Log-Log, Log-Lin, and Lin-Log

TL;DR

• Log–Lin: 1 unit of x → 100e^β change in y
• Lin–Log: 1% change in x → β·ln(x₂/x₁)in y.
• Log–Log: 1% change in x → β% change in y (elasticity).
• Interactions let these effects vary by group or with another variable’s level.

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1. Outcome is log-transformed (Log–Lin)

Model

Interpretation

• Semi-elasticity: a 1-unit increase in x changes y by ≈ 100 β₁ %.
• Exact % change for Δx: %Δy = 100 (e^{β₁ Δx} − 1).

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2. Predictor is log-transformed (Lin–Log)

Model

Interpretation

• 1 % increase in x → ≈ 0.01 β₁ units change in y.
• Exact finite change: Δy = β₁ ln(x₂/x₁).

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3. Both outcome and predictor log-transformed (Log–Log)

Model

Interpretation

• 1 % increase in x → ≈ β₁ % increase in y.
• Exact finite change: ln(y₂/y₁) = β₁ ln(x₂/x₁) ⇒ y₂/y₁ = (x₂/x₁)^{β₁}.

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📊 Quick Reference Table

Model	Equation	Coefficient meaning
Log–Lin	ln y = β₀ + β₁ x	1 unit x → ≈ 100 β₁ % change in y (exact 100(e^{β₁}−1) %)
Lin–Log	y = β₀ + β₁ ln x	1 % x → 0.01 β₁ units change in y (Δy = β₁ ln(x₂/x₁))
Log–Log	ln y = β₀ + β₁ ln x	1 % x → β₁ % change in y (y₂/y₁ = (x₂/x₁)^{β₁})

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