7.5  Further Reading

Jarrow (1998), Mills (1999), Franses and van Dijk (2000), Alexander (2001), and Tsay (2013) discuss time series analysis for financial markets.

Bai and Shi (2011) review a growing literature on large covariance matrix estimation techniques that improve on the direct use of sample covariance matrices.

Opschoor et al. (2013) extend DCC-GARCH to include indices that reflect evolving financial conditions.

Empirical studies suggest that market correlations are more pronounced during periods of heightened volatility. Campbell, Koedijk and Kofman (2002), Cotter and Longin (2007), and Mittnik (2014) propose the use of “tail” correlations implied by the historicalvalue-at-risk of standardized portfolios calculated at various confidence levels.

Sometimes efforts to construct a covariance matrix yield a result that is far from positive semidefinite. This might occur if a risk manager manually overrides one or more calculated covariances in the matrix. It might also occur if, due to data availability issues, covariances are calculated with different amounts of historical data for certain key factor pairs. A standard solution is to find the “nearest” covariance matrix to the non-positive-semidefinite matrix. This is an optimization problem for which there is a growing literature. See Finger (1997), Rebonato and Jäckel (1999), Higham (2002), Li, Li and Qi (2010) and Qi and Sun (2010).