Simple and multiple regression: fitting lines, reading coefficients, and R².
Warming up the engine…
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)
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.
Interpret coefficients in units, with ceteris paribus stated, and never confuse statistical significance with size or causality.
Now prove you have it
Move the curve to where you think it lands, and get told exactly which part you got right.
Real-world scenarios on this model
Also in Econometrics
Equilibrium at X 50, Y 45.
Equilibrium: X* = 50.0, Y* = 45.0
Current equations
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