Building Econometric Models

Chapter 9

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Modelling volatility and correlation

An Excursion into Non-linearity Land

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• 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).

A Sample Financial Asset Returns Time Series

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Daily S&P 500 Returns for August 2003 – August 2013

Non-linear Models: A Definition

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• 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”.

Types of non-linear models

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• 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, e.g.
• - ARCH / GARCH
• - switching models
• - bilinear models