Chapter 9 - Modelling volatility and correlation. In this chapter, you will learn how to: Discuss the features of data that motivate the use of GARCH models, explain how conditional volatility models are estimated, test for ‘ARCH-effects’ in time series data, produce forecasts from GARCH models, contrast various models from the GARCH family,. | ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 Chapter 9 Modelling volatility and correlation ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 An Excursion into Non-linearity Land Motivation: the linear structural (and time series) models cannot explain a number of important features common to much financial data - leptokurtosis - volatility clustering or volatility pooling - leverage effects Our “traditional” structural model could be something like: yt = 1 + 2x2t + . + kxkt + ut, or more compactly y = X + u We also assumed that ut N(0, 2). ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 A Sample Financial Asset Returns Time Series Daily S&P 500 Returns for August 2003 – August 2013 ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 Non-linear Models: A Definition Campbell, Lo and MacKinlay (1997) define a non-linear data generating process as one that can be written yt = f(ut, ut-1, ut-2, ) where ut is | ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 Chapter 9 Modelling volatility and correlation ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 An Excursion into Non-linearity Land Motivation: the linear structural (and time series) models cannot explain a number of important features common to much financial data - leptokurtosis - volatility clustering or volatility pooling - leverage effects Our “traditional” structural model could be something like: yt = 1 + 2x2t + . + kxkt + ut, or more compactly y = X + u We also assumed that ut N(0, 2). ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 A Sample Financial Asset Returns Time Series Daily S&P 500 Returns for August 2003 – August 2013 ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 Non-linear Models: A Definition Campbell, Lo and MacKinlay (1997) define a non-linear data generating process as one that can be written yt = 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”. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 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 - switching models - bilinear models ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 Testing for Non-linearity – The RESET Test The “traditional” tools of time series analysis (acf’s, spectral analysis) may find no evidence that we could use a linear model, but the
