Regression intercept

Regression intercept is the value a model predicts for the outcome when every explanatory variable equals zero, and it is where the fitted line crosses the vertical axis.

Also known asB0 · constant term

ByHoang TruongUpdated June 30, 2026

FrameworkOrdinary least squares (OLS)

See it move

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A scatter-line chart plots advertising spend in thousands of euros on the horizontal axis against revenue in thousands of euros on the vertical axis, with the fitted line following the equation ŷ = 14.5 + 2.8x. The line crosses the vertical axis at 14.5, which is the intercept — the revenue the model predicts when advertising spend equals zero.

Where it fits

SubjectData Analysis & StatisticsCoreTopicSimple Linear Regression & OLSCore

The formula

LaTeX

\hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \bar{X}

Variables

$\hat{β}_{0}$: Estimated OLS intercept
$\overset{ˉ}{Y}$: Sample mean of the outcome variable Y
$\hat{β}_{1}$: Estimated OLS slope
$\overset{ˉ}{X}$: Sample mean of the explanatory variable X

Derived from the OLS condition that the fitted line passes through the point (X̄, Ȳ).

Check yourself

PracticeCORE

A firm regresses monthly customer enquiries (Y) on the number of online advertisements placed (X) and obtains: Ŷ = 45 + 12X. Which statement about the intercept correctly interprets this equation?

Select an answer to check your understanding.