Part III

Statistical Models for Financial Markets

Part III takes an econometric approach to modelling relationships between financial asset prices. Chapter 11 introduces time series models, where relationships are modelled without the confines of economic or financial theories. These models aim to find the most appropriate statistical model for the data and use this model for prediction. In this chapter autoregressive moving average (ARMA) models, unit root tests and Granger causality are introduced as essential background for the following chapter, on cointegration.

Cointegration refers not to co-movements in returns, but to co-movements in asset prices: if spreads are mean-reverting, asset prices are tied together in the long term by a common stochastic trend, and we say that the prices are cointegrated. Since the seminal work of Engle and Granger (1987), cointegration has become the prevalent tool for applied economic analysis. Cointegration also arises naturally in many financial systems: within term structures, between spot and futures prices, and between international equity and bond market indices. The application of cointegration to modelling these relationships is explored in Chapter 12, and following this a new long-short equity hedge fund model based on cointegration is described in some detail.

The final chapter is about high-frequency data: time series properties and nonlinear prediction models based on neural networks and nearest neighbour algorithms. This chapter has been quite selective in its coverage, to present the concepts from an advanced academic literature that are most relevant to practitioners.

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Time Series Models

Some financial models such as the capital asset pricing model and the arbitrage pricing model described in Chapter 8 are well established for use with time series data. However, there is much to be said for analysing data without imposing constraints that are implied by a theory. The aim of time series analysis is to find the most appropriate statistical model for the data and to use this model for prediction. In this way the variables are allowed to speak for themselves, without the confines of economic or financial theories.

This chapter is about the statistical models that are used for the analysis of financial time series. In any type of data analysis it is always necessary to assume some sort of model. We have already encountered several time series models in Chapters 3 and 4. Statistical volatility and correlation models only assume that the data were generated by a process with given statistical properties; they assume nothing about the behaviour of financial markets.

The chapter begins by describing the statistical properties of univariate time series, and particular attention is drawn to the fundamentally different models that apply to stationary and non-stationary processes. In most financial markets returns are stationary and prices are non-stationary. Unit root tests that distinguish between stationary and non-stationary data are described in §11.1.4, and §11.1 concludes by explaining why it is not usually appropriate to transform prices by taking deviations from a trend line.

Section 11.2 describes autoregressive moving average (ARMA) models for stationary univariate time series and derives their autocorrelation properties. If returns are autocorrelated then it will be possible to forecast market prices at least to some extent, although the degree of forecastability may not be sufficient to make trading profits. Section 11.3 shows how to identify the right ARMA model for the data, estimate it and then use it for forecasting. The Box-Pierce test for autocorrelation is described in §11.3.2, and in §11.3.3 the testing down methodology that is normally used to specify the final form of the ARMA model is described.

Section 11.4 extends the analysis to multivariate systems of time series and introduces the concept of joint stationarity that is a necessary property for statistical models of correlation. It concludes with a discussion of Granger

Variables are allowed to speak for themselves, without the confines of economic or financial theories

causality, which is causality in the sense that one series leads or lags another (Granger, 1969). Evidence of Granger causality provides many insights into the dynamics of returns in different markets and may be used as a basis for predictive time series models.

11.1 Basic Properties of Time Series

A time series {y,} is a discrete time continuous state process where the variable is identified by the value that it takes at time t, denoted yt. For example, a time trend, yt - t, is a very simple deterministic time series. A basic stochastic time series is white noise, y, - e„ where e, is an independent and identically distributed (i.i.d) variable with mean 0 and variance a2 for all t, written 8, ~ i.i.d.(0, a2). A special case is Gaussian white noise, where the ef are independent and normally distributed variables with mean 0 and variance a2 for all t, written e, ~ NID(0, a2). Usually time is taken at equally spaced intervals1 from -oo to +oo and the finite sample size T of data on is for t = 1, 2, 3, . . ., T.

In financial markets the modelling procedures for return data and for price data are different. To understand why, one needs to draw the basic distinction between stationary and non-stationary time series. Daily return data on most financial markets are generated by stationary processes and consequently returns are mean-reverting in the sense defined in §11.1.2. In fact they are often rapidly mean-reverting since there is very little autocorrelation in many financial market returns. The statistical concepts and methods that apply to return data do not apply to price data. For example, volatility and correlation are concepts that only apply to stationary processes. It makes no sense to try to estimate volatility or correlation on price data. Daily (log) price data are commonly assumed to be generated by a non-stationary stochastic process. Random walks are non-stationary processes that are very often applied to log prices, or to prices themselves.

11.1.1 Time Series Operators

It will facilitate the analysis to introduce some operators that are specific to time series and to provide an understanding of the way that standard operators such as addition and multiplication are applied to time series. The first difference operator is defined by

Ar, = V, - v, ,.

Note that powers of the first difference operator, such as

A\v, = Av, - Av, , = v, - 2y, i +y,-2,

Time deformation mappings are possible, see §13.1.4.

The modelling procedures for return data and for price data are different. To understand why, one needs to draw the basic distinction between stationary and non-stationary time series