Interpretation of Coefficients

ECON 3209 · Week 10, Lecture 2 · Kerala Agricultural University

Department of Development Economics, KAU

Autumn 2026

Learning Outcomes

By the end of this lecture, you will be able to:

  1. interpret slopes in a multiple regression as partial effects
  2. use ceteris paribus language for economic explanations
  3. distinguish raw and standardized coefficients
  4. compare R² and adjusted R² when regressors are added
  5. avoid common mistakes when interpreting intercepts and partial slopes

What Does a Coefficient Mean in MLR?

In the model

\[Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \cdots + \beta_k X_{ki} + \varepsilon_i\]

the coefficient \(\beta_j\) measures the expected change in \(Y\) from a one-unit increase in \(X_j\), holding all other regressors fixed.

This is the core idea of partial effect and ceteris paribus interpretation.

Kerala Example: Loan Default Risk

Suppose we estimate

\[Default_i = \beta_0 + \beta_1 Income_i + \beta_2 Collateral_i + \beta_3 LoanSize_i + \varepsilon_i\]

  • If \(\hat\beta_1=-0.012\), then a ₹1,000 increase in monthly income is associated with a 1.2 percentage-point lower default probability, holding collateral and loan size fixed.
  • If \(\hat\beta_3=0.021\), larger loans are associated with greater default risk, other things equal.

Always say what is held constant. Without that phrase, the interpretation is incomplete.

Intercepts, Units, and Economic Meaning

Intercept \(\beta_0\)

  • Predicted value of \(Y\) when all regressors equal zero
  • Sometimes meaningful
  • Sometimes only a mathematical anchor

Slopes \(\beta_j\)

  • Must be interpreted in the units of \(X_j\) and \(Y\)
  • Unit changes matter: kilograms, millimetres, rupees, hectares
  • Rescaling changes the number but not the economic content

Standardized Coefficients

A standardized coefficient asks: what happens to \(Y\) in standard deviation units when \(X_j\) changes by one standard deviation?

\[\hat{\beta}_j^{std}=\hat\beta_j\left(\frac{s_{X_j}}{s_Y}\right)\]

Use standardized coefficients when: - regressors are measured in very different units - you want a rough comparison of relative importance

Do not use them to replace substantive interpretation in original units.

Python Demo: Fit and Interpret an MLR

Python Demo: Standardize Variables

R² versus Adjusted R²

\[R^2 = 1 - \frac{SSR}{SST}\]

  • \(R^2\) never falls when you add regressors.
  • A useless regressor can still increase \(R^2\) slightly.

\[\bar{R}^2 = 1 - \frac{SSR/(n-k-1)}{SST/(n-1)}\]

  • Adjusted \(R^2\) penalizes adding weak regressors.
  • It is often more informative for comparing models with different numbers of covariates.

Python Demo: Reading the OLS Summary Table

Python Demo: Writing Economic Interpretations

Once you have the estimates, write out what each number means:

Common Interpretation Mistakes

  • Mistake 1: forgetting the phrase “holding other variables constant”.
  • Mistake 2: interpreting the intercept when zero values are impossible or unrealistic.
  • Mistake 3: comparing raw coefficients measured in different units.
  • Mistake 4: treating a high \(R^2\) as proof of causality.
  • Mistake 5: saying a coefficient is “important” without checking its scale, sign, and precision.

Econometric interpretation is about meaning + units + conditioning set.

🏋️ Exercise

  1. Fit a model for repayment_score using income, collateral, and loan size.
  2. Write a one-sentence interpretation for each slope.
  3. Standardize the variables and compare raw and standardized slopes.
  4. Add a random noise variable and report the change in \(R^2\) and adjusted \(R^2\).

Summary

✅ MLR coefficients are partial effects: they describe one regressor’s effect holding the rest fixed.

✅ Interpretation always depends on the units of both the dependent variable and the regressor.

✅ Standardized coefficients are useful for rough comparisons, not as a substitute for economic meaning.

✅ \(R^2\) always rises weakly with extra regressors, but adjusted \(R^2\) penalizes unnecessary complexity.

✅ Clear reporting uses ceteris paribus language and avoids overclaiming.

Next Lecture

Lecture 3 — Partial Effects in Python

We will cover: - building MLR models in statsmodels - reading every part of the summary table - plotting partial effects - computing scenario-based marginal effects