Greqam

Jamel Jouini
email: jamel.jouini@fsegn.rnu.tn.

Bootstrap Tests in Bivariate VAR Process with Single Structural Change: Power versus Corrected Size and Empirical Illustration

Keywords: Break date, Bootstrap techniques, Graphical methods, Selection procedures. JEL Classification: C20

This paper evaluates the finite-sample performance of single structural change tests based on the asymptotic distribution and bootstrap procedures. In addition to the conventional case of stationary regressors, we consider nonstationary regressors and others characterized by the presence of a break in their structure. While our paper borrows the idea of assessing the performance of structural break tests from an other paper, ours is the first to examine and compare the power of such tests on the basis of corrected size by using graphical methods. We endeavour to see whether some conclusions, obtained for some tests, remain again valid for others based on an other type of processes. Some bootstrap procedures quasi-perfectly correct the size distortions of their asymptotic counterparts and have the same power performance as them on the corrected size basis; property often difficult to obtain. We finally propose a modelling strategy to study the relationship between U.S. interest rates. The results show that such relationship has been altered by a regime-shift located at the beginning of the 1980s.

Introduction

In diverse fields of application, many time series might exhibit the phenomenon of structural change which is of capital importance in economics and econometrics. The problem consists in testing the null hypothesis of structural stability against the alternative of instability. Under the alternative hypothesis, we can have one break or more. The literature addressing the first issue is huge. Indeed, Chow (1960) was the first who considered a classic F test allowing to test for single structural Davidson and MacKinnon (1998) whose interest is the principle of size correction that allows to show the true power of tests instead of the nominal power, and hence to examine accurately the properties of the tests. We consider, in addition to the conventional case of stationary regressors, nonstationary regressors and others characterized by the presence of regime-shift in their structure. While some single structural change tests enjoy certain optimality properties in many cases, little is known about their comparative power properties under other conditions. In this paper, we attempt to study this point based on a bivariate VAR(2) process. To illustrate empirically the usefulness of all the procedures, we propose a modelling strategy to study the relationship between U.S. interest rates, and to detect an eventual break date in this relationship. [START_REF] Bai | Estimating and testing linear models with multiple structural changes[END_REF] The remainder of the paper is organized as follows. Section 2 presents the basic model and the estimation method allowing to estimate the regression coefficients and the break date. Section 3 defines the "supremum" test of Andrews (1993), and the related "exponential" and "average" tests of Andrews and Ploberger (1994). In section 4, we introduce the bootstrap technique and the relating basic concepts being useful to carry out the simulation experiments. Section 5 defines the graphical methods of Davidson and MacKinnon (1998), namely the P value Plots, P value Discrepancy Plots and Size-Power Curves. Monte Carlo evidence on the size and power performance of both asymptotic and bootstrap approximations is given in section 6. The results indicate that the parametric and nonparametric bootstrap techniques outperform the asymptotic approximation since the bootstrap tests show more accurate size and the same power properties on the corrected size basis as their asymptotic counterparts. Thus, getting the size right and achieving high power are possible tasks. Note that this property is often difficult to obtain when evaluating the performance of tests in terms of size and power. Another feature of substantial importance is that the Hansen's (2000) bootstrap procedures are sometimes inadequate. This then calls these procedures into question. We also study the effect of change of the error distribution on the performance of the testing procedures. Section 7 reports an empirical application using U.S. interest rates to illustrate the usefulness of all the procedures in terms of selection of a break date in the data. The results show that the relationship between the interest rates has been altered by a break located in 1981:5. Some economic explanations are provided to highlight the underlying factors that fostered the change and altered the relationship between the considered series. Section 8 concludes the paper. The different graphs of the simulation study are reported in Appendices A and B, and the graphics of the interest rates are presented in Appendix C.

The model and estimation method

Let us consider the following linear regression model:

y t =      x 0 t δ + u t , x 0 t (δ + θ) + u t , if 1 ≤ t ≤ k 0 , if k 0 < t ≤ T, (2.1)
where y t is the dependent variable, x t is a m-vector of regressors, δ is the vector of regression coefficients, θ represents the magnitude of change, k 0 is the break date, and u t is the disturbance that has a distribution D ¡ 0, σ 2 ¢ , with σ 2 < ∞. The break point k 0 is explicitly treated as unknown and is contained in the interval [k 1 , k 2 ] which is defined to restrict the break date to be bounded from the boundaries of the sample, as there are not enough observations to identify the parameters of the two subsamples. When testing for single structural change, we are not concerned with the distribution of x t . Thus, the problem of structural change in the model (2.1) does not take into account the nature of the regressors; stationary or nonstationary. We are then interested in testing H 0 : θ = 0 versus H 1 : θ 6 = 0. In other words, we test the null hypothesis of stability against the alternative hypothesis of a single structural change.

Under the null hypothesis, the model (2.1) becomes

y t = x 0 t δ + u t , t= 1, . . . , T, (2.2)
which does not depend on the break date k 0 . The estimation method considered is that based on the least-squares principle (Bai, 1997). Indeed, under the null hypothesis the ordinary least-squares (OLS) estimates are δ, the residuals ût , and the estimator of the variance σ2 = (Tm) -1 P T t=1 û2 t . Under the alternative hypothesis, the model can be written as

y t = x 0 t δ + x 0 t θI (t > k 0 ) + u t , (2.3)
where I (•) is an indicator function that takes the value 1 if its argument is true and 0 otherwise. (2.4)

Finally, the OLS estimates of the regression coefficients and the residuals are respectively δ = δk , θ = θk , and ũt = ût k.

The structural change tests

Chow (1960) was the first who considered a classic F test allowing to test for single structural change when the break date is known a priori. He uses the following Wald statistic:

F k = (T -m) σ2 -(T -2m) σ2 k σ2 k . (3.1)
This test statistic is equivalent to the likelihood ratio statistic when the errors u t are i.i.d. N ¡ 0, σ 2 ¢ . In this paper, we are interested in testing the null hypothesis of stability when the true break date k 0 is unknown a priori. Quandt (1960) extends the analysis of Chow (1960) and proposes a likelihood ratio test equivalent to

SupF T = sup k∈[k 1 ,k 2 ] F k . (3.2)
Note that the break date corresponding to the maximal value of the sequence of the statistics F k for all the possible break dates contained in the interval [k 1 , k 2 ] constitutes an estimator of the break point. [START_REF] Aïssa | Bai and Perron's and spectral density methods for structural change detection in the US inflation process[END_REF] The asymptotic distribution of the test statistic SupF T has been derived in Andrews (1993) for stationary regressors. Hansen (2000) derives the asymptotic distribution of the test statistic SupF T when the marginal distribution of the regressors is affected by a structural change. He finds the same distribution as Andrews (1993), but it depends on

λ * = π * 2 (1 -π * 1 ) / [π * 1 (1 -π * 2 )] , (3.3)
rather than

λ = π 2 (1 -π 1 ) / [π 1 (1 -π 2 )] , (3.4)
as found by Andrews (1993).

Note that π * 1 = υ (π 1 ), π * 2 = υ (π 2 )
, and υ is a function such that

υ (r) = [r + ω (r -κ) I (r ≥ κ)] / [1 + ω (1 -κ)] , (3.5)
where ω is the magnitude of change in the regressors, κ = [ T ] is the corresponding break date, and π 1 and π 2 are such that π 1 = k 1 /T and π 2 = k 2 /T . Note that λ * is a function only of π * 1 and π * 2 , the relative information cumulated in the regressors at the times π 1 and π 2 . From Table 1 of Andrews (1993), the asymptotic critical values for SupF T are increasing in λ. Thus, if λ * > λ, then the statistic SupF T has high tendency to reject the null hypothesis if Andrews' critical values are used. On the other hand, if λ * < λ, then this statistic tends to reject too infrequently, reducing power. [START_REF] Aïssa | Structural breaks in the US inflation process[END_REF] In our simulation experiments presented in section 6 (π 1 = 0.15, π 2 = 0.85, ω = 3 and = 0.5), λ * = 49.61 > λ = 32.11 and thus the statistic SupF T using the Andrews' critical values has severe size distortions as shown by the P value plots in Figure 8.

Andrews and Ploberger (1994) suggest two related tests, namely the "exponential" and "average" tests

ExpF T = ln Z k exp (F k /2) dw (k) , (3.6
)

AveF T = Z k F k dw (k) , (3.7)
where w is a measure putting weight

1/ (k 2 -k 1 ) on each integer k in the interval [k 1 , k 2 ].
The asymptotic distributions of all the above-mentioned test statistics are derived under the regressor stationarity hypothesis. [START_REF] Beran | Simulating power functions[END_REF] Consequently, the nonstationarity and structural change in the marginal distribution of the regressors affect the asymptotic distributions of these statistics, and then the critical values tabulated based on these distributions are not valid. This then justifies the recourse to the bootstrap as an approximation method of the distributions in such cases.

In this paper, we are interested in comparing the size and power of the asymptotic and bootstrapping versions of the SupF T , ExpF T and AveF T tests. For the power, the attention is particularly paid to the location of the break and distance of the alternative from the null. Although, for example, the ExpF T and AveF T tests enjoy certain optimality properties in large samples, little is known about their comparative power properties under other conditions.

Bootstrap tests

Since the asymptotic tests may, in finite-samples, be biased in the sense that they have empirical sizes that differ from their nominal ones, an alternative approach is necessary. One such distribution-free method is the bootstrap method, introduced by Efron (1979). The bootstrap is based on the idea that the sample is a good representation of the underlying population distribution. As a result, the population distribution can be approximated by drawing samples repeatedly (so-called, resamples) from the original sample. [START_REF] Beran | Bootstrap tests and confidence regions for functions of a covariance matrix[END_REF] It is well known that the bootstrap procedure can successfully approximate the finite-sample distribution especially when the statistic is asymptotically pivotal (Horowitz, 1997; and Hall, 1992).

A main feature of bootstrap tests is that, under certain conditions, their actual sizes will converge to the true ones faster than asymptotic tests and at times converge considerably faster. This paper proposes the following procedures for the structural change tests, to construct the bootstrapping distribution [START_REF] Chow | Tests of equality between sets of coefficients in two linear regressions[END_REF] under the null hypothesis of stability. [START_REF] Christiano | Searching for a break in GNP[END_REF]

Standard procedures

The standard bootstrap procedure is as follows:

1. Compute the asymptotic test statistic in the usual way.

2. Estimate the model under the null hypothesis by the least-squares principle and construct the following bootstrap DGP: [START_REF] Beran | Simulating power functions[END_REF] The work of Ploberger and Krämer (1996) is a notable exception since the authors show that in the presence of general trends in the regressors, the asymptotic distribution of the CUSUM test is bounded by the asymptotic distribution in the case of stationary regressors. [START_REF] Beran | Bootstrap tests and confidence regions for functions of a covariance matrix[END_REF] See Jeong and Maddala (1993) or Horowitz (1994) for a review of bootstrap methods in econometrics. [START_REF] Chow | Tests of equality between sets of coefficients in two linear regressions[END_REF] As we mentioned above the nonstationarity and structural change in the marginal distribution of the regressors affect the asymptotic distributions of the test statistics in complicated ways. An alternative solution consists in using bootstrap methods to yield approximations to the distributions of the considered test statistics. [START_REF] Christiano | Searching for a break in GNP[END_REF] The idea of constructing the bootstrap DGP under the null hypothesis is proposed by Beran (1986), and Beran and Srivastava (1985).

y * t = x 0 t δ + u * t , (4.1)
where δ is the restricted parameter estimate. This DGP is just an element of the model (2.2) using the parameters estimated under the null hypothesis. [START_REF] Davidson | Graphical methods for investigating the size and power of test statistics[END_REF] For the parametric bootstrap, we have u * t ∼ D ¡ 0, σ2 ¢ where σ2 = (Tm) -1 P T t=1 û2 t . Note that without the normality assumption, it does not make sense to generate bootstrap errors from the normal distribution. Instead, we want to generate them nonparametrically by resampling with replacement from the residual vector ȗ = {ȗ t } T t=1 ;

ȗt = r T T -m à ût - 1 T T X s=1 ûs ! . (4.2)
3. Based on these data, compute a bootstrap test statistic in exactly the same way as the real sample was used to compute the asymptotic statistic.

Hansen's (2000) procedures

The standard bootstrap procedures need the joint modelling of y t and x t which require the correct specification of the marginal distribution (including any structural change). To ignore the marginal distribution of the regressors when generating bootstrap data, Hansen (2000) proposes two alternative bootstrap procedures, called the homoskedastic bootstrap and the heteroskedastic bootstrap to yield approximations to the distributions of test statistics in the presence of nonstationarity and structural change in the regressors. These techniques treat the regressors x t as if they are fixed (exogenous) even when they contain lagged dependent variables, and allow for structural change in the regressors and heteroskedastic errors. The Hansen's bootstrap techniques replicate the correct first-order asymptotic distribution, but it is easy to see that in general the bootstrap does not (in any way) replicate the finite sample distribution of the data or the test statistics. The only exception is when the regressors x t are strictly exogenous and the errors i.i.d. normal, in which case the homoskedastic bootstrap described below will yield exact inference. [START_REF] Davidson | The size distortion of bootstrap tests[END_REF] Consequently, caution should be applied when interpreting the bootstrap tests.

For the homoskedastic bootstrap, the data y * are generated by drawing observations from the N (0, 1) distribution. [START_REF] Davidson | Bootstrap tests: how many bootstraps?[END_REF] To compute a realization of the bootstrap test statistic, we regress y * on x t to get σ * 2 , and we regress y * on x t and x t I (t > k) to get σ * 2 k . The heteroskedastic bootstrap consists in generating bootstrap data y * according to e * t ũt , where e * t ∼ i.i.d. N (0, 1) and ũt are the regression residuals defined in section 2. The bootstrap Wald statistic is then the true population parameters. [START_REF] Davidson | The size distortion of bootstrap tests[END_REF] This is a well known property of bootstrap inference with an exactly pivotal statistic. For more details, see Hall (1994). [START_REF] Davidson | Bootstrap tests: how many bootstraps?[END_REF] It may be possible to use alternative distributions such as the empirical distribution function (EDF) of the residuals.

F * k = (T -m) σ * 2 -(T -2m) σ * 2 k σ * 2 k .
The method applies as well for the other tests.

Computation of bootstrap P value

The bootstrap P value is computed as follows:

1. Draw B bootstrap samples from the above DGPs so as to obtain B bootstrap statistics.

2. The bootstrap P value, say p * , is then estimated by the proportion of bootstrap samples that yield a statistic greater than the asymptotic statistic. [START_REF] Davidson | The power of bootstrap and asymptotic tests[END_REF] We reject the null hypothesis if the bootstrap P value p * is less than α, the significance level of the test.

The bootstrap when the null hypothesis is false

We know that to obtain an exact finite-sample distribution, it is preferred to construct the bootstrap DGP under the null hypothesis. When this hypothesis is false, a reasonable choice is the pseudo-true null, which is the DGP that satisfies the null hypothesis using pseudo-true values for the unknown parameters obtained under the incorrect assumption that the null hypothesis is true. [START_REF] Diebold | Testing structural stability with endogenous break point: a size comparison of analytic and bootstrap procedures[END_REF] Consequently, bootstrap sampling when the null hypothesis is false is equivalent to sampling from the null hypothesis model with pseudo-true parameter values; for more details, see Horowitz (1994, 1997).

Graphical methods

We use the graphical methods developed in Davidson and MacKinnon (1998), namely the P value plots, the P value discrepancy plots and the size-power curves to obtain evidence on the finitesample properties of hypothesis testing procedures. Let us consider a Monte Carlo experiment in which N P values {p i } N i=1 ,17 are generated using a DGP that is a special case of the null hypothesis. All of the graphs we report are based on the EDF of the P values {p i } N i=1 which is then defined, at any point α k in the (0, 1) interval, by (5.2)

F (α k ) ≡ 1 N N X i=1 I (p i ≤ α k ) . ( 5

P value plots and P value discrepancy plots

The P value plot that we will use is a plot of F (α k ) against α k . Each of the P values {p i } N i=1 should be distributed as U (0, 1) and the resulting graph should be close to the 45 degree line. If the tests show size distortions, then the plots must be over or under the 45 degree line. The P value discrepancy plot is a graph of

³ F (α k ) -α k ´against α k .
The resulting graph must be close to the horizontal axis y = 0 if the tests don't show size distortions. These plots convey a lot more information than P value plots for test statistics that are well behaved but some of this information is spurious, simply reflecting experimental randomness.

Size-power curves

It can be difficult to compare the power of alternative test statistics if they don't have the correct size. The size-power curves are constructed with two EDFs F and F 0 . The first EDF is for an experiment in which the DGP satisfies the null hypothesis while for the second EDF, it does not. Note that it is preferable to use the same sequence of random numbers to reduce the experimental error. To generate the size-power curve on a correct size-adjusted basis, we plot the points ³ F (α k ) , F 0 (α k ) ´when α k describes the (0, 1) interval. Given several tests, if any test shows better power properties than the others, then the size-power curve of this test converges more rapidly to the horizontal axis y = 1 than those of the others. Note that there is an infinite number of DGPs that satisfy the null hypothesis. When the test statistics are pivotal we can use any DGP to generate F . However, when the test statistics are nonpivotal the choice of the DGP used to correct the size can matter greatly. Davidson and MacKinnon (1998) argue that a reasonable choice is the pseudo-true null, which is the DGP that satisfies the null hypothesis and is as close as possible, in the sense of the Kullback-Leibler information criterion, to the DGP used to generate F 0 ; see also Horowitz (1994, 1997).

Monte Carlo design

Diebold and Chen (1996) compare the size performance of two approximations to the finite-sample distributions of the "supremum" test statistics developed in Andrews (1993), one based on the asymptotic distributions and the other based on the nonparametric bootstrap. Their results show that for an AR(1) process, the bootstrap technique improves on the asymptotic approximation and quasi-perfectly solves the inference problem. In this paper, through Monte Carlo analysis, we endeavour to see whether such conclusions remain again valid for the related "exponential" ExpF T and "average" AveF T tests of Andrews and Ploberger (1994) for a bivariate VAR(2) process and by using, in addition, the Hansen's (2000) procedures. We also examine accurately the power properties of all the tests on the corrected size basis using the graphical methods outlined above. Thus, while our paper borrows the idea of evaluating the performance of single structural change tests from the paper of Diebold and Chen (1996), ours is the first to examine and compare the power of such tests on the basis of corrected size by using the size-power curves of Davidson and MacKinnon (1998).

We consider several ways of approximating the finite-sample distributions of the statistics. The first approximation is the Andrews (1993), and Andrews and Ploberger (1994) asymptotic distributions, the second approximation is the parametric bootstrap distribution (bootstrap 0), the third approximation is the nonparametric bootstrap distribution using the EDF of the residuals (4.2) (bootstrap 1), [START_REF] Efron | Bootstrap methods; another look at the Jackknife[END_REF] the fourth approximation is the homoskedastic bootstrap (bootstrap 2), and the last one is the heteroskedastic bootstrap (bootstrap 3).

The number of Monte Carlo replications is set at N = 1000 and we choose B = 999. [START_REF] Hall | On the number of bootstrap simulations required to construct a confidence interval[END_REF] Note that when computing the probabilities of rejecting the null hypothesis when it is true, we don't require a large number of bootstrap simulations and the choice of a small number does not materially affect the simulation results because the experimental errors tend to cancel out across replications. [START_REF] Hall | The bootstrap and edgeworth expansion[END_REF] However, when the null hypothesis is false, we must choose a large number so as to avoid the power loss which can be generated by the tests in many circumstances. For the computation of the test statistics, the possible break dates are contained in the interval [0.15T, 0.85T ]. P value functions (PVFs) constructed using the asymptotic distributions are plotted. They show the frequencies of rejecting the null hypothesis at the 5% significance level when it is true or false as function of some characteristics of the DGP.

The Monte Carlo experiments were carried out using programs written in Gauss with Gauss pseudo-random number generators.

Inference under the null hypothesis

The theoretical results of Davidson and MacKinnon (1999) show that the bootstrap tests should perform better. But in many circumstances, in small samples, these tests can show size distortions. Here we show that, using the P value plots, some bootstrap procedures correct the size of the tests and solve the inference problem since the error in rejection probability committed by bootstrap tests is very minimal. [START_REF] Efron | Bootstrap methods; another look at the Jackknife[END_REF] We must confirm the fact that the nonparametric bootstrap gives results indistinguishable from those of the parametric bootstrap in some circumstances. [START_REF] Hall | On the number of bootstrap simulations required to construct a confidence interval[END_REF] The number of bootstrap simulations is chosen to satisfy the condition that α (B + 1) is an integer, where α is the level of the bootstrap test (see, Davidson and MacKinnon (2000) for more details). This choice of B deletes all eventual bias of the bootstrap estimation of a P value. Globally, when the bootstrap tests are the sole concern, we recommend to consider large values of B. [START_REF] Hall | The bootstrap and edgeworth expansion[END_REF] From Hall (1986), the error in rejection probability made by a test using bootstrap-based critical values is

O ³ T -(j+1)/2
´(for some integer j ≥ 1), regardless of the number of bootstrap samples used to estimate the bootstrap critical values. Consequently, the bootstrap methods provide an asymptotic refinement even with small numbers of bootstrap samples.

To evaluate the performance of all the procedures in finite samples, we consider the following bivariate VAR(2) process:

y t = δ 1 + δ 2 y t-1 + δ 3 y t-2 + δ 4 x t-1 + δ 5 x t-2 + u t , 1 ≤ t ≤ T, (6.1)
where T = 40 and u t ∼ i.i.d. N (0, 1). The regression coefficients are δ 1 = 0.80, δ 2 = 0.90, δ 3 = -0.20, δ 4 = 1.00, and δ 5 = -0.50. This process satisfies the null hypothesis and is called the no-break model, so the P value plots should be ideally close to the 45 degree line. For a given significance level α, the asymptotic standard error of the empirical size α is (α (1α) /N ) 1/2 which can be estimated by (α (1 -α) /N ) 1/2 . This last quantity decreases as N increases.

To justify the use of the bootstrap and to highlight the validity of all the approximations, we consider three different distributions for the regressors x t . Let ϕ t be the variable generated from the N (1, 1) distribution;

1. Stationary (i.i.d.) regressors: x t = ϕ t .

2. Nonstationary (mean trend) regressors: x t = 2 + 0.05t + e t , where e t ∼ i.i.d. N (0, 1).

Mean break in the regressors

: x t = ϕ t if t ≤ κ and x t = ω + ϕ t if t > κ.
The random regressors x t are independently generated for each Monte Carlo replication. This means that the presented results are unconditional, rather than conditional on a particular set of regressors. For the first distribution, the asymptotic approximations are not supposed to present size distortions asymptotically since the asymptotic distributions of the test statistics have been derived under the hypothesis of stationary regressors. The other distributions then justify the recourse to the bootstrap to reduce the error in rejection probability committed by the asymptotic tests.

The bootstrap samples must be constructed recursively because of the presence of lagged dependent variables. This is necessary because y * t must depend on y * t-1 and y * t-2 , and not on y t-1 and y t-2 from the observed data. The recursive rule for generating a bootstrap sample is

y * t = δ1 + δ2 y * t-1 + δ3 y * t-2 + δ4 x t-1 + δ5 x t-2 + u * t , 1 ≤ t ≤ T, (6.2)
where y * 0 = y 0 and y * -1 = y -1 . δ1 , δ2 , δ3 , δ4 and δ5 are consistent least-squares estimates of δ 1 , δ 2 , δ 3 , δ 4 and δ 5 . For the parametric bootstrap u * t ∼ i.i.d.N ¡ 0, σ2 ¢ , where σ2 is a consistent estimator of σ 2 whereas for the nonparametric one u * t ∼ EDF(ȗ t ), where ȗt is given by (4.2). As we see every bootstrap sample is conditional on the observed value of y 0 and y -1 . This initialization is certainly the most convenient and it may, in many circumstances, be the only method that is feasible.

Figures 12) show the PVFs constructed using the asymptotic distributions and for the case where there is a mean break in the regressors. They show the percentages of rejecting the null hypothesis at the 5% level. In Figure 1, we present the PVFs as function of ω for κ = T/2 = 20, and Figure 2 presents the ones as function of κ for ω = 3. We observe that the AveF T test is more accurate than the other tests especially the ExpF T test which presents quite severe over-rejection in the overwhelming majority of cases. Another feature of substantial importance is that the AveF T test performs quite well in the first case. However, the other tests are slightly more adequate in the second case in comparison with the first case. For the second case and unlike the AveF T test, the size distortions of the SupF T and ExpF T tests are severer when the break date is far from the boundaries of the sample.

P value plots for the case where ω = 3 and κ = T/2 = 20 are presented in Figures 678(Appendix B). From the ones of tests based on the conventional asymptotic approximation, we first observe that the error in rejection probability committed by all the tests is severe especially when there is mean trend and mean break in the regressor process since the P value plots are above the 45 degree line, i.e. the actual size is greater than the nominal rejection frequency. [START_REF] Hall | Methodology and theory for the bootstrap[END_REF] This is not surprising since the asymptotic distributions of the test statistics have been derived under the hypothesis of stationary regressors. In comparison with the other tests, the ExpF T test substantially over-reject more than the others, and the AveF T test is slightly more adequate than the SupF T test for the last regressor process.

Tests based on the Hansen's bootstrap procedures are poor since they don't allow to correct the size distortions of the asymptotic tests as they also over-reject relative to the nominal size. The heteroskedastic bootstrap AveF T test has higher tendency to over-reject than the one based on the conventional asymptotic approximation. This calls the usefulness and the validity of this bootstrap procedure into question. The parametric and nonparametric bootstrap procedures give similar results, [START_REF] Hansen | Tests for parameter instability in regressions with I(1) processes[END_REF] correct the size distortions of the asymptotic tests and allow to do correct inference since the P value plots are very close to the 45 degree line, i.e. these tests achieve correct size. These procedures are then more accurate than the Hansen's procedures which don't completely solve the inference problem.

We now consider another case in which we set T = 50, δ 2 = 0.50, δ 3 = -0.75, and the other characteristics of the DGP are kept fixed. The P value plots are presented in Figures 91011. We first consider the tests based on the conventional asymptotic distribution. We observe that the SupF T test presents quite satisfactory results. The size distortions of the ExpF T test are quite severe especially when there is structural change in the regressor process. The AveF T test works well for the first two regressor processes since the P value plots are very close to the 45 degree line, i.e. the empirical size is very close to the correct rejection frequency. Thus, the error in rejection probability committed by this test is almost equal to zero ( ¯F (α k )α k ¯' 0, ∀ k), which implies that the discrepancy P value plots are very close to the horizontal axis y = 0.

The bootstrapping versions of the SupF T test almost provide the same results as the conventional asymptotic distribution for the first two regressor processes. The performance of the heteroskedastic bootstrap is quite significantly poor when there is mean break in the regressors. The bootstrap procedures quasi-perfectly correct the size distortions of the asymptotic ExpF T test although there is slight under-rejection by the heteroskedastic bootstrap for the third regressor distribution. For the AveF T test, except for the heteroskedastic bootstrap which has tendency to over-reject, the other procedures are adequate. The parametric and nonparametric bootstrap procedures seem to be slightly more accurate than the Hansen's procedures.

For the two considered cases, the theoretical results of Davidson and MacKinnon (1999) showing that the bootstrap tests should perform better are well proved for the parametric and nonparametric procedures. But, those showing that in many circumstances and in small samples, the bootstrap tests can show size distortions are well proved for the Hansen's procedures in the overwhelming majority of cases.

From the obtained results of the two DGPs, we observe that there is difference in the results, and this may be explained by the fact that the performance of the tests may depend on how the data were actually generated. Thus, a test that works well against certain DGPs may not work well against others. This can be viewed in our Monte Carlo experiments for some testing procedures.

Inference when the null hypothesis is false

The anxiety for the size is understandable because a test with uncontrolled size is not useful regardless of the power. However, the fact that the bootstrap is able to eliminate the size distortions makes the test appropriate for a power study. In this case, we are interested in estimating the frequency of rejecting the null hypothesis when it is false. The main result in the analysis of power of the tests is found in Davidson and MacKinnon (2006). This result stipulates that the bootstrap tests with correct sizes can also often be shown to have the same power properties as their asymptotic counterparts. Consequently, if bootstrapping does result in a power loss when B is large, that loss arises simply because bootstrapping corrects the tendency of the asymptotic tests to over-reject.

To compare the performance of the different procedures we advocate the size-power curves constructed on the corrected size basis. We then require to carry out two experiences so as to generate F and F 0 using the same sequence of random numbers to reduce the experimental error. Note that for the bootstrap tests, we don't require to specify a priori the break date because the DGPs are constructed under the null hypothesis of no structural change.

Under the alternative hypothesis of structural change, the DGP is as follows:

y t = ( δ 11 + δ 21 y t-1 + δ 31 y t-2 + δ 41 x t-1 + δ 51 x t-2 + u t , 1 ≤ t ≤ T/2, δ 12 + δ 22 y t-1 + δ 32 y t-2 + δ 42 x t-1 + δ 52 x t-2 + u t , T/2 < t ≤ T, (6.3)
where T = 100 and u t ∼ i.i.d. N (0, 1). The regression coefficients are δ 11 = 0.80, δ 21 = 0.50, δ 31 = -0.75, δ 41 = 1.00, δ 51 = -0.50, δ 12 = 0.30, δ 22 = 0.00, δ 32 = -0.25, δ 42 = 0.50, and δ 52 = 0.00. [START_REF] Hansen | Approximate asymptotic p-values for structural change tests[END_REF] The parametric and nonparametric bootstrap DGPs are then constructed using the recursive rule given by (6.2) with the modification that in this case the regression coefficient estimates ( δ1 , δ2 , δ3 , δ4 and δ5 ) are obtained under the incorrect assumption that the null hypothesis is true (see, section 4.5). [START_REF] Hansen | Testing for structural change in conditional models[END_REF] We keep the same regressor processes considered in the study of the size performance. Figures 345(Appendix A) show the PVFs constructed using the asymptotic distributions for all the regressor processes, respectively. They show the rejection proportions of the null hypothesis at the 5% level as function of the sample size. We first observe that the ExpF T test is more powerful than the others whatever the value of the sample size, and the SupF T test is in turn more powerful than the AveF T test except for the case T = 100 and the stationary regressor process. This power discrepancy between all the tests tends to zero as the sample size increases. Another feature of substantial importance is that the probability of rejecting the null hypothesis by all the tests tends to unity as the sample size increases. [START_REF] Horowitz | Bootstrap-based critical values for the information matrix test[END_REF] Thus, the tests are consistent.

The size-power curves are presented in Figures 12-14 for the case where ω = 3 and κ = T/2 = 50. For all the tests, the procedures show perfect power properties for all the regressor processes with very slight superiority for the stationary distribution since the size-power curves are too close to the horizontal axis y = 1. The bootstrap tests don't bring out any power loss since the intrinsic size-power curves are confused with those obtained for the asymptotic distribution. Thus, the theory that the test bootstrap power would be very similar to that of the asymptotic test on the basis of size correction is well proved.

The optimality properties of the ExpF T and AveF T tests obtained in large samples are maintained under the conditions imposed here. The presence of mean trend and mean break in the marginal distribution of the regressors has not strongly affected the results since almost the same power performance as for the stationary distribution is obtained.

The intuition often suggests that the bootstrap tests are less powerful than their asymptotic counterparts. This is because the formers have much less severe size distortions and then have low tendency to over-reject the null hypothesis unlike the asymptotic tests. The main result of this analysis is that in some cases, many bootstrap tests work very well since they have correct size and high power; property often difficult to obtain when evaluating the performance of the tests in terms of size and power.

We now consider another DGP for which we set δ 21 = 0.90, δ 31 = -0.20, δ 22 = 0.40, δ 32 = 0.30, and the other characteristics of the DGP are kept fixed. This is done to see whether the power performance of the tests depends on how the data are actually generated. The size-power curves are plotted in Figures 151617. In comparison with the results of the previous case, the procedures show lower power for all the regressor processes. As for the first DGP, there is very slight superiority for the stationary distribution since the corresponding size-power curves are closer to the horizontal axis y = 1 than those of the other regressor processes. The size-power curves of the bootstrap procedures are confused with those obtained for the asymptotic distribution. This proves once again the theory that the power of the bootstrap tests would be very similar to the one of the asymptotic tests on the corrected size basis. The results of this experiment show that change of some factors affects the power performance of the tests, but globally it remains satisfactory.

The samples used in the applications are often larger than those used in our Monte Carlo experiments. However, all the procedures are accurate with a sample size of 100 in the cases considered here. Thus, there is not much to gain by carrying out experiments with larger sample sizes. To that effect, we have carried out two other Monte Carlo experiments in which we set |θ| = 0.7 for the first and T = 150 for the second, and we kept fixed, for the two experiments, the other characteristics of the first considered DGP. [START_REF] Horowitz | Bootstrap methods in econometrics: theory and numerical performance[END_REF] The tests become more powerful for a larger magnitude of change. Thus, what is important to improve the power performance of the tests is the magnitude of jump and not the sample size.

Case of Student error terms

We now study the effect of change of the error distribution on the performance of the testing procedures. To that effect, we simulate leptokurtic data using the Student distribution with 5 degrees of freedom for the error terms. The same other characteristics of all the considered DGPs as well as the number of bootstrap and Monte Carlo replications are kept fixed. We claim that the number of bootstrap simulations, B = 999, is sufficient to capture the probability excess in the distribution tails.

For the study of the size performance, the P value plots are reported in Figures 18-20 for the first DGP (model (6.1)) and in Figures 212223for the second DGP (model (6.1) with T = 50, δ 2 = 0.50 and δ 3 = -0.75). We observe that there is a slight deterioration in the size performance of the asymptotic tests for the two DGPs with respect to the Gaussian case. The correction of the size distortions by the parametric and nonparametric bootstrap procedures is quasi-perfect for the two DGPs as in the Gaussian case. The Hansen's procedures show a slight degradation with respect to the Gaussian case for the SupF T and ExpF T tests only for the first DGP. For the study of the power performance, the size-power curves are reported in Figures 242526for the first DGP (model (6.3)) and in Figures 272829for the second DGP (model (6.3) with δ 21 = 0.90, δ 31 = -0.20, δ 22 = 0.40 and δ 32 = 0.30). The examination of these curves suggests that for the two DGPs, there is a power loss with respect to the case of normal errors for all the testing procedures and whatever the regressor distribution.

We can then conclude that as the change of the DGP characteristics, the change of the error distribution affects the size and power properties of the asymptotic and bootstrap testing procedures of single structural change.

Structural change analysis for U.S. interest rates

The importance of structural change in the U.S. inflation is considered by Ben Aïssa et al. (2004), Ben Aïssa and Jouini (2003), and Jouini and Boutahar (2003). Indeed, they use some selection procedures based on information criteria, sequence of tests for multiple breaks and a test similar to that based on Kolmogorov-Smirnov statistic applied to the evolutionary spectrum to determine the number of changes and their locations in the monthly U.S. inflation series. Both they find significant results since the breaks coincide with important facts and economic events such that the two Oil-Price Shocks and the major events in the International Monetary System.

The theory of the term structure of interest rates suggests that, if interest rates can be characterized as I (1) processes, then they should be cointegrated. Stock and Watson (1988) carry out cointegration tests for three monthly U.S. interest rates covering the period 1960:1-1979:8, [START_REF] Jeong | A perspective on application of bootstrap methods in econometrics[END_REF] and find evidence of two cointegrating vectors. In the same context, Hansen (1992) tests the hypothesis that relationships between U.S. interest rates are stable over the period 1960:1-1990:3. Indeed, he reports results of two fully modified regressions indicating that the change in the Federal Reserve's operating procedures in 1979 altered the relationship between some interest rates. He finds that this regime-shift only appears to have affected the relationship between the treasury-bill rates and the federal funds rate but not the relationship between the treasury-bill rates of different maturities.

The modelling strategy and the results

In the same context of studying the relationship between U.S. interest rates, we propose a different modelling strategy and apply the estimation and testing procedures discussed above using the monthly 90-day treasury-bill rate and the federal funds rate-plotted in Figure 30 (Appendix C)covering the period 1957:1-2002:10 (yielding 550 observations) and obtained from the St. Louis Reserve Federal Bank database. Thus, we adopt a bivariate VAR(3) process:

y t = δ 1 + δ 2 y t-1 + δ 3 y t-2 + δ 4 y t-3 + δ 5 x t-1 + δ 6 x t-2 + δ 7 x t-3 + u t , (7.1)
where y t is the 90-day treasury-bill rate and x t is the federal funds rate. The number of bootstrap replications is set at B = 9999, a large value that we would recommend using in practice to improve the inference accuracy. The results of the different testing procedures are as follows: The AveF T test fails to reject the null hypothesis of parameter stability for all the procedures even at the 10% significance level. For the other tests, only the heteroskedastic bootstrap does not lead to reject the null hypothesis. The fact that this procedure does not allow a rejection of the null hypothesis is not surprising because the reported simulation results show that this procedure has sometimes tendency to under-reject the null hypothesis. However, the rejection of the null by the other procedures is obvious since the simulation results show that these procedures reject the null hypothesis with high proportions when there is a break in the data.

The relationship between the two series is then unstable and the break date thus selected by the SupF T test is located in 1981:5. Looking at the graphs of the series in Figure 30, we confirm the choice of this break since we observe that these series may be affected by structural breaks as there is a turbulence period at the beginning of the 1980s. The fact to find this regime-shift confirms the results of Hansen (1992) who found that the relationship between the 90-day treasury-bill rate and the federal funds rate has been affected by a regime-shift in 1980. Thus, we see that with two different modelling strategies and two different techniques we find the same regime-shift though the sample sizes are not the same. This gives us more information about the fact that the regime-shift is strongly evident and can be easily detected. The corresponding 95% confidence interval of the break date computed using the asymptotic distribution derived by Bai and Perron (1998) covers the quite large period stretching from 1979:3 to 1984:2, indicating that the break date is imprecisely estimated.

The break date 1981:5 may be explained by the following events and facts. In 1981, the United States of America have known the disinflation wave and which is known as "disinflation of Volcker " concerning a new control of the monetary policy. Indeed, before the year 1970, the repercussions of the real shocks on the economy were regarded as tiny. The considerable fluctuations of oil prices modified this point of view and led the Federal Reserve of the United States to reconsider the implementation of the monetary policy. In 1977, unemployment was running at 7%. Total labour costs to employers however, rose both in January 1978 and in 1979 because of increases in payroll taxes and the minimum wage, steps taken by administrative decision not market process but widely regarded as constituting almost as powerful an inflationary influence as were cash wage increases themselves. By 1979, spare capacity was becoming limited and production costs were rising at an annual rate of some 8%. During 1979-1980, moreover, the price of oil and oil-related products also rose sharply under the spur of supply restrictions and rising world demand. In some ways the restrictive policies initially adopted to counter the stagflation of 1979-1980, like some of the main causes at work, followed lines similar to those observed in 1974-1975. At a deeper level, however, the two sets of policies differed substantially. Couched primarily in terms of monetary aggregates, the post-1979 policies seemed to embody a willingness to permit, and even to bring about, greater rises in interest rates than did the policies five years before, and the Federal Reserve's discount rate at once edged up to 12%. After the election of Ronald Reagan in November 1980, the discount rates were raised to 14% in May 1981. The tax proposals of the president were totally different than the procedures taken in 1974-1975. President Reagan projected great tax cuts. The total effects of these last measures should lead to the restoration of the federal budget balance by 1984-1985. The year 1981 was marked by the rise of the tax deficit whereas the interest rates were on a rather high level of 9% and unemployment kept the margin of 8%. Actuated by the bad news coming from Mexico which had borrowed heavily and which stipulated that it doubts to refund its loans to the United States at the limits envisaged, all the American economy embarked on a productivity growth phase of the supported by a decrease of oil prices and the reduction of the inflation rate of the with approximately 5%, but also by the deceleration of the raising of wages whereas the dollar, under the influence of the raised interest rates, attracted the flow of foreign capital, which appreciated approximately 30%.

Conclusion

The paper has presented a detailed evaluation of the performance of some tests for single structural change based on the graphical methods of Davidson and MacKinnon (1998) whose interest is the principle of size correction that allows to show the true power of tests instead of the nominal power. The bootstrap methods improve the performance of the tests. The exception is that the Hansen's heteroskedastic bootstrap has sometimes tendency to under-reject. Moreover, this procedure seems less precise than the asymptotic approximation for the AveF T test since the error in rejection probability committed by this bootstrapping version of the test is severer. The parametric and nonparametric bootstrap procedures are slightly more accurate than the hansen's procedures. We found that on the basis of the corrected size, the bootstrap procedures have the same power properties than their asymptotic counterparts. The empirical illustration highlights the practical importance of the simulation results and indicates that the relationship between the interest rates has been altered by a regime-shift located at the beginning of the 1980s.

  For a given k, the model (2.3) can be estimated by OLS to get the estimates ³ δk , θk ´, the residuals ûtk , and the estimator of the variance σ2 k = (T -2m) -1 P T t=1 û2 tk . The break date estimator k is such that k = arg min k σ2k .

Figure 1 .

 1 Figure 1. Size of the Asymptotic Tests as Function of the Jump Size of the Regressors

Figure 2 .Figure 3 .

 23 Figure 2. Size of the Asymptotic Tests as Function of the Location of the Break for the Regressors

Figure 4 .Figure 5 .

 45 Figure 4. Power of the Asymptotic Tests as Function of the Sample Size for the Nonstationary Regressor Process

  .1) Davidson and MacKinnon suggest the use of the following set of {α k } K k=1 : 18 α k = .001, .002, . . . , .010, .015, . . . , .990, .991, . . . , .999 (K = 215) .

It is well known that a proper treatment of parameter changes can be useful in uncovering the underlying factors that fostered the changes and altered a relationship between economic variables.

This estimator is the same as that obtained by minimizing the sum of squared residuals.

[START_REF] Hansen | Testing for structural change in conditional models[END_REF] plots λ * as a function of κ for four positive values of ω, π1 = 0.15 and π2 = 0.85. For each ω, λ * is maximized at = π1 and minimized at = π2. As ω increases, λ * can become arbitrarily large. The sampling implications is that the SupF T statistic can have arbitrarily large size distortion.

Note that all the parameter estimates are kept fixed during the bootstrap replications, i.e. they are treated as

In the case of fixed exogenous regressors and independent normal errors, the statistic p * can be interpreted as an exact Monte Carlo P value[START_REF] Dwass | Modified randomization tests for nonparametric hypotheses[END_REF].[START_REF] Dufour | Exact tests for structural change in first-order dynamic models[END_REF] use this motivation in their analysis of the CUSUM test in regression models with fixed exogenous regressors and independent normal errors, and use this property to develop bounds for similar models with an added lagged dependent variable.

The pseudo-true values for the unknown parameters are simply the OLS estimates of these parameters obtained under the incorrect assumption that the null hypothesis is true.

Let pi denote the asymptotic P value which is computed using the methods of[START_REF] Hansen | Approximate asymptotic p-values for structural change tests[END_REF].

Let p * i be the bootstrap P value. The EDF of the bootstrap P values is obtained by replacing pi by p * i in (5.1).

This implies that F (α k )α k > 0 for all k, which means that the discrepancy P value plots are above the horizontal axis y = 0.

This confirms the fact that the nonparametric bootstrap gives results indistinguishable from those of the parametric bootstrap.

The magnitude of change θ is then equal to 0.50.

Note that δ1, δ2, δ3, δ4 and δ5 cannot be consistent because when there exists a break date in the DGP while the estimation is done without breaks, the obtained coefficient regression estimates are not consistent because of the misspecification.

This is obvious since as the intuition suggests, a test statistic having tendency to over-reject (section 6.2) can have high power.

The results of these experiments are not reported here and are available upon request from the author.

They used data covering this period, presumably to exclude an eventual regime-shift in the term structure due to the change in the Federal Reserve's operating procedures in 1979.

Acknowledgements

The author is grateful to Emmanuel FLACHAIRE and the participants in the "Journées Doctorales de l'ADRES", Paris, February 2005 for useful comments and remarks.

Appendix A: P value Functions


Ph.D Bahram Sheikhi
email: sheikhibahram@gmail.com

Ph.D Amir Letafatkar
email: letafatkaramir@yahoo.com

Ph.D Paulo Henrique Marchetti
email: paulo.marchetti@csun.edu

Ph.D Fereshteh Eftekhari
email: eftekhary8fereshteh@gmail.com

Ph.D Brian J Wallace
email: wallaceb@uwosh.edu

Ph.D Filippo Maselli
email: masellifilippo76@gmail.com

Ph.D Giacomo Rossettini
email: giacomo.rossettini@gmail.com

PhD Urs Granacher
email: urs.granacher@sport.uni-freiburg.de

PhD Hassane Zouhal
email: hassane.zouhal@univ-rennes2.fr

Bahram Sheikhi

Ph.D Fereshteh Eftekhari

Effects of kinesio taping on tuck jump performance in competitive male athletes

Keywords: Kinesiology taping, tuck jump, neuromuscular control

Introduction

A combination of anatomical, neuromuscular, and biomechanical factors have been recognized to contribute to non-contact knee injuries, some of which are modifiable [START_REF] Nguyen | A preliminary multifactorial approach describing the relationships among lower extremity alignment, hip muscle activation, and lower extremity joint excursion[END_REF][START_REF] Gehring | Gender and fatigue have influence on knee joint control strategies during landing[END_REF][START_REF] Prieske | Sex-specific effects of surface instability on drop jump and landing biomechanics[END_REF]. A number of biomechanical factors during landing such as ipsilateral trunk motion, pelvic drop, hip adduction and internal rotation, knee valgus, tibial internal or external rotation and external hip adduction moment have been reported as risk factors for knee injury [START_REF] Davey | Risk Factors Associated With a Noncontact Anterior Cruciate Ligament Injury to the Contralateral Knee After Unilateral Anterior Cruciate Ligament Injury in High School and College Female Athletes: A Prospective Study[END_REF][START_REF] Gans | Epidemiology of recurrent anterior cruciate ligament injuries in National Collegiate Athletic Association sports: the Injury Surveillance Program, 2004-2014[END_REF][START_REF] Hewett | Biomechanical measures of neuromuscular control and valgus loading of the knee predict anterior cruciate ligament injury risk in female athletes: a prospective study[END_REF][START_REF] Kaeding | Epidemiology and Diagnosis of Anterior Cruciate Ligament Injuries[END_REF][START_REF] Kimura | Mechanisms for anterior cruciate ligament injuries in badminton[END_REF][START_REF] Koga | Mechanisms for noncontact anterior cruciate ligament injuries: knee joint kinematics in 10 injury situations from female team handball and basketball[END_REF][START_REF] Labella | Anterior cruciate ligament injuries: diagnosis, treatment, and prevention[END_REF].

Dynamic sports-related movements can result in lower extremity injuries if athletes exhibit neuromuscular performance deficits [START_REF] Fulton | Injury risk is altered by previous injury: a systematic review of the literature and presentation of causative neuromuscular factors[END_REF]. Researchers have classified and outlined neuromuscular deficits into four groups using video analysis. These groups include ligament dominance, quadriceps dominance, leg dominance, and trunk dominance [START_REF] Hewett | Understanding and preventing acl injuries: current biomechanical and epidemiologic considerations -update 2010[END_REF]. Specifically, ligament dominance is an imbalance between the neuromuscular and ligamentous controls of dynamic knee joint stability [START_REF] Hewett | Understanding and preventing acl injuries: current biomechanical and epidemiologic considerations -update 2010[END_REF][START_REF] Myer | Real-time assessment and neuromuscular training feedback techniques to prevent ACL injury in female athletes[END_REF]. This imbalance is represented by an inability to control lower extremity frontal plane motion during landing and cutting [START_REF] Hewett | Understanding and preventing acl injuries: current biomechanical and epidemiologic considerations -update 2010[END_REF][START_REF] Myer | Real-time assessment and neuromuscular training feedback techniques to prevent ACL injury in female athletes[END_REF]. Quadriceps dominance is also an imbalance between knee extensor and flexor strength, recruitment, and coordination [START_REF] Hewett | Understanding and preventing acl injuries: current biomechanical and epidemiologic considerations -update 2010[END_REF][START_REF] Myer | Real-time assessment and neuromuscular training feedback techniques to prevent ACL injury in female athletes[END_REF]. Leg dominance is reported as another imbalance between the lower extremities in strength, coordination, and control. Finally, trunk dominance is considered one of these very imbalances, which occurs between the inertial demands of the trunk and control and coordination to resist it [START_REF] Hewett | Understanding and preventing acl injuries: current biomechanical and epidemiologic considerations -update 2010[END_REF][START_REF] Myer | Real-time assessment and neuromuscular training feedback techniques to prevent ACL injury in female athletes[END_REF]. These four potential neuromuscular control deficits have been declared as risk factors that significantly contribute to injuries in athletes [START_REF] Myer | Real-time assessment and neuromuscular training feedback techniques to prevent ACL injury in female athletes[END_REF]. Tuck jump assessment was designed as a clinician-friendly tool to provide practitioners and clinicians with a novel direction to identify neuromuscular imbalances highly associated with injury [START_REF] Fort-Vanmeerhaeghe | Intra-and Inter-Rater Reliability of the Modified Tuck Jump Assessment[END_REF].

Kinesio taping (KT) is an elastic therapeutic tape used to prevent and treat sports injuries and various musculoskeletal conditions and may help alleviate some neuromuscular control deficits [START_REF] Wallis | Clinical therapeutic applications of the kinesio taping method[END_REF][START_REF] Williams | Kinesio taping in treatment and prevention of sports injuries: a meta-analysis of the evidence for its effectiveness[END_REF]. KT is adopted to improve function, stability, and proprioception [START_REF] Fernandes De | Kinesio taping effect on quadriceps strength and lower limb function of healthy individuals: A blinded, controlled, randomized, clinical trial[END_REF][START_REF] Mendez-Rebolledo | Short-Term Effects of Kinesio Taping on Muscle Recruitment Order During a Vertical Jump: A Pilot Study[END_REF][START_REF] Rajasekar | Does Kinesio taping correct exaggerated dynamic knee valgus? A randomized double blinded sham-controlled trial[END_REF] and enhance muscle contraction and force production in painful musculoskeletal conditions [START_REF] Wallis | Clinical therapeutic applications of the kinesio taping method[END_REF][START_REF] Rajasekar | Does Kinesio taping correct exaggerated dynamic knee valgus? A randomized double blinded sham-controlled trial[END_REF][START_REF] Lyman | Investigating the effectiveness of kinesio taping space correction method in healthy adults on patellofemoral joint and subcutaneous space[END_REF]. By improving proprioception and muscle activation, KT may favorably influence injury risk [START_REF] Rajasekar | Does Kinesio taping correct exaggerated dynamic knee valgus? A randomized double blinded sham-controlled trial[END_REF][START_REF] Sheikhi | The influence of kinesio taping on trunk and lower extremity motions during different landing tasks: implications for anterior cruciate ligament injury[END_REF][START_REF] Kirmizigil | The effectiveness of kinesio taping in recovering from delayed onset muscle soreness: A crossover study[END_REF]. Applying KT at a stretch greater than 50% can help correct the knee valgus position [START_REF] Rajasekar | Does Kinesio taping correct exaggerated dynamic knee valgus? A randomized double blinded sham-controlled trial[END_REF][START_REF] Sheikhi | The influence of kinesio taping on trunk and lower extremity motions during different landing tasks: implications for anterior cruciate ligament injury[END_REF] and improve jump performance [START_REF] Mendez-Rebolledo | Short-Term Effects of Kinesio Taping on Muscle Recruitment Order During a Vertical Jump: A Pilot Study[END_REF][START_REF] De Jesus | The effects of varied tensions of kinesiology taping on quadriceps strength and lower limb function[END_REF]. Other studies have reported limited, inconclusive evidence regarding the effectiveness of KT on muscle strength and functional performance [START_REF] Fernandes De | Kinesio taping effect on quadriceps strength and lower limb function of healthy individuals: A blinded, controlled, randomized, clinical trial[END_REF][START_REF] De Jesus | The effects of varied tensions of kinesiology taping on quadriceps strength and lower limb function[END_REF][START_REF] Ataabadi | The effects of facilitatory and inhibitory kinesiotaping of Vastus Medialis on the activation and fatigue of superficial quadriceps muscles[END_REF]. Some studies noted increased electromyographic activity of the muscles at 24 and 72 hours after applying KT [START_REF] Yeung | Acute effects of kinesio taping on knee extensor peak torque and electromyographic activity after exhaustive isometric knee extension in healthy young adults[END_REF][START_REF] Lins | Delayed effect of Kinesio Taping on neuromuscular performance, balance, and lower limb function in healthy individuals: a randomized controlled trial[END_REF]. The results of these studies raise the hypothesis of possible delayed effects of KT application on neuromuscular performance [START_REF] Reneker | Effectiveness of kinesiology tape on sports performance abilities in athletes: A systematic review[END_REF][START_REF] Wang | Kinesio taping is superior to other taping methods in ankle functional performance improvement: a systematic review and meta-analysis[END_REF], offering the possibility for new research. Generally, the working mechanism of KT remains unknown. The effects of KT on motor performance may be due to the interplay between cutaneous afferent stimulation and motor unit firing in both central and peripheral nervous systems [START_REF] Kase | Clinical therapeutic applications of the Kinesio (! R) taping method[END_REF]. An increase in peripheral nerve stimulation was shown to promote the excitability of the motor cortex [START_REF] Kase | Clinical therapeutic applications of the Kinesio (! R) taping method[END_REF]. Therefore, reduction of motor neuron threshold may be induced by cutaneous stimulation, resulting in easier recruitment of the motor units, and in turn, leading to an improved performance [START_REF] De Almeida Lins | Kinesio Taping does not alter neuromuscular performance of femoral quadriceps or lower limb function in healthy subjects: Randomized, blind, controlled, clinical trial[END_REF][START_REF] Firth | The effect of kinesiotape on function, pain, and motoneuronal excitability in healthy people and people with Achilles tendinopathy[END_REF]. However, previous studies indicate that applying KT to the muscles does not alter the neuromuscular performance or lower limb function in healthy participants [START_REF] De Almeida Lins | Kinesio Taping does not alter neuromuscular performance of femoral quadriceps or lower limb function in healthy subjects: Randomized, blind, controlled, clinical trial[END_REF][START_REF] Firth | The effect of kinesiotape on function, pain, and motoneuronal excitability in healthy people and people with Achilles tendinopathy[END_REF].

In the context of improving neuromuscular performance, less attention has been devoted to the influence of KT on neuromuscular control deficits during jump tasks. The hypothesis that KT, leading to improved neuromuscular control deficits, will improve athlete's performance needs to be formally verified. Therefore, we designed a randomized controlled trial (RCT) to evaluate the acute effects of KT at different stretches on neuromuscular performance in athletes. The present study aimed to compare the acute effects of Kinesio taping at different stretches on tuck jump performance in active individuals. We hypothesized that KT would improve neuromuscular performance in athletes.

Materials & Methods

Seventy-five competitive males participated in this RCT retrospectively registered at [UMIN000044935, date of first registration 21/07/2021]. Sample size calculations were performed using G*Power (version 3.1.9.2, Dusseldorf, Germany). In accordance with previous evidence [START_REF] Sheikhi | The influence of kinesio taping on trunk and lower extremity motions during different landing tasks: implications for anterior cruciate ligament injury[END_REF][START_REF] Santos Glória | Kinesio taping does not alter muscle torque, muscle activity or jumping performance in professional soccer players: A randomized, placebocontrolled, blind, clinical trial[END_REF][START_REF] Kang | Sample size determination and power analysis using the G*Power software[END_REF], a medium effect size (f= 0.25), an alpha level of 0.05, and a power of 0.80 were considered. The calculation revealed that 22 participants were required in each group. Twenty-five competitive males were included in each group to account for possible missing data and a 15% loss from participants missing follow-ups.

The research ethics committee of the Sport Sciences Research Institute (Approval ID: SSRI.REC-2106-1072) approved this study. All participants were informed of the study procedures and signed an informed consent form before participating, in conformity with the Declaration of Helsinki [START_REF]World Medical Association Declaration of Helsinki: ethical principles for medical research involving human subjects[END_REF].

Inclusion criteria included: no history of surgery in the lower extremity in the previous before study participation, no musculoskeletal injury that could interfere with or contraindicate the assessment procedures [START_REF] De Jesus | The effects of varied tensions of kinesiology taping on quadriceps strength and lower limb function[END_REF][START_REF] Taylor | Biomechanical Differences of Multidirectional Jump Landings Among Female Basketball and Soccer Players[END_REF], and no allergy to the adhesive material used [START_REF] De Almeida Lins | Kinesio Taping does not alter neuromuscular performance of femoral quadriceps or lower limb function in healthy subjects: Randomized, blind, controlled, clinical trial[END_REF]. In addition, the exclusion criteria were: current pain in the lumbar spine and lower limb, cardiac disorders, vestibular conditions and neurological abnormalities [START_REF] De Jesus | The effects of varied tensions of kinesiology taping on quadriceps strength and lower limb function[END_REF][START_REF] Taylor | Biomechanical Differences of Multidirectional Jump Landings Among Female Basketball and Soccer Players[END_REF][START_REF] Cruz | The effects of three jump landing tasks on kinetic and kinematic measures: implications for ACL injury research[END_REF].

Procedures

Demographic information, including age, body height and mass, was collected during the assessment. Before randomization, the demographic data of all participants were collected and recorded by a blinded assessor. After the assessment, the participants were randomly assigned into one of three groups: KT without stretch (KT0), KT with approximately 50% and 75% stretch (KT50 and KT75, respectively) using the website http://randomizer.org/ (Social Psychology Network, Connecticut, USA). Concealed allocation was performed using a computer-generated block randomized table of numbers (1; KT0 group, 2; KT50 group and 3; KT75 group). All participants were evaluated daily at the same time in a laboratory. The experimental conditions were controlled for the ambient temperature (24°C), lighting and humidity. All participants were asked to refrain from training, maintain a regular diet and sleep routine, and avoid smoking, caffeine and alcohol for 24 hours before testing sessions. All participants' tuck jump performance was assessed using different KT stretches (three groups: KT0, KT50 and KT75) at baseline, immediately after the application of KT, 24 and 72 h after the application of KT (Figure 1). The randomization was performed by a researcher who was not involved in the recruitment or interventions of participants.

-insert Figure 1 here-

Kinesiology Taping Application

The examiner measured the distances between the muscle origin and insertion of all participants with a meter to determine the stretch imposed during the KT (Kinesio Tex Gold, FP, 5 cm wide) application. After the measure, the mathematical rule of three was used to determine and individualize the length of the KT [START_REF] De Jesus | The effects of varied tensions of kinesiology taping on quadriceps strength and lower limb function[END_REF]. Percentages are listed as the percentage of stretch to be applied based on 100% of the available stretch. For example, 50-75%. The meaning of this is 50-75% of the available stretch, with 100% being the maximum stretch. Finally, KT was applied to the participants' muscles according to the established stretch, bilaterally on the lower extremity muscles, by the same trained researcher who was a certified KT practitioner with 8 years of experience using KT. Moreover, the KT was applied to all major muscles responsible for jump landing [START_REF] Llurda-Almuzara | Relationship between lower limb EMG activity and knee frontal plane projection angle during a single-legged drop jump[END_REF] (gastrocnemius [GT], biceps femoris [BF], vastus lateralis [VL], vastus medialis

[VM], rectus femoris [RF] and gluteus medius [Gmed]) (Figure 2). Before applying the KT adhesive, hairs were removed, and the skin was cleaned with alcohol (70% isopropyl alcohol pads) to ensure KT adherence. In addition, the participants' skin sensitivity was controlled with a KT test patch for 24 h [START_REF] De Jesus | The effects of varied tensions of kinesiology taping on quadriceps strength and lower limb function[END_REF].

The KT was applied to the muscles as follows.

For the GT, the participant was relaxed in the prone position; the KT was split into a Y-strip so that each side could be longitudinally taped along the origin of the medial and lateral GT muscles.

Both proximal ends of the Y-strip were placed, without stretch, 4 cm below the popliteal line, with the ankle in the neutral position. The proximal half of the strip was then stretched and placed on the calf up to the marked midpoint, with the participant's ankle at maximum dorsiflexion. The distal half of the strip was also stretched and placed from the midpoint to the upper part of the posterior calcaneus tuberosity, with the participants' ankles still at maximum dorsiflexion. The distal end of the Y-strip was then placed, without stretch, with the ankle back in the neutral position [START_REF] Nunes | Effect of kinesio taping on jumping and balance in athletes: a crossover randomized controlled trial[END_REF].

For the BF, the participants were positioned lying on their sides with the hip flexed and internally rotated, the knee extended, and the contralateral leg slightly bent for stability. In addition, the KT was applied from the ischial tuberosity to the posterior region of the fibular head.

For the VL muscle, the KT was applied 10 cm distal to the greater trochanter, extending to the lateral edge of the patella. Next, the KT was applied to the RF from 10 cm below the anterior superior iliac spine to the upper edge of the patella, with the participant in a relaxed supine position [START_REF] De Jesus | The effects of varied tensions of kinesiology taping on quadriceps strength and lower limb function[END_REF]. The strip was fixed on the VL muscle from the greater trochanter to the lateral edge of the patella. Finally, the KT was applied to the middle third of the medial region of the thigh to the medial edge of the patella for the VM. This application was performed with the participants standing on one foot, with the hip of the dominant limb at 0° and the knee flexed at 90° [START_REF] De Almeida Lins | Kinesio Taping does not alter neuromuscular performance of femoral quadriceps or lower limb function in healthy subjects: Randomized, blind, controlled, clinical trial[END_REF].

Moreover, the KT was applied to the quadriceps from the proximal attachment to the distal one [START_REF] Wallis | Clinical therapeutic applications of the kinesio taping method[END_REF].

For the Gmed, the KT was applied from the iliac crest to the greater trochanter in the side-lying position. The participants were in a side-lying position with 90° hip flexion, adduction, and internal rotation. The KT was applied from muscle insertion to origin using the Y-strip taping technique. The base of the Y strip was additionally applied on the lateral surface of the greater trochanter with no stretch. The anterior tail was applied towards the anterior superior iliac spine, with light or paper-off-technique stretch and the last 1-2 inches with no stretch. The posterior tail was also applied towards the posterior superior iliac spine with a similar stretch as mentioned above [START_REF] Wallis | Clinical therapeutic applications of the kinesio taping method[END_REF][START_REF] Rajasekar | Does Kinesio taping correct exaggerated dynamic knee valgus? A randomized double blinded sham-controlled trial[END_REF].

-insert Figure 2 here-

Two-Dimensional Video Analysis

The performance was captured with two standard digital video cameras (Sony HDR-PJ675) sampling at 48 Hz on tripods with a height set to that of the participant's waist. One was aligned 2 m away in the sagittal plane, and the other was aligned 2 m away in the frontal plane. The video recordings were analyzed using a commercial software package (Kinovea version 0.8.15) [START_REF] Dingenen | Can two-dimensional measured peak sagittal plane excursions during drop vertical jumps help identify three-dimensional measured joint moments[END_REF]. The participants were required to wear shorts with the hem approximately mid-thigh [START_REF] Fort-Vanmeerhaeghe | Intra-and Inter-Rater Reliability of the Modified Tuck Jump Assessment[END_REF] to allow visible tracking of the knees. Markers were appropriately positioned on the acromioclavicular joint, manubrium sterni, anterior superior iliac spine, greater trochanter, and medial and lateral femoral epicondyles, medial and lateral malleoli. The ankle joint center was defined at the midpoint of the medial and lateral malleoli markers. The knee joint center was also identified at the midpoint of the medial and lateral femoral epicondyle markers [START_REF] Dingenen | Can two-dimensional measured peak sagittal plane excursions during drop vertical jumps help identify three-dimensional measured joint moments[END_REF]. The knee valgus angle was described as the angle between the line formed by the knee joint center and the ankle joint center and the line formed by the anterior superior iliac spine (ASIS) and the knee joint center. The reliability of two-dimensional video assessment has been previously confirmed (ICC= 0.72 to 0.91) [START_REF] Munro | Reliability of 2-dimensional video assessment of frontal-plane dynamic knee valgus during common athletic screening tasks[END_REF].

Tuck Jump Assessment

All participants completed the same 15 min post-activation potentiation (PAP) warm-up strategies [START_REF] Blazevich | Post-activation potentiation versus post-activation performance enhancement in humans: historical perspective, underlying mechanisms, and current issues[END_REF][START_REF] Hodgson | Post-activation potentiation[END_REF]. It consisted of multidirectional movements combined with strength and dynamic stretching exercises (5 min), maximal-and progressive-intensity displacements (5 min), including change of directions (2 min), jumps (2 × 5 repetitions), and acceleration/deceleration movements (2 × 5 repetitions) [START_REF] Blazevich | Post-activation potentiation versus post-activation performance enhancement in humans: historical perspective, underlying mechanisms, and current issues[END_REF][START_REF] Hodgson | Post-activation potentiation[END_REF]. During the familiarization session, the participants received instructions on performing the test, including cues to lift the knees to hip height and land on the same spot. A demonstration of the exercise was also provided. The participants were allowed to practice no more than two trials of the tuck jump prior to data collection. Furthermore, the participants were instructed to place their feet in the middle of a rectangle marked on the floor while performing the assessment. This square consisted of four smaller rectangles (41 cm in length and 35 cm in width). The number of jumps in 10 seconds was recorded. Three trials were completed with a 2-minute rest between trials. These values were averaged across the 3 successful trials for each task and used for statistical analysis. The tuck jump assessment score has been shown to be reliable (ICC= 0.85 to 0.88) for assessing neuromuscular control in healthy adult athletes [START_REF] Gokeler | Between-session and inter-rater reliability of the modified tuck jump assessment in healthy adult athletes[END_REF].

After the test, all videos were reviewed, and 10 flaws a participant may have displayed during the jumps were scored (Table 1). The overall score of the tuck jumps was the sum of the total number of flaws observed [START_REF] Myer | Real-time assessment and neuromuscular training feedback techniques to prevent ACL injury in female athletes[END_REF][START_REF] Fort-Vanmeerhaeghe | Intra-and Inter-Rater Reliability of the Modified Tuck Jump Assessment[END_REF]. Evidence suggests that these neuromuscular deficits may be associated with anterior cruciate ligament (ACL) injury [START_REF] Hewett | Understanding and preventing acl injuries: current biomechanical and epidemiologic considerations -update 2010[END_REF][START_REF] Myer | Real-time assessment and neuromuscular training feedback techniques to prevent ACL injury in female athletes[END_REF]. Therefore, these deficits should be examined and identified in individual athletes [START_REF] Hewett | Understanding and preventing acl injuries: current biomechanical and epidemiologic considerations -update 2010[END_REF][START_REF] Myer | Real-time assessment and neuromuscular training feedback techniques to prevent ACL injury in female athletes[END_REF]. A second investigator performed all tuck jump quality assessments and was blinded to the specific condition being evaluated.

Statistical Analysis

The Kolmogorov-Smirnov test was used to confirm the normality of the data; Levene's test evaluated the homogeneity of variances. Group demographics were compared using a one-way analysis of variance (ANOVA). The mean, standard deviation (SD), and 95% confidence intervals (95%CI) were calculated. Linear mixed models were used for each KT stretch to compare differences between the tuck jump test score and the four assessment time points (3 conditions [Tape without stretch, Tape at 50%, Tape at 75%] × 4 Tests [pretest, posttest-1, posttest-2, posttest-3]). Furthermore, the magnitudes of the differences were examined using the standardized difference based on Hedges g units using effect sizes. The effect size results were qualitatively interpreted using the following thresholds: standardized mean differences of 0.2, 0.5, and 0.8 for small, moderate, and large effect sizes, respectively [START_REF] Turner | Assessing group-based changes in high-performance sport. Part 2: Effect sizes and embracing uncertainty through confidence intervals[END_REF]. Statistical significance was set a priori at p < 0.05. All analyses were conducted using the SPSS Version 23.0 (Chicago, IL, USA).

Results

Seven participants (KT0 group [n = 3], KT50 group [n = 2], and KT75 group [n = 2]) did not complete the assessment protocol since they missed one of the assessments. Therefore, 68 participants were analyzed. No adverse events occurred in any KT group.

The mean and standard deviation of the participant's demographic characteristics are presented in Table 2. At baseline, there were no significant differences in the demographic characteristics and the scores of tuck jump between the three groups; all the tested variables showed p > 0.05. The frequency distribution of individual tuck jump assessment criteria for the three groups at the fourtime point is displayed in Figure 3.

-insert Table 2 here--insert Figure 3 here-No significant effect of KT on tuck jump performance was found (p>0.05). General linear mixed model analysis showed no significant group × time interaction effect for tuck jump performance (F= 0.08, p = 0.99). The effect size ranged from small (g = 0.28) to moderate (g = 0.69). The participants' tuck jump performance was not improved immediately after the application of KT, 24 h and 72 h. No significant differences were found between the KT groups at any of the time points (p>0.05) (Table 3).

-insert Table 3 here-

Discussion

This study aimed to investigate the effects of KT at different stretches on the tuck jump performance of competitive male athletes. As the main finding, our RCT revealed that applying a KT 0%, 50%, and 75% stretch did not improve tuck jump performance.

To the best of our knowledge, this is the first study to investigate KT's effect at different stretches on jumping performance despite growing research investigating the KT effect on lower limb functions in injured and healthy individuals [START_REF] Fernandes De | Kinesio taping effect on quadriceps strength and lower limb function of healthy individuals: A blinded, controlled, randomized, clinical trial[END_REF][START_REF] Mendez-Rebolledo | Short-Term Effects of Kinesio Taping on Muscle Recruitment Order During a Vertical Jump: A Pilot Study[END_REF][START_REF] De Jesus | The effects of varied tensions of kinesiology taping on quadriceps strength and lower limb function[END_REF][START_REF] De Almeida Lins | Kinesio Taping does not alter neuromuscular performance of femoral quadriceps or lower limb function in healthy subjects: Randomized, blind, controlled, clinical trial[END_REF]. For example, Jesus et al. [START_REF] De Jesus | The effects of varied tensions of kinesiology taping on quadriceps strength and lower limb function[END_REF] conducted a similar study. Their findings demonstrated that different types of KT stretches did not modulate the quadriceps isometric strength or single-hop tests in the short or the long term in healthy participants [START_REF] Fernandes De | Kinesio taping effect on quadriceps strength and lower limb function of healthy individuals: A blinded, controlled, randomized, clinical trial[END_REF][START_REF] De Jesus | The effects of varied tensions of kinesiology taping on quadriceps strength and lower limb function[END_REF]. Furthermore, current studies have shown that the direction of KT cannot reduce or enhance muscle activity [START_REF] Ataabadi | The effects of facilitatory and inhibitory kinesiotaping of Vastus Medialis on the activation and fatigue of superficial quadriceps muscles[END_REF]. A recent study also examined vertical jump performance in healthy young adults failing to find significant differences in jump performance [START_REF] Nakajima | The effect of kinesio tape on vertical jump and dynamic postural control[END_REF]. Opposite to our finding, Limroongreungrat et al. [START_REF] Limroongreungrat | Immediate effect of ACL kinesio taping technique on knee joint biomechanics during a drop vertical jump: a randomized crossover controlled trial[END_REF] showed that using the ACL-KT technique with 75% stretch can change the drop vertical jump task pattern in healthy people.

While the potential working mechanism of KT may be reflected after a longer application time [START_REF] Mendez-Rebolledo | Short-Term Effects of Kinesio Taping on Muscle Recruitment Order During a Vertical Jump: A Pilot Study[END_REF], our results demonstrated that the KT application did not affect the tuck jump score after 24 h, 72 h, thus leaving open the debate on its effect. KT has been proposed to provide tactile stimulation [START_REF] Konishi | Tactile stimulation with kinesiology tape alleviates muscle weakness attributable to attenuation of Ia afferents[END_REF][START_REF] Yam | Effects of Kinesio tape on lower limb muscle strength, hop test, and vertical jump performances: a meta-analysis[END_REF]. However, KT tactile inputs may not be strong enough to modulate muscular strength and biomechanical/technical aspects of jumping in healthy athletes [START_REF] Fu | Effect of Kinesio taping on muscle strength in athletes-a pilot study[END_REF]. In addition, it has been suggested that KT can add proprioceptive information, stimulating Golgi tendon organs and muscle spindles during voluntary movement and functional tasks. Muscle spindles and Golgi tendon organs may send sensory information to the central nervous system to facilitate an early activation response from the muscles (e.g., of the trunk and lower extremity) [START_REF] Mendez-Rebolledo | Short-Term Effects of Kinesio Taping on Muscle Recruitment Order During a Vertical Jump: A Pilot Study[END_REF][START_REF] Aydoğdu | Clinical outcomes of kinesio taping applied in patients with knee osteoarthritis: a randomized controlled trial[END_REF][START_REF] Yoshida | The effect of kinesio taping on lower trunk range of motions[END_REF][START_REF] Hadamus | Assessment of the Impact of Kinesiology Taping Application Versus Placebo Taping on the Knee Joint Position Sense. Preliminary Report[END_REF][START_REF] Inglés | Effect of Kinesio Taping and balance exercises on postural control in amateur soccer players: A randomised control trial[END_REF][START_REF] Magalhães | Prolonged use of Kinesiotaping does not enhance functional performance and joint proprioception in healthy young males: Randomized controlled trial[END_REF][START_REF] Bae | The effects of kinesio taping on potential in chronic low back pain patients anticipatory postural control and cerebral cortex[END_REF]. Recently, the contextual effects and the rituality presented during the application of the KT (e.g., the rituality of the colour, the shape and the length of the tape) were also proposed to explain its effect [START_REF] Rossettini | Context matters: the psychoneurobiological determinants of placebo, nocebo and context-related effects in physiotherapy[END_REF].

Nonetheless, the basic KT application mechanisms have yet to be thoroughly investigated, offering opportunities for future research.

This study has some limitations that should be considered when interpreting its results. First, this study had no control group (without KT) or a sham group (with a fake KT), thus preventing the possibility of checking the role of confounders and placebo effects [START_REF] Rossettini | Manual therapy RCTs: should we control placebo in placebo control?[END_REF]. Second, the athletes assessed herein were not blinded to their KT stretches applied to the adhesive tape. Third, this study did not investigate sex-specific neuromuscular performance changes. Finally, this study did not measure changes in muscle activation. Thus, future studies should explore the effect of KT application on the electromyographic activity of lower extremity muscles during sports associated with ACL injury occurrences and other musculoskeletal conditions (e.g., low back pain) [START_REF] Maselli | The diagnostic value of Red Flags in thoracolumbar pain: a systematic review[END_REF]59].

In summary, this study demonstrated no effects on the neuromuscular jump performance after the different stretch applications of KT. Therefore, compelling evidence is lacking to support using KT to enhance neuromuscular performance. Further research is needed to investigate the efficacy of using KT on neuromuscular performance in the athlete population, especially females.

Tables captions:

Table 1. The identifiable flaws during tuck jump assessment [START_REF] Gehring | Gender and fatigue have influence on knee joint control strategies during landing[END_REF][START_REF] Davey | Risk Factors Associated With a Noncontact Anterior Cruciate Ligament Injury to the Contralateral Knee After Unilateral Anterior Cruciate Ligament Injury in High School and College Female Athletes: A Prospective Study[END_REF].

Table 2. Mean (SD) age, height, and mass of the three groups evaluated.

Table 3. Neuromuscular performance (tuck jump assessment) differences between the "KT 0", "KT 50" and "KT 75" group.

Accepted manuscript

Table 1. The identifiable flaws during tuck jump assessment [START_REF] Hewett | Understanding and preventing acl injuries: current biomechanical and epidemiologic considerations -update 2010[END_REF][START_REF] Myer | Real-time assessment and neuromuscular training feedback techniques to prevent ACL injury in female athletes[END_REF].

Flaws during tuck jump assessment Score

(1) Lower extremity valgus at landing

Total Score

Accepted manuscript

  Storari L, Barbari V et al. Prevalence and incidence of low back pain among runners: a systematic review. BMC Musculoskelet Disord 2020; 21: 343.

Figure captions:Figure 1 .

 1 Figure captions: Figure 1. Study flow chart.

Figure 2 .

 2 Figure 2. Kinesio taping applications: gastrocnemius (GT), biceps femoris (BF), vastus lateralis (VL), vastus medialis (VM), rectus femoris (RF), gluteus medius (Gmed) muscles.

Figure 3 .

 3 Figure 3. Tuck jump assessment scoring frequencies for (A) KT 0, (B) KT 50, and (C) KT75 groups.

Figure 3 A

 3 Figure 3 A. Tuck jump assessment scoring frequencies for KT 0 group.

Figure 3 B

 3 Figure 3 B.Tuck jump assessment scoring frequencies for KT 50 group.

Figure 3 C

 3 Figure 3 C. Tuck jump assessment scoring frequencies for KT 75 group.

( 2 )( 3 )( 4 ) 5 ) 8 ) 9 )

 234589 Thighs not parallel (peak of jump) Thighs not equally positioned side to side (during flight) Feet not positioned at the level of the shoulders (Feet not parallel (front to back) (6) Different foot contact timing (asymmetrical landing) (7) Excessive landing contact noise (Pause between jumps (Technique worsens within 10 seconds (10) The participant did not land in the same footprint (excessive inflight motion).

Table 2 .

 2 Mean (SD) age, height, and mass of the three groups evaluated. KT, Kinesio taping; KT0, Kinesio taping without tension; KT50, 50% tension applied to the Kinesio taping; KT75, 75% tension applied to the Kinesio taping; †, p-value of ANOVA; cm, centimeter; kg, kilogram; SD, standard deviation; m, meter; h, hours; wk, week

	Characteristic KT0 (n = 22)	KT50 (n = 23)	KT75 (n = 23)	p-value †
	Age (year)	23.59 ± 1.99	22.70 ± 2.54	22.78 ± 2.11	0.34
	Height (cm)	179.77 ± 8.84	177.87 ± 8.29	177.48 ± 7.37	0.61
	Mass (kg)	74.05 ± 5.67	72.80 ± 7.28	70.63 ± 6.49	0.21
	BMI (kg/m 2 )	22.93 ± 1.15	23.02 ± 1.76	22.41 ± 1.32	0.31
	Training experience (year)	7.05 ± 2.89	7.83 ± 2.62	7.61 ± 3.09	0.65
	Training (h/wk)	8.45 ± 2.63	7.09 ± 2.35	8.17 ± 1.97	0.12
	Abbreviation:

Table 3 .

 3 Neuromuscular performance (tuck jump assessment) differences between the "KT 0", "KT 50" and "KT 75" group. KT, Kinesio taping; KT0, Kinesio taping without tension; KT50, 50% tension applied to the Kinesio taping; KT75, 75% tension 20 applied to the Kinesio taping; 95% CI, 95% confidence interval.

	19
					"KT 0" vs "KT 50"	"KT 0" vs "KT	"KT 50" vs "KT 75"					Group × time
	Tuck jump assessment	KT 0 KT 50 KT 75	group differences: effect size (95%CI) and P value	75" group differences: effect size (95%CI) and P value	group differences: effect size (95%CI) and P value	Group Score F value P-	Time Score F value P-	Interaction Effect F P-Score value
	Baseline	4.59 ± 1.10	4.22 ± 1.13	3.96 ± 0.98	NA (-0.26 to 0.92) p = 0.74	NA (0.004 to 1.20) p = 0.15	NA (-0.34 to 0.82) p = 0.99
	Immediately after the application of KT	4.36 ± 0.79	4.09 ± 0.90	3.83 ± 0.72	0.31 (-0.27 to 0.90) p = 0.76	0.69 (0.10 to 1.30) p = 0.09	0.31 (-0.26 to 0.90) p = 0.83
	Twenty-four hours after the application of KT	4.23 ± 1.15	3.91 ± 0.79	3.61 ± 0.89	0.32 (-0.26 to 0.91) p = 0.82	0.59 (0.003 to 1.20) p = 0.10	0.35 (-0.23 to 0.94) p = 0.85	5.36	0.007	2.6	0.06	0.08	0.99
	Seventy-two hours after applying KT	4.09 ± 0.81	3.87 ± 0.76	3.65 ± 0.71	0.28 (-0.31 to 0.87) p = 0.99	0.57 (-0.02 to 1.17) p = 0.17	0.29 (-0.28 to 0.88) p = 0.99
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Abbreviation:

Grant support: none Financial disclosures or conflicts of interest: none IRB approval: Prior to data collection, this study was approved by the research ethics committee of Sport Sciences Research Institute, Tehran-Iran (Approval ID: IR.SSRI.REC.1400.1149).