Autocorrelation

Autocorrelation (also called serial Correlation) measures how a time series is related to a lagged version of itself. Helps identify repeating patterns, seasonality, or persistence in data.

Formally, for a time series $y_t$, the autocorrelation at lag $k$ is:

$${\rho }_k=\frac{\text{Cov}(y_t,y_{t-k})}{\sqrt{\text{Var}(y_t)\text{Var}(y_{t-k})}}$$

It takes values between $-1$ and $1$:

• ${\rho }_k>0$: Positive correlation → if $y_t$ is above its mean, $y_{t-k}$ tends to be above its mean too.
• ${\rho }_k<0$: Negative correlation → if $y_t$ is above its mean, $y_{t-k}$ tends to be below its mean.
• ${\rho }_k\approx 0$: No linear relationship between $y_t$ and its past at lag $k$.

Intuition

Autocorrelation tells you how predictable a series is from its past:

• Stock returns: low/no autocorrelation (mostly random).
• Daily temperatures: high autocorrelation at lag 1 (yesterday’s temperature is a good predictor of today’s).
• Strong seasonal effects: autocorrelation spikes at seasonal lags (e.g., 7 days for weekly seasonality).

Why it matters

• Used to detect Decomposition in Time Series & Stationary Time Series.
• Helps decide AR and MA terms in ARIMA.
• Checked via Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots (ACF, PACF Plots) to select lagged values.

Hurst exponents

• The Hurst exponent ($H$) is a measure used to characterize the long-term memory of time series.
• It helps to determine the presence of autocorrelation or persistence in the data.
• The goal of the Hurst exponent is to provide us with a scalar value that will help us to identify whether a series is
  • random walking,
  • trending,
  • Mean reverting.