What is a lag in time series analysis?

In time series analysis, a lag refers to a past value of a variable at a previous time step. It represents the idea of looking backward in the data sequence to analyze how past observations influence the current or future values. For example, in a daily temperature dataset, the temperature from yesterday (time t-1) is a lag of 1 relative to today’s temperature (time t). Lags are created by shifting the time series data by a specific number of periods, effectively aligning past values with the current timestamp. This allows models to incorporate historical patterns, such as trends or seasonality, into predictions or analyses.

Lags are fundamental in time series modeling because many patterns depend on prior observations. For instance, autoregressive (AR) models predict future values using linear combinations of past values, where each term in the model corresponds to a lag. A common example is the ARIMA model, which combines autoregressive terms (lags of the target variable) with moving averages of past forecast errors. Lags also play a role in measuring autocorrelation—the correlation of a time series with its own lagged values. By calculating autocorrelation at different lags, developers can identify repeating patterns, such as weekly seasonality in daily sales data (e.g., a lag of 7 days) or hourly peaks in website traffic (e.g., a lag of 24 hours).

Developers often use lags in feature engineering for machine learning models. For example, predicting tomorrow’s stock price might require features like the past 5 days’ closing prices (lags 1 to 5). In Python, creating lagged features can be done with libraries like Pandas using the shift() method. For instance, df['lag_1'] = df['value'].shift(1) creates a column where each row contains the value from the prior row. However, handling lags requires care: missing values at the start of the shifted series must be addressed (e.g., by truncating or imputing), and overusing lags can lead to redundant features or overfitting. When applied thoughtfully, lags enable models to capture temporal dependencies critical for accurate forecasts.