_{1}

The present study highlights the drawback of using Sanso, Arago and Carrion’s (2004) AIT-ICSS algorithm in detecting sudden changes in the unconditional volatility when long memory is present in volatility. Simulation experiments show that the AIT-ICSS test is severely oversized and exhibits low power when long memory is present in volatility.

Volatility modeling and forecasting play a crucial role in financial market and have been well-researched in the area of economics and finance because of its importance in capital market theories. Volatility of asset returns highlights the risk or uncertainty associated with the asset and, hence, exploring the behavior of volatility of asset returns is relevant for the pricing of financial assets, risk management, portfolio selection, trading strategies and the pricing of derivative instruments [

However, in some instances, it has been applied even when long memory is present in the conditional volatility series, such as by Malik et al. [

The remainder of this paper is organized as follows: Section 2 introduces the AIT-ICSS algorithm and the associated problem when long memory is incorporated in the series. In Section 3, we undertake Monte Carlo simulation experiments to assess the AIT-ICSS algorithm. Section 4 describes the problem of AIT-ICSS when applied on data and Section 5 concludes with a summary of our main findings.

In the AIT-ICSS test, suppose_{T} is the total number of variance changes in T observations, and

A cumulative sum of squares procedure is used to estimate the number of change points and is given as:

where

where

where C_{T} is the sum of squared residuals from the whole sample period.

It needs to be noted that when we make use of Equation (4.2) we are implicitly assuming that the spectral density at zero of the squared returns is well behaved and, in particular, that as m gets large, the right hand side of Equation (4.2) converges to the following:

In practice, the lag truncation parameter m in Equation (4.2) is estimated using the procedure given in Newey and West [

This section presents the performance of the AIT-ICSS algorithm in the presence of long memory in conditional volatility via simulation. The sample size, the number of Monte Carlo trials and the significance level are taken to be (T = 100, T = 200, T = 500 and T = 1000), 10,000 and 5%^{1} respectively. We consider different specifications of FIGARCH (p, d, q) (i.e., FIGARCH (0, d, 0), FIGARCH (1, d, 0) and FIGARCH (1, d, 1)) models. The FIGARCH (p, d, q) model is given as:

where

We consider the following cases under (6).

FIGARCH (1, d, 1):

FIGARCH (1, d, 0):

FIGARCH (0, d, 0)

Under the null hypothesis, there is no sudden change in^{2} of the series to compute the power of the test. To generate _{t} using the respective FIGARCH specification and keep first part of the series as it is and multiply the second half by (1 + l), where l indicates the percentage change in the unconditional volatility of the series. Here the results for FIGARCH

(1, d, 1) and FIGARCH (1, d, 0) specifications are very similar and due to lack of space we only report the results of all FIGARCH (1, d, 1) specifications. We also report the power of the test for FIGARCH (0, d, 0) for different values of d (0, 0.5 and 0.75).

(0, d, 0) model and FIGARCH (1, d, 1) model (for β = 0.10) when d = 0.

FIGARCH (1, d, 1) (d = 0 & f = 0.975) | FIGARCH (0, d, 0) for different values of d | ||||||||
---|---|---|---|---|---|---|---|---|---|

β | T = 100 | T = 200 | T = 500 | T = 1000 | d | T = 100 | T = 200 | T = 500 | T = 1000 |

0.10 | 0.044 | 0.042 | 0.052 | 0.055 | 0.00 | 0.017 | 0.024 | 0.033 | 0.035 |

0.30 | 0.095 | 0.088 | 0.113 | 0.115 | 0.25 | 0.106 | 0.179 | 0.332 | 0.464 |

0.50 | 0.191 | 0.202 | 0.266 | 0.263 | 0.50 | 0.149 | 0.256 | 0.484 | 0.629 |

0.70 | 0.275 | 0.403 | 0.489 | 0.541 | 0.75 | 0.103 | 0.170 | 0.253 | 0.303 |

FIGARCH (1, d, 1) (d = 0.5 & f = 0.25) | FIGARCH (1, d, 0) (d = 0.5 & f = 0) | ||||||||

β | T = 100 | T = 200 | T = 500 | T = 1000 | β | T = 100 | T = 200 | T = 500 | T = 1000 |

0.10 | 0.129 | 0.214 | 0.366 | 0.520 | 0.10 | 0.179 | 0.299 | 0.491 | 0.652 |

0.30 | 0.145 | 0.311 | 0.484 | 0.646 | 0.30 | 0.213 | 0.391 | 0.621 | 0.756 |

0.50 | 0.190 | 0.365 | 0.615 | 0.753 | 0.45 | 0.235 | 0.407 | 0.649 | 0.806 |

0.70 | 0.156 | 0.366 | 0.649 | 0.793 | - | - | - | - | - |

FIGARCH (1, d, 1) (d = 0.75 & f = 0.05) | FIGARCH (1, d, 0) (d = 0.75 & f = 0) | ||||||||

β | T = 100 | T = 200 | T = 500 | T = 1000 | β | T = 100 | T = 200 | T = 500 | T = 1000 |

0.10 | 0.129 | 0.188 | 0.298 | 0.349 | 0.10 | 0.151 | 0.192 | 0.331 | 0.349 |

0.30 | 0.193 | 0.293 | 0.436 | 0.498 | 0.30 | 0.192 | 0.286 | 0.443 | 0.533 |

0.50 | 0.275 | 0.422 | 0.616 | 0.700 | 0.50 | 0.294 | 0.441 | 0.657 | 0.744 |

0.70 | 0.326 | 0.563 | 0.773 | 0.890 | 0.70 | 0.319 | 0.525 | 0.784 | 0.914 |

FIGARCH (1, d, 1) with d = 0 & f = 0.975 | FIGARCH (0, d, 0) with d = 0 | |||||||
---|---|---|---|---|---|---|---|---|

Lambda | T = 100 | T = 200 | T = 500 | T = 1000 | T = 100 | T = 200 | T = 500 | T = 1000 |

0.10 | 0.301 | 0.412 | 0.515 | 0.526 | 0.036 | 0.064 | 0.206 | 0.415 |

0.30 | 0.346 | 0.466 | 0.583 | 0.615 | 0.170 | 0.482 | 0.939 | 1.000 |

0.50 | 0.373 | 0.502 | 0.665 | 0.732 | 0.413 | 0.880 | 0.999 | 1.000 |

1.00 | 0.479 | 0.681 | 0.826 | 0.900 | 0.857 | 1.000 | 1.000 | 1.000 |

3.00 | 0.774 | 0.887 | 0.943 | 0.976 | 0.996 | 1.000 | 1.000 | 1.000 |

FIGARCH (1, d, 1) with d = 0.5 & f = 0.25 | FIGARCH (0, d, 0) with d = 0.5 | |||||||

Lambda | T = 100 | T = 200 | T = 500 | T = 1000 | T = 100 | T = 200 | T = 500 | T = 1000 |

0.10 | 0.194 | 0.400 | 0.631 | 0.798 | 0.171 | 0.274 | 0.462 | 0.600 |

0.30 | 0.290 | 0.497 | 0.759 | 0.854 | 0.161 | 0.278 | 0.517 | 0.625 |

0.50 | 0.467 | 0.713 | 0.859 | 0.919 | 0.209 | 0.397 | 0.607 | 0.745 |

1.00 | 0.773 | 0.921 | 0.984 | 0.998 | 0.374 | 0.593 | 0.783 | 0.875 |

3.00 | 0.984 | 0.999 | 1.000 | 1.000 | 0.693 | 0.895 | 0.976 | 0.989 |

FIGARCH (1, d, 1) with d = 0.75 & f = 0.05 | FIGARCH (0, d, 0) with d = 0.75 | |||||||

Lambda | T = 100 | T = 200 | T = 500 | T = 1000 | T = 100 | T = 200 | T = 500 | T = 1000 |

0.10 | 0.336 | 0.522 | 0.765 | 0.900 | 0.109 | 0.168 | 0.259 | 0.303 |

0.30 | 0.371 | 0.547 | 0.783 | 0.884 | 0.111 | 0.157 | 0.283 | 0.382 |

0.50 | 0.458 | 0.636 | 0.812 | 0.914 | 0.142 | 0.220 | 0.372 | 0.465 |

1.00 | 0.634 | 0.788 | 0.894 | 0.937 | 0.219 | 0.367 | 0.547 | 0.635 |

3.00 | 0.925 | 0.984 | 0.992 | 0.997 | 0.451 | 0.645 | 0.824 | 0.868 |

sudden change in the unconditional volatility of 100% or more and the power of the test increases as sample size increases even for sudden change of 30% or more. On the other hand, for non-zero d, we observe good power for other specifications only when there is sudden change of 300% or more for sample size greater than 200. The FIGARCH (0, 0.75, 0) model exhibits the worst power of all the specifications under study.

This indicates that when long memory is present in the conditional volatility of a time series, the AIT-ICSS algorithm breaks down and is no longer suitable for detecting sudden changes in its unconditional volatility.

We apply the AIT-ICSS algorithm on weekly data of four major benchmark series having long memory in the conditional volatility. The data period is from January 1996 to March 2015.

S&P_500 | FTSE_100 | Nikkei_225 | CAC_40 | |
---|---|---|---|---|

d | 0.527^{#} | 0.512^{#} | 0.424^{#} | 0.588^{#} |

SE | (0.162) | (0.188) | (0.142) | (0.182) |

^{#} and ^{*} mean significant at 1% and 5% level of significance. SE represents the standard error.

Index (period) | Number of change points | Time period | Events |
---|---|---|---|

S&P_500 | 4 | 05-05-1982 to 03-08-1988 10-08-1988 to 22-04-1992 29-04-1992 to 15-07-1998 22-07-1998 to 11-07-2007 18-07-2007 to 28-11-2012 | - - - Russian financial crisis Global financial crisis |

FTSE_100 | 3 | 04-01-1984 to 08-07-1998 15-07-1998 to 16-03-2003 23-03-2003 to 17-10-2007 24-10-2007 to 28-11-2012 | - - - Global financial crisis |

Nikkei_225 | 0 | - | No break detected |

CAC_40 | 4 | 06-01-1988 to 05-08-1998 12-08-1998 to 10-07-2002 17-07-2002 to 19-03-2003 16-03-2003 to 09-01-2008 16-01-2008 to 28-11-2012 | - Russian financial crisis - - Global financial crisis |

Our simulation experiment indicates that the AIT-ICSS test is severely oversized and exhibits low power when long memory is present in the volatility of a time series. The empirical analysis also indicates that most of the breaks detected by the AIT-ICSS test cannot be related to major macroeconomic and political events and, hence, are probably spurious.

Dilip Kumar, (2016) On Detecting Sudden Changes in the Unconditional Volatility of a Time Series. Theoretical Economics Letters,06,256-261. doi: 10.4236/tel.2016.62028