3. Autocorrelation: does today resemble yesterday?

Level 5: Time series: why the past does not repeat itself the same way

If today's return says nothing about tomorrow's, it seems reasonable to conclude that nothing at all about tomorrow can be anticipated.

That is false. You can fail to know whether the price will rise or fall and still have a fairly good idea of how much it is going to move.

This module measures exactly that difference.

What autocorrelation measures

Autocorrelation measures exactly what its name suggests: the correlation of a series with itself, comparing each value with the value the series itself had a certain number of periods earlier.

That number of periods is called the lag:

• The lag 1 autocorrelation compares today's return with yesterday's return across the whole sample.
• The lag 5 one compares today's return with that of five sessions ago.
• And so on, lag by lag.

The calculation is formally the same correlation coefficient you already know from the previous level, and it takes values between -1 and +1. The only difference is that instead of comparing two different variables, such as a share's return against the market's return, it compares the same variable with itself shifted in time.

What each sign means

The interpretation of each sign is straightforward, but it deserves precision:

• Positive and sizable at lag 1: a positive return today tends to be followed, on average, by another positive return tomorrow, and a negative one tends to be followed by another negative one. It is a pattern of short-term persistence or momentum.
• Negative: a rise tends to be followed by a fall and vice versa. It is a pattern of short-term mean reversion.
• Close to zero: knowing today's return provides, on average, no linear information about the sign or the size of tomorrow's return. As you will see in the next module, that is the statistical signature compatible with a random walk.

The autocorrelation function

The set of autocorrelations computed for different lags (lag 1, lag 2, lag 3 and so on) is known as the autocorrelation function.

Plotting it makes it possible to see at a glance whether there is detectable temporal structure, and at what horizon it shows up.

For example, a series might show nothing relevant at one day but a weak negative autocorrelation at five days, suggesting a short-term reversion dynamic that is not apparent when looking only one day at a time.

It never comes out exactly zero

There is one point that separates rigorous analysis from superficial analysis. An estimated autocorrelation is never exactly zero, even when the real process has no temporal structure at all. Out of pure sampling noise, any calculation on a finite sample is going to throw up a number slightly different from zero.

So the relevant question is not whether the estimated autocorrelation is exactly zero. The question is whether it is different enough from zero not to be explainable by simple sampling noise.

The reference band depends on the sample size

There is a simple rule of thumb. Under the hypothesis that there is no real autocorrelation, the standard error of an autocorrelation estimated with n observations is approximately 1 divided by the square root of n.

Look at what that implies depending on how much data you have in front of you:

• With n=252, that is, one year of daily data, that gives a standard error of approximately 0.063. As an approximate reference band, values below 0.13 in absolute terms (about two standard deviations) are fairly compatible with pure noise, whereas larger values start to be hard to explain by chance alone.
• With a larger sample, say n=2,500, about ten years of daily data, the standard error falls to approximately 0.02. So even a small autocorrelation, of 0.05, starts to be statistically notable with that amount of data.

The sample size completely changes which autocorrelation is considered relevant. Reporting an autocorrelation figure without saying how many observations it was computed with, or without comparing it with this reference band, is a common error that gives a false sense of precision.

It measures linear dependence only

There is a crucial nuance that connects with two later modules of this same level. Autocorrelation as defined so far measures only linear dependence between the values of the series itself, the returns.

It is perfectly possible for these two things to happen at the same time:

• That a share's returns have practically zero autocorrelation, compatible with a random walk as far as the direction of the price is concerned.
• That the square of those returns, a measure of the size of the movement regardless of sign, shows a strong and clearly significant autocorrelation.

This is not a contradiction. It means that although it cannot be predicted whether the price will rise or fall tomorrow, there is predictable information about how large the movement is likely to be, whatever its sign.

That is precisely the operational definition of the volatility clustering dealt with in module 6 of this level: a day with a large return, positive or negative, tends to be followed by another day with a large return, even though the sign of each one is, separately, hard to anticipate.

The arithmetic of the two autocorrelations

A numerical example illustrates this distinction well.

Imagine you compute the lag 1 autocorrelation of a share's daily returns over the last ten years and obtain 0.03. With an approximate standard error of 0.02 for that sample size, that value is within the range of what is explainable by noise, so there is no solid evidence of directional predictability.

But now compute the lag 1 autocorrelation of the squared returns of that same share, and suppose you obtain 0.25.

That value is enormously larger than the standard error of 0.02: a difference of more than ten standard deviations, compelling statistical evidence that the size of the movements is indeed serially correlated even though its direction is not.

This distinction between the autocorrelation of the returns and the autocorrelation of their size is one of the most useful and least intuitive ideas in the whole of financial time series analysis.

And it underpins much of why risk management models take recent volatility seriously, even though nobody claims to predict whether the market rises or falls tomorrow.