Autocorrelation

What Is Autocorrelation?

Autocorrelation measures how strongly a time series relates to lagged versions of itself. A lag is the number of periods separating two observations. A lag-1 autocorrelation, for example, compares each value with the value from the previous period. The prefix "auto" means self, so the measure compares a series against itself.

Autocorrelation is usually expressed as a coefficient between -1 and 1, similar to a standard correlation coefficient. A value near 1 means high values tend to be followed by high values, and low values by low values. A value near -1 means the series tends to alternate. A value near zero means past values carry little linear information about future ones.

How It Is Calculated

For a series x and a lag k, the sample autocorrelation is commonly computed as:

r_k = sum of (x_t - mean) x (x_(t+k) - mean) / sum of (x_t - mean)^2

The mean is the average of the series. Statistical software performs this calculation automatically and often plots results for several lags in a correlogram, which shows where the series has significant dependence.

Why It Matters to Investors

Autocorrelation is central to debates about the efficient market hypothesis. If asset returns showed strong, persistent autocorrelation, past returns could in principle forecast future ones, which would conflict with the idea that prices quickly reflect available information. In practice, daily stock returns typically show little autocorrelation, which is one reason they are often described as close to a random walk. Small short-term patterns have been documented in some markets, but they are often modest.

Autocorrelation also matters when analyzing investment data. Price levels are usually highly autocorrelated because today's price is close to yesterday's. This can make a naive regression analysis look more meaningful than it is. Analysts often work with returns instead of prices and use tests such as the Ljung-Box test to check for remaining dependence.

Advantages and Limitations

Measuring autocorrelation helps reveal trends, cycles, and forecastable structure in a series. However, it captures only linear relationships, can be unstable in short samples, and does not guarantee that a past pattern will continue. Ignoring autocorrelation can also lead to overstated confidence in statistical results.