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] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166]


42

0.6-1-1-1-1-1-1-1-1-1-1

(b) Jan-97 Apr-97 Jul-97 Oct-97 Jan-98 Apr-98 Jul-98 Oct-98 Jan-99

Figure 4.15 (a) GARCH(1, 1) volatilities of NYMEX and KCBOT natural gas futures; (b) optimal time-varying hedge ratio.

- . + E/( for / = 1, . . ., n.

The multivariate GARCH models that are described below all assume that the conditional distributions of returns are normal and differ only in the form assumed for the conditional variance and covariance equations.

The diagonal vech model described in §4.5.1 imposes severe cross equation restrictions because each equation is a separate GARCH(1,1). The -dimensional vech model is written as

vech(H() = A + Bvech& ,5( ,) + Cvech(H, ,), (4.22)

200 t-1



In some markets these restrictions can lead to substantial differences between the vech estimates and those from other multivariate GARCH parameteriza-tions, so the vech model should be employed with caution

where is the conditional covariance matrix at time f and vech(H() is the vector that stacks all the elements of the covariance matrix. In the bivariate diagonal vech \t = (zu ,e2(), A = ( co2, co3), = diag (ab a2, a3) and = diag(P,, p2, p3), but more general forms of the coefficient matrices are possible, provided that restrictions are imposed on their parameters to ensure positive definiteness. In some markets these restrictions can lead to substantial differences between the vech estimates and those from other multivariate GARCH parameterizations, so the vech model should be employed with caution.

A general parameterization that involves the minimum number of parameters while imposing no cross equation restrictions and ensuring positive definiteness for any parameter values is the BEKK model, after Baba, Engle, Kraft and Kroner who wrote the preliminary version of Engle and Kroner (1993). The conditional covariance matrix has the following multivariate GARCH(1,1) parameterisation:

H, = AA + B , ,B + CH,-iC,

(4.23)

where , and are nxn matrices and A is triangular.

The BEKK parameterization for a bivariate model involves 11 parameters, only two more than the vech parameterization, but for higher-dimensional systems the extra number of parameters in the BEKK increases, and completely free estimation becomes very difficult indeed. Often it is necessary to impose restrictions and so reduce the number of parameters to estimate (Bollerslev et cti, 1994). It may be assumed that all elements of are the same if the market reaction coefficients of all variables in the system are assumed to be identical. If persistence in volatility and correlation are the same throughout, all elements of could be imposed to be identical. This is the scalar BEKK model. The forecasting performance of a scalar BEKK is often inferior to that of the diagonal BEKK model, where and are assumed to be diagonal (Engle, 2000b). Engle and Mezrich (1995) is a very accessible reference on multivariate GARCH model parameterization, and a general framework for asymmetric multivariate GARCH models is described in Kroner and Ng (1998).

4.5.3 Time-Varying Covariance Matrices Based on Univariate GARCH Models

The conditional variance of an -dimensional multivariate process is a time-series oi nxn covariance matrices, one matrix for each point in time, denoted H,. Each of the n(n+ l)/2 distinct elements of these covariance matrices has its own GARCH model. If there are just three parameters in each conditional variance or conditional covariance equation, as in the diagonal vech and BEKK models described above, there will be 1n(n + l)/2 parameters in total, plus the n parameters, one in each conditional mean equation. Thus the three-variate GARCH model has a minimum of 21 parameters. This is already a



large number, but just think of a 10-dimensional system, with at least 175 parameters to estimate!

It is therefore not surprising that estimation of multivariate GARCH models can pose problems. Since all parameters are estimated simultaneously the convergence problems outlined in §4.3 can become insurmountable. Even in relatively low-dimensional systems parameterizations of multivariate GARCH models should be as parsimonious as possible. The lack of robustness to alternative parameterizations in conjunction with the inevitable computational difficulties in systems with more than five or so variables, even with the most parsimonious of parameterizations, casts considerable doubt on the practicalities of full multivariate GARCH for modelling large covariance matrices. However, there are some approximations that allow multivariate GARCH covariance matrices H, to be generated by univariate GARCH models alone. Three of these models are described here.

The lack of robustness to alternative parameterizations in conjunction with the inevitable computational difficulties casts considerable doubt on the practicalities of full multivariate GARCH for modelling large covariance matrices

The first model, introduced by Bollerslev (1990), approximates the time-varying covariance matrix as a product of time-varying volatilities and a correlation matrix that does not vary over time. The constant GARCH correlation may be written:

H, = D,CD„

where D, is a diagonal matrix of time-varying GARCH volatilities, and is the constant correlation matrix. Individual return data are used to estimate GARCH volatilities, using one of the models described in §4.2, and the correlation matrix is estimated by taking equally weighted moving averages over the whole data period.

Recently this model has been generalized by Engle (2000b) to the case where the correlation matrix is time-varying. Obviously it would defeat the point if now a multivariate GARCH model were used to estimate this time-varying correlation matrix, so to keep the model as simple as possible, Engle advocates using a GARCH(1,1) model with the same parameters for all the elements of the correlation matrix - or, as in the RiskMetrics matrices (§7.4) using an EWMA with the same smoothing constant throughout.

Another method for generating -dimensional GARCH covariance matrices using only univariate GARCH is to use a capital asset pricing model framework (§8.1). These factor GARCH models allow all the individual asset volatilities and correlations to be generated from univariate GARCH models of the market volatility and the specific risk (Engle et al., 1990). In the CAPM individual asset returns are related to market returns Xt by the regression equation

= H,

for / - 1, 2,

(4.24)

Factor GARCH models allow all the individual asset volatilities and correlations to be generated from univariate GARCH models of the market volatility and the specific risk



[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] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166]