Forecasting Financial Markets – Part 1: Time Series Analysis

The presentation in this post covers a number of important topics in forecasting, including:

  • Stationary processes and random walks
  • Unit roots and autocorrelation
  • ARMA models
  • Seasonality
  • Model testing
  • Forecasting
  • Dickey-Fuller and Phillips-Perron tests for unit roots

Also included are a number of detailed worked examples, including:

  1. ARMA Modeling
  2. Box Jenkins methodology
  3. Modeling the US Wholesale Price Index
  4. Pesaran & Timmermann study of excess equity returns
  5. Purchasing Power Parity



    Long Memory and Regime Shifts in Asset Volatility

    This post covers quite a wide range of concepts in volatility modeling relating to long memory and regime shifts and is based on an article that was published in Wilmott magazine and republished in The Best of Wilmott Vol 1 in 2005.  A copy of the article can be downloaded here.

    One of the defining characteristics of volatility processes in general (not just financial assets) is the tendency for the serial autocorrelations to decline very slowly.  This effect is illustrated quite clearly in the chart below, which maps the autocorrelations in the volatility processes of several financial asssets.

    Volatility Autocorrelations

     Thus we can say that events in the volatility process for IBM, for instance, continue to exert influence on the process almost two years later. 

    This feature in one that is typical of a black noise process – not some kind of rap music variant, but rather:

     “a process with a 1/fβ spectrum, where β > 2 (Manfred Schroeder, “Fractalschaos, power laws“). Used in modeling various environmental processes. Is said to be a characteristic of “natural and unnatural catastrophes like floods, droughts, bear markets, and various outrageous outages, such as those of electrical power.” Further, “because of their black spectra, such disasters often come in clusters.”" [Wikipedia].

    Because of these autocorrelations, black noise processes tend to reinforce or trend, and hence (to some degree) may be forecastable.  This contrasts with a white noise process, such as an asset return process, which has a uniform power spectrum, insignificant serial autocorrelations and no discernable trending behavior:

    White Noise Power Spectrum

    An econometrician might describe this situation by saying that a  black noise process is fractionally integrated order d, where d = H/2, H being the Hurst Exponent.  A way to appreciate the difference in the behavior of a black noise process vs. a white process is by comparing two fractionally integrated random walks generated using the same set of quasi random numbers by Feder’s (1988) algorithm (see p 32 of the presentation on Modeling Asset Volatility).

    Fractal Random Walk - White Noise

    Fractal Random Walk - Black Noise Process

     As you can see. both random walks follow a similar pattern, but the black noise random walk is much smoother, and the downward trend is more clearly discernible.  You can play around with the Feder algorithm, which is coded in the accompanying Excel Workbook on Volatility and Nonlinear Dynamics .  Changing the Hurst Exponent parameter H in the worksheet will rerun the algorithm and illustrate a fractal random walk for a black noise (H > 0.5), white noise (H=0.5) and mean-reverting, pink noise (H<0.5) process. 

    One way of modeling the kind of behavior demonstrated by volatility process is by using long memory models such as ARFIMA and FIGARCH (see pp 47-62 of the Modeling Asset Volatility presentation for a discussion and comparison of various long memory models).  The article reviews research into long memory behavior and various techniques for estimating long memory models and the coefficient of fractional integration d for a process.

    But long memory is not the only possible cause of long term serial correlation.  The same effect can result from structural breaks in the process, which can produce spurious autocorrelations.  The article goes on to review some of the statistical procedures that have been developed to detect regime shifts, due to Bai (1997), Bai and Perron (1998) and the Iterative Cumulative Sums of Squares methodology due to Aggarwal, Inclan and Leal (1999).  The article illustrates how the ICSS technique accurately identifies two changes of regimes in a synthetic GBM process.

    In general, I have found the ICSS test to be a simple and highly informative means of gaining insight about a process representing an individual asset, or indeed an entire market.  For example, ICSS detects regime shifts in the process for IBM around 1984 (the time of the introduction of the IBM PC), the automotive industry in the early 1980′s (Chrysler bailout), the banking sector in the late 1980′s (Latin American debt crisis), Asian sector indices in Q3 1997, the S&P 500 index in April 2000 and just about every market imaginable during the 2008 credit crisis.  By splitting a series into pre- and post-regime shift sub-series and examining each segment for long memory effects, one can determine the cause of autocorrelations in the process.  In some cases, Asian equity indices being one example, long memory effects disappear from the series, indicating that spurious autocorrelations were induced by a major regime shift during the 1997 Asian crisis. In most cases, however, long memory effects persist.

    Excel Workbook on Volatility and Nonlinear Dynamics 

    There are several other topics from chaos theory and nonlinear dynamics covered in the workbook, including:

    More on these issues in due course.

    Modeling Asset Volatility

    I am planning a series of posts on the subject of asset volatility and option pricing and thought I would begin with a survey of some of the central ideas. The attached presentation on Modeling Asset Volatility sets out the foundation for a number of key concepts and the basis for the research to follow.

    Perhaps the most important feature of volatility is that it is stochastic rather than constant, as envisioned in the Black Scholes framework.  The presentation addresses this issue by identifying some of the chief stylized facts about volatility processes and how they can be modelled.  Certain characteristics of volatility are well known to most analysts, such as, for instance, its tendency to “cluster” in periods of higher and lower volatility.  However, there are many other typical features that are less often rehearsed and these too are examined in the presentation.

    Long Memory
    For example, while it is true that GARCH models do a fine job of modeling the clustering effect  they typically fail to capture one of the most important features of volatility processes – long term serial autocorrelation.  In the typical GARCH model autocorrelations die away approximately exponentially, and historical events are seen to have little influence on the behaviour of the process very far into the future.  In volatility processes that is typically not the case, however:  autocorrelations die away very slowly and historical events may continue to affect the process many weeks, months or even years ahead. 

    Volatility Direction Prediction Accuracy

    There are two immediate and very important consequences of this feature.  The first is that volatility processes will tend to trend over long periods – a characteristic of Black Noise or Fractionally Integrated processes, compared to the White Noise behavior that typically characterizes asset return processes.  Secondly, and again in contrast with asset return processes, volatility processes are inherently predictable, being conditioned to a significant degree on past behavior.  The presentation considers the fractional integration frameworks as a basis for modeling and forecasting volatility.

    Mean Reversion vs. Momentum
    A puzzling feature of much of the literature on volatility is that it tends to stress the mean-reverting behavior of volatility processes.  This appears to contradict the finding that volatility behaves as a reinforcing process, whose long-term serial autocorrelations create a tendency to trend.  This leads to one of the most important findings about asset processes in general, and volatility process in particular: i.e. that the assets processes are simultaneously trending and mean-reverting.  One way to understand this is to think of volatility, not as a single process, but as the superposition of two processes:  a long term process in the mean, which tends to reinforce and trend, around which there operates a second, transient process that has a tendency to produce short term spikes in volatility that decay very quickly.  In other words, a transient, mean reverting processes inter-linked with a momentum process in the mean.  The presentation discusses two-factor modeling concepts along these lines, and about which I will have more to say later.

    One of the most striking developments in econometrics over the last thirty years, cointegration is now a principal weapon of choice routinely used by quantitative analysts to address research issues ranging from statistical arbitrage to portfolio construction and asset allocation.  Back in the late 1990′s I and a handful of other researchers realized that volatility processes exhibited very powerful cointegration tendencies that could be harnessed to create long-short volatility strategies, mirroring the approach much beloved by equity hedge fund managers.  In fact, this modeling technique provided the basis for the Caissa Capital volatility fund, which I founded in 2002.  The presentation examines characteristics of multivariate volatility processes and some of the ideas that have been proposed to model them, such as FIGARCH (fractionally-integrated GARCH).

    Dispersion Dynamics
    Finally, one topic that is not considered in the presentation, but on which I have spent much research effort in recent years, is the behavior of cross-sectional volatility processes, which I like to term dispersion.  It turns out that, like its univariate cousin, dispersion displays certain characteristics that in principle make it highly forecastable.  Given an appropriate model of dispersion dynamics, the question then becomes how to monetize efficiently the insight that such a model offers.  Again, I will have much more to say on this subject, in future.