_{1}

This paper compares the statistical properties of time-varying causality tests when errors of variables have multivariate stochastic volatility (SV). The time-varying causal-ity tests in this paper are based on a logistic smooth transition autoregressive model. The compared time-varying causality tests include asymptotic tests, heteroskedasticity-robust tests, and tests using wild bootstrap. Our simulation results show that asymptotic tests and heteroskedasticity-robust counterparts have size distortions under multivariate SV, whereas tests using wild bootstrap have better size properties regardless of type of error. In particular, the time-varying causality test with first-order Taylor approximation using wild bootstrap has better statistical properties.

Granger causality is one of most representative methods to analyze causality between economic variables. It is based on linear vector autoregressive (VAR) models and investigates whether past information is effective for prediction. Although Granger causality is used for various studies, it can be applied to examine only stable linear relationships in the long run. The relationship between economic variables is not necessarily stable in the long run and frequently has time-varying properties. This implies that a causality relationship can also be time-varying, and hence we should take into account the time-varying properties when analyzing a causality relationship.

One method to introduce time-varying properties to Granger causality is through the use of a logistic smooth transition (LST) function. By using an LST function with time as the transition variable, we can test for both smooth and abrupt causalities. When a causality has such nonlinearity, the usual Granger causality tests based on a linear VAR model have low power and tend to give the misleading result of having no causalities in the system. [

While time-varying causality is significant for the precise analysis of variables, heteroskedastic variances influence the tests for causality and nonlinearity such as time- varying properties. For example, [

This paper investigates the statistical properties of time-varying causality tests when the disturbance terms have SV. The investigated tests include asymptotic tests based on first-order and third-order Taylor approximation and their counterparts with the heteroskedasticity-consistent covariance matrix estimators (HCCME) as introduced by [

Our simulation results provide evidence that asymptotic time-varying causality tests and their counterparts with HCCME over-reject the null hypothesis of no causality in the presence of SV. This implies that their tests tend to yield misleading and unreliable results. In particular, their tests based on third-order Taylor approximation have larger distortions than those based on first-order Taylor approximation. In contrast, we find that time-varying causality tests using wild bootstrap have reasonable empirical sizes and sufficient power. The results of this paper would enable appropriate and reliable time-varying causality tests.

The rest of this paper is organized as follows. Section 2 presents time-varying causality tests. Section 3 provides the size and power properties of tests. Finally, Section 4 concludes the paper.

We consider the following bivariate vector autoregressive system to test for time-varying causality relationship.

where

where

have

when

The null and alternative hypotheses to test for time-varying causality in the system are

If

The regression models for (1) using the first-order and third-order Taylor series approximation are given by

where

where

Testing for time-varying causality is expressed as

The Wald statistics to test for time-varying causality are derived as

where

where

Wild bootstrap is also used for regression models with heteroskedastic variances to obtain reliable results. The method can simply resample heteroskedastic variances like SV. This paper employs the recursive-design wild bootstrap. The testing procedure is as follows.

Step 1. Compute test statistics (11) and (12) by applying (7) and (8) to the data.

Step 2. Estimate the system using the restricted model with

Step 3. Obtain the estimates

Step 4. Generate the bootstrapped sample as

where

Step 5. Compute test statistics (11) and (12), denoted as WB1 and WB3, by applying (7) and (8) to the generated bootstrap sample.

Step 6. Repeat the bootstrap iterations M for steps 4 and 5. We obtain M statistics WB1 and WB3.

Step 7. Compute the bootstrap p-values as follows:

This section conducts Monte Carlo simulations to compare the size and power properties of causality tests under multivariate SV. The nominal size of the tests is 0.05, and we consider sample sizes

We first investigate the size properties based on data generating process (DGP) given as

where

The correlation parameter

We next examine the empirical sizes of tests under multivariate SV. The property of SV is that volatility is influenced by an error and changes stochastically. Multivariate SV allows for the correlation between errors of volatilities.

where

Here,

where

The regression parameter

F0 | F1 | F3 | HC0 | HC1 | HC3 | WB0 | WB1 | WB3 | |
---|---|---|---|---|---|---|---|---|---|

0.046 | 0.053 | 0.066 | 0.068 | 0.071 | 0.138 | 0.048 | 0.042 | 0.030 | |

0.051 | 0.048 | 0.063 | 0.063 | 0.065 | 0.112 | 0.050 | 0.043 | 0.032 | |

0.049 | 0.059 | 0.090 | 0.074 | 0.078 | 0.153 | 0.046 | 0.042 | 0.031 | |

0.048 | 0.062 | 0.076 | 0.061 | 0.071 | 0.123 | 0.053 | 0.040 | 0.033 | |

0.043 | 0.053 | 0.061 | 0.068 | 0.074 | 0.135 | 0.052 | 0.042 | 0.032 | |

0.046 | 0.046 | 0.055 | 0.060 | 0.067 | 0.108 | 0.050 | 0.045 | 0.032 | |

0.051 | 0.059 | 0.095 | 0.068 | 0.071 | 0.151 | 0.047 | 0.040 | 0.030 | |

0.051 | 0.059 | 0.076 | 0.060 | 0.069 | 0.120 | 0.048 | 0.044 | 0.028 |

F0 | F1 | F3 | HC0 | HC1 | HC3 | WB0 | WB1 | WB3 | |
---|---|---|---|---|---|---|---|---|---|

0.048 | 0.054 | 0.072 | 0.075 | 0.074 | 0.145 | 0.050 | 0.041 | 0.027 | |

0.049 | 0.052 | 0.067 | 0.069 | 0.071 | 0.119 | 0.052 | 0.047 | 0.031 | |

0.050 | 0.088 | 0.127 | 0.109 | 0.129 | 0.225 | 0.049 | 0.068 | 0.062 | |

0.050 | 0.086 | 0.125 | 0.108 | 0.134 | 0.235 | 0.052 | 0.073 | 0.066 | |

0.057 | 0.053 | 0.068 | 0.076 | 0.073 | 0.148 | 0.053 | 0.041 | 0.029 | |

0.063 | 0.055 | 0.066 | 0.066 | 0.064 | 0.114 | 0.056 | 0.049 | 0.033 | |

0.131 | 0.140 | 0.203 | 0.083 | 0.124 | 0.235 | 0.039 | 0.058 | 0.053 | |

0.160 | 0.159 | 0.229 | 0.067 | 0.111 | 0.223 | 0.041 | 0.064 | 0.047 |

F0 | F1 | F3 | HC0 | HC1 | HC3 | WB0 | WB1 | WB3 | |
---|---|---|---|---|---|---|---|---|---|

0.049 | 0.053 | 0.066 | 0.062 | 0.068 | 0.134 | 0.048 | 0.039 | 0.028 | |

0.050 | 0.049 | 0.063 | 0.065 | 0.061 | 0.108 | 0.049 | 0.049 | 0.032 | |

0.051 | 0.066 | 0.093 | 0.077 | 0.092 | 0.190 | 0.051 | 0.051 | 0.041 | |

0.054 | 0.071 | 0.088 | 0.043 | 0.091 | 0.166 | 0.053 | 0.057 | 0.051 | |

0.038 | 0.044 | 0.045 | 0.062 | 0.063 | 0.114 | 0.046 | 0.044 | 0.028 | |

0.036 | 0.043 | 0.046 | 0.067 | 0.064 | 0.100 | 0.050 | 0.049 | 0.037 | |

0.072 | 0.078 | 0.099 | 0.067 | 0.095 | 0.189 | 0.045 | 0.057 | 0.041 | |

0.076 | 0.083 | 0.097 | 0.061 | 0.084 | 0.157 | 0.046 | 0.054 | 0.046 |

Asymmetric multivariate SV also results in size distortions for causality tests. We set

We next investigate the power properties based on DGP, given as

where c is the point at which a regime changes from one to another. We set c to

This paper investigated the statistical properties of time-varying causality tests when the errors of variables have multivariate SV. It is important to clarify the statistical properties of time-varying causality tests under SV, because economic variables often have SV and the relationship between them is time-varying. The tests we compared include the standard linear Granger causality and the time-varying causality tests, their tests with HCCME, and their tests using wild bootstrap. Simulation results indicate that time-varying causality tests and their counterparts with HCCME have size distortions

F0 | F1 | F3 | HC0 | HC1 | HC3 | WB0 | WB1 | WB3 | |
---|---|---|---|---|---|---|---|---|---|

0.045 | 0.071 | 0.079 | 0.068 | 0.081 | 0.140 | 0.048 | 0.054 | 0.037 | |

0.049 | 0.162 | 0.117 | 0.065 | 0.168 | 0.182 | 0.048 | 0.150 | 0.077 | |

0.046 | 0.239 | 0.175 | 0.070 | 0.248 | 0.290 | 0.048 | 0.212 | 0.110 | |

0.050 | 0.413 | 0.310 | 0.062 | 0.435 | 0.418 | 0.049 | 0.416 | 0.262 | |

0.047 | 0.246 | 0.183 | 0.068 | 0.259 | 0.303 | 0.047 | 0.218 | 0.122 | |

0.049 | 0.423 | 0.316 | 0.065 | 0.438 | 0.425 | 0.054 | 0.420 | 0.262 | |

0.052 | 0.108 | 0.096 | 0.073 | 0.100 | 0.181 | 0.046 | 0.076 | 0.041 | |

0.063 | 0.368 | 0.295 | 0.075 | 0.380 | 0.429 | 0.056 | 0.341 | 0.218 | |

0.097 | 0.509 | 0.471 | 0.113 | 0.556 | 0.698 | 0.056 | 0.471 | 0.359 | |

0.110 | 0.818 | 0.829 | 0.106 | 0.831 | 0.913 | 0.065 | 0.802 | 0.755 | |

0.104 | 0.519 | 0.525 | 0.109 | 0.563 | 0.732 | 0.065 | 0.486 | 0.394 | |

0.104 | 0.811 | 0.827 | 0.099 | 0.831 | 0.916 | 0.066 | 0.793 | 0.760 | |

0.058 | 0.184 | 0.138 | 0.078 | 0.187 | 0.251 | 0.050 | 0.148 | 0.073 | |

0.067 | 0.737 | 0.612 | 0.080 | 0.752 | 0.741 | 0.056 | 0.733 | 0.531 | |

0.106 | 0.866 | 0.805 | 0.116 | 0.887 | 0.924 | 0.067 | 0.856 | 0.732 | |

0.111 | 0.991 | 0.988 | 0.104 | 0.990 | 0.996 | 0.068 | 0.988 | 0.971 | |

0.112 | 0.874 | 0.847 | 0.120 | 0.896 | 0.945 | 0.065 | 0.860 | 0.755 | |

0.117 | 0.991 | 0.989 | 0.109 | 0.991 | 0.996 | 0.069 | 0.989 | 0.972 |

F0 | F1 | F3 | HC0 | HC1 | HC3 | WB0 | WB1 | WB3 | |
---|---|---|---|---|---|---|---|---|---|

0.129 | 0.148 | 0.200 | 0.083 | 0.133 | 0.243 | 0.041 | 0.062 | 0.052 | |

0.159 | 0.204 | 0.255 | 0.062 | 0.168 | 0.262 | 0.039 | 0.085 | 0.062 | |

0.132 | 0.229 | 0.255 | 0.085 | 0.236 | 0.356 | 0.043 | 0.118 | 0.088 | |

0.162 | 0.300 | 0.334 | 0.069 | 0.281 | 0.391 | 0.040 | 0.153 | 0.115 | |

0.134 | 0.241 | 0.272 | 0.089 | 0.250 | 0.361 | 0.043 | 0.126 | 0.093 | |

0.149 | 0.294 | 0.346 | 0.072 | 0.283 | 0.409 | 0.044 | 0.156 | 0.116 | |

0.145 | 0.174 | 0.227 | 0.097 | 0.148 | 0.275 | 0.042 | 0.069 | 0.058 | |

0.196 | 0.330 | 0.366 | 0.100 | 0.279 | 0.422 | 0.059 | 0.173 | 0.126 | |

0.236 | 0.410 | 0.464 | 0.176 | 0.413 | 0.615 | 0.083 | 0.256 | 0.215 | |

0.282 | 0.561 | 0.660 | 0.169 | 0.531 | 0.713 | 0.101 | 0.401 | 0.378 | |

0.252 | 0.423 | 0.489 | 0.190 | 0.435 | 0.645 | 0.098 | 0.275 | 0.238 | |

0.300 | 0.581 | 0.665 | 0.176 | 0.551 | 0.718 | 0.099 | 0.394 | 0.397 | |

0.141 | 0.214 | 0.250 | 0.094 | 0.190 | 0.330 | 0.042 | 0.108 | 0.077 | |

0.202 | 0.474 | 0.479 | 0.107 | 0.451 | 0.555 | 0.059 | 0.324 | 0.218 | |

0.253 | 0.605 | 0.629 | 0.194 | 0.620 | 0.767 | 0.095 | 0.472 | 0.381 | |

0.286 | 0.791 | 0.824 | 0.175 | 0.761 | 0.855 | 0.105 | 0.660 | 0.582 | |

0.265 | 0.626 | 0.657 | 0.208 | 0.631 | 0.791 | 0.095 | 0.487 | 0.410 | |

0.301 | 0.785 | 0.827 | 0.182 | 0.754 | 0.860 | 0.111 | 0.652 | 0.585 |

under highly persistent SV. Standard linear Granger causality tests perform relatively well under SV but has low power under time-varying causality. In contrast, time-varying causality tests using wild bootstrap have better size properties regardless of type of error. In particular, the time-varying causality test with first-order Taylor approximation and wild bootstrap has better statistical properties. These results indicate that the time-varying causality test with first-order Taylor approximation and wild bootstrap is reliable and useful to test for time-varying causality.

Maki, D. (2016) Properties of Time-Varying Causality Tests in the Presence of Multivariate Stochastic Volatility. Open Journal of Statistics, 6, 777- 788. http://dx.doi.org/10.4236/ojs.2016.65064

*This research was supported by KAKENHI (Grant number: 15K03527).