"Frontmatter". In: Analysis of Financial Time Series

A NEW APPROACH TO VAR 291feature is to verify the adequacy of the exceedance rate, the second feature is toexamine the distribution of exceedances, and the final feature is to check the independenceassumption of the model. We discuss briefly some statistics that are usefulfor checking these three features. These statistics are based on some basic statisticaltheory concerning distributions and stochastic processes.Exceedance RateA fundamental property of univariate Poisson processes is that the time durationsbetween two consecutive events are independent and exponentially distributed. Toexploit a similar property for checking a two-dimensional process model, Smithand Shively (1995) propose to examine the time durations between consecutiveexceedances. If the two-dimensional Poisson process model is appropriate for theexceedance times and excesses, the time duration between the ith and (i − 1)thexceedances should follow an exponential distribution. More specifically, lettingt 0 = 0, we expect thatz ti =∫ tit i−11T g(η; k s,α s ,β s )ds, i = 1, 2,...are independent and identically distributed (iid) as a standard exponential distribution.Because daily returns are discrete-time observations, we employ the time durationsz ti = 1 Tt i ∑t=t i−1 +1S(η; k t ,α t ,β t ) (7.38)and use the quantile-to-quantile (QQ) plot to check the validity of the iid standardexponential distribution. If the model is adequate, the QQ-plot should show a straightline through the origin with unit slope.Distribution of ExcessesUnder the two-dimensional Poisson process model considered, the conditional distributionof the excess x t = r t − η over the threshold η is a generalized Pareto distribution(GPD) with shape parameter k t and scale parameter ψ t = α t − k t (η − β t ).Therefore, we can make use of the relationship between a standard exponential distributionand GPD, and define⎧⎪⎨w ti =⎪⎩(r ti − ηln 1 − k tiψ ti)+if k ti ̸= 0r ti − ηψ tiif k ti = 0.−1k ti(7.39)If the model is adequate, {w ti } are independent and exponentially distributed withmean 1; see also Smith (1999). We can then apply the QQ-plot to check the validityof the GPD assumption for excesses.