4.9.2 Orthogonal GARCH
Ding (1994), Alexander and Chibumba (1997), and Klaassen (2000) propose an n-dimensional GARCH model based upon the principal components of a constant unconditional covariance matrix Σ of tW. Set
[4.82]
where v is an n×n matrix with columns equal to orthonormal eigenvectors of Σ. The tDi are modeled as conditionally uncorrelated univariate GARCH(p,q) processes:
[4.83]
[4.84]
By construction, the tDi are necessarily unconditionally uncorrelated. The model makes a simplifying assumption that they are also conditionally uncorrelated. Essentially, orthogonal GARCH is CCC-GARCH with a change of coordinates. Instead of assuming that tW has a conditional correlation matrix that is constant over time, it assumes that tD does.
An orthogonal GARCH process is estimated from a time series {–αw, … , –1w, 0w} by first constructing the unconditional covariance matrix Σ. This can be set equal to the data’s sample covariance matrix. From this, construct v. Decompose data points:
[4.85]
to obtain a time series {–αdi, … , –1di, 0di} for each i. Univariate GARCH(p,q) processes [4.83] are estimated from these using a separate maximum likelihood analysis for each.