Time series regression models a dependent series as a function of one or more explanatory (predictor) variables, using observations that are ordered in time. The regression relationship describes how the typical value of the dependent variable changes when the predictors change. Time itself, a trend, seasonal indicators, or lagged information can be among those predictors.
Time series regression models a dependent series as a function of one or more explanatory (predictor) variables, using observations that are ordered in time. The regression relationship describes how the typical value of the dependent variable changes when the predictors change. Time itself, a trend, seasonal indicators, or lagged information can be among those predictors.
Daily electricity demand can be modelled using temperature (hot days raise cooling load) together with calendar information such as weekday vs weekend. The aim is to describe demand, not only to follow last week's demand in isolation.
The dependent variable (response) is the series we want to explain or forecast, written y. An explanatory or predictor variable is an x that we treat as given when we write the equation. In simple regression there is one x. In multiple regression there are several predictors (temperature, holiday indicator, trend, and so on).
Time can be an explanatory factor: a time index t can capture a slow trend, and month or weekday indicators can capture a repeating seasonal calendar. Previous values of y can also appear as lagged predictors when that is justified by the problem.
ARIMA-style models use mainly the past of the same series. Regression is useful when an external series (temperature, price, advertising) is believed to drive the response. It can also organise trend and seasonal dummy variables in one equation. The fitted equation then supports interpretation (“one extra degree of temperature”) and prediction of new observations when future x values can be supplied.
| Ordinary (cross-section) regression | Time series regression | |
|---|---|---|
| Data order | Observations are often treated as separate cases | Observations have a time order that must be kept |
| Typical assumption | Errors independent of each other | Neighbouring errors may be correlated |
| Extra structure | Usually no lag or season by default | Trend, season, lags and residual autocorrelation matter |
Ordinary least squares (OLS) formulas can still be used to fit a line, but time series data may violate ordinary regression assumptions because residuals can be autocorrelated. That does not make regression useless; it means inference and prediction need extra care. Methods such as GLS and WLS, and residual checking, appear later in this unit. They are not required to define time series regression.