Understanding volatility as a process

Quantitative Finance, April 2004

Robert F Engle
Michael Armellino Professor in the Management of Financial Services, New York University Stern School of Business; Nobel Laureate (2003), economic sciences

Robert Engle laid the foundations for the fast-growing field of financial econometrics by developing some of the field's fundamental tools, in work recognised by the award of the 2003 economics Nobel which Engle shared with his sometime colleague Clive Granger.

Engle's Nobel citation was "for methods of analysing economic time series with time-varying volatility", specifically the concept known as ARCH – autoregressive conditional heteroskedasticity. ARCH models can accurately capture the long-term properties of many time series, and have become an indispensable tool for researchers and analysts studying the financial markets and problems of risk evaluation.

The ARCH concept aims to bridge the gap between what Engle has called the great workhorse of applied econometrics[1], the least squares model, with the real-life data sets encountered in financial and economic applications. The basic assumption of the least squares model is homoskedasticity – the assumption that the expected value of all error terms, when squared, is constant. Data sets where error terms may reasonably be expected to vary are said to suffer from heteroskedasticity. Rather than considering this as a problem to be corrected, ARCH models treat heteroskedasticity as a variance to be modelled.

The concept developed in 1982 while Engle was on sabbatical from his professorship at the University of California, San Diego, at the London School of Economics. Engle joined UCSD in 1975 after five years at MIT, following an MS in physics and PhD in economics at Cornell University, New York.

Engle was then working on problems of macroeconomics, particularly a question raised by an earlier Nobel laureate. "I was interested in a conjecture that Milton Friedman had made, that one of the causes of business cycles was uncertainty of information – when future prices and wage costs were uncertain then investors would be less likely to build a plant than otherwise," Engle recalls. "If that was true, that would have to be changing over time. I was looking for methods to measure time varying volatilities. The breakthrough was figuring out that you could build a predictive model. That's what the word conditional is – you could estimate the distribution of future volatilities conditional on what we see today."

ARCH basically does for variance in time series what the common factor method did for expected values, by taking additional current information into account. "Recognising that I could build a forecasting model for variance and make it a well defined process was the breakthrough," Engle notes. "We often think of variance as a number that we're trying to estimate, but the breakthrough came through treating it as a process rather than a number."

Once the concept was in place, Engle put together a test programme with David Hendry at the LSE, and applied it to UK inflation data. The data was revealed to be time varying in certain ways, even though it didn't look like the business cycle model, Engle recalls.

A more generalised version of ARCH, GARCH, was formulated by Engle's graduate student Tim Bollerslev in 1986. Although ARCH's first applications were in macroeconomics, they soon found a natural home in financial applications where the variance of a return represents the risk level. Obviously some time periods are riskier for investments than others – ARCH models could provide a volatility measure, akin to a standard deviation, that can be used in decisions concerning risk analysis, portfolio selection and derivative pricing.

"The macroeconomic interpretation was that uncertainty would change people's behaviour," Engle notes. "That might be true but it's not a very big effect compared to other effects, but in finance uncertainty and risk are the key driving features for what happens. Finance turned out to be a natural place to apply this, and I was happy to develop finance applications. There was a paper published by French, Schwert and Stambaugh applying these models to financial questions which was very influential to me and to other people working in this area. That was in the late 1980s, so I've spent the last 10 or 15 years remaking myself as a financial econometrician."

Cointegration, exogeneity and beyond
As well as ARCH, Engle has contributed to the development of several other milestones in financial economics. He collaborated at a key stage in the development of cointegration with his fellow Nobel laureate, Clive Granger. Cointegration describes systems of non-stationary variables which are driven by some underlying common trend, and can thus be combined to allow correct statistical inference. The concept has provided powerful tools for dealing with the dynamics of an economic system over various time periods.

"This was an idea that Clive was working on for some while," Engle says. "He proposed a definition of cointegration in an early paper, then he and I talked about it a lot in San Diego and came up with some empirical tests to see if a pair of series did satisfy this condition of cointegration." The work was initially presented as two separate papers, on the theory and empirical results, but these were combined for publication, resulting in a joint credit on this landmark paper[2]. "We labelled the theoretical part the Granger representation theorem to make it clear this was his development," Engle notes.

Another landmark was the 1983 paper with David Hendry and Jean-Francois Richard on exogeneity[3]. "The idea behind exogeneity is trying to figure out what are the key factors of data in models that we need to think about in deciding whether we can take one set of variables as given while we consider another set," Engle says. "The exogeneity paper was trying to figure out under what circumstances can you model a subset of variables and take the rest as given. It means you really don't lose any information if you don't model the variables you're conditional on."

The groundwork on defining exogeneity was done during an earlier sabbatical at LSE in the late 1970s, but the paper took several years to be published. "It was controversial at the time," Engle says. "I think it was instrumental in resolving some questions about whether you could really test for exogeneity and in what circumstances. There was a large literature that involved testing whether some variable was exogenous or not and it turned out you couldn't really do that – it was the causality concept they were testing, not exogeneity. This paper really clarified that whole concept."

Engle's current work is concentrated on applying ARCH models to more complex systems, particularly high dimension systems involving variances and correlations for a large number of assets, and those involving high frequency data. "I'm interested in these high dimension volatility models because they have applications for asset allocation – there's some substantial work going on there," he says. "They also have implications for problems like credit risk, which is a problem in that multiple firms will have extreme negative shocks at the same time. That's a model of tail dependence which is interesting to statisticians as well as financial economists."

An understanding of the ultimate cause of volatility remains elusive, however. "There's lots of papers written looking at causes of volatility, but basically I think it's going to be hard to really quantify a large proportion of the causes," Engle says. "These effects are so complicated and difficult to specify that we're unlikely to be able to write down a specification that encompasses all the factors that individual investors take into account. I do think there is more interesting work that can be done in associating major changes like business cycle effects and monetary regimes with volatility, and I think we will see that increasingly with the interaction between macroeconomics and finance."

Unanswered or unanswerable?
Engle is also addressing what he sees as one of the key unanswered questions in financial economics, the problem of optimal hedging. "If you have a risky position and you want to reduce the risk by taking an offsetting position, how big a position you should take is a complex statistical problem and requires knowing precisely what the stochastic situation of the whole problem is," he points out. "The most interesting aspects of this that I'm studying is how do we do this with derivatives on multiple underlying assets, as you need models of the underlying joint distribution. If we could figure out what the joint distribution of the assets look like and how much the volatilities move together, the tools are in place – but people haven't really figured out how to do this yet. I think in the next five years we'll make a lot of progress on this."

Progress on other key questions might not materialise, however. "There's a question that many people are interested in that I don't find interesting, which is the asset pricing question – what's the best way to measure expected returns and what do they depend on?" he notes. "My feeling is these are more or less unanswerable questions because they require either very long data sets or you have to assume the environment is constant over long periods of time – that's a big unanswered problem but I think it is unanswerable."

Engle began his career as a physicist before turning to economics as a post-graduate – so what does he think of the emergence of the field of econophysics? "From what I've seen, it hasn't come of age yet," he says. "The physicists would be well advised to look a little more closely at what has been done in the finance literature and the way finance people approach the problems. Physics has a great deal to offer to this kind of research, but they have to do their homework. I feel like it's taken me 15 years to understand better and better the way the finance profession works, and I think the econophysicists will find their contributions become more and more important as they learn more and more about the way finance questions are posed and answered."

Physics does provide a great model for combining experimental and theoretical research, he adds. "The idea that empirical research and theoretical research have a common goal is an important thing for us to think about. They need to make sense together and economics often doesn't do this. Econometrics is the place where those two frontiers come together."

REFERENCES
[1] Engle, R. F., "GARCH101, The Use of ARCH/GARCH Models in Applied Econometrics", Journal of Economic Perspectives, Vol 15, No 4, (2001), 157-168.
[2] Engle, R. F. and Granger, C. W. J.: "Cointegration and error correction: Representation, estimation and testing", Econometrica 55 (1987), 251276.
[3] Engle, R. F., Hendry, D. F. and Richard, J. F.: "Exogeneity", Econometrica 51 (1983), 277304.