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1.10 Univariate ARCH and GARCH ModelsIn the time series we have considered so far, the disturbances or errors {Zt} are assumed to be homoskedastic, that is, the variance of Zt is assumed to be independent of t. Autoregressive Conditional Heteroskedasticity (ARCH) models and Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models are used to model changes in the variance of the errors as a function of time. An ARCH process of order q, ARCH( q), is given by (see Engle (1982)) where {t} is an independently distributed Gaussian random sequence with zero mean and unit variance; ht is the conditional variance of Zt conditional on all the information up to time t-1, It-1: GARCH models are generalizations of ARCH models where ht, the conditional variance at time t, depends on earlier variances. That is, a GARCH( p, q) process is given by ( 10.1) with (see Bollerslev (1986)) When p=0 we have an ARCH( q) model; when both p and q are zero, Zt is simply white noise.
An ARCH(q) model and a GARCH(p, q) model are represented in this package by ARCHModel[{0, 1, ..., q}] respectively. Note that since the variance is positive, we usually have 0>0, i≥0, and i≥0 for i>0.
To generate an ARCH or a GARCH process, we can again use the function TimeSeries. Example 10.1 Generate a time series of length 10 according to the ARCH(2) model where . We load the package first. The random number generator is seeded. This generates a time series of length 10 according to the given ARCH(2) model. Out[3]= | |
Note that the initial values of zt ( t=-q+1, ..., -1, 0) are needed to generate a time series from an ARCH( q) model. As in the case of generating a time series from an ARMA model, a third argument can be given to TimeSeries to specify the initial values {z-q+1, z-q+2, ..., z0} (or {{h-p+1, h-p+2, ..., h0}, {z-q+1, z-q+2, ..., z0}} in the case of a GARCH model). If the third argument is omitted, the initial values are all set to zero as was tacitly done in Example 10.1. Example 10.2 Generate a time series of length 10 according to the GARCH( 1, 1) model with 0=0.05, 1=0.5, and 1=0.32 and with the initial values z0=0.1andh0=0.3. This generates a time series of length 10 according to the given GARCH( 1, 1) model. Out[4]= | |
The so-called ARCH- or GARCH-regression model is a regression model with the disturbances following an ARCH process (see ( 10.1) and ( 10.2)) or a GARCH process (see ( 10.1) and ( 10.3)), respectively. That is, where "x""t" is a known column vector that can contain lagged values of Y (i.e., yt-1, yt-2, ... etc.), and "b" is a column vector of unknown parameters. The first term on the right-hand side of ( 10.4) is the conditional mean of Yt; that is, , and the conditional variance of Yt is that of Zt and is given by ( 10.2) or ( 10.3).
An AR-ARCH model has {Yt} following an AR process and the disturbances {Zt} following an ARCH process. It is easy to see that in this case, and "b"={1, 2, ..., p}. If the AR process has a constant term (i.e., nonzero mean), then we have and "b"={, 1, 2, ..., p}. Example 10.3 Generate a time series of length 100 according to the AR(2)-ARCH(1, 1) model with 1=0.2, 2=-0.5, 0=0.06, 1=0.4, and 1=0.7. This series is generated from the given GARCH(1, 1) model. This will be used as the disturbances to generate the AR(2) series below. This generates the AR(2) time series with the given GARCH disturbances. Note that the last entry {0, 0} contains the initial values of y, {y-1, y0}. The series is plotted here. Out[7]= | |
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