back start next


[start] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [ 86 ] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134]


86

8.3.3 Generalization of the Process Equation

In Equation 8.13, all the returns considered by the variance equation are observed over "recent" intervals ending at time t - 1. This strong limitation will be justified by its empirical success, but we can also formulate a more general process equation with observation intervals ending in the past, before t - l,

r, = a, s,

2 (8.25)

<>} = co+ E-=,ELi (ELkrt-i) + E?=i

where

c0>0, cjk>0 for j = l...n, k=]...j;

bt > 0 for i = 1 ... q 4

This generalized process equation considers all returns between any pair of two time points in the period between t - n and t - l. It covers the case of HARCH (all Cjk - 0 except some cj\) as well as that of ARCH and GARCH (all = 0 except some cjj). The last term of the variance equation is a "GARCH term," which may contribute to a parsimonious model formulation. Such a GARCH term may partially model the fading volatility memory of several market components together, but therefore miss the chance of clearly indicating the actual component structure. The main idea of HARCH, taking intervals of different sizes, may also be combined with other ideas from the recent literature about GARCH variations. HARCH can also be regarded as a special case of the Quadratic ARCH model suggested by Sentana (1991). The results obtained for QARCH also apply to HARCH. However, QARCH has been developed in a very different context. Sentana (1991) gives neither a concept of volatilities observed over long intervals nor the stationarity and moment conditions as in Section 8.3.1.

For HARCH, the simple form of Equation 8.13 is preferred. This HARCH is successful in empirical studies, but its computation and estimation is tedious because of the large number of coefficients c-j. This can be strongly improved by introducing the EMA-HARCH process with its partial volatility concept in the next section.

8.3.4 EMA-HARCH Model

In HARCH, the coefficients c\ ... cn are not regarded as free parameters of the model. The heterogeneous market approach leads to a low number of free model parameters, which determine a much higher number n of dependent coefficients modeling the long memory of volatility.

The approach is to keep in the equation only a handful of representative interval sizes instead of keeping all of them, and replace the influence of the neighboring interval sizes by an exponential moving average (EMA) of the returns measured on each interval. This also has the advantage of including a memory of the past



intervals. Let us now introduce the concept of partial volatility ?, which can be regarded as the contribution of the jh component to the total market volatility 2. Here the volatility ? is defined as the volatility observed over an interval of size kj. We can reformulate the HARCH equation in terms of , as follows:

r, = a, et

of = + J2Ci°h (8-27)

where n is now the number of time components in the model. The notation is slightly changed to Cj instead of Cj used in the old formulation to reflect the different meaning of the coefficients. Unlike the standard HARCH, the partial volatility ? has a memory of the volatility of past intervals of size ;. The formal definition of a? is

Pj o-2r , + (1 - ytj) J2n-i (8.28)

where kj is the aggregation factor of the returns and takes n possible values, following the relation

kj = pj~2 + 1 for j > 1 with ki = 1 (8.29)

When p = 4,kj can only take the values 1,2,5, 17,65, 257, 1025, ,4" 2 + 1. For a 5-min data series, this would mean that the horizons would correspond to 5 min, 10 min, 25 min and so on. The construction of Equation 8.29 ensures that the time components (kjs) are economically meaningful. Equation 8.28 is the iterative formula for an exponentially weighted moving average, a special application of Equation 3.51. The volatility memory is defined as a moving average of recent volatility. The depth of the volatility memory is determined by the constant pj\

Pj = e M{ki) (8.30)

where the memory decay time constant of the component is given as the function M of the components volatility interval kj. Without introducing new parameters, M(kj) can be defined as

M(kj) = (kj+ik (8.-31)

This definition is based on the start and the end point of the component interval kj and makes sure that that the EMA kernel is centered at the characteristic time horizon of the component.



It is easy to prove that a necessary stationarity condition for the new formulation is

«

YkjCj < 1 (8.32)

The proof relies on the fact that the expectation of the exponential moving average is the same as the expectation of the underlying time series and that the expectation of cross terms is zero. A similar proof as in Dacorogna et al. (1998a) can be given for the sufficiency of this condition.

We can now define the impact 1 j of each component,

lj = kjCj (8.33)

Every component with a coefficient Cj has an impact Jj on the conditional volatility process. The stationarity condition, Equation 8.32, can be re-formulated using the sum of impacts:

«

Jlj < 1 (8-34)

An iterative formula needs an initial value for aj at the beginning of the time series. A reasonable assumption of that initial value is the unconditional expectation of ajr Here the first value is computed from a data sample prior to the model estimation sample. We term this sample the "build-up" sample.

8.3.5 Estimating HARCH and EMA-HARCH Models

HARCH and EMA-HARCH models are applied to and estimated for empirical FX data here. The time series are homogeneous in #-time, which removes the seasonal pattern of intraday volatility. The time interval is 30 min and the sample includes 10 years of data from January 1, 1987, to December 31, 1996.

For the computation and estimation of both HARCH and EMA-HARCH, we use seven components. For HARCH, the components of Table 8.3 are used. For EMA-HARCH, one component is built from only one time interval but includes, according to Equation 8.28, a moving average that extends over a certain range, which should account for the neighboring time intervals. In fact, there are now two parameters controlling the component definition. The time interval size over which returns are computed, kj, and the range of the moving average, M(kj). Both of them are fixed and the optimization is carried out to solve the Cj parameters. The optimization is implemented by searching for the maximum of the log-likelihood function. The procedure we follow to find the maximum is described at the end of Section 8.2.1. It combines two methods: a genetic algorithm (GA) search (Pictet et al., 1995) and the Berndt, Hall, Hall, and Hausman (BHHH) algorithm (Berndt etal., 1974).



[start] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [ 86 ] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134]