One model, many specifications. See how functional form, a squared term, and a dummy variable each change what a coefficient means and how the fitted line bends. The same seeded data sit under all three views.
slope β̂₁ = 0.630
Discussion. The fitted curve barely moves, yet the slope's units change with every form. When would you report an elasticity rather than a dollar effect, and why is 100·β̂ only an approximation to the exact 100·(e^β̂ − 1)?
OLS is re-fit by least squares in the chosen form's transformed space, then mapped back to the raw axes. For log–level, a one-unit rise in x gives %Δy = 100·(e^β̂ − 1) exactly, with 100·β̂ as the first-order approximation. For log–log, β̂ is the elasticity directly (the % change in y per % change in x), exact at the margin.