Testing for Cointegration in Multivariate Time Series

Home , Cointegration

Örebro University Örebro University School of Business Statistics, Paper, Second level, 15 Credits Supervisor: Panagiotis Mantalos Examiner: Thomas Laitila Spring 2013

An evaluation of the Johansens trace test and three different bootstrap tests when testing for cointegration

Jonas Englund 880131

Abstract In this paper we examine, by Monte Carlo simulation, size and power of the Johansens trace test when the error covariance matrix is nonstationary, and we also investigate the properties of three different bootstrap cointegration tests. Earlier studies indicate that the Johansen trace test is not robust in presence of heteroscedasticity, and tests based on resampling methods have been proposed to solve the problem. The tests that are evaluated is the Johansen trace test, nonparametric bootstrap test and two different types of wild bootstrap tests. The wild bootstrap test is a resampling method that attempts to mimic the GARCH model by multiplying each residual by a stochastic variable with an expected value of zero and unit variance. The wild bootstrap tests proved to be superior to the other tests, but not as good as earlier indicated. The more the error terms differs from white noise, the worse these tests are doing. Although the wild bootstrap tests did not do a very bad job, the focus of further investigation should be to derive tests that does an even better job than the wild bootstrap tests examined here.

Key words: Johansen trace test, wild bootstrap, cointegration, heteroscedasticity, simulation.

Table of contents

1. INTRODUCTION ...... 1 2. THEORY ...... 3 2.1 Random walk ...... 3 2.2 The vector autoregressive model ...... 3 2.3 The generalized autoregressive conditional heteroscedasticity model ...... 3 2.4 Integration and cointegration ...... 4 2.5 The vector error correction model ...... 5 2.6 The Johansen trace statistic ...... 5 2.7 Bootstrap ...... 6 2.7.1 Nonparametric bootstrap ...... 6 2.7.2 Wild bootstrap ...... 7 3 METHOD ...... 9 3.1 Data generating processes ...... 9 3.1.1 Standard random walk ...... 9 3.1.2 Random walk with a break in the variance...... 9 3.1.3 Random walk with GARCH effects ...... 9 3.1.4 Vector error correction model ...... 10 3.1.5 Vector error correction model with a break in the variance ...... 10 3.1.6 Vector error correction model with GARCH effects...... 10 3.1.7 Comparative simulations ...... 10 3.2 Monte Carlo algorithms ...... 10 3.2.1 Algorithm for evaluating the Johansens trace test ...... 11 3.2.2 Algorithm for evaluating the bootstrap tests ...... 11 3.2.3 Rescaling of the residuals ...... 11 4. RESULTS ...... 12 4.1 Size ...... 12 4.2 Power ...... 14 4.3 Comparative results ...... 16 5. DISCUSSION ...... 18 5.1 Conclusion ...... 18 6. REFERENCES ...... 19

Appendix: P-value plots ...... 20 1

1. INTRODUCTION In this paper we examine the size and power of the Johansens trace test and three bootstrap tests when testing for cointegration when the error terms in the series exhibit certain behavior. In earlier studies, this “certain behavior” has been accompanied by using a simple break in the variance as well as a generalized autoregressive conditional heteroscedasticity model (Mantalos, 2001; Cavaliere & Taylor, 2008); this kind of certain behavior of the error terms is also used in this study. To begin with, a short introduction of time series analysis along with its methods and models used through the rest of the paper will be brought into attention.

In the usual concept of regression analysis, we often attempt to explain the variation in one variable by estimating a model consisting of independent or explaining variables. Another useful tool of regression analysis is that it can provide predictions of a dependent variable. In this case, the data are collected from several entities at one point in time (Stock & Watson, 2012). In a time series though, as Stock and Watson (2012) describes it, the data consists of several measurements over time on one or several entities. Time series analysis with all of its applications is a useful asset mainly when examination of macro-data is to be carried out, such as inflation and stock prices. One main condition that has to be fulfilled in order to make good predictions of the future, is that the series has to be stationary (Stock & Watson, 2012). For a thorough introduction of time series analysis, see Greene (2008) or Tsay (2005).

There are several tests derived for testing for stationarity in a time series, such as the Dickey Fuller (DF) test, the Augmented DF test and the KPSS test (Greene, 2008). This paper will not consider univariate time series though, but multivariate ditto. In a multivariate time series setting, it is often of interest whether two or more series are cointegrated and of which rank. If two or more nonstationary series are said to be cointegrated then there exists a linear combination of them that is stationary. A few tests derived for this purpose have been suggested through the years such as the Johansens trace test, Johansens max test and the DOLS estimator (Stock & Watson, 2012; Greene, 2008).

The Johansen trace test was derived by Johansen (1991) in order to test for cointegration in multivariate time series. This test tests the null hypothesis of at most 푟 cointegration relationships in multivariate time series, against the alternative that there are more than 푟 cointegration relationships. The test statistic derived by Johansen follows a distribution that is a function of standard Brownian motions, and thus has critical values that has to be found via simulation (Tsay, 2005). The critical values used by most statistical software are simulated asymptotical critical values, which in turn may make this test sensitive to situations where small samples are used. Moreover, this test does not seem to be robust when the series are heteroscedastic; or with other words, when the error terms are not white noise (Mantalos, 2001; Cavaliere, Rahbek & Taylor, 2007).

The wild bootstrap test has been suggested by as an attempt to solve the problem with the nonrobust Johansen trace test in presence of heteroscedasticity. The first version of the wild bootstrap was suggested by Wu (1986), and refined by Liu (1988), as a suggestion to how to use bootstrapping effectively when the innovation terms in a model are heteroscedastic. They proposed a solution by forcing the residuals to follow a distribution with expectation 0 2

and constant variance. Cavaliere and Taylor (2008) studied the properties of the wild bootstrap test for cointegration, and found it to be robust to heteroscedasticity. Mantalos (2001) also studied the properties of the wild bootstrap cointegration test and got similar results as Cavaliere and Taylor. The methodology of this simulation study is very similar to the methodology used in Cavaliere and Taylor as well as in Mantalos. A key difference in this study is that different parameter values are used in the data generating processes, compared to those parameter values used in the earlier mentioned studies.

2. THEORY In the following section, models and concepts considered in this paper will be introduced.

2.1 Random walk A random walk is a series that follows a stochastic trend, and can, in its simplest form, be 푡 2 written as 푦푡 = 푦푡−1 + 푒푡 = 푦0 + ∑푖=1 푒푖, where 푒푡~푖. 푖. 푑. 푁(0, 휎 ) ∀ 푡. Further on this series will be referred to as the “standard random walk”.

2.2 The vector autoregressive model A vector autoregressive (VAR) model is a model where 푦푗,푡, 푗 = 1, 2, … , 푀, is modeled as a function of past values of 푦푗 ∀ 푗. A bivariate VAR(1)-model can be is written as

푦1,푡 = 휑1 + 훿1,1푦1,푡−1 + 훿1,2푦2,푡−1 + 푒1,푡 and

푦2,푡 = 휑2 + 훿2,1푦1,푡−1 + 훿2,2푦2,푡−1 + 푒2,푡, where 푒1,푡 and 푒2,푡 are innovation terms with covariance matrix

휎 휎 [ 11 12]. 휎21 휎22

In matrix form, according to Greene (2008), the general VAR(p)-model with 푀 variables is written as 풚푡 = 흋 + 휹1풚푡−1 + ⋯ + 휹푟풚푡−푝 + 풆푡, where 풆푡 has covariance matrix

휎11 ⋯ 휎1푀 [ ⋮ ⋱ ⋮ ]. 휎푀1 ⋯ 휎푀푀

2.3 The generalized autoregressive conditional heteroscedasticity model The generalized autoregressive conditional heteroscedasticity (GARCH) model is a model where the innovations, 푒푡, can be described by the process presented next. This form of heteroscedasticity is used in order to make this simulation study consistent with earlier work, see Cavaliere and Taylor (2008), Flachaire (2003), Davidson and Flachaire (2001), and Mantalos (2001) for example. Consider the model 푦푡 = 휇푡 + 푒푡, where 휇푡 is the mean equation which might, for example, be an AR(1) process, and where 푒푡 is the innovation term. In the univariate GARCH-model, the innovations are modeled in the following way:

1/2 푒푡 = ℎ푡 푣푡, where

푞 2 푚 ℎ푡 = 푐 + ∑푖=1 휃푖푒푡−푖 + ∑푖=1 훾푖ℎ푡−푖. 4

Thus ℎ푡 follow an ARMA(q,m)-process. The GARCH-model can also be extended to the multivariate case, which can be written as

풚푡 = 흁푡 + 풆푡, where 풚푡 is a column vector of length 푀 (number of variables); 흁푡 is the mean equation; and 1/2 1 풆푡 = 풉푡 ⨀풗푡 (where ⨀ denotes the Hardamard product) is a vector of innovation terms with covariance matrix

휎11 ⋯ 휎1푀 [ ⋮ ⋱ ⋮ ], 휎푀1 ⋯ 휎푀푀 where 풗푡 is a vector with expected value ퟎ and covariance matrix

1 ⋯ 0 [⋮ ⋱ ⋮]. 0 ⋯ 1

In turn, 풉푡 is modeled by the following process:

푞 2 푚 풉푡 = 푐푗 + ∑푖=1 휃푗,푖푒푗,푡−푖 + ∑푖=1 훾푗,푖ℎ푗,푡−푖, 푗 = 1,2, … 푀, where 푗 corresponds to the 푗:th row in the column vector 풉푡. This is known as the general multivariate GARCH(q,m) model (Greene, 2008; Tsay, 2005). A bivariate GARCH1(1,1) model can then be written as

1/2 풚푡 = 흁푡 + 풉푡 ⨀풗푡, where

2 푐1 휃1 0 푒1,푡−1 훾1 0 ℎ1,푡−1 풉푡 = [푐 ] + [ ] [ 2 ] + [ ] [ ]. 2 0 휃2 푒2,푡−1 0 훾2 ℎ2,푡−1

In this case, 풗푡 has expected value ퟎ and covariance matrix 푰 ∀ 푡 where 푰 has dimension (2,2).

2.4 Integration and cointegration A series that is integrated is a series that is stationary in the mean at order d, denoted I(d). For example, the random walk described earlier is an I(1)-process since its first difference is stationary, which can be written as

∆푦푡 = 푦푡 − 푦푡−1 = 푒푡.

1 Hardamard multiplication is carried out by performing an element-by-element multiplication. For example, consider the following equation with two vectors of dimension (2,1): 푨⨀푩 = (푎1푏1, 푎2푏2). 5

In the multivariate case, the series may each be an integrated process of some order. There are other possibilities as well, some series may, for example, be integrated of order d and some of order d-1. In this paper we will only consider processes where each series in the multivariate case are integrated of order one or not integrated at all. If each or some of the series in a multivariate time series is integrated of order d, then some or all series may follow a common stochastic trend.

2.5 The vector error correction model The vector error correction model is abbreviated VECM. Consider the VAR(p) model described earlier; Engel and Granger (1987) showed that any VAR model can be written as

푝−1 ∗ ∆풚푡 = 풚푡 − 풚푡−1 = 흋 + 흅풚푡−1 + ∑ 휹푖 ∆풚푡−푖 + 풆푡 푖=1

∗ 푝 where 풆푡 is defined as before and 흅 = −(푰 − 휹1 − 휹2 − ⋯ − 휹푝) and 휹푗 = − ∑푖=푗+1 휹푖. Further, 흅 can be written as 휶휷′ where 휷 contains the cointegration vectors and 휶 contains the speed of adjustment parameters (Tsay, 2005). In order to describe the cointegration vectors we can consider two cointegrated series, then the following series (where both vectors has dimension (M,1))

′ 휷 풚푡 is stationary in the mean. The speed of adjustment parameters determines the speed at which the series are returning to the common stochastic trend after a shock. In the bivariate case, by assuming no autocorrelation, this representation is written as

∆풚푡 = 흋 + 흅풚푡−1 + 풆푡, which is what is going to be considered in this paper.

2.6 The Johansen trace statistic To carry out the Johansen trace test, consider the following bivariate VAR(1) model

풚푡 = 흋 + 휹1풚푡−1 + 풆푡.

Now, by estimating two models using ordinary least squares (OLS) we can acquire four canonical correlations between the columns in 푫 and 푬; where 푫 and 푬 are two matrices of dimensions (푇, 2), where 푇 is the number of observations, containing the estimated residuals after estimation of the parameters in the following models, respectively

∆풚푡 = 퐛0 + 퐛1∆풚푡−1 + 풂1 and

풚푡−1 = 퐛2 + 퐛3∆풚푡−1 + 풂2.

According to Johnson and Wichern (2007), the canonical correlations are then obtained by computing the ordered characteristic roots, or eigenvalues, of the matrix

−1/2 −1 −1/2 푹퐷퐷 푹퐷퐸푹퐸퐸푹퐸퐷푹퐷퐷 , where “푹푖,푗 is the (cross-) correlation matrix between variables in set 푖 and 푗, for 푖, 푗 = 퐷, 퐸” (Greene 2008, p. 763). The eigenvalues are obtained by solving the equation (for lambda)

−1/2 −1 −1/2 |푹퐷퐷 푹퐷퐸푹퐸퐸푹퐸퐷푹퐷퐷 − 휆푰| = 0.

Canonical correlation can be thought of as the strength of association between two sets of variables (Johnson & Wichern, 2007). The Johansen trace statistic can then be attained by computing

푀 ∗ 2 푇푟푎푐푒 푠푡푎푡푖푠푡푖푐 = −푇 ∑푖=푟+1 ln [1 − (푟푖 ) ], where the null hypothesis is that there are at most 푟 cointegrating vectors and the alternative ∗ hypothesis is the complement of the null hypothesis. The term 푟푖 is the i:th canonical correlation. "Large" values of the trace statistic indicates that the null hypothesis is false, for a given alfa level. For critical values of this test, see MacKinnon, Haug and Michelis (1998). For further reading on canonical correlation, see chapter ten in Johnson and Wichern (2007).

2.7 Bootstrap Bootstrap techniques are referred to as resampling methods (Casella & Berger, 2002). The general idea of bootstrap techniques is to obtain an estimate of the sampling distribution of an estimator. Efron (1979) first proposed the nonparametric bootstrap in his paper “Bootstrap Methods: Another look at the Jackknife”. For a discussion of why and when bootstrapping works, see Janssen and Pauls (2003); also see Cavaliere and Taylor (2008), Flachaire (2003), Davidson and Flachaire (2001), and Mantalos (2001) for a thorough review on how different bootstrapping methods works in situations partly similar to those discussed here. The general nonparametric bootstrap has been proven to work well under the condition of identically and independent distributed error terms, which implies homoscedasticity and stationarity in the volatility (Wu, 1986). In presence of nonstationary volatility though, Wu proposed the wild bootstrap for dealing with the problem of heteroscedasticity. How the bootstrapped residuals are used in this simulation study is described in 3.2.2.

2.7.1 Nonparametric bootstrap As an introductory example of the use of nonparametric bootstrap: Suppose that we wish to 2 푛 estimate the distribution of the usual arithmetic sample mean, 푥̅ = ∑푖=1 푥푖⁄푛, where the sample mean serves as an estimate of the population mean, 휇; also, the sample has been obtained by a simple random sampling procedure. By resampling 풙 = (푥1, 푥2, … , 푥푛) and

2 In “large” samples there is really no need to use bootstrap in order to estimate the sampling distribution of the sample mean since, according to the central limit theorem, the distribution of 푥̅ is normally distributed with 퐸[푥̅] = 휇 and 푉[푥̅] = 휎2 1 , where 휎2 can be estimated by 푠2 = ∑푛 (푥 − 푥̅)2 (Casella & Berger, 2002; Wackerly, Mendenhall & Scheaffer, 푛 푛−1 푖=1 푖 2002). 7

∗ ∗ ∗ ∗ ∗ acquire the bootstrap sample 풙 = (푥1, 푥2, … , 푥푛), we can estimate 푥̅ . The resample procedure is carried out by randomly draw 푛 observations from 풙 with replacement, where each observation 푖, 푖 = 1, 2, … , 푛, has inclusion probability 푛−1. By repeating this procedure a number of times we can obtain an estimate of the distribution of the sample mean. This method is useful when it is difficult or impossible to obtain an estimate of the distribution of an estimator analytically.

2.7.2 Wild bootstrap The wild bootstrap was first suggested by Wu (1986), and refined by Liu (1988), as a suggestion to how to use bootstrapping effectively when the innovation terms in a model are heteroscedastic. He proposed a solution by forcing the residuals to follow a distribution with expectation 0 and constant variance. To carry out the wild bootstrap, first an estimation of the error terms is made by estimating a model and then calculate the residuals, in the following way

푒̂푖 = 푦푖 − 푦̂푖.

At the second step the error terms are forced to follow a distribution with properties as previously described. This is made by the step considered next

∗ ∗ 푒̂푖 = 푒̂푖푣푖 ,

∗ ∗ ∗ where 푣푖 is a random variable with 퐸[푣푖 ] = 0 and 푉[푣푖 ] = 1 ∀ 푖. Another assumption often stated about 푣∗ is that it should have a third moment equal to unity as well (Davidson & Flachaire, 2001). In order to accompany such properties, a number of suggestions have been made through the years where the most common is a distribution proposed by Mammen (1993). The desired properties can be obtained by using a two point distribution and solving the following system of equations3

∗ ∗ 푝푣1 + (1 − 푝)푣2 = 0

∗2 ∗2 푝푣1 + (1 − 푝)푣2 = 1

∗3 ∗3 푝푣1 + (1 − 푝)푣2 = 1.

Mammen (1993) ended up with the following distribution which has the desired properties

∗ −(√5 − 1)/2 푤푖푡ℎ 푝푟표푏푎푏푖푙푖푡푦 (√5 + 1)/(2√5) 푣푖 = { . (√5 + 1)/2 푤푖푡ℎ 푝푟표푏푎푏푖푙푖푡푦 (√5 − 1)/(2√5)

Further on, this wild bootstrap method will be called “Mammen wild bootstrap”. Even though this distribution obtains a desirable third moment property, Davidson and Flachaire (2001) points out that the wild bootstrap with the Rademacher distribution gives smaller

3 The first equation is an expression for the expected value; the second equation is the second moment and in this case it is also the variance since the first moment is equal to zero; and the third equation is an expression for the third moment. 8

error rejection probabilities; the Mammen wild bootstrap seem better, compared to the Rademacher wild bootstrap, when the innovation terms have a nonsymmetrical distribution though. The Rademacher distribution is very similar to the Bernoulli distribution, where the only difference is that the variable takes the value minus one instead of zero, with probability 1/2. Results from both of these types of wild bootstrap are presented in this paper. 9

3 METHOD In order to examine the properties, i.e. size and power, of the Johansen trace test and the bootstrap tests, simulations will be carried out in STATA 11.2, a software for computational and statistical analysis. The results will be displayed with tables and p-value plots with either 120 or 480 observations, and with nominal significance level set to either five or one percent.

3.1 Data generating processes The number of observations used in the samples are set to 480 and 120, which corresponds to 40 and 10 years of monthly data, respectively. The processes from where the series are to be simulated from are described next.

3.1.1 Standard random walk Bivariate standard random walk series are generated in order to examine the size properties of each test. The following processes describes how this kind of series are generated

푡 푦1,푡 = 푦1,푡−1 + 푒1,푡 = 푦1,0 + ∑푖=1 푒1,푖, (I)

푡 푦2,푡 = 푦2,푡−1 + 푒2,푡 = 푦2,0 + ∑푖=1 푒2,푖, (II) where the innovation terms are independent of each other. Further, 퐸[푒푗,푡] = 0 ∀ 푡 and 푉[푒푗,푡] = 1 ∀ 푡 for 푗 = 1, 2. Without loss of generality, 푦1,0 and 푦2,0 are both set to zero.

3.1.2 Random walk with a break in the variance Two different types of breaks are going to be considered when simulating time series, where both of them have a mean equation equal to (I) and (II). The first to be considered is when the standard deviation of the error term is one before time 푇/2, and two from time 푇/2,

휏 = 휎퐼(푇≥푇/2)⁄휎퐼(푇<푇/2) = 2. (III)

The second “break in the variance”-series has innovation terms with standard deviation one before time 푇/2 and four from time 푇/2; also written as

휏 = 휎퐼(푇≥푇/2)⁄휎퐼(푇<푇/2) = 4. (IV)

3.1.3 Random walk with GARCH effects These series has a mean equation equal to the random walk, and the error terms are modeled as GARCH(1,1). Three different series with error terms that follows a GARCH process are going to be generated, in accordance with Mantalos (2001); one with high persistence, one with medium and one with low. The parameter values in the high persistence GARCH model are

0.001 0.199 0 0.8 0 풄 = [ ], 휽 = [ ], 휸 = [ ]. (V) 0.001 0 0.199 0 0.8

The parameter values in the medium persistence GARCH model are 10

0.05 0.05 0 0.9 0 풄 = [ ], 휽 = [ ], 휸 = [ ]. (VI) 0.05 0 0.05 0 0.9

And the parameter values in the low persistence GARCH model are

0.2 0.05 0 0.75 0 풄 = [ ], 휽 = [ ], 휸 = [ ]. (VII) 0.2 0 0.05 0 0.75

Since the parameter values are the same for both series in the bivariate model, 풄, for example, will further on only be referred to as 푐.

3.1.4 Vector error correction model The series generated by this process has parameter values

0 −0.01 1 흋 = [ ], 휶 = [ ], 휷 = [ ]. (VIII) 0 −0.01 −1

3.1.5 Vector error correction model with a break in the variance This series is almost equal to the one described above, with the only difference that the error terms both has 휏 times as large standard deviation from time 푇/2.

3.1.6 Vector error correction model with GARCH effects The series generated by this process has a mean equation equal to the one above, and the parameters in the GARCH model is equal to (V), (VI) and (VII), respectively.

3.1.7 Comparative simulations Two simulations where the series has parameter values equal to those used in Mantalos (2001) as well as in Cavaliere and Taylor (2008) are going to be carried out. The series in Mantalos is a high persistence GARCH series with 푐 = 0.01, 휃 = 0.09 푎푛푑 훾 = 0.9; and the DGP in Cavaliere and Taylor is a break in the variance-series with break set at 0.1푇 with 휏 = 5. The sample size used in these two simulations are set to 480 and the test used is the Rademacher wild bootstrap.

3.2 Monte Carlo algorithms When estimating the size of the tests, 푁 is set to 10000; which give size estimates that is expected to be between 0.0457 and 0.0543 95 percent of the times, given that size is equal to 0.05. The following data generating process is used to evaluate the size of the tests; that is, under the null hypothesis of no cointegration:

풚푡 = 풚푡−1 + 풆푡.

For power estimation, the data generating process described next is used

′ ∆풚푡 = 흋 + 휶휷 풚푡−1 + 풆푡.

Be aware of that the error terms in the two last equations either follow (I) and (II), (III) or (IV), or a GARCH process described by (V), (VI) or (VII). Two slightly different algorithms 11

are used for estimating the actual size and power of the Johansens trace test and the bootstrap tests. Firstly, the scheme for the Johansens trace test evaluation is explained.

3.2.1 Algorithm for evaluating the Johansens trace test (1) Simulate the time series (2) Estimate a VECM (3) Calculate the trace statistic (4) Repeat (1)-(3) 푁 times (5) Observe the percentage of times the null hypothesis is rejected

The estimated actual size or power of the test is then estimated by the procedure in (5).

3.2.2 Algorithm for evaluating the bootstrap tests Below, “bootstrap sample” is to be viewed as either nonparametric bootstrap or wild bootstrap.

(1) Simulate the time series (2) Estimate a VECM (3) Calculate the trace statistic (4) Estimate residuals (5) Rescale residuals (6a) Draw a bootstrap sample (6b) Generate two new series under the null hypothesis with the bootstrapped residuals (6c) Calculate the trace* statistic (6d) Repeat (6a)-(6c) 199 times (7) Observe percentage of times trace* is less than trace (8) Repeat (1)-(7) 푁 times (9) Observe the percentage of times the null hypothesis is rejected

The actual size or power of the tests is then estimated by the procedure in (9). In (6c), trace* is the trace statistic calculated for the bootstrap series. The step described in (7) can be viewed as the bootstrap tests p-value. For estimation of the size, 푁 = 10000; and for estimation of the power, 푁 = 1000.

3.2.3 Rescaling of the residuals It is common to rescale the estimated residuals to force them to have zero mean. The rescaling scheme used here is carried out by simply demeaning the residuals; that is, subtract the residual arithmetic average from every estimated residual. 12

4. RESULTS Results will be displayed in tables and with p-value plots. The null hypothesis is stated as: there are no cointegration present between the two series.

4.1 Size Sizes of the tests are displayed in Table 1 and 2 for time series with 푇 = 480 and 푇 = 120 observations, respectively. As indicated by the results displayed in the two tables below, the Johansens trace test and nonparametric bootstrap test is very unreliable when time series volatility exhibits a “large” break (휏 = 4) at time 푇/2, whereas the wild bootstrap tests are fairly robust. The size of the wild bootstrap tests are estimated to be below the nominal level of 0.05, both in series with 120 and 480 observations (when there is a “large” break in the volatility). For p-value plots, see section one in the Appendix.

The Mammen and Rademacher wild bootstrap seem to be doing equally well. These tests do not seem to be as robust as the results in Mantalos (2001) and Cavaliere and Taylor (2008) suggests though. They tend to reject the null hypothesis too seldom in face of a break in volatility, and too often in face of high persistent GARCH effects. The size of these tests is between 0.030 and 0.078, so they are at least more robust than the Johansens trace test and the nonparametric bootstrap test.

Table 1: Size of the tests when 푇 = 480 with a nominal significance level of 0.05 Volatility model Johansens trace Nonparametric Rademacher wild Mammen wild test bootstrap bootstrap bootstrap

휏 = 1 .049 .054 .052 .052

휏 = 2 .074 .083 .040 .039

휏 = 4 .174 .182 .037 .040

(푉) .121 .123 .073 .076

(푉퐼) .049 .058 .053 .052

(푉퐼퐼) .048 .055 .049 .051

Table 2: Size of the tests when 푇 = 120 with a nominal significance level of 0.05 Volatility model Johansens trace Nonparametric Rademacher wild Mammen wild test bootstrap bootstrap bootstrap

휏 = 1 .052 .051 .051 .052

휏 = 2 .064 .068 .035 .033

휏 = 4 .135 .140 .030 .031

(푉) .097 .105 .078 .078

(푉퐼) .056 .058 .047 .053

(푉퐼퐼) .047 .052 .050 .052

In the following two tables, results are presented with nominal alfa level set to one percent. We can see that the pattern is similar to the results previously displayed. The size of the wild bootstrap tests are between .006 and .026, and again more robust than both the nonparametric bootstrap test and the Johansens trace test.

Table 3: Size of the tests when 푇 = 480 with a nominal significance level of 0.01 Volatility model Johansens trace Nonparametric Rademacher wild Mammen wild test bootstrap bootstrap bootstrap

휏 = 1 .011 .009 .010 .011

휏 = 2 .023 .023 .009 .008

휏 = 4 .065 .062 .007 .008

(푉) .055 .054 .022 .025

(푉퐼) .012 .012 .012 .010

(푉퐼퐼) .011 .010 .011 .011

Table 4: Size of the tests when 푇 = 120 with a nominal significance level of 0.01 Volatility model Johansens trace Nonparametric Rademacher wild Mammen wild test bootstrap bootstrap bootstrap

휏 = 1 .011 .009 .011 .011

휏 = 2 .017 .019 .007 .006

휏 = 4 .051 .049 .007 .006

(푉) .040 .039 .024 .026

(푉퐼) .013 .013 .010 .012

(푉퐼퐼) .012 .012 .011 .010

4.2 Power For parameter values of the VECM, see (VIII). The power of the wild bootstrap tests does not seem to be lower than for the Johansens trace test when the nominal alfa level is attained. The power of the Mammen wild bootstrap test is greater than the power for the Rademacher wild bootstrap test in eleven out of twelve cases when a significance level of 5 percent is used. The power of the tests when 푇 = 120 is very close to the nominal alfa level because of the somewhat low number of observations.

Table 5: Power of the tests when 푇 = 480 with a nominal significance level of 0.05 Volatility model Johansens trace Nonparametric Rademacher wild Mammen wild test bootstrap bootstrap bootstrap

휏 = 1 .186 .201 .201 .224

휏 = 2 .254 .294 .134 .136

휏 = 4 .425 .422 .104 .117

(푉) .480 .480 .370 .397

(푉퐼) .200 .245 .212 .224

(푉퐼퐼) .207 .074 .198 .229

Table 6: Power of the tests when 푇 = 120 with a nominal significance level of 0.05 Volatility model Johansens trace Nonparametric Rademacher wild Mammen wild test bootstrap bootstrap bootstrap

휏 = 1 .054 .070 .056 .057

휏 = 2 .069 .089 .031 .049

휏 = 4 .183 .185 .049 .047

(푉) .249 .237 .201 .233

(푉퐼) .067 .061 .057 .065

(푉퐼퐼) .057 .055 .056 .066

The following two tables yields a similar pattern as the two previous ones. And in this case, the Mammen wild bootstrap test has power estimates greater than the Rademacher wild bootstrap test eight out of twelve times.

Table 7: Power of the tests when 푇 = 480 with a nominal significance level of 0.01 Volatility model Johansens trace Nonparametric Rademacher wild Mammen wild test bootstrap bootstrap bootstrap

휏 = 1 .051 .046 .053 .052

휏 = 2 .102 .099 .036 .038

휏 = 4 .190 .197 .031 .025

(푉) .300 .293 .198 .195

(푉퐼) .052 .074 .047 .054

(푉퐼퐼) .069 .048 .058 .061

Table 8: Power of the tests when 푇 = 120 with a nominal significance level of 0.01 Volatility model Johansens trace Nonparametric Rademacher Mammen Wild test bootstrap Wild bootstrap bootstrap

휏 = 1 .010 .011 .007 .008

휏 = 2 .014 .020 .005 .012

휏 = 4 .062 .063 .009 .015

(푉) .148 .149 .116 .113

(푉퐼) .010 .013 .007 .013

(푉퐼퐼) .012 .013 .008 .012

4.3 Comparative results In this section size of the Rademacher wild bootstrap is presented for series with parameter values used in two other studies, see section 3.1.7 for more information. The results in this section is presented with p-value plots. As indicated by the figure below, the results attained here is very similar to those attained in Mantalos (2001).

Figure 1: Replicated p-value plot from Mantalos (2001) Rademacher wild bootstrap test with high persistence GARCH effects; the solid line is the estimated actual size while the dashed line is the nominal size

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We can also see in the next figure that the results are similar to those in Cavaliere and Taylor (2008) as well. 17

Figure 2: Replicated p-value plot from Cavaliere and Taylor (2008) Rademacher wild bootstrap test with break at 0.1푇 and 휏 = 5; the solid line is the estimated actual size while the dashed line is the nominal size

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5. DISCUSSION Despite the very impressive results in Mantalos (2001) and Cavaliere and Taylor (2008), this simulation study does not seem to give results that are as positive as the results found in these studies. Mantalos found the size to be very close to the nominal level when the volatility exhibited high persistence GARCH effects, which was not the case in this study. As indicated, the Rademacher wild bootstrap test does a very good job with the parameter values used in Mantalos (2001), while doing worse with parameter values in the high persistence volatility model used in this study. See figure 1 in section 4.3 and figure 11 and 12 in the appendix for a comparison. A replication of the results in Cavaliere and Taylor (2008) were also made. They investigated the size of the Rademacher wild bootstrap test when the break were set at time 0.1푇, with 휏 = 5. Also they found the wild bootstrap test to be very robust to heteroscedastic volatility, which was not the case in this study where other parameter values were used. A “phenomenon” that needs further investigation is whether the results found in this study, compared to those found in Mantalos as well as in Cavaliere and Taylor, is due to different sample sizes.

A drawback of this Monte Carlo simulation study is that only VAR of order 1 has been used as data generating process for estimation of power. Also, in a real situation we do not know the order of the VAR model, which in such a situation has to be chosen before the cointegration testing procedure. The VAR model can, for instance, be chosen by an information criterion such as Akaikes or Schwarz information criterion.

In further studies of this kind, a method developed by MacKinnon and Davidson (2001) for choosing appropriately many bootstrap samples, could be used. They argue that such a method is more advantageous than to choose the number of bootstrap samples somewhat arbitrarily. Another suggestion, one made by Flachaire (2003), to take into consideration is how to rescale the residuals. He suggests a different rescaling scheme than the one used in this study.

5.1 Conclusion The wild bootstrap tests seem to be somewhat robust in various situations of heteroscedastic series, as also indicated in previous studies. The results in this simulation study do not suggest that the wild bootstrap is quite as robust as earlier indicated though. The Mammen wild bootstrap test seem to have higher power than the Rademacher wild bootstrap test. Further investigation and proposal of tests that are even more robust is of great importance. 19

6. REFERENCES Casella, G., & Berger, R. (2002). Statistical Inference: Second Edition. Duxbury Press. Cavaliere, G., Rahbek, A., & Taylor, R. (2007). Testing for co-integration in vector autoregressions with non-stationary volatility. Granger Centre Discussion Paper, 07/02. Cavaliere, G., & Taylor, R. (2008). Bootstrap Unit Root Tests for Time Series with Nonstationary Volatility. Econometric Theory, 24, 43-71. Davidson, R., & Flachaire, E. (2001). The Wild Bootstrap, Tamed at Last. Department of Economics: Queen’s University. Efron, B. (1979). Bootstrap Methods: Another look at the Jackknife. The Annals of Statistics, 7, 1-26. Engle, R., & Granger, C. (1987). Cointegration and Error Correction: Representation, Estimation, and Testing. Econometrica, 35, 251-276. Flachaire, E. (2003). Bootstrapping heteroscedastic regression models: wild bootstrap vs. pairs bootstrap. Eurequa: Universit’e Paris 1 Panth’eon-Sorbonne. Greene, W. (2008). Econometric analysis: Fourth edition. Prentice-Hall. Janssen, A., & Pauls, T. (2003). How do Bootstrap and Permutation tests work? The Annals of Statistics, 31, 768-806. Johansen, S. (1991). Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models. Econometrica, 59, 1551-1580. Johnson, R., & Wichern, D. (2007). Applied Multivariate Statistical Analysis: Sixth Edition. Pearson Education, Inc. Liu, R, Y. (1988). Bootstrap procedure under some non-i.i.d. models. Annals of Statistics 16, 1696–1708. MacKinnon, J., & Davidson, R. (2001). Bootstrap Tests: How Many Bootstraps? Working Paper. Department of Economics: Queen’s University. MacKinnon, J., Haug, A., & Michelis, L. (1998). Numerical Distribution Functions of Likelihood Ratio Tests for Cointegration. Economics Publications and Research, Paper 7. Mammen, E. (1993). Bootstrap and Wild Bootstrap for High Dimensional Linear Models. The Annals of Statistics, 21, 255-285. Mantalos, P. (2001). ECM-Cointegration test with GARCH(1,1) Errors. Department of Science and Health: Blekinge Institute of Technology. Mantalos, P., & Shukur, G. (1999). Testing for cointegration relations – A bootstrap approach. Department of Statistics: Göteborg University. Mantalos, P., & Shukur, G. (2001). Bootstrapped johansen tests for cointegration relationships: a graphical analysis. Journal of Statistical Computation and Simulation, 68, 351-371. Stock, J., & Watson, M. (2012). Introduction to Econometrics: Third edition. Pearson Education Limited. Tsay, R. (2005). Analysis of Financial Time Series: Second Edition. John Wiley & Sons, Inc. Wackerly, D., Mendenhall, W., & Scheaffer, R. (2002). Mathematical Statistics with Applications: Sixth edition. Duxbury Press. Wu, C, F, J. (1986). Jackknife, Bootstrap and Other Resampling Methods in Regression Analysis. The Annals of Statistics, 14, 1261-1295. 20

Appendix: P-value plots Only p-value plots for series with 480 observations are displayed in this appendix. Plots when 푇 = 120 are available as well and can be shown given on request. For each figure, the solid line is the estimated actual size while the dashed line is the nominal size.

1. Size of the tests Figure 1: Nonparametric bootstrap, 휏 = 1

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Figure 2: Rademacher wild bootstrap, τ = 1

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Figure 3: Mammen wild bootstrap, 휏 = 1

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Figure 4: Nonparametric bootstrap, 휏 = 2

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Figure 5: Rademacher wild bootstrap, 휏 = 2

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Figure 6: Mammen wild bootstrap, 휏 = 2

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Figure 7: Nonparametric bootstrap, 휏 = 4

Actual size Actual

0 .05 .1 .15 .2 Nominal size

Figure 8: Rademacher wild bootstrap, 휏 = 4

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Figure 9: Mammen wild bootstrap, 휏 = 4

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Figure 10: Nonparametric bootstrap, 푐 = 0.001, 휃 = 0.199, 훾 = 0.8

Actual size Actual

0 .05 .1 .15 .2 Nominal size

Figure 11: Rademacher wild bootstrap, 푐 = 0.001, 휃 = 0.199, 훾 = 0.8

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Figure 12: Mammen wild bootstrap, 푐 = 0.001, 휃 = 0.199, 훾 = 0.8

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Figure 13: Nonparametric bootstrap, 푐 = 0.05, 휃 = 0.05, 훾 = 0.9

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Figure 14: Rademacher wild bootstrap, 푐 = 0.05, 휃 = 0.05, 훾 = 0.9

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Figure 15: Mammen wild bootstrap, 푐 = 0.05, 휃 = 0.05, 훾 = 0.9

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Figure 16: Nonparametric bootstrap, 푐 = 0.2, 휃 = 0.05, 훾 = 0.75

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Figure 17: Rademacher wild bootstrap, 푐 = 0.2, 휃 = 0.05, 훾 = 0.75

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Figure 18: Mammen wild bootstrap, 푐 = 0.2, 휃 = 0.05, 훾 = 0.75

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2. Power of the tests Notice here that p-value plots are only displayed for tests that attain the nominal alfa level. In each figure displayed in this section, the solid line is the estimated power while the dashed line is the nominal alfa level.

Figure 19: Nonparametric bootstrap, 휏 = 1

Power

0 .2 .4 .6 .8 1 Alfa

Figure 20: Rademacher wild bootstrap, 휏 = 1

Power

0 .2 .4 .6 .8 1 Alfa

Figure 21: Mammen wild bootstrap, 휏 = 1

Power

0 .2 .4 .6 .8 1 Alfa

Figure 22: Rademacher wild bootstrap, 푐 = 0.05, 휃 = 0.05, 훾 = 0.9

Power

0 .2 .4 .6 .8 1 Alfa

Figure 23: Mammen wild bootstrap, 푐 = 0.05, 휃 = 0.05, 훾 = 0.9

Power

0 .2 .4 .6 .8 1 Alfa

Figure 24: Nonparametric bootstrap, 푐 = 0.2, 휃 = 0.05, 훾 = 0.75

Power

0 .2 .4 .6 .8 1 Alfa

Figure 25: Rademacher wild bootstrap, 푐 = 0.2, 휃 = 0.05, 훾 = 0.75

Power

0 .2 .4 .6 .8 1 Alfa

Figure 26: Mammen wild bootstrap, 푐 = 0.2, 휃 = 0.05, 훾 = 0.75

Power

0 .2 .4 .6 .8 1 Alfa