Does Tail Dependence Make A Difference In the ... - Boston College

4 Estimation
4.1 Modeling the Marginal Dynamics
Even though we concentrate on the dependence structure of tail risk, this is by no means to imply that
marginal distribution is of no importance. As demonstrated by Fermanian and Scaillet (2005), misspecification of
the marginal distribution may lead to spurious results for the dependence measure estimation. Furthermore, the
measures of systemic risk (CoVaR or MES) fundamentally depend on the estimation of marginal VaR or Expected
Shortfall (ES) for either the market or firms’ return. Therefore we must first model the conditional marginal
distributions. 20
The time series of equity data usually exhibit time varying volatility and heavy-tailedness, we model each
marginal series i for simplicity by a univariate AR(1) and GJR-GARCH(1,1,1) model:
Y i,t = φ i 0 + φ i 1Y i,t−1 + e i,t where e i,t = σ i,t ɛ i,t and ɛ i,t ∼ iid(0, 1)
σ 2 i,t = ω i + α i e 2 i,t−1 + γ i e 2 i,t−1I (ei,t−1