Answer First
In an OLS regression, a coefficient measures the expected change in the dependent variable when the regressor increases by one unit, holding all other variables constant. The interpretation depends on whether the variables are linear, logged, or categorical.
Problem Setup
Linear regression: \[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + u. \] Interpretation:
- \(\beta_1\): effect of a one‑unit increase in \(x_1\)
- Holding other regressors constant
- Assuming the model is correctly specified
Special cases:
- Log‑linear
- Linear‑log
- Log‑log
- Dummy variables
Step-by-Step Explanation
1. Linear–linear model
\(\beta_1\) = change in \(y\) for a one‑unit increase in \(x_1\). Example: If \(\beta_1 = 2.5\), then increasing \(x_1\) by 1 increases \(y\) by 2.5 units.
2. Log–linear model
\(\ln(y) = \beta_0 + \beta_1 x\). Interpretation: a one‑unit increase in \(x\) changes \(y\) by approximately \(100\beta_1\%\).
3. Linear–log model
\(y = \beta_0 + \beta_1 \ln(x)\). Interpretation: a 1% increase in \(x\) changes \(y\) by \(\beta_1/100\) units.
4. Log–log model
\(\ln(y) = \beta_0 + \beta_1 \ln(x)\). Interpretation: \(\beta_1\) is an elasticity.
5. Dummy variables
If \(D\) is 0/1, then \(\beta_D\) is the difference in mean outcomes between the two groups, holding other variables constant.
Intuition
OLS coefficients describe how the dependent variable responds to changes in each regressor while keeping everything else fixed. The scale of the variables determines the correct interpretation.
Common Exam Mistakes
- Interpreting coefficients without holding other variables constant.
- Mixing up log‑linear and linear‑log interpretations.
- Forgetting that dummy variables measure group differences.
- Interpreting coefficients causally without checking assumptions.
Final Summary
Regression coefficients measure how the dependent variable changes when a regressor changes, holding other variables constant. Interpretations differ for linear, logged, and dummy variables, but the core idea remains the same.
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