文档介绍：8-1Chapter 8Modelling volatility and correlation8-21 An Excursion into Non-linearity Land?Motivation: the linear structural (and time series) models cannot explain a number of important mon to much financial data- leptokurtosis:尖峰性,厚尾- volatility clustering or volatility pooling波动性集群- leverage effects与价格同幅上升相比,价格大幅下降后,波动性上升较多?Our “traditional” structural model could be something like:yt = ?1 + ?2x2t + ... + ?kxkt + ut,or y = X?+ also assumed ut? N(0,?2).8-3A Sample Financial Asset Returns Time SeriesDaily S&P 500 Returns for January 1990 – December 1999-----4Non-linear Models: A Definition?Campbell, Lo and MacKinlay (1997) define a non-linear data generating process as one that can be writtenyt= f(ut, ut-1, ut-2, …)where ut is an iid error term and f is a non-linear function.?They also give a slightly more specific definition as yt= g(ut-1, ut-2, …)+ ut?2(ut-1, ut-2, …) where g is a function of past error terms only and ?2 is a variance term.?Models with nonlinear g(?) are “non-linear in mean”, while those with nonlinear ?2(?) are “non-linear in variance”. ?Models can be linear in mean and variance(CLRM,ARMA), or linear in mean but non-linear in variance(GARCH).8- Types of non-linear models?The linear paradigm is a useful one. Many apparently non-linear relationships can be made linear by a suitable transformation. On the other hand, it is likely that many relationships in finance are intrinsically non-linear.?There are many types of non-linear models, .- ARCH / GARCH for modelling and forecasting volatility- switching models : allow the behaviour of a series to follow different processes at different points in time.- bilinear models8- Testing for Non-linearity?The “traditional” tools of time series analysis (acf’s, spectral analysis) may find no evidence that we could use a linear model, but the data may still not be independent.?General test (Portmanteau多用途t