Interactive modelEconometricsintermediate

Linear Regression & OLS

Simple and multiple regression: fitting lines, reading coefficients, and R².

Equilibrium: X* = 50.0, Y* = 45.0

Current equations

Fitted line ŷ = a + bX: Y = 20 + 0.5·XMean of X: X = 50

Fourteen data points, one line through them. Slide the slope and intercept and try to fit the cloud by eye. OLS does this automatically: it picks the line that makes the squared vertical gaps between points and line as small as possible.

Source: Wooldridge, Introductory Econometrics, ch. 2-3

Shift the curves

Slope estimate (b)0

Tilt the line: OLS picks the slope that makes the squared residuals smallest.

Intercept estimate (a)0

Slide the line up or down. Watch how far you can drift before the fit is obviously worse.

The Linear Regression & OLS model, in writing

Definition

Fitting a line (or hyperplane) by minimizing squared residuals: coefficients read as the change in y per unit x, holding other included variables constant.

β̂ = (X′X)⁻¹X′y; simple case: β̂₁ = cov(x,y)/var(x)

The intuition

OLS answers 'what's the average relationship?', which is only causal if x is uncorrelated with everything omitted. R² measures fit, not truth; a coefficient's meaning changes with every control you add or drop.

Exam tip

Interpret coefficients in units, with ceteris paribus stated, and never confuse statistical significance with size or causality.

Related models in Econometrics

Difference-in-Differences →Probit & Logit Models →