_{1}

By means of Monte Carlo experiments using the weighted bootstrap, we evaluate the size and power properties in small samples of Chow and Denning’s [1] multiple variance ratio test and the automatic variance ratio test of Choi [2]. Our results indicate that the weighted bootstrap tests exhibit desirable size properties and substantially higher power than corresponding conventional tests.

The foundation of the efficient market hypothesis lies in the ground-breaking works of Bachelier [

The existing literature provides several methods to investigate whether a given time series is a martingale or not. The variance ratio test is one of the most commonly employed procedures to study this property of the time series. The Lo and Mac Kinlay’s [

In this paper, the weighted bootstrap procedure is proposed as an alternative to improve the small sample properties of the Chow and Denning [

Section 2 presents the methodology used in this study. Section 3 presents the results of the Monte Carlo experiments. Section 4 provides conclusion of the study.

Suppose P_{t} is an asset price at time t, where_{t} be ln(P_{t}), the log price series. The first order autoregressive model is given by:

where _{t}) is k times the variance of a one-period return.

Suppose y_{t} is an asset return at time t (

where

which follows the standard normal distribution asymptotically if y_{t} is a martingale difference sequence, where

Chow and Denning [_{i}) = 1 for

The decision to reject the null hypothesis is based on the maximum of the absolute value of the individual variance ratio statistics.

Choi [

where

and

and

where m(z) is the quadratic spectral kernel. Choi [_{y}(0), where f_{y}(.) denotes the normalized spectral density of the time series {y_{t}}. Choi [_{0}: 2πf_{y}(0) = 1) the AVR(k) statistic is defined as

as_{t} is IID (independent and identically distributed) with finite fourth moment [

The following steps define the procedure of using the weighted bootstrap on variance ratio test statistics:

1. Find normalized returns

2. For each t, draw a weighting factor _{t}.

3. Form a bootstrap sample of T observations

4. Calculate the required test statistic (suppose VRS^{*}(k^{*})), the VRS statistic obtained from

5. Repeat steps 1 to 4 sufficiently many (say m) times to form a bootstrap distribution of the test statistics

The two tailed p-value of the test can be obtained as the proportion of absolute values of

To evaluate the quality of Chow and Denning’s [^{*} test, we set holding periods (k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) = (2, 5, 10, 20, 40, 80). The following models are considered to evaluate the size properties of the tests used:

Model 1: GARCH(1,1)

Model 2: Stochastic volatility

In these model, we use two types of random errors: the standard normal distribution (ε_{t} ~ N(0,1)) and the Student-t distribution with 3 degree of freedom (as suggested by White (2000)). To evaluate the power properties of the MVR and AVR test statistics, we use model 3 and model 4 which take the error term from model 1 and model 2 (that is, u_{t} term from model 1 and model 2 also act as error term in model 3 and model 4).

Model 3: AR(1) model

Model 4: Long memory (ARFIMA (0, 0.1, 0)) model

For all the cases, the number of Monte Carlo trials is set to 1000 and the significance level is set at 5%. In the following tables for evaluating size and power properties, GARCH_N and SV_N represents model 1 and model 2 with error term from Standard Normal distribution; and GARCH_t and SV_t represents model 1 and model 2 with error term from the Student-t distribution with 3 degrees of freedom. To modify the size and power properties of MVR and AVR tests for smaller samples (N = 100, 500 and 1000), we propose the weighted bootstrap procedure. The number of bootstrap iterations is set to 500.

^{*} and AVR^{*} test. We find severe size distortion across all data generating processes for all sample sizes for MVR and AVR test. But even after applying weighted bootstrap procedure on MVR and AVR test statistics, we find size distortion for sample sizes of 100 and 500. We find the size distortion to be less of a problem for MVR^{*} and AVR^{*} test statistics for a sample size of 1000.

GARCH_1 | GARCH_2 | SV_1 | SV_2 | |
---|---|---|---|---|

MVR | ||||

100 | 0.020 | 0.026 | 0.025 | 0.033 |

500 | 0.029 | 0.027 | 0.037 | 0.036 |

1000 | 0.032 | 0.028 | 0.041 | 0.047 |

AVR | ||||

100 | 0.019 | 0.028 | 0.012 | 0.006 |

500 | 0.020 | 0.064 | 0.007 | 0.016 |

1000 | 0.024 | 0.111 | 0.021 | 0.006 |

MVR^{*} | ||||

100 | 0.043 | 0.032 | 0.049 | 0.026 |

500 | 0.054 | 0.061 | 0.048 | 0.052 |

1000 | 0.054 | 0.053 | 0.054 | 0.049 |

AVR^{*} | ||||

100 | 0.057 | 0.071 | 0.052 | 0.053 |

500 | 0.047 | 0.069 | 0.056 | 0.059 |

1000 | 0.044 | 0.043 | 0.043 | 0.046 |

MVR^{*} and AVR^{*} represent the MVR and AVR tests with weighted bootstrap.

^{*} and AVR^{*} tests when model 3 (AR(1) model) is the alternative. We find a significant improvement in the power properties of MVR and AVR tests by the application of the weighted bootstrap procedure on the conventional tests used. When we compare the power properties of MVR^{*} and AVR^{*} test statistics, we can see that the power of AVR^{*} test statistic is higher than that of MVR^{*} test statistic for most of the cases against the AR(1) model alternative.

^{*} and AVR^{*} tests when model 4 (long memory) is employed as the alternative. We find higher power for MVR^{*} and AVR^{*} test statistics for sample size 1000. For other sample sizes, we find improvement in power properties of MVR and AVR test.

GARCH_1 | GARCH_2 | SV_1 | SV_2 | |
---|---|---|---|---|

MVR | ||||

100 | 0.082 | 0.071 | 0.067 | 0.076 |

500 | 0.285 | 0.272 | 0.407 | 0.398 |

1000 | 0.542 | 0.445 | 0.753 | 0.738 |

AVR | ||||

100 | 0.089 | 0.105 | 0.077 | 0.060 |

500 | 0.345 | 0.343 | 0.317 | 0.352 |

1000 | 0.582 | 0.566 | 0.636 | 0.650 |

MVR^{*} | ||||

100 | 0.113 | 0.073 | 0.099 | 0.091 |

500 | 0.389 | 0.355 | 0.476 | 0.459 |

1000 | 0.650 | 0.520 | 0.808 | 0.749 |

AVR^{*} | ||||

100 | 0.157 | 0.201 | 0.185 | 0.186 |

500 | 0.418 | 0.419 | 0.553 | 0.561 |

1000 | 0.641 | 0.579 | 0.811 | 0.820 |

Long Memory | GARCH_1 | GARCH_2 | SV_1 | SV_2 |
---|---|---|---|---|

MVR | ||||

100 | 0.128 | 0.134 | 0.151 | 0.161 |

500 | 0.654 | 0.685 | 0.407 | 0.398 |

1000 | 0.948 | 0.951 | 0.936 | 0.957 |

AVR | ||||

100 | 0.137 | 0.144 | 0.127 | 0.145 |

500 | 0.563 | 0.555 | 0.569 | 0.588 |

1000 | 0.888 | 0.889 | 0.885 | 0.897 |

MVR^{*} | ||||

100 | 0.159 | 0.167 | 0.166 | 0.166 |

500 | 0.750 | 0.732 | 0.742 | 0.722 |

1000 | 0.950 | 0.963 | 0.956 | 0.963 |

AVR^{*} | ||||

100 | 0.257 | 0.272 | 0.285 | 0.288 |

500 | 0.748 | 0.755 | 0.731 | 0.750 |

1000 | 0.952 | 0.951 | 0.957 | 0.945 |

In this study, we evaluate the small sample size and power properties of the Chow and Denning’s [

Dilip Kumar, (2016) Weighted Bootstrap Approach for the Variance Ratio Tests: A Test of Market Efficiency. Theoretical Economics Letters,06,426-431. doi: 10.4236/tel.2016.63048