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Why Moving Averages

January 8, 2010

Moving averages are a ubiquitous tool in financial econometrics, especially dominant in both technical analysis and high-frequency trading. Given the sophistication of both disciplines, one is inclined to beg the question why such a seemingly trivial statistical technique as moving averages form a core of their foundation.

One explanation is due to a beautiful mathematical result, which has significant implication for building trading systems.

Consider the standard moving average of order one:

x_{t} = \epsilon_{t} + \theta\epsilon_{t-1}

This model has zero covariance with any time-shifted version of itself beyond x_{t-1} (formally, x_{t} has zero autocovariance with non-unity lags). Combine this with a linearly deterministic process \kappa_{t}, whose value may be zero:

x_{t} = \epsilon_{t} + \theta\epsilon_{t-1}  + \kappa_{t}

Now, extend the moving average lag to be of infinite-order, while retaining zero autocovariance:

x_{t} = \displaystyle\sum_{j=0}^ \infty \theta_{j}\epsilon_{t-j}  + \kappa_{t}

Now comes the punchline, given those minimal assumptions: many price series for financial instruments can be represented by such x_{t}, particularly those in high-frequency finance (namely, those which are zero-mean, autocovariance stationary white noise). In other words, many price series are generated by processes which are precisely moving averages. This is an elegant and unexpected theoretical result.

Note, somewhat to the detriment of technical analysis credibility: this same conclusion does not tend to apply to low-frequency price series, such as daily data. This difference between low- and high-frequency data again harks back to the stationarity principle introduced by the Three Horsemen.

For readers interested in diving deeper, the primary results from which the above originate are the Wold theorem and Ansley, Spivey, and Wrobleski (On the Structure of Moving Average Prices). For those interested in high-frequency trading, Hasbrouck covers this result and its relationship to the Roll microstructure model nicely in Empirical Market Microstructure.

Technical readers should note one reasonable additional technical constraint is required of x_{t} for convergence; namely, the moving average weights \epsilon_{t} must be finitely summable:

\displaystyle\sum_{j=0}^ \infty \theta_{j} < \infty

4 Comments leave one →
  1. January 8, 2010 7:25 am

    One of my “guess-explanations” as to why moving averages are so ubiquitous is also that they are much “easier” to compute (i.e. just an addition and multiplication) compared to the median – although the latter is a more robust statistical tool.

    I am more into the technical analysis side of things (rather than high-frequency trading) and am interested/planning in running tests to experiment with replacing a moving average by a median in a technical trading system and see if results/performance/robustness is enhanced.

    Really liked your “Three Horsemen” post also!

    • quantivity permalink
      January 8, 2010 8:43 am

      @Jez: thanks for your comment, and complement. I agree re “easy”, both in the computational and intuition senses–given both technical analysis and econometrics were developed in the pre-cheap computing era, thus neither compute nor rich graphical visualization were available.

      I look forward to following your blog to read about your results, as they evolve. I suggest you might also want to consider other quantiles, in addition to median. I discuss a bit of thinking why that might be useful in the Stability by Quantile post.


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