# Transfer Entropy for Nonparametric Granger Causality Detection: An Evaluation of Different Resampling Methods

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Transfer Entropy and Its Estimator

**Theorem**

**1.**

**Proof.**

#### 2.2. Density Estimation and Bandwidth Selection

**Theorem**

**2.**

**Theorem**

**3.**

**Proof.**

#### 2.3. Resampling Methods

- Time-Shifted Surrogates
- (TS.a) The first resampling method only deals with the driving variable X. Suppose we have observations {${x}_{1},\dots ,{x}_{n}$}, the time-shifted surrogates are generated by cyclically time-shifting the components of the time series. Specifically, an integer d is randomly generated within the interval $([0.05n],\phantom{\rule{3.33333pt}{0ex}}[0.95n])$, and then the first d values of {${x}_{1},\dots ,{x}_{n}$} would be moved to the end of the series, to deliver the surrogate sample ${X}^{*}=\{{x}_{d+1},\dots ,{x}_{n},{x}_{1},\dots ,{x}_{d}\}$. Compared with the traditional surrogates based on phase randomization of the Fourier transform, the time-shifted surrogates can preserve the whole statistical structure in X. The couplings between X and Y are destroyed, although the null hypothesis of X not causing Y is imposed.
- (TS.b) The second scheme resamples both the driving variable X and the response variable Y separately. Similar to (TS.a), ${Y}^{*}=\{{y}_{c+1},\dots ,{y}_{n},{y}_{1},\dots ,{y}_{c}\}$ is created given another random integer c from the range $([0.05n],\phantom{\rule{3.33333pt}{0ex}}[0.95n])$. In contrast with the standard time-shifted surrogates described in (TS.a), in this setting we add more noise to the coupling between X and Y.

- Smoothed Local BootstrapThe smoothed bootstrap selects samples from a smoothed distribution instead of drawing observations from the empirical distribution directly. See [42] for a discussion of the smoothed bootstrap procedure. Based on rather mild assumptions, Neumann and Paparoditis [43] show that there is no need to reproduce the whole dependence structure of the stochastic process to get an asymptotically correct nonparametric dependence estimator. Hence a smoothed bootstrap from the estimated conditional density is able to deliver a consistent statistic. Specifically, we consider two versions of the smoothed bootstrap that are different in dependence structure to some extent.
- (SMB.a) In the first setting, ${Y}^{*}$ is resampled without replacement from the smoothed local bootstrap. Given the sample $Y=\{{y}_{1},\dots {y}_{n}\}$, the bootstrap sample is generated by adding a smoothing noise term ${\epsilon}_{i}^{Y}$ such that ${\tilde{y}}_{i}^{*}={y}_{i}^{*}+{h}_{b}{\epsilon}_{i}^{Y}$, where ${h}_{b}>0$ is the bandwidth used in bootstrap procedure, ${\epsilon}_{i}^{Y}$ represents a sequence of i.i.d. $N(0,1)$ random variables. Without random replacement from the original time series, this procedure does not disturb the original dynamics of $Y=\{{y}_{1},\dots {y}_{n}\}$ at all. After ${Y}^{*}$ is resampled, both ${X}^{*}$ and ${Z}^{*}$ are drawn from the smoothed conditional densities $f(x|{Y}^{*})$ and $f(z|{Y}^{*})$ as described in [44].
- (SMB.b) Secondly, we implement the smoothed local bootstrap as in [7]. The only difference between this setting and (SMB.a) is that the bootstrap sample ${Y}^{*}$ is drawn with replacement from the smoothed kernel density.

- Stationary BootstrapPolitis and Romano [38] propose the stationary bootstrap to maintain serial dependence within the bootstrap time series. This method replicates the time dependence of original data by resampling blocks of the data with randomly varying block length. The lengths of the bootstrap blocks follows a geometric distribution. Given a fixed probability p, the length ${L}_{i}$ of block i is decided as $P({L}_{i}=k)={(1-p)}^{k-1}p$ for $k=1,2,\dots $, and the starting points of block i are randomly and uniformly drawn from the original n observations. To restore the dependence structure exactly under the null, we combine the stationary bootstrap with the smoothed local bootstrap for our simulations.
- (STB) In short, firstly ${y}_{1}^{*}$ is picked randomly from the original n observations of $Y=\{{y}_{1},\dots {y}_{n}\}$, denoted as ${y}_{1}^{*}={y}_{s}$ where $s\in [1,n]$. With probability p, ${y}_{2}^{*}$ is picked at random from the data set; and with probability $1-p$, ${y}_{2}^{*}={y}_{s+1}$, so that ${y}_{2}^{*}$ would be the next observation to ${y}_{s}$ in original series $Y=\{{y}_{1},\dots {y}_{n}\}$. Proceeding in this way, $\{{y}_{1}^{*},\dots ,{y}_{n}^{*}\}$ can be generated. If ${y}_{i}^{*}={y}_{s}$ and $s=n$, the “circular boundary condition” would kick in, so that ${y}_{i+1}^{*}={y}_{1}$. After ${Y}^{*}=\{{y}_{1}^{*},\dots ,{y}_{n}^{*}\}$ is generated, both ${X}^{*}$ and ${Z}^{*}$ are randomly drawn from the smoothed conditional densities $f(x|{Y}^{*})$ and $f(z|{Y}^{*})$ as in (SMB.b).

## 3. Simulation Study

- Linear vector autoregressive process (VAR).$$\begin{array}{ccc}\hfill {X}_{t}& =a{Y}_{t-1}+{\epsilon}_{x,t},\phantom{\rule{4pt}{0ex}}& \phantom{\rule{1.em}{0ex}}{\epsilon}_{x,t}\sim N(0,1)\hfill \\ \hfill {Y}_{t}& =a{Y}_{t-1}+{\epsilon}_{y,t},\phantom{\rule{4pt}{0ex}}& \phantom{\rule{1.em}{0ex}}{\epsilon}_{y,t}\sim N(0,1).\hfill \end{array}$$
- Nonlinear VAR. This process is considered in [47] to show the failure of linear Granger causality test.$$\begin{array}{ccc}\hfill {X}_{t}& =a{X}_{t-1}{Y}_{t-1}+{\epsilon}_{x,t},\phantom{\rule{4pt}{0ex}}\hfill & \phantom{\rule{1.em}{0ex}}{\epsilon}_{x,t}\sim N(0,1)\hfill \\ \hfill {Y}_{t}& =0.6{Y}_{t-1}+{\epsilon}_{y,t},\phantom{\rule{4pt}{0ex}}\hfill & \phantom{\rule{1.em}{0ex}}{\epsilon}_{y,t}\sim N(0,1).\hfill \end{array}$$
- Bivariate ARCH process.$$\begin{array}{cc}\hfill {X}_{t}& \sim N(0,1+a{Y}_{t-1}^{2})\hfill \\ \hfill {Y}_{t}& \sim N(0,1+a{Y}_{t-1}^{2}).\hfill \end{array}$$
- Bilinear process considered in [48].$$\begin{array}{ccc}\hfill {X}_{t}& =& 0.3{X}_{t-1}+a{Y}_{t-1}{\epsilon}_{y,t-1}+{\epsilon}_{x,t},\phantom{\rule{4pt}{0ex}}\phantom{\rule{1.em}{0ex}}{\epsilon}_{x,t}\sim N(0,1)\hfill \\ \hfill {Y}_{t}& =& 0.4{Y}_{t-1}+{\epsilon}_{y,t},\phantom{\rule{4pt}{0ex}}\phantom{\rule{1.em}{0ex}}{\epsilon}_{y,t}\sim N(0,1).\hfill \end{array}$$
- Bivariate AR(2)-GARCH process.$$\begin{array}{ccc}\hfill {X}_{t}& =& 0.5{X}_{t-1}-0.2{X}_{t-2}+{\epsilon}_{x,t},\hfill \\ \hfill {\epsilon}_{x,t}& =& \sqrt{{h}_{x,t}}{\upsilon}_{x,t},\phantom{\rule{1.em}{0ex}}{\upsilon}_{x,t}\sim t(5),\hfill \\ \hfill {h}_{x,t}& =& 0.9+a{\epsilon}_{y,t-1}^{2}+0.2{h}_{x,t-1};\phantom{\rule{5.69046pt}{0ex}}\hfill \\ \hfill {Y}_{t}& =& 0.3{Y}_{t-1}+0.2{Y}_{t-2}+{\epsilon}_{y,t},\hfill \\ \hfill {\epsilon}_{y,t}& =& \sqrt{{h}_{y,t}}{\upsilon}_{y,t},\phantom{\rule{1.em}{0ex}}{\upsilon}_{y,t}\sim t(5),\hfill \\ \hfill {h}_{y,t}& =& 0.3+0.1{\epsilon}_{y,t-1}^{2}+0.2{h}_{y,t-1}.\hfill \end{array}$$
- Bivariate ARMA-GARCH process.$$\begin{array}{ccc}\hfill {X}_{t}& =& 0.3{X}_{t-1}+0.3{\epsilon}_{x,t-1}+{\epsilon}_{x,t},\hfill \\ \hfill {\epsilon}_{x,t}& =& \sqrt{{h}_{x,t}}{\upsilon}_{x,t},\phantom{\rule{1.em}{0ex}}{\upsilon}_{x,t}\sim t(5),\hfill \\ \hfill {h}_{x,t}& =& 0.5+a{\epsilon}_{y,t-1}^{2}+0.3{h}_{x,t-1};\phantom{\rule{5.69046pt}{0ex}}\hfill \\ \hfill {Y}_{t}& =& 0.4{Y}_{t-1}-0.2{\epsilon}_{y,t-1}+{\epsilon}_{y,t},\hfill \\ \hfill {\epsilon}_{y,t}& =& \sqrt{{h}_{y,t}}{\upsilon}_{y,t},\phantom{\rule{1.em}{0ex}}{\upsilon}_{y,t}\sim t(5),\hfill \\ \hfill {h}_{y,t}& =& 0.8+0.05{\epsilon}_{y,t-1}^{2}+0.4{h}_{y,t-1}.\hfill \end{array}$$
- Bivariate AR(1)-EGARCH process.$$\begin{array}{ccc}\hfill {X}_{t}& =& 0.5{X}_{t-1}+{\epsilon}_{x,t},\hfill \\ \hfill {\epsilon}_{x,t}& =& \sqrt{{h}_{x,t}}{\upsilon}_{x,t},\phantom{\rule{1.em}{0ex}}{\upsilon}_{x,t}\sim t(5),\hfill \\ \hfill log({h}_{x,t})& =& -0.5+a|\frac{{\epsilon}_{y,t-1}^{2}}{\sqrt{{h}_{x,t-1}}}|+0.2\frac{{\epsilon}_{y,t-1}^{2}}{\sqrt{{h}_{x,t-1}}}+0.9log({h}_{x,t-1});\phantom{\rule{5.69046pt}{0ex}}\hfill \\ \hfill {Y}_{t}& =& 0.6{Y}_{t-1}+{\epsilon}_{y,t},\hfill \\ \hfill {\epsilon}_{y,t}& =& \sqrt{{h}_{y,t}}{\upsilon}_{y,t},\phantom{\rule{1.em}{0ex}}{\upsilon}_{y,t}\sim t(5),\hfill \\ \hfill log({h}_{y,t})& =& -0.6+0.05|\frac{{\epsilon}_{y,t-1}^{2}}{\sqrt{{h}_{x,t-1}}}|+0.02\frac{{\epsilon}_{y,t-1}^{2}}{\sqrt{{h}_{x,t-1}}}+0.8log({h}_{x,t-1}).\hfill \end{array}$$
- VECM process. Note that in this situation both $\{{X}_{t}\}$ and $\{{Y}_{t}\}$ are not stationary.$$\begin{array}{ccc}\hfill {X}_{t}& =& 1.2+0.6{Y}_{t-1}+{\epsilon}_{x,t}\hfill \\ \hfill {\epsilon}_{x,t}& =& \sqrt{1-{a}^{2}}{\upsilon}_{x,t}+a{\epsilon}_{Y,t-1},\phantom{\rule{1.em}{0ex}}{\upsilon}_{x,t}\sim N(0,1)\hfill \\ \hfill {Y}_{t}& =& {Y}_{t-1}+{\epsilon}_{y,t},\phantom{\rule{1.em}{0ex}}{\epsilon}_{y,t}\sim N(0,1).\hfill \end{array}$$
- Threshold AR(1) process.$$\begin{array}{ccc}\hfill {X}_{t}& =& \left(\right)open="\{"\; close>\begin{array}{c}0.2{X}_{t-1}+{\epsilon}_{x,t}\phantom{\rule{4pt}{0ex}}{\epsilon}_{x,t}\sim N(0,1),\phantom{\rule{1.em}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}{Y}_{t-1}0\hfill \\ 0.9{X}_{t-1}+{\epsilon}_{x,t}\phantom{\rule{4pt}{0ex}}{\epsilon}_{x,t}\sim N(0,1),\phantom{\rule{1.em}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}{Y}_{t-1}\ge 0\hfill \end{array}\hfill \end{array}$$
- Two-way VAR process.$$\begin{array}{ccc}\hfill {X}_{t}& =0.7{X}_{t-1}+b{Y}_{t-1}+{\epsilon}_{x,t},\phantom{\rule{4pt}{0ex}}& \phantom{\rule{1.em}{0ex}}{\epsilon}_{x,t}\sim N(0,1)\hfill \\ \hfill {Y}_{t}& =c{X}_{t-1}+0.5{Y}_{t-1}+{\epsilon}_{y,t},\phantom{\rule{4pt}{0ex}}& \phantom{\rule{1.em}{0ex}}{\epsilon}_{y,t}\sim N(0,1).\hfill \end{array}$$

## 4. Application

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Size-size and size-power plots of Granger non-causality tests, based on 500 replications and smoothed local bootstrap (

**a**). The data generating process (DGP) is the bivariate VAR process in Equation (9), with Y affecting X. The left (right) column shows observed rejection rates under the null (alternative) hypothesis. The sample size varies from $n=200$ to $n=2000$.

**Figure 2.**Size-size and size-power plots of Granger non-causality tests, based on 500 replications and smoothed local bootstrap (

**a**). The DGP is the bivariate non-linear VAR process in Equation (10), with Y affecting X. The left (right) column shows observed rejection rates under the null (alternative) hypothesis. The sample size varies from $n=200$ to $n=2000$.

**Figure 3.**Size-size and size-power plots of Granger non-causality tests, based on 500 replications and smoothed local bootstrap (

**a**). The DGP is the bivariate ARCH process in Equation (11), with Y affecting X. The left (right) column shows observed rejection rates under the null (alternative) hypothesis. The sample size varies from $n=200$ to $n=2000$.

**Figure 4.**Size-size and size-power plots of Granger non-causality tests, based on 500 replications and smoothed local bootstrap (

**a**). The DGP is the bilinear process in Equation (12), with Y affecting X. The left (right) column shows observed rejection rates under the null (alternative) hypothesis. The sample size varies from $n=200$ to $n=2000$.

**Figure 5.**Size-size and size-power plots of Granger non-causality tests, based on 500 replications and smoothed local bootstrap (

**a**). The DGP is the bivariate AR2-GARCH process in Equation (13), with Y affecting X. The left (right) column shows observed rejection rates under the null (alternative) hypothesis. The sample size varies from $n=200$ to $n=2000$.

**Figure 6.**Size-size and size-power plots of Granger non-causality tests, based on 500 replications and smoothed local bootstrap (

**a**). The DGP is the bivariate ARMA-GARCH process in Equation (14), with Y affecting X. The left (right) column shows observed rejection rates under the null (alternative) hypothesis. The sample size varies from $n=200$ to $n=2000$.

**Figure 7.**Size-size and size-power plots of Granger non-causality tests, based on 500 replications and smoothed local bootstrap (

**a**). The DGP is the bivariate AR1-EGARCH process in Equation (15), with Y affecting X. The left (right) column shows observed rejection rates under the null (alternative) hypothesis. The sample size varies from $n=200$ to $n=2000$.

**Figure 8.**Size-size and size-power plots of Granger non-causality tests, based on 500 replications and smoothed local bootstrap (

**a**). The DGP is the VECM process in Equation (16), with Y affecting X. The left (right) column shows observed rejection rates under the null (alternative) hypothesis. The sample size varies from $n=200$ to $n=2000$.

**Figure 9.**Size-size and size-power plots of Granger non-causality tests, based on 500 replications and smoothed local bootstrap (

**a**). The DGP is the bivariate threshold AR(1) process in Equation (17), with Y affecting X. The left (right) column shows observed rejection rates under the null (alternative) hypothesis. The sample size varies from $n=200$ to $n=2000$.

**Figure 10.**Size-power plots of Granger non-causality tests, based on 500 replications and smoothed local bootstrap (

**a**). The DGP is the two-way VAR process in Equation (18), with X affecting Y and Y affecting X. The left (right) column shows observed rejection rates for testing X (Y) causing Y (X). The sample size varies from $n=200$ to $n=2000$.

**Figure 11.**Graphical representation of pairwise causalities on global stock returns and volatilities. All “→” in the graph indicate a significant directional causality at the 5% level.

**Figure 12.**Time-varying p-values for the TE-based Granger causality test in Return series. The causal linkages from DJIA to other markets, as well as the linkages from other markets to DJIA are tested.

Size | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}=\mathbf{0.05}$ | $\mathit{\alpha}=\mathbf{0.10}$ | ||||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | |||

$C=4.8$ | 200 | 0.0560 | 0.0500 | 0.0460 | 0.0420 | 0.0440 | 0.0820 | 0.0740 | 0.0740 | 0.0780 | 0.0780 | ||

500 | 0.0740 | 0.0680 | 0.0660 | 0.0700 | 0.0660 | 0.1160 | 0.1120 | 0.1200 | 0.1160 | 0.1220 | |||

1000 | 0.0620 | 0.0560 | 0.0600 | 0.0560 | 0.0560 | 0.0940 | 0.0920 | 0.0980 | 0.0960 | 0.0980 | |||

2000 | 0.0380 | 0.0340 | 0.0380 | 0.0460 | 0.0460 | 0.0940 | 0.0920 | 0.0960 | 0.0980 | 0.0980 | |||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | ||||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | |||

$C=8$ | 200 | 0.0500 | 0.0440 | 0.0460 | 0.0460 | 0.0440 | 0.1120 | 0.1000 | 0.0960 | 0.0960 | 0.0920 | ||

500 | 0.0840 | 0.0760 | 0.0760 | 0.0720 | 0.0660 | 0.1360 | 0.1300 | 0.1060 | 0.1160 | 0.1140 | |||

1000 | 0.0720 | 0.0680 | 0.0620 | 0.0560 | 0.0580 | 0.1280 | 0.1280 | 0.1160 | 0.1260 | 0.1200 | |||

2000 | 0.0880 | 0.0780 | 0.0820 | 0.0760 | 0.0820 | 0.1420 | 0.1380 | 0.1320 | 0.1340 | 0.1440 | |||

Power | |||||||||||||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | ||||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | |||

$C=4.8$ | 200 | 0.1880 | 0.1980 | 0.1900 | 0.1920 | 0.1900 | 0.2780 | 0.2780 | 0.2880 | 0.2860 | 0.2920 | ||

500 | 0.3460 | 0.3460 | 0.3400 | 0.3480 | 0.3420 | 0.4520 | 0.4460 | 0.4580 | 0.4500 | 0.4500 | |||

1000 | 0.5440 | 0.5340 | 0.5320 | 0.5340 | 0.5280 | 0.6400 | 0.6520 | 0.6460 | 0.6480 | 0.6460 | |||

2000 | 0.7500 | 0.7420 | 0.7500 | 0.7460 | 0.7460 | 0.8160 | 0.8080 | 0.8100 | 0.8120 | 0.8120 | |||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | ||||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | |||

$C=8$ | 200 | 0.1660 | 0.1680 | 0.1640 | 0.1680 | 0.1700 | 0.2660 | 0.2640 | 0.2680 | 0.2740 | 0.2740 | ||

500 | 0.2900 | 0.2900 | 0.3020 | 0.3040 | 0.3020 | 0.4020 | 0.3960 | 0.3940 | 0.4020 | 0.3980 | |||

1000 | 0.4980 | 0.4980 | 0.5000 | 0.4900 | 0.4980 | 0.6040 | 0.6120 | 0.6120 | 0.6140 | 0.6120 | |||

2000 | 0.8420 | 0.8380 | 0.8400 | 0.8340 | 0.8460 | 0.8960 | 0.8880 | 0.8900 | 0.8900 | 0.8900 |

**Table 2.**Observed size and power of the TE-based test for the nonlinear VAR process in Equation (10).

Size | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}=\mathbf{0.05}$ | $\mathit{\alpha}=0.10$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.0500 | 0.0360 | 0.0340 | 0.0380 | 0.0360 | 0.0760 | 0.0720 | 0.0780 | 0.0760 | 0.0720 | |

500 | 0.0580 | 0.0580 | 0.0580 | 0.0580 | 0.0580 | 0.0960 | 0.0980 | 0.1040 | 0.1040 | 0.0980 | ||

1000 | 0.0340 | 0.0360 | 0.0380 | 0.0360 | 0.0380 | 0.0620 | 0.0580 | 0.0740 | 0.0700 | 0.0720 | ||

2000 | 0.0380 | 0.0320 | 0.0440 | 0.0460 | 0.0420 | 0.0780 | 0.0700 | 0.0960 | 0.0920 | 0.0880 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.0560 | 0.0480 | 0.0460 | 0.0520 | 0.0480 | 0.1000 | 0.0800 | 0.0940 | 0.0940 | 0.0940 | |

500 | 0.0640 | 0.0620 | 0.0500 | 0.0480 | 0.0540 | 0.1120 | 0.1100 | 0.1080 | 0.1040 | 0.1040 | ||

1000 | 0.0440 | 0.0360 | 0.0320 | 0.0300 | 0.0280 | 0.0900 | 0.0860 | 0.0820 | 0.0780 | 0.0800 | ||

2000 | 0.0260 | 0.0300 | 0.0280 | 0.0280 | 0.0260 | 0.0700 | 0.0640 | 0.0740 | 0.0660 | 0.0640 | ||

Power | ||||||||||||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.1400 | 0.1360 | 0.1380 | 0.1400 | 0.1440 | 0.2480 | 0.2420 | 0.2300 | 0.2360 | 0.2300 | |

500 | 0.3380 | 0.3360 | 0.3340 | 0.3360 | 0.3320 | 0.4400 | 0.4400 | 0.4440 | 0.4300 | 0.4360 | ||

1000 | 0.6060 | 0.6040 | 0.6240 | 0.6220 | 0.6220 | 0.7180 | 0.7260 | 0.7140 | 0.7180 | 0.7160 | ||

2000 | 0.8760 | 0.8760 | 0.8780 | 0.8740 | 0.8780 | 0.9440 | 0.9300 | 0.9320 | 0.9300 | 0.9280 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.0940 | 0.0900 | 0.0900 | 0.0900 | 0.0880 | 0.1800 | 0.1740 | 0.1700 | 0.1680 | 0.1760 | |

500 | 0.1880 | 0.1800 | 0.1800 | 0.1760 | 0.1780 | 0.2960 | 0.2960 | 0.3000 | 0.2980 | 0.3020 | ||

1000 | 0.3800 | 0.3940 | 0.3900 | 0.3840 | 0.3820 | 0.5520 | 0.5440 | 0.5480 | 0.5520 | 0.5480 | ||

2000 | 0.8340 | 0.8300 | 0.8280 | 0.8220 | 0.8300 | 0.9040 | 0.9040 | 0.9060 | 0.8980 | 0.9040 |

**Table 3.**Observed size and power of the TE-based test for the bivariate ARCH process in Equation (11).

Size | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}=\mathbf{0.05}$ | $\mathit{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.0660 | 0.0580 | 0.0620 | 0.0640 | 0.0620 | 0.1420 | 0.1280 | 0.1260 | 0.1340 | 0.1280 | |

500 | 0.0640 | 0.0560 | 0.0560 | 0.0560 | 0.0580 | 0.1120 | 0.1060 | 0.1080 | 0.1020 | 0.1000 | ||

1000 | 0.0480 | 0.0460 | 0.0400 | 0.0420 | 0.0340 | 0.0740 | 0.0700 | 0.0700 | 0.0660 | 0.0620 | ||

2000 | 0.0220 | 0.0200 | 0.0080 | 0.0080 | 0.0080 | 0.0460 | 0.0500 | 0.0220 | 0.0260 | 0.0180 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.0920 | 0.0760 | 0.0760 | 0.0720 | 0.0840 | 0.1540 | 0.1360 | 0.1280 | 0.1260 | 0.1320 | |

500 | 0.1100 | 0.0900 | 0.0720 | 0.0760 | 0.0800 | 0.1980 | 0.1860 | 0.1620 | 0.1480 | 0.1600 | ||

1000 | 0.0920 | 0.0960 | 0.0740 | 0.0720 | 0.0800 | 0.1500 | 0.1500 | 0.1220 | 0.1180 | 0.1240 | ||

2000 | 0.0780 | 0.0740 | 0.0660 | 0.0620 | 0.0600 | 0.1260 | 0.1180 | 0.1140 | 0.1160 | 0.1120 | ||

Power | ||||||||||||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.2320 | 0.2240 | 0.2180 | 0.2240 | 0.2160 | 0.3320 | 0.3340 | 0.3320 | 0.3520 | 0.3460 | |

500 | 0.3520 | 0.3420 | 0.3520 | 0.3540 | 0.3540 | 0.4800 | 0.4800 | 0.4740 | 0.4780 | 0.4780 | ||

1000 | 0.5020 | 0.5060 | 0.5120 | 0.5120 | 0.5000 | 0.6340 | 0.6320 | 0.6340 | 0.6280 | 0.6300 | ||

2000 | 0.6020 | 0.6060 | 0.5940 | 0.5920 | 0.5960 | 0.7340 | 0.7280 | 0.7360 | 0.7300 | 0.7280 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.2940 | 0.2880 | 0.3220 | 0.3180 | 0.3080 | 0.4360 | 0.4320 | 0.4380 | 0.4400 | 0.4340 | |

500 | 0.5520 | 0.5500 | 0.5620 | 0.5720 | 0.5680 | 0.6880 | 0.6940 | 0.6880 | 0.6900 | 0.6920 | ||

1000 | 0.7720 | 0.7720 | 0.7820 | 0.7800 | 0.7740 | 0.8600 | 0.8560 | 0.8640 | 0.8600 | 0.8640 | ||

2000 | 0.9780 | 0.9720 | 0.9780 | 0.9740 | 0.9760 | 0.9900 | 0.9900 | 0.9920 | 0.9920 | 0.9920 |

Size | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}=\mathbf{0.05}$ | $\mathit{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.0460 | 0.0420 | 0.0420 | 0.0420 | 0.0460 | 0.0820 | 0.0780 | 0.0900 | 0.0860 | 0.0900 | |

500 | 0.0540 | 0.0480 | 0.0480 | 0.0500 | 0.0480 | 0.0980 | 0.0940 | 0.1040 | 0.1040 | 0.1040 | ||

1000 | 0.0440 | 0.0440 | 0.0440 | 0.0420 | 0.0500 | 0.1020 | 0.1020 | 0.1060 | 0.1000 | 0.1040 | ||

2000 | 0.0460 | 0.0480 | 0.0480 | 0.0480 | 0.0520 | 0.0940 | 0.0940 | 0.1000 | 0.1000 | 0.1020 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.0440 | 0.0400 | 0.0460 | 0.0460 | 0.0460 | 0.0940 | 0.0860 | 0.0940 | 0.0900 | 0.0940 | |

500 | 0.0500 | 0.0480 | 0.0440 | 0.0420 | 0.0420 | 0.0840 | 0.0880 | 0.0820 | 0.0820 | 0.0820 | ||

1000 | 0.0660 | 0.0620 | 0.0540 | 0.0520 | 0.0560 | 0.1220 | 0.1200 | 0.1180 | 0.1140 | 0.1100 | ||

2000 | 0.0360 | 0.0380 | 0.0400 | 0.0340 | 0.0360 | 0.0880 | 0.0920 | 0.1000 | 0.0980 | 0.0960 | ||

Power | ||||||||||||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.1800 | 0.1800 | 0.2040 | 0.2000 | 0.1960 | 0.3000 | 0.2900 | 0.3020 | 0.3000 | 0.3060 | |

500 | 0.4620 | 0.4580 | 0.4680 | 0.4460 | 0.4640 | 0.5920 | 0.5940 | 0.6020 | 0.6060 | 0.6100 | ||

1000 | 0.7700 | 0.7800 | 0.7780 | 0.7820 | 0.7820 | 0.8620 | 0.8560 | 0.8640 | 0.8580 | 0.8600 | ||

2000 | 0.9780 | 0.9820 | 0.9800 | 0.9800 | 0.9780 | 0.9860 | 0.9880 | 0.9880 | 0.9880 | 0.9880 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.1440 | 0.1380 | 0.1540 | 0.1440 | 0.1460 | 0.2340 | 0.2320 | 0.2540 | 0.2420 | 0.2440 | |

500 | 0.3240 | 0.3200 | 0.3320 | 0.3320 | 0.3320 | 0.4620 | 0.4680 | 0.4680 | 0.4700 | 0.4700 | ||

1000 | 0.6400 | 0.6360 | 0.6400 | 0.6240 | 0.6260 | 0.7520 | 0.7480 | 0.7460 | 0.7500 | 0.7460 | ||

2000 | 0.9620 | 0.9580 | 0.9620 | 0.9600 | 0.9640 | 0.9880 | 0.9900 | 0.9860 | 0.9900 | 0.9840 |

Size | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}=\mathbf{0.05}$ | $\mathit{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.0600 | 0.0540 | 0.0840 | 0.0780 | 0.0740 | 0.1280 | 0.1280 | 0.1520 | 0.1500 | 0.1520 | |

500 | 0.0580 | 0.0560 | 0.0740 | 0.0720 | 0.0700 | 0.1100 | 0.1060 | 0.1380 | 0.1320 | 0.1300 | ||

1000 | 0.0420 | 0.0480 | 0.0440 | 0.0540 | 0.0540 | 0.0900 | 0.0840 | 0.0920 | 0.0920 | 0.0900 | ||

2000 | 0.0480 | 0.0440 | 0.0320 | 0.0340 | 0.0320 | 0.0800 | 0.0800 | 0.0660 | 0.0680 | 0.0660 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.0680 | 0.0620 | 0.0920 | 0.0880 | 0.0860 | 0.1280 | 0.1200 | 0.1440 | 0.1400 | 0.1340 | |

500 | 0.0880 | 0.0880 | 0.1120 | 0.1060 | 0.1040 | 0.1400 | 0.1400 | 0.1480 | 0.1520 | 0.1520 | ||

1000 | 0.0600 | 0.0620 | 0.0660 | 0.0680 | 0.0680 | 0.1120 | 0.1140 | 0.1080 | 0.1160 | 0.1140 | ||

2000 | 0.0820 | 0.0800 | 0.0800 | 0.0840 | 0.0760 | 0.1380 | 0.1380 | 0.1360 | 0.1300 | 0.1300 | ||

Power | ||||||||||||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.0860 | 0.0800 | 0.0860 | 0.0840 | 0.0840 | 0.1420 | 0.1460 | 0.1500 | 0.1440 | 0.1400 | |

500 | 0.0980 | 0.0920 | 0.0980 | 0.0840 | 0.0920 | 0.1700 | 0.1700 | 0.1500 | 0.1480 | 0.1580 | ||

1000 | 0.0760 | 0.0800 | 0.0520 | 0.0540 | 0.0620 | 0.1500 | 0.1480 | 0.1220 | 0.1160 | 0.1080 | ||

2000 | 0.0540 | 0.0480 | 0.0280 | 0.0260 | 0.0260 | 0.1080 | 0.1200 | 0.0460 | 0.0460 | 0.0500 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.1120 | 0.1020 | 0.1320 | 0.1300 | 0.1360 | 0.1900 | 0.1860 | 0.2200 | 0.2120 | 0.2140 | |

500 | 0.1540 | 0.1560 | 0.1740 | 0.1660 | 0.1680 | 0.2560 | 0.2480 | 0.2680 | 0.2640 | 0.2560 | ||

1000 | 0.2180 | 0.2100 | 0.2080 | 0.2000 | 0.2020 | 0.3080 | 0.3140 | 0.3020 | 0.2960 | 0.3020 | ||

2000 | 0.2740 | 0.2820 | 0.2540 | 0.2500 | 0.2560 | 0.3900 | 0.3900 | 0.3420 | 0.3460 | 0.3540 |

Size | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}=0.05$ | $\mathit{\alpha}=0.10$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.0580 | 0.0480 | 0.0600 | 0.0560 | 0.0600 | 0.1080 | 0.0980 | 0.1160 | 0.1160 | 0.1080 | |

500 | 0.0640 | 0.0640 | 0.0660 | 0.0640 | 0.0700 | 0.1060 | 0.0960 | 0.1120 | 0.1100 | 0.1080 | ||

1000 | 0.0540 | 0.0500 | 0.0460 | 0.0420 | 0.0440 | 0.0940 | 0.0980 | 0.0760 | 0.0800 | 0.0820 | ||

2000 | 0.0240 | 0.0260 | 0.0160 | 0.0180 | 0.0200 | 0.0660 | 0.0720 | 0.0280 | 0.0300 | 0.0280 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.0540 | 0.0380 | 0.0780 | 0.0760 | 0.0700 | 0.1160 | 0.1000 | 0.1520 | 0.1480 | 0.1520 | |

500 | 0.0900 | 0.0820 | 0.1020 | 0.1060 | 0.0940 | 0.1480 | 0.1240 | 0.1600 | 0.1520 | 0.1520 | ||

1000 | 0.0540 | 0.0580 | 0.0600 | 0.0620 | 0.0660 | 0.1220 | 0.1140 | 0.1220 | 0.1220 | 0.1260 | ||

2000 | 0.0620 | 0.0640 | 0.0640 | 0.0660 | 0.0680 | 0.1220 | 0.1220 | 0.1180 | 0.1140 | 0.1080 | ||

Power | ||||||||||||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.2680 | 0.2620 | 0.2560 | 0.2600 | 0.2640 | 0.3840 | 0.3820 | 0.3800 | 0.3720 | 0.3760 | |

500 | 0.3540 | 0.3460 | 0.3540 | 0.3600 | 0.3580 | 0.4580 | 0.4500 | 0.4600 | 0.4640 | 0.4520 | ||

1000 | 0.3380 | 0.3420 | 0.3280 | 0.3240 | 0.3260 | 0.4200 | 0.4260 | 0.3880 | 0.3860 | 0.3900 | ||

2000 | 0.2740 | 0.2800 | 0.2300 | 0.2320 | 0.2320 | 0.3360 | 0.3380 | 0.2820 | 0.2820 | 0.2880 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.3860 | 0.3800 | 0.4140 | 0.4060 | 0.4100 | 0.5180 | 0.5000 | 0.5320 | 0.5260 | 0.5180 | |

500 | 0.7260 | 0.7280 | 0.7220 | 0.7200 | 0.7180 | 0.8060 | 0.8020 | 0.8040 | 0.8000 | 0.7980 | ||

1000 | 0.8500 | 0.8440 | 0.8400 | 0.8380 | 0.8320 | 0.8780 | 0.8740 | 0.8700 | 0.8700 | 0.8700 | ||

2000 | 0.8900 | 0.8960 | 0.8860 | 0.8880 | 0.8900 | 0.9240 | 0.9200 | 0.9160 | 0.9160 | 0.9140 |

**Table 7.**Observed size and power of the TE-based test for the AR(1)-EGARCH process in Equation (15).

Size | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}=\mathbf{0.05}$ | $\mathit{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.0420 | 0.0420 | 0.0740 | 0.0740 | 0.0740 | 0.1000 | 0.0940 | 0.1560 | 0.1500 | 0.1480 | |

500 | 0.0640 | 0.0520 | 0.0820 | 0.0860 | 0.0860 | 0.1040 | 0.0940 | 0.1460 | 0.1460 | 0.1460 | ||

1000 | 0.0400 | 0.0360 | 0.0460 | 0.0440 | 0.0480 | 0.0760 | 0.0740 | 0.0880 | 0.0880 | 0.0900 | ||

2000 | 0.0360 | 0.0400 | 0.0320 | 0.0380 | 0.0360 | 0.0680 | 0.0680 | 0.0520 | 0.0520 | 0.0580 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.0540 | 0.0480 | 0.0860 | 0.0880 | 0.0860 | 0.1140 | 0.1060 | 0.1460 | 0.1440 | 0.1460 | |

500 | 0.0900 | 0.0740 | 0.1020 | 0.1020 | 0.1020 | 0.1440 | 0.1340 | 0.1760 | 0.1660 | 0.1580 | ||

1000 | 0.0720 | 0.0640 | 0.0800 | 0.0800 | 0.0800 | 0.1240 | 0.1200 | 0.1380 | 0.1320 | 0.1340 | ||

2000 | 0.0660 | 0.0640 | 0.0680 | 0.0700 | 0.0720 | 0.1020 | 0.1080 | 0.1100 | 0.1100 | 0.1160 | ||

Power | ||||||||||||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.1660 | 0.1460 | 0.2200 | 0.2220 | 0.2280 | 0.2580 | 0.2520 | 0.3280 | 0.3280 | 0.3260 | |

500 | 0.2500 | 0.2420 | 0.2940 | 0.2960 | 0.2960 | 0.3740 | 0.3720 | 0.4140 | 0.4080 | 0.4140 | ||

1000 | 0.2860 | 0.2860 | 0.3040 | 0.3100 | 0.3000 | 0.3840 | 0.3820 | 0.4000 | 0.4020 | 0.3960 | ||

2000 | 0.3180 | 0.3020 | 0.2900 | 0.2900 | 0.2900 | 0.3780 | 0.3840 | 0.3520 | 0.3440 | 0.3600 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.2160 | 0.2000 | 0.2740 | 0.2660 | 0.2640 | 0.3000 | 0.2880 | 0.3560 | 0.3580 | 0.3600 | |

500 | 0.3660 | 0.3520 | 0.4040 | 0.4080 | 0.4020 | 0.4900 | 0.4720 | 0.5720 | 0.5600 | 0.5580 | ||

1000 | 0.6200 | 0.6020 | 0.6600 | 0.6520 | 0.6520 | 0.7280 | 0.7140 | 0.7420 | 0.7440 | 0.7460 | ||

2000 | 0.8300 | 0.8360 | 0.8460 | 0.8420 | 0.8380 | 0.8840 | 0.8820 | 0.8880 | 0.8860 | 0.8920 |

Size | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}=\mathbf{0.05}$ | $\mathit{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.0060 | 0.0060 | 0.0040 | 0.0040 | 0.0040 | 0.0220 | 0.0200 | 0.0220 | 0.0220 | 0.0200 | |

500 | 0.0220 | 0.0220 | 0.0240 | 0.0240 | 0.0220 | 0.0460 | 0.0480 | 0.0500 | 0.0480 | 0.0560 | ||

1000 | 0.0400 | 0.0400 | 0.0400 | 0.0400 | 0.0420 | 0.0840 | 0.0860 | 0.0900 | 0.0840 | 0.0860 | ||

2000 | 0.0440 | 0.0420 | 0.0380 | 0.0360 | 0.0360 | 0.0800 | 0.0780 | 0.0820 | 0.0740 | 0.0700 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.0080 | 0.0100 | 0.0100 | 0.0080 | 0.0100 | 0.0340 | 0.0300 | 0.0300 | 0.0280 | 0.0340 | |

500 | 0.0240 | 0.0220 | 0.0220 | 0.0200 | 0.0240 | 0.0680 | 0.0660 | 0.0680 | 0.0680 | 0.0660 | ||

1000 | 0.0480 | 0.0420 | 0.0420 | 0.0440 | 0.0440 | 0.0860 | 0.0840 | 0.0840 | 0.0820 | 0.0880 | ||

2000 | 0.0200 | 0.0220 | 0.0200 | 0.0200 | 0.0220 | 0.0560 | 0.0540 | 0.0520 | 0.0560 | 0.0540 | ||

Power | ||||||||||||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.2540 | 0.2520 | 0.2600 | 0.2600 | 0.2640 | 0.3260 | 0.3300 | 0.3400 | 0.3380 | 0.3440 | |

500 | 0.5140 | 0.5120 | 0.5220 | 0.5280 | 0.5120 | 0.6120 | 0.5920 | 0.5980 | 0.6060 | 0.6080 | ||

1000 | 0.7840 | 0.7840 | 0.7800 | 0.7840 | 0.7800 | 0.8340 | 0.8320 | 0.8420 | 0.8440 | 0.8420 | ||

2000 | 0.9240 | 0.9260 | 0.9280 | 0.9260 | 0.9280 | 0.9560 | 0.9560 | 0.9520 | 0.9540 | 0.9520 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.2500 | 0.2440 | 0.2500 | 0.2560 | 0.2480 | 0.3100 | 0.3000 | 0.3140 | 0.3160 | 0.3140 | |

500 | 0.4600 | 0.4640 | 0.4680 | 0.4700 | 0.4740 | 0.5600 | 0.5560 | 0.5540 | 0.5600 | 0.5580 | ||

1000 | 0.7620 | 0.7660 | 0.7780 | 0.7680 | 0.7760 | 0.8220 | 0.8180 | 0.8240 | 0.8240 | 0.8280 | ||

2000 | 0.9520 | 0.9560 | 0.9560 | 0.9540 | 0.9520 | 0.9780 | 0.9700 | 0.9780 | 0.9780 | 0.9780 |

**Table 9.**Observed size and power of the TE-based test for the threshold AR(1) process in Equation (17).

Size | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}=0.05$ | $\mathit{\alpha}=0.10$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.0420 | 0.0360 | 0.0420 | 0.0360 | 0.0360 | 0.0780 | 0.0680 | 0.0720 | 0.0700 | 0.0720 | |

500 | 0.0460 | 0.0420 | 0.0440 | 0.0460 | 0.0480 | 0.0880 | 0.0880 | 0.0860 | 0.0900 | 0.0940 | ||

1000 | 0.0380 | 0.0420 | 0.0420 | 0.0400 | 0.0360 | 0.0980 | 0.0900 | 0.0760 | 0.0840 | 0.0800 | ||

2000 | 0.0440 | 0.0480 | 0.0420 | 0.0400 | 0.0400 | 0.0860 | 0.0900 | 0.0780 | 0.0800 | 0.0780 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.0400 | 0.0400 | 0.0380 | 0.0400 | 0.0400 | 0.0800 | 0.0800 | 0.0920 | 0.0800 | 0.0860 | |

500 | 0.0420 | 0.0380 | 0.0440 | 0.0560 | 0.0560 | 0.1080 | 0.1020 | 0.1020 | 0.1040 | 0.1060 | ||

1000 | 0.0480 | 0.0440 | 0.0440 | 0.0440 | 0.0500 | 0.1020 | 0.0980 | 0.0960 | 0.1080 | 0.1000 | ||

2000 | 0.0440 | 0.0460 | 0.0480 | 0.0440 | 0.0460 | 0.0940 | 0.0860 | 0.0840 | 0.0880 | 0.0920 | ||

Power | ||||||||||||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.8520 | 0.8460 | 0.8200 | 0.8280 | 0.8120 | 0.9080 | 0.9120 | 0.8940 | 0.8960 | 0.8920 | |

500 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

1000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

2000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.3900 | 0.3740 | 0.3120 | 0.3160 | 0.3180 | 0.4980 | 0.5000 | 0.4420 | 0.4460 | 0.4440 | |

500 | 0.9460 | 0.9440 | 0.9220 | 0.9240 | 0.9200 | 0.9760 | 0.9760 | 0.9600 | 0.9640 | 0.9660 | ||

1000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

2000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

$\mathit{X}\to \mathit{Y}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}=\mathbf{0.05}$ | $\mathit{\alpha}=0.10$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.2780 | 0.2600 | 0.3540 | 0.3520 | 0.3560 | 0.3980 | 0.3760 | 0.4560 | 0.4560 | 0.4660 | |

500 | 0.5640 | 0.5360 | 0.6260 | 0.6240 | 0.6260 | 0.6780 | 0.6680 | 0.7420 | 0.7400 | 0.7460 | ||

1000 | 0.8380 | 0.8320 | 0.8800 | 0.8740 | 0.8780 | 0.9040 | 0.8980 | 0.9200 | 0.9240 | 0.9200 | ||

2000 | 0.9900 | 0.9900 | 0.9980 | 0.9960 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.2480 | 0.2240 | 0.3080 | 0.3180 | 0.3100 | 0.3540 | 0.3360 | 0.3900 | 0.3940 | 0.3920 | |

500 | 0.4180 | 0.4020 | 0.4740 | 0.4700 | 0.4700 | 0.5400 | 0.5320 | 0.5880 | 0.5920 | 0.5800 | ||

1000 | 0.7960 | 0.7840 | 0.8140 | 0.8140 | 0.8160 | 0.8560 | 0.8540 | 0.8740 | 0.8720 | 0.8720 | ||

2000 | 0.9800 | 0.9780 | 0.9820 | 0.9800 | 0.9800 | 0.9900 | 0.9880 | 0.9900 | 0.9900 | 0.9920 | ||

$\mathit{Y}\to \mathit{X}$ | ||||||||||||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=4.8$ | 200 | 0.5140 | 0.5060 | 0.5480 | 0.5560 | 0.5480 | 0.6120 | 0.5960 | 0.6920 | 0.6920 | 0.6940 | |

500 | 0.9400 | 0.9340 | 0.9480 | 0.9440 | 0.9460 | 0.9620 | 0.9580 | 0.9720 | 0.9700 | 0.9740 | ||

1000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

2000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

$\mathbf{\alpha}=\mathbf{0.05}$ | $\mathbf{\alpha}=\mathbf{0.10}$ | |||||||||||

n | TS.a | TS.b | SMB.a | SMB.b | STB | TS.a | TS.b | SMB.a | SMB.b | STB | ||

$C=8$ | 200 | 0.4040 | 0.3700 | 0.4420 | 0.4400 | 0.4280 | 0.5060 | 0.4900 | 0.5400 | 0.5400 | 0.5320 | |

500 | 0.8260 | 0.8220 | 0.8220 | 0.8200 | 0.8160 | 0.8740 | 0.8740 | 0.8700 | 0.8720 | 0.8680 | ||

1000 | 0.9820 | 0.9840 | 0.9820 | 0.9800 | 0.9800 | 0.9940 | 0.9940 | 0.9940 | 0.9920 | 0.9940 | ||

2000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Return | ||||||
---|---|---|---|---|---|---|

DJIA | Nikkei | Hangseng | FTSE | DAX | CAC | |

Mean | 0.1433 | −0.0126 | 0.1294 | 0.0823 | 0.1535 | 0.0790 |

Median | 0.2911 | 0.1472 | 0.2627 | 0.2121 | 0.4029 | 0.1984 |

Maximum | 10.6977 | 11.4496 | 13.9169 | 12.5845 | 14.9421 | 12.4321 |

Minimum | −20.0298 | −27.8844 | −19.9212 | −23.6317 | −24.3470 | −25.0504 |

Std. Dev. | 2.2321 | 3.0521 | 3.3819 | 2.3367 | 3.0972 | 2.9376 |

Skewness | −0.8851 | −0.6978 | −0.3773 | −0.8643 | −0.6398 | −0.6803 |

Kurtosis | 10.8430 | 8.9250 | 5.9522 | 13.2777 | 7.9186 | 8.0780 |

LB Test | 49.9368${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 15.4577 | 28.7922 | 61.0916${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 28.0474 | 43.5004${}^{\phantom{\rule{0.166667em}{0ex}}**}$ |

ADF Test | −38.9512${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | −37.1989${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | −35.4160${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | −38.8850${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | −37.2015${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | −38.7114${}^{\phantom{\rule{0.166667em}{0ex}}**}$ |

Volatility | ||||||

DJIA | Nikkei | Hangseng | FTSE | DAX | CAC | |

Mean | 4.5983 | 7.7167 | 9.6164 | 5.5306 | 8.5698 | 8.2629 |

Median | 2.2155 | 4.6208 | 4.5827 | 2.7161 | 4.0596 | 4.7122 |

Maximum | 208.2227 | 265.9300 | 379.4385 | 149.1572 | 175.0968 | 179.8414 |

Minimum | 0.0636 | 0.1882 | 0.1554 | 0.1154 | 0.1263 | 0.2904 |

Std. Dev. | 9.9961 | 13.5154 | 21.3838 | 10.1167 | 15.2845 | 12.7872 |

Skewness | 10.9980 | 9.6361 | 10.2868 | 6.7179 | 5.3602 | 6.0357 |

Kurtosis | 180.0844 | 140.7606 | 152.4263 | 67.0128 | 42.0810 | 58.3263 |

LB Test | 1924.0870${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 933.3972${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 1198.6872${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 1970.7366${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 2770.5973${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 1982.4141${}^{\phantom{\rule{0.166667em}{0ex}}**}$ |

ADF Test | −16.0378${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | −17.9044${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | −17.4896${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | −14.0329${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | −14.1928${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | −13.6136${}^{\phantom{\rule{0.166667em}{0ex}}**}$ |

To | DJIA | Nikkei | Hangseng | FTSE | DAX | CAC | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

From | ||||||||||||||

$\mathit{C}=\mathbf{4.8}$ | $\mathit{C}=\mathbf{8}$ | $\mathit{C}=\mathbf{4.8}$ | $\mathit{C}=\mathbf{8}$ | $\mathit{C}=\mathbf{4.8}$ | $\mathit{C}=\mathbf{8}$ | $\mathit{C}=\mathbf{4.8}$ | $\mathit{C}=\mathbf{8}$ | $\mathit{C}=\mathbf{4.8}$ | $\mathit{C}=\mathbf{8}$ | $\mathit{C}=\mathbf{4.8}$ | $\mathit{C}=\mathbf{8}$ | |||

TS.a | DJIA | - | 0.367 | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.972 | 0.273 | 0.609 | 0.009${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.533 | 0.012${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.277 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | ||

Nikkei | 0.231 | 0.081 | - | 0.997 | 0.951 | 0.883 | 0.059 | 0.868 | 0.242 | 0.004${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | |||

Hangseng | 0.898 | 0.197 | 0.963 | 0.407 | - | 0.701 | 0.035${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.969 | 0.174 | 0.640 | 0.005${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | |||

FTSE | 0.483 | 0.004${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.004${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.419 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | 0.839 | 0.185 | 0.917 | 0.615 | |||

DAX | 0.977 | 0.149 | 0.027${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.009${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.004${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | 0.695 | 0.025${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | |||

CAC | 0.995 | 0.713 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.294 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.918 | 0.741 | 0.639 | 0.203 | - | |||

TS.b | DJIA | - | 0.402 | 0.004${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.967 | 0.341 | 0.595 | 0.020${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.569 | 0.049${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.288 | 0.012${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | ||

Nikkei | 0.219 | 0.110 | - | 0.999 | 0.956 | 0.853 | 0.103 | 0.855 | 0.255 | 0.009${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.006${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | |||

Hangseng | 0.899 | 0.269 | 0.975 | 0.485 | - | 0.696 | 0.088 | 0.971 | 0.231 | 0.664 | 0.023${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | |||

FTSE | 0.477 | 0.016${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.006${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.003${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.404 | 0.018${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | - | 0.847 | 0.245 | 0.896 | 0.642 | |||

DAX | 0.976 | 0.221 | 0.027${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.009${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.005${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.011${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.010${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | 0.692 | 0.065 | |||

CAC | 0.993 | 0.729 | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.346 | 0.016${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.907 | 0.763 | 0.650 | 0.244 | - | |||

SMB.a | DJIA | - | 0.564 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.999 | 0.349 | 0.796 | 0.003${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.817 | 0.014${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.425 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | ||

Nikkei | 0.321 | 0.147 | - | 0.988 | 0.946 | 0.957 | 0.085 | 0.944 | 0.321 | 0.006${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | |||

Hangseng | 0.946 | 0.273 | 0.967 | 0.483 | - | 0.860 | 0.016${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.994 | 0.188 | 0.793 | 0.004${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | |||

FTSE | 0.579 | 0.005${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.021${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.701 | 0.003${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | 0.947 | 0.297 | 0.943 | 0.739 | |||

DAX | 0.988 | 0.240 | 0.044${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.012${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.006${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | 0.788 | 0.020${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | |||

CAC | 0.993 | 0.762 | 0.006${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.594 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.946 | 0.861 | 0.842 | 0.270 | - | |||

SMB.b | DJIA | - | 0.583 | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.994 | 0.334 | 0.797 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.855 | 0.014${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.413 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | ||

Nikkei | 0.351 | 0.155 | - | 0.992 | 0.940 | 0.952 | 0.088 | 0.965 | 0.322 | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.003${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | |||

Hangseng | 0.945 | 0.276 | 0.970 | 0.506 | - | 0.866 | 0.015${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.997 | 0.215 | 0.829 | 0.004${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | |||

FTSE | 0.637 | 0.006${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.008${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.714 | 0.003${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | 0.954 | 0.345 | 0.953 | 0.789 | |||

DAX | 0.980 | 0.256 | 0.034${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.011${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.009${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | 0.831 | 0.042${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | |||

CAC | 0.993 | 0.825 | 0.003${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.591 | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.980 | 0.898 | 0.862 | 0.304 | - | |||

STB | DJIA | - | 0.645 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.996 | 0.334 | 0.787 | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.823 | 0.015${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.430 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | ||

Nikkei | 0.363 | 0.114 | - | 0.989 | 0.944 | 0.946 | 0.079 | 0.955 | 0.296 | 0.003${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | |||

Hangseng | 0.964 | 0.272 | 0.984 | 0.491 | - | 0.859 | 0.013${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.987 | 0.197 | 0.786 | 0.004${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | |||

FTSE | 0.652 | 0.005${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.016${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.688 | 0.003${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | 0.940 | 0.262 | 0.965 | 0.761 | |||

DAX | 0.982 | 0.234 | 0.048${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.017${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.006${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | 0.828 | 0.029${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | |||

CAC | 0.996 | 0.814 | 0.007${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.578 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.967 | 0.835 | 0.799 | 0.265 | - |

To | DJIA | Nikkei | Hangseng | FTSE | DAX | CAC | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

From | ||||||||||||||

$\mathit{C}=\mathbf{4.8}$ | $\mathit{C}=\mathbf{8}$ | $\mathit{C}=\mathbf{4.8}$ | $\mathit{C}=\mathbf{8}$ | $\mathit{C}=\mathbf{4.8}$ | $\mathit{C}=\mathbf{8}$ | $\mathit{C}=\mathbf{4.8}$ | $\mathit{C}=\mathbf{8}$ | $\mathit{C}=\mathbf{4.8}$ | $\mathit{C}=\mathbf{8}$ | $\mathit{C}=\mathbf{4.8}$ | $\mathit{C}=\mathbf{8}$ | |||

TS.a | DJIA | - | 0.997 | 0.998 | 0.998 | 0.994 | 0.974 | 0.853 | 0.828 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.005${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | ||

Nikkei | 0.998 | 0.995 | - | 0.996 | 1.000 | 0.997 | 0.994 | 0.971 | 0.950 | 1.000 | 0.999 | |||

Hangseng | 0.943 | 0.003${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.989 | 0.992 | - | 0.822 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.973 | 0.953 | |||

FTSE | 0.010${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.955 | 0.944 | 0.997 | 1.000 | - | 0.806 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.985 | 0.946 | |||

DAX | 0.975 | 0.931 | 0.934 | 0.898 | 0.999 | 0.995 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | 0.072 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | |||

CAC | 0.996 | 0.944 | 0.988 | 0.987 | 0.999 | 0.997 | 0.003${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.054 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | |||

TS.b | DJIA | - | 0.993 | 0.994 | 0.996 | 0.992 | 0.958 | 0.857 | 0.806 | 0.010${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.018${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.005${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | ||

Nikkei | 0.986 | 0.994 | - | 0.996 | 0.997 | 0.988 | 0.988 | 0.942 | 0.926 | 0.998 | 0.996 | |||

Hangseng | 0.919 | 0.011${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.976 | 0.966 | - | 0.819 | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.965 | 0.941 | |||

FTSE | 0.019${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.003${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.950 | 0.927 | 0.994 | 0.997 | - | 0.785 | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.982 | 0.932 | |||

DAX | 0.976 | 0.904 | 0.943 | 0.890 | 0.997 | 0.999 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | 0.113 | 0.009${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | |||

CAC | 0.998 | 0.951 | 0.979 | 0.983 | 0.999 | 0.998 | 0.009${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.077 | 0.003${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | |||

SMB.a | DJIA | - | 0.823 | 0.786 | 0.984 | 0.968 | 0.468 | 0.189 | 0.210 | 0.004${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.027${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | ||

Nikkei | 0.957 | 0.941 | - | 1.000 | 1.000 | 0.843 | 0.800 | 0.743 | 0.614 | 0.998 | 0.993 | |||

Hangseng | 0.458 | 0.030${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.888 | 0.808 | - | 0.165 | 0.007${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.839 | 0.736 | |||

FTSE | 0.034${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.557 | 0.417 | 1.000 | 1.000 | - | 0.358 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.967 | 0.793 | |||

DAX | 0.702 | 0.255 | 0.448 | 0.327 | 1.000 | 1.000 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | 0.030${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | |||

CAC | 0.887 | 0.241 | 0.673 | 0.643 | 1.000 | 1.000 | 0.005${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.012${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | |||

SMB.b | DJIA | - | 0.851 | 0.806 | 0.973 | 0.969 | 0.563 | 0.269 | 0.226 | 0.004${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.032${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | ||

Nikkei | 0.950 | 0.940 | - | 1.000 | 1.000 | 0.888 | 0.849 | 0.792 | 0.675 | 0.999 | 0.993 | |||

Hangseng | 0.451 | 0.022${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.919 | 0.869 | - | 0.209 | 0.005${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.868 | 0.793 | |||

FTSE | 0.023${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.561 | 0.457 | 1.000 | 1.000 | - | 0.434 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.974 | 0.840 | |||

DAX | 0.819 | 0.442 | 0.491 | 0.356 | 1.000 | 1.000 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | 0.030${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | |||

CAC | 0.958 | 0.554 | 0.781 | 0.696 | 1.000 | 1.000 | 0.009${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.014${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | |||

STB | DJIA | - | 0.847 | 0.813 | 0.972 | 0.964 | 0.602 | 0.278 | 0.221 | 0.009${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.045${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.004${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | ||

Nikkei | 0.967 | 0.956 | - | 1.000 | 1.000 | 0.835 | 0.789 | 0.777 | 0.624 | 0.998 | 0.997 | |||

Hangseng | 0.527 | 0.033${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.933 | 0.887 | - | 0.222 | 0.010${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.003${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.860 | 0.762 | |||

FTSE | 0.024${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.002${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.635 | 0.497 | 1.000 | 1.000 | - | 0.405 | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.977 | 0.826 | |||

DAX | 0.795 | 0.392 | 0.561 | 0.410 | 1.000 | 1.000 | 0.004${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - | 0.027${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | |||

CAC | 0.903 | 0.561 | 0.809 | 0.748 | 1.000 | 1.000 | 0.020${}^{\phantom{\rule{0.166667em}{0ex}}*}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.010${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | 0.001${}^{\phantom{\rule{0.166667em}{0ex}}**}$ | - |

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## Share and Cite

**MDPI and ACS Style**

Diks, C.; Fang, H.
Transfer Entropy for Nonparametric Granger Causality Detection: An Evaluation of Different Resampling Methods. *Entropy* **2017**, *19*, 372.
https://doi.org/10.3390/e19070372

**AMA Style**

Diks C, Fang H.
Transfer Entropy for Nonparametric Granger Causality Detection: An Evaluation of Different Resampling Methods. *Entropy*. 2017; 19(7):372.
https://doi.org/10.3390/e19070372

**Chicago/Turabian Style**

Diks, Cees, and Hao Fang.
2017. "Transfer Entropy for Nonparametric Granger Causality Detection: An Evaluation of Different Resampling Methods" *Entropy* 19, no. 7: 372.
https://doi.org/10.3390/e19070372