5 and d v < 0.5 is restrictive.

Monte Carlo Study
The asymptotic properties of all tests and rank estimates presented in Section 3 are derived by the respective authors, and some of them also present simulations to explore the finite sample results of the test statistics. This however is not the case for all tests and a comprehensive comparative study suited to guide the choice of appropriate methods in practical applications is entirely missing. To close this gap, we conduct an extensive Monte Carlo study. In addition to general results, we are particularly interested in answering two empirically motivated questions.
i.) How does correlation between the underlying short-run components influence the size of the tests? This question is particularly important, since applied researchers will generally want to test for fractional cointegration if two related series seem to be co-moving. Similar trajectories however, can also be generated by persistent processes with highly correlated innovations. Tests for the null hypothesis of no fractional cointegration should therefore be robust to a relatively high degree of correlation between the short-run components of the series.
ii.) Is there a notable difference in the power of the tests depending on whether the data is generated from a triangular model or from a common-components model? Both models are used in the literature to motivate and construct testing procedures, but to our knowledge simulation results are typically based on the triangular representation. In practice, either model could be justified -depending on the application. For example, if one is considered with potential fractional cointegrating relationships between stock prices, it is not clear why one of the stock prices should be seen as a perturbed version of the other one (as it is the case in the triangular model that treats the series in an asymmetric way) so that the common-components model is more suitable. In contrast to that, in the case of the potential parity between implied volatility and the expected average realized volatility over the next month (the socalled implied-realized parity analyzed by Christensen & Prabhala (1998), Christensen & Nielsen (2006b), and Nielsen (2007), among others), there is theoretical reason to assume that the implied volatility is a perturbed version of the expected average future realized volatility, since it contains a variance-risk premium (cf. Chernov (2007)). Therefore, a triangular model is more suitable.
We focus on three data generating processes (DGPs) based on the general model from equations (3) to (5). For simplicity we set c 1 = c 2 = 0 and b = b 1 = b 2 , so that the processes are mean zero and have a common memory reduction parameter. A simple bivariate model without fractional cointegration is constructed by setting ξ 1 = ξ 2 = 0. This model -referred to as DGP1 -is given by where correlation between u 1t and u 2t is allowed. This is our size-DGP.
For the power simulations, we consider a triangular model and a common-components model. In both cases we set ξ 1 = ξ 2 = 1, which implies a cointegrating vector of β = (1, −1) .
The triangular model DGP2 is given by and the common-components model DGP3 is defined by In both DGP2 and DGP3 we have Y t = ∆ d e t 1{t > 0}. The underlying short-run components u 1t and u 2t , or u 1t and e t -depending on the DGP -have unit variance and correlation ρ. This makes the test applicable in stationary time series as well as in non-stationary ones.
Since the limiting distributions of the non-pivotal test statistics of Marmol & Velasco (2004) and Nielsen (2010) (2002)) and Zhang et al. (2018) are not tests but rank estimates. Instead of the rejection frequency, we therefore report the ratio of correctly estimated cointegrating ranks. Therefore, the results cannot be interpreted as size or power and in the size table and graphs the estimates should yield 0 instead of 0.05.   Size with varying correlation and bandwidth, d=1 The group of procedures that is only applicable to non-stationary systems consists of Marmol & Velasco (2004), Nielsen (2010), and Zhang et al. (2018). It can be observed that the procedure of Marmol & Velasco (2004) behaves similar to that of Hualde & Velasco (2008)  Finally, the test of Robinson (2008a) is only applicable for stationary systems. Here, it can be observed that the test does not hold its size for ρ = 0. This is because the Hausman-testing principle requires one of the estimates of the memory parameter to be more efficient than the other one, but the multivariate estimate is not more efficient in absence of correlation. For other values of ρ, however, the test has good size properties. Interestingly, the test also has good size properties if d = 1, even though it assumes stationarity. The intermediate value of d = 0.7, on the other hand, leads to a moderately oversized test. Overall, it can be observed that the bandwidth choice has a limited effect on the performance of the procedures. A notable exception is the test by Wang et al. (2015), where the size improves considerably as the bandwidth is increased. In general, correlation in the underlying short-run component is mistaken for cointegration more often in stationary systems than in non-stationary ones.
As a robustness check for the finite sample analysis of the size properties conducted here, we consider two alternative size DGPs in Tables 4 and 5 in the appendix. These results are obtained from DPG2 and DGP3 where we set b = 0, so that the time series are not cointegrated. In this case, the processes share a common trend Y t , but the linear combination v t = X 1t − X 2t does not have reduced memory so that the definition of cointegration is not fulfilled. The results show that the tests that are already heavily affected by correlation show even more distorted size results if there is a common component in the DGP. The general picture, however, remains unaffected.
Overall, in terms of size for bivariate systems and taking the range of admissible parameter values into account, we find that the test of Souza et al. (2018) has the best performance, followed by those of Chen & Hurvich (2006) and Wang et al. (2015).
To analyze the power of the procedures, we focus on the triangular representation in DGP2 with b = d so that the memory reduces to zero in the cointegrating relation. Again δ m is set to 0.75. The results are shown in Table 2. In the following, we focus on the results for parameter constellations for which the tests have reasonable size properties. It should be noted that the procedure of Marmol & Velasco (2004) is not theoretically justified for any of the parameter constellations considered, since it assumes that the series are non-stationary, and d v > 1/2.
It can be seen that the rank estimate of Nielsen & Shimotsu (2007) correctly identifies the presence of fractional cointegration even in relatively small samples. Since the estimate works well under the null hypothesis if ρ is low, it clearly outperforms its competitors in this situation. The power of the test of Hualde & Velasco (2008) is also high, but it suffers from similar size issues in case of strongly correlated short-run components.
Among the tests that are more widely applicable the approach of Souza et al. (2018) generates higher power than that of Wang et al. (2015) (except for ρ = 0.99), which in turn outperforms the approach of Chen & Hurvich (2006). Furthermore, it can be seen that the test of Souza et al. (2018) also outperforms more restrictive approaches such as those of Robinson (2008a) and Nielsen (2010). For the test of Chen & Hurvich (2006) we can observe that the power is lower for d = 0.7 than for other values of d. Furthermore, the power becomes non-monotonic in T in some cases. This effect is likely to be caused by the fact that the order of differentiation required may be estimated incorrectly for intermediate values of d. The approach of Zhang et al. (2018) performs similar to that of Nielsen (2010).
With regard to the test of Robinson (2008a), it is noteworthy that the power is considerably lower for ρ = 0.9 than it is for ρ = 0.45 or ρ = 0.99. Further simulation results on this V-shaped dependence pattern between the power of the test and ρ (not reported here) show that the test has no power if ρ = 0.8 and its power is very low in a neighborhood of this point. The size of this neighborhood shrinks to zero as the sample size increases.
The test of Marmol & Velasco (2004) develops good power for non-stationary values of d, even though its theoretical properties are not derived under this alternative.
Overall, we find that the rank estimation of Nielsen & Shimotsu (2007) performs best in identifying the correct order of fractional cointegration if the correlation between the series is low. Among the more broadly applicable methods the test of Souza et al. (2018) clearly performs best in terms of size and power.
The same experiment is repeated for a weakly cointegrated scenario where we set b = d/3. In this case the test of Marmol & Velasco (2004) becomes applicable for d = 1 and those of Nielsen (2010) and Zhang et al. (2018) are no longer applicable for d = 1. Table 6 in the appendix shows the results. It can be seen that the general ordering of the tests in terms of power remains the same in the weakly cointegrated case.
Both tables so far are generated based on the triangular model (DGP2), but we are also interested in the performance based on the common-components model (DGP3). Those results are displayed in Table   3. It can be seen that there is a number of striking differences in the relative performance of the tests. NS07* 100 0.989 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 500 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2500 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000   With respect to the other tests, it can be seen that the variance ratio approach of Nielsen (2010) needs larger samples to develop power, but the test of Hualde & Velasco (2008) has very good power properties -also for low values of ρ where it maintains its size. The procedure of Zhang et al. (2018) turns out to perform better than that of Nielsen (2010) in very small samples.
As for the triangular model, the same analysis is repeated for a weakly cointegrated commoncomponents model where b = d/3. The results are shown in Table 7 in the appendix. As before, the general ordering remains the same. However, it can be seen that the non-monotonicity of the Chen  In addition to that, the rank estimation procedure of Nielsen & Shimotsu (2007) completely loses its ability to identify the cointegrating relationship for low values of ρ.
To analyze the effect of the bandwidth choice on the power of the procedures, we conduct a similar analysis to that for the size in Figure  To explore the behavior of the tests in higher dimensional systems, we conduct another set of simulations in a triangular system of dimension p = 3 to determine the size, the power and the precision of estimates of the cointegrating rank r. The results are shown in Tables 8, 9 and 10 in the appendix.
In general, it can be seen that general patterns observed for p = 2 are magnified. For example, the test of Chen & Hurvich (2006) becomes oversized in stationary systems also if the series are correlated.
Noteworthy is that the test of Robinson (2008a) shows very good size properties across all sample sizes, but requires larger sample sizes to accurately estimate the rank if r > 0, and it does overestimate the rank if r = 1 and ρ = 0.99. Furthermore, in larger samples of T = 1000 or more the rank estimation procedure of Zhang et al. (2018) shows a very good performance for all values of r.
Overall, in the bivariate setup we find that the methods of Robinson & Yajima (2002) and Hualde & Velasco (2008) have the highest power but they have size issues in case of strongly correlated short-run components. The test of Souza et al. (2018) tends to have the best power among the methods that have satisfactory size properties across all scenarios. In stationary systems with common components the test of Chen & Hurvich (2006) also has good power properties. For p = 3, where the test of Souza et al. (2018) is no longer applicable and that of Chen & Hurvich (2006) becomes liberal in stationary (triangular) systems, especially the rank estimation procedure of Zhang et al. (2018) for non-stationary systems can be recommended due to its robustness.

Conclusion
This review is written with the objective to provide guidance for the selection of methods in practical applications. We judge the methods based on i.) the range of values of d and b that are allowed, ii.) the ability to distinguish correctly between common trends and correlated innovations, and iii.) the performance across different DGPs -namely triangular systems as well as common-components models.
Based on our Monte Carlo studies, we find that some of the proposed approaches have weaknesses in their finite sample behavior in some empirically relevant scenarios -especially in presence of correlated short-run components. This concerns mostly the methods of Nielsen & Shimotsu (2007) (or Robinson & Yajima (2002)), Marmol & Velasco (2004), and Hualde & Velasco (2008). With regard to iii.), we find that the size properties of the tests in the triangular case and the common-components model is generally comparable. For the power of the tests, however, there are important differences between the two cases.
In particular, the test of Chen & Hurvich (2006) has much better power for stationary systems under the common components specification, whereas the methods of Robinson & Yajima (2002) and Hualde & Velasco (2008) become worse in their ability to detect fractional cointegration.
Overall, we conclude that the test of Souza et al. (2018) for bivariate systems has the best properties, both theoretically and empirically, and is a good choice for the applied econometrician. It allows for the whole empirically relevant range of d and b, it is robust to correlation, and it provides comparable performance in both -triangular systems and common-components models.
Although the methods of Robinson (2008a), Nielsen (2010), and Zhang et al. (2018) turn out to be robust to short-run correlation and are appealing due to their simplicity, they impose practically relevant restrictions on the permissible range of d and b, and they are outperformed by their competitors in terms of power in bivariate systems.