We use autocorrelation to quantify market inefficiency. The autocorrelation function is an old, common knowledge method which anybody with access to the data can in principle apply. Therefore it is perfect for demonstration in that we do not reveal any proprietary know-how by using it, yet it is quite convincing as it relates directly to the concept of a martingale.
The autocorrelation is defined as the expectation value of the product of the elements of the time series separated by time lag, and is a function of this time lag:
A(T)=E[x(t),x(t+T)]|over all available t
where T denotes the time lag and E is the expectation (averaging operator). Most often we will be dealing with autocorrelation of logarithmic returns. Unless stated otherwise, the time lag we show is the lag in “business time” or in other words, week-end and holiday periods (periods with no data) are excluded.
By construction, an autocorrelation is symmetric around 0. Therefore, plotting only one side (either positive or negative lags) is sufficient. Because in most contexts we talk about prediction, it is more intuitive to plot the negative lag side — that way one can interpret the axis of lags as a time axis, keeping in mind that what is about to happen (and is being predicted) is located at the 0 bin. Although in reality — and this needs to be said for the more rigorous reader — this axis is a diagonal direction of fixed time sums in the two-dimensional space of pairs of time points.
The time processes we know from experience to be predictable, such as the beating of our heart, variation of atmospheric temperature with season, ocean surf and tides, and the like, have informative autocorrelation values at non-zero time lags. Predictability does not imply causality, nor is causality always needed — even though winter does not cause spring, once you know you are in the middle of winter, you can predict that the temperature will be much higher in just a couple of months with good degree of confidence. It is often possible to transform the time series representing the process so that “informative” means non-zero autocorrelation, and non-zero means “informative”. Sometimes this can be done by replacing the original time series by that of increments or ratios (such as logarithmic returns), sometimes by subtracting an autocorrelation of a suitably constructed reference process — a synthetic, usually computer-simulated model of reality which incorporates the features we know about and consider trivial, but not the ones we want to learn about.
Forex Automaton™ has accumulated a collection of non-trivial correlation data for the forex markets. We regard their existence as a sufficient, but not a necessary condition of predictability. In other words, the market can be still predictable with completely trivial two-point autocorrelations (but with e.g. non-trivial genuine three-point correlations — although this does sound like one of those artificial math concoctions). If two-point autocorrelations happen to be non-trivial, that’s a sure sign of predictability — but you can’t count on that. Therefore, we do not suggest building a trading system bottom-up on the basis of autocorrelations (otherwise we would not make them public), or at least this is not how our own trading system was designed and built. But once the autocorrelations are found, ignoring them would be foolish.