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139

Technical Appendices

These appendices have been provided with the aim of making the book self-contained for readers with no background in statistical inference and time series modelling. They are very concise and are only concerned with concepts that are fundamental to other parts of the book. Therefore they should not be viewed as a substitute for the many comprehensive texts on statistics and econometrics that provide a much more detailed and complete coverage of these topics; Greene (1998), Hamilton (1994) and Hendry (1996) are particular favourites of mine and Brooks (2002), will be specific to the econometrics of financial markets.

Appendix 1 outlines the theory of linear statistical models, from the basic principles of regression to the properties of ordinary least squares estimators under different statistical assumptions. Appendix 2 introduces basic concepts for statistical inference: hypothesis testing, confidence intervals and classical statistical tests for the parameters of regression models. Appendix 3 shows how to test the specification of a linear model using a residual analysis, and how to estimate parameters of a linear model when diagnostic tests indicate that the classical assumption of non-spherical disturbances does not hold. Appendix 4 discusses how to apply linear models to financial data that are highly collinear, measured with error, missing on some variables or subject to structural breaks. Appendix 5 shows how to test the predictive ability of statistical models with backtests, using statistical or operational performance measures. Appendix 6 describes the likelihood methods that are used to fit the parameters of many probability distributions.

Appendix 1 Linear Regression

Linear regression models are based on a relationship of the form

r=p,z1 + p2z2+ ... + pkxk.

(A.l.l)

On the left-hand side is the dependent variable, often denoted Y, and on the right there are independent variables, Xx, X2, , Xk. These are also called



explanatory variables or regressors. The coefficients Rj, R2, . . ., p\ are model parameters and each one measures the effect that its associated independent variable has upon Y. It is conventional to assume that X{ = 1, so that the model has coefficients, including a constant P The purpose of regression is to find estimates of the true parameter values and predictions of the dependent variable using data on the dependent and independent variables.

The data, which must consist of an equal number of observations on each variable, may be time series (indexed by the subscript ?), cross-sectional (indexed by the subscript V), or panel data (a mixture of cross-section and time series, indexed by the subscript/, ?). Most of the market models in this book have a statistical rather than an economic foundation and are estimated on daily (or even higher-frequency) time series data.

A.l.l The Simple Linear Model

The simplest case of a linear relationship (A.l.l) is when = 2 and Xx - 1 for all t so that there is a constant term in the model. For convenience the constant is denoted by a (the intercept of the line with the vertical axis), p2 is denoted by P (the slope of the line) and X2 is replaced simply by X. This gives the equation of a line in the two-dimensional plane:

A scatter plot of the data is a plot of each pair (X„ Yt), for t = 1, . . ., T, as a point in the (X, Y) plane. It may indicate the existence of certain statistical relationships between Zand Y. The points (Xt, Yt) will not all lie along a line so one needs to add an error process to the right-hand side of the equation, giving the simple linear model

Consider a linear model for the returns on a single stock Eletrobras, Y, in the Brazilian index Ibovespa, X. Daily closing prices on the stock and index from August 1994 to December 1997 are shown in Figure A. 1, along with a scatter plot of their returns. A line through the scatter plot gives a predicted or fitted value of for each value of X as

where a and P denote the estimates of the line intercept a and slope p. The difference between the actual value of Y and the fitted value of Y at time t is denoted e, and called the residual at time t: e, - Yt - Yt. So the actual data point for Y is the fitted model value plus the residual:

Y, = a+pZ,.

Y, = a + fix, + e,.

(A. 1.2)

Y, = ci+px„

Y, = a + $Xl + el.

(A. 1.3)

Comparing the theoretical model (A. 1.2) with the estimated model (A. 1.3), note that the error process e, is a discrete time continuous state stochastic



Aug-94 Dec-94 May-95 Oct-95 Mar-96 Aug-96 Jan-97 Jun-97 Oct-97

800-

0-1-1-1-1-1-1-1-1-1-

(b) Aug-94 Dec-94 May-95 Oct-95 Mar-96 Aug-96 Jan-97 Jun-97 Oct-97

-0.4J

(c) Ibovespa

Figure A.l Daily closing prices of (a) Ibovespa and (b) Eletrobras, from August 1994 to December 1997; (c) scatter plot of returns to Ibovespa against returns to Eletrobras.



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