The aim of this book is to provide the researcher in financial markets with the techniques necessary to undertake the empirical analysis of financial time series. To accomplish this aim we introduce and develop both univariate modelling techniques and multivariate methods, including those regression techniques for time series that seem to be particularly relevant to the finance area.

Why do we concentrate exclusively on time series techniques when, for example, cross-sectional modelling plays an important role in empirical investigations of the capital asset pricing model (CAPM; see, as an early and influential example, Fama and MacBeth, 1973)? Moreover, why do we not address the many issues involved in modelling financial time series in continuous time and the spectral domain, although these approaches have become very popular, for example, in the context of derivative asset pricing? Our answer is that, apart from the usual considerations of personal expertise and interest plus constraints on manuscript length, it is because time series analysis, in both its theoretical and empirical aspects, has been for many years an integral part of the study of financial markets.

The first attempts to study the behaviour of financial time series were undertaken by financial professionals and journalists rather than by academics. Indeed, this seems to have become a long-standing tradition, as, even today, much empirical research and development still originates from the financial industry itself. This can be explained by the practical nature of the problems, the need for specialised data and the potential gains from such analysis.

As we have seen in previous chapters, financial time series often appear to be well approximated by random walks. The relationship between random walks and the theory of efficient capital markets was briefly discussed in chapter 1, where it was argued that the random walk assumption that asset price changes are independent is usually too restrictive to be consistent with a reasonably broad class of optimising models; what is in fact required is that a variable related to the asset price be a martingale (see, for example, Andreou, Pittis and Spanos, 2001).

Martingales and random walks are discussed formally in section 5.1, with tests of the random walk hypothesis being the subject of section 5.2. The relaxation of the assumption that changes in a time series must be independent and identically distributed allows the possibility of examining non-linear stochastic processes, and the remainder of the chapter therefore introduces various non-linear models that are now used regularly in analysing the volatility of financial time series – a fundamental concern of financial modellers. Ways of measuring volatility itself are introduced in section 5.3, before formal stochastic models are considered. Stochastic volatility (SV) models are discussed in section 5.4, ARCH processes in section 5.5, further models related to ARCH in section 5.6, and section 5.7 briefly looks at the forecasting performance of alternative volatility models.

The following series can be obtained from http://lboro.ac.uk/departments/ec/cup.

1. RS: 91-day Treasury bill rate, monthly, March 1952 to December 2005 (648 observations).

2. R20: Yield on 20-year UK gilts, monthly, March 1952 to December 2005 (648 observations).

3. RSQ: 91-day Treasury bill rate, quarterly, 1952Q1 to 2005Q4 (216 observations).

4. R20Q: Yield on 20-year UK gilts, quarterly, 1952Q1 to 2005Q4 (216 observations).

5. RSQREAL: Real 91-day Treasury bill rate, quarterly, 1952Q1 to 2005Q (216 observations).

6. FTAPRICE: FTA All Share price index, monthly, January 1965 to December 2005 (492 observations).

7. FTADIV: FTA All Share dividend index, monthly, January 1965 to December 2005 (492 observations).

8. FTARET: FTA All Share nominal returns, monthly, January 1965 to December 2005 (492 observations).

9. RPI: U.K. Retail Price Index, monthly, January 1965 to December 2005 (492 observations).

10. USTB: U.S. 3-month Treasury bill rate, monthly, April 1953 to February 2005 (623 observations).

11. GIASE: Absolute returns on the General Index of the Athens Stock Exchange, daily, 1 June 1998 to 10 September 1998 (12,117 observations).

12. NORD: Nord Pool Exchange electricity prices, daily, 22 March 2002 to 3 December 2004 (988 observations)

13. VIX: VIX prices, daily, January 1990 to September 2005 (4130 observations)

14. EXCHD: Dollar/sterling exchange rate, daily, 1974 to 1994 (5192 observations).

15. EXCHQ: Dollar/sterling exchange rate, quarterly, 1972Q1 to 1996Q4 (100 observations).

16. S&P500: S&P 500 index, annual, 1871 to 2006 (136 observations).

17. S&P500R: S&P 500 real returns, annual 1872 to 2006 (135 observations).

18. S&P500D: S&P 500 index, daily, 1928 to 1991 (17,054 observations).

19. FT30: FT 30 index, daily, 1935 to 1994 (15,003 observations).

20. FTSE100: FTSE 100 index, weekly, 1984 to 1993 (521 observations).

21. CTLD: Courtaulds share price, weekly, 1984 to 1993 (521 observations).

22. LGEN: Legal and General share price, weekly, 1984 to 1993 (521 observations).

23. PRU: Prudential share price, weekly, 1984 to 1993 (521 observations).

One of the earliest problems that faced time series analysts was the modelling of long-term persistence, or trends, in the observed data. A major motivation for studying the trend properties of time series was the belief that long-term components should be removed in order to uncover any remaining short-term regularities. Until the 1980s the dominant view was that these properties could be well described by the deterministic functions of a time index. This approach was pioneered by Jevons in the mid-nineteenth century and was popularised fifty years later by Persons with the celebrated ‘Harvard A-B-C curve’ methodology of stock market prediction (see Samuelson, 1987, and Morgan, 1990). At present, the dominant paradigm in economic and financial time series modelling builds upon the random walk model, first introduced into finance by Bachelier (1900), where, as we have seen in chapter 2, stochastic trends arise from the accumulation of random changes.

These two approaches constitute the main apparatus for the analysis of non-stationary time series – i.e. of time series that, broadly speaking, wander without bound and origin and do not have well-defined or finite unconditional moments. The first approach deals with trend stationary processes, a class of non-autonomous processes, which can be made stationary by removing a deterministic trend in the form of a polynomial in time. The second approach deals with random walks, which are produced from the accumulation (or integration in continuous time) of white-noise random variables.

The choice of unconditional distribution is something that has always puzzled academics and practitioners in finance. The standard assumption since the 1960s has been that financial prices are geometric Brownian motions and, therefore, logarithmic returns follow a normal distribution. This assumption has profound implications for a variety of theoretical and practical problems in finance, as expected returns and risks in a multivariate normal financial world can be fully described probabilistically using just means, variances and covariances. Much of finance analysis, such as primary and derivative asset pricing, portfolio optimisation and risk management, is built upon the assumption of normally distributed returns. The normal distribution also has important implications for econometric and statistical analysis, since most of the underlying theory is parametric and has been developed on the basis of normality.

The popularity of the normal distribution is easily justifiable from a practical point of view because it offers tractability and computational simplicity. Furthermore, the normality assumption is supported theoretically by the central limit theorem (CLT), which states that the sum of iid random variables with finite mean and variance will asymptotically converge to a normal distribution. Under these assumptions, the normal distribution will offer the best approximation to empirical return distributions in samples of reasonable size.

The analysis of the general linear regression model forms the basis of every standard econometrics text and we see no need to repeat such a development here. Models relating to financial time series, however, often cannot be analysed within the basic framework of ordinary least squares regression, or even its extensions incorporating generalised least squares or instrumental variables techniques. This chapter therefore develops a general theory of regression, based on the original work of Hansen (1982), White (1984) and White and Domowitz (1984), that builds upon the univariate time series techniques of the previous chapters and is applicable to many, but by no means all, of the regression problems that arise in the analysis of the relationships between financial time series.

Section 8.1 thus sets out the basic dynamic linear regression model, while section 8.1 incorporates ARCH error effects into the framework. Misspecification testing is the topic of section 8.3, and section 8.4 discusses robust estimation techniques and generalised method of moments (GMM) estimation, which may be used when the standard assumptions of regression are found to be invalid. The multivariate linear regression model is briefly introduced in section 8.5. This paves the way for more general multivariate regression techniques, and the remaining sections of the chapter deal with vector autoregressions and its various extensions, including a discussion of the concepts of exogeneity and causality.

Chapter 8 has developed regression techniques for modelling relationships between non-integrated time series. As we have seen in earlier chapters, however, many financial time series are integrated, often able to be characterised as I(1) processes, and the question thus arises as to whether the presence of integrated variables affects our standard regression results and conventional procedures of inference. This question is long-standing, since it has been known since Yule (1897) that an unremoved deterministic time trend could produce erroneous regression results by acting as a common but non-causal influence behind otherwise independent time series. Later, Yule (1926) was the first to explore directly the problem of ‘nonsense correlations’, arguing that these resulted from violations of the assumptions behind linear correlation, in particular that of serial independence. Through analytical examples, Yule showed that estimated correlations can be significantly biased if the underlying variables are polynomials of time. He also performed a set of impressive hand-calculated Monte Carlo experiments that demonstrated that nonsense correlations could also arise when analysing the relationships between pairs of I(1) or I(2) variables. Soon afterwards, Slutsky (1937) and Working (1934) were able to argue that random walk processes could produce conspicuous, yet erroneous, cyclical behaviour. Indeed, Working (1934, p. 11) expressed a view that, unfortunately, was ignored for many years: ‘Economic theory has fallen far short of recognising the full implications of the resemblance of many economic time series to random-difference series; and methods of statistical analysis in general use have given these implications virtually no recognition.’

In the nine years since the manuscript for the second edition of The Econometric Modelling of Financial Time Series was completed there have continued to be many advances in time series econometrics, some of which have been in direct response to features found in the data coming from financial markets, while others have found ready application in financial fields. Incorporating these developments was too much for a single author, particularly one whose interests have diverged from financial econometrics quite significantly in the intervening years! Raphael Markellos has thus become joint author, and his interests and expertise in finance now permeate throughout this new edition, which has had to be lengthened somewhat to accommodate many new developments in the area.

Chapters 1 and 2 remain essentially the same as in the second edition, although examples have been updated. The material on unit roots and associated techniques has continued to expand, so much so that it now has an entire chapter, 3, devoted to it. The remaining material on univariate linear stochastic models now comprises chapter 4, with much more on fractionally differenced processes being included in response to developments in recent years. Evidence of non-linearity in financial time series has continued to accumulate, and stochastic variance models and the many extensions of the ARCH process continue to be very popular, along with the related area of modelling volatility. This material now forms chapter 5, with further non-linear models and tests of non-linearity providing the material for chapter 6.