Introduction

In the realm of time series analysis, understanding the relationships between observations at different time points is crucial for model building and forecasting. Two fundamental concepts that aid in uncovering these relationships are sample autocorrelation and partial autocorrelation. Sample autocorrelation measures the linear relationship between an observation and its lagged values, while partial autocorrelation quantifies the correlation between two observations while controlling for the influence of intermediate observations. Let’s delve into these concepts to grasp their significance in analyzing and modeling time series data.

Sample Autocorrelation

Sample autocorrelation, often denoted as $r_k$, assesses the correlation between an observation at time $t$ and its lagged counterpart at time $t-k$. Mathematically, it is computed as:

$$r_k=\frac{\sum_{t=k+1}^n\left(y_t-\bar{y}\right)\left(y_{t-k}-\bar{y}\right)}{\sum_{t=1}^n\left(y_t-\bar{y}\right)^2}$$

Where:

• $y_t$ represents the observation at time $t$
• $\bar{y}$ denotes the mean of the observations
• $n$ is the total number of observations

Sample autocorrelation values range between -1 and 1. A positive autocorrelation $(r_k \gt 0)$ indicates a tendency for observations to move in the same direction with a lag, while a negative autocorrelation $(r_k \lt 0)$ suggests an inverse relationship.

Partial Autocorrelation

Partial autocorrelation measures the correlation between two observations while controlling for the influence of intermediate observations. It helps identify the direct relationship between observations at different time lags, excluding the indirect impact of intervening observations. The partial autocorrelation function is often denoted as $\phi_{kk}$ and is closely related to autoregressive models.

Example: Consider a stock price time series. To compute the sample autocorrelation at lag $k=1$, we would calculate the correlation between the stock prices at time $t$ and time $t−1$. A high positive autocorrelation might indicate that the stock’s price tends to follow a trend from day to day, while a low or negative autocorrelation suggests more erratic movement.