MLR Setup & Estimation

ECON 3209 · Week 10, Lecture 1 · Kerala Agricultural University

Department of Development Economics, KAU

Autumn 2026

Learning Outcomes

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

1. write the multiple linear regression model in matrix and scalar form
2. explain why adding regressors helps us isolate ceteris paribus effects
3. compute the OLS estimator using both matrix algebra and statsmodels
4. compare simple and multiple regression estimates in a Kerala-style dataset
5. state the key assumptions needed for OLS estimation in MLR

Why Move Beyond Simple Regression?

Simple regression limitation

• A one-variable model is often too narrow for economic data.
• Crop yield depends on fertilizer, rainfall, irrigation, soil quality, and more.
• Loan default depends on income, collateral, loan size, repayment history.

Multiple regression advantage

• Controls for several factors at once
• Reduces omitted-variable bias
• Produces partial effects: the effect of one variable holding others fixed
• Improves empirical relevance for policy analysis

In agricultural economics, most relationships are multifactor relationships. MLR is therefore the standard workhorse model.

The MLR Model in Scalar and Matrix Form

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

\[\mathbf{Y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}\]

• \(\mathbf{Y}\) is an \(n \times 1\) vector of outcomes.
• \(\mathbf{X}\) is an \(n \times (k+1)\) matrix containing a column of ones and all regressors.
• \(\boldsymbol{\beta}\) is a \((k+1) \times 1\) vector of unknown parameters.
• \(\boldsymbol{\varepsilon}\) collects unobserved factors.

Dimensions Matter

Object	Meaning	Dimension
\(\mathbf{Y}\)	dependent variable vector	\(n \times 1\)
\(\mathbf{X}\)	design matrix	\(n \times (k+1)\)
\(\boldsymbol{\beta}\)	coefficient vector	\((k+1) \times 1\)
\(\boldsymbol{\varepsilon}\)	error vector	\(n \times 1\)

The intercept is included by placing a column of ones inside \(\mathbf{X}\). Without that column, the fitted plane is forced through the origin.

OLS Estimation

We choose \(\hat{\boldsymbol{\beta}}\) to minimise the sum of squared residuals:

\[S(\boldsymbol{\beta})=(\mathbf{Y}-\mathbf{X}\boldsymbol{\beta})'(\mathbf{Y}-\mathbf{X}\boldsymbol{\beta})\]

The first-order condition gives the normal equations:

\[\mathbf{X}'\mathbf{X}\hat{\boldsymbol{\beta}}=\mathbf{X}'\mathbf{Y}\] \[\hat{\boldsymbol{\beta}}=(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{Y}\]

• We need \(\mathbf{X}'\mathbf{X}\) to be invertible.
• This fails under perfect multicollinearity.

Adding Regressors Changes the Question

Simple model

\[Yield_i = \alpha_0 + \alpha_1 Fertilizer_i + u_i\]

Here \(\alpha_1\) mixes together: - direct fertilizer effect - rainfall differences - irrigation differences

Multiple model

\[Yield_i = \beta_0 + \beta_1 Fertilizer_i + \beta_2 Rainfall_i + \beta_3 Irrigation_i + \varepsilon_i\]

Now \(\beta_1\) is the fertilizer effect holding rainfall and irrigation constant.

MLR changes the interpretation from a raw association to a conditional association.

Python Demo: Simulate Kerala Farm Data

Python Demo: Matrix Calculation of \(\hat{\beta}\)

Python Demo: Comparing Matrix OLS and statsmodels

Both methods solve the same normal equations. statsmodels adds standard errors, tests, confidence intervals, and fitted values.

Key Assumptions Behind MLR Estimation

1. Linearity in parameters: the model is linear in the unknown coefficients.
2. Random sampling: observations are drawn from the relevant population.
3. No perfect multicollinearity: no regressor is an exact linear combination of the others.
4. Zero conditional mean: \(E(\varepsilon_i \mid X_{1i},\ldots,X_{ki})=0\).

Under these conditions, OLS is unbiased. For multiple-regression inference, we later revisit homoskedasticity and normality in the MLR setting.

Python Demo: Nested Models and Fit

🏋️ Exercise

1. Use a simulated farm dataset with fertilizer, rainfall, and irrigation.
2. Compute \(\hat{\boldsymbol{\beta}}\) using the matrix formula.
3. Estimate the same model with statsmodels.
4. Explain why the fertilizer coefficient changes when rainfall is added.

Summary

✅ MLR writes the outcome as \(\mathbf{Y}=\mathbf{X}\boldsymbol{\beta}+\boldsymbol{\varepsilon}\).

✅ OLS chooses \(\hat{\boldsymbol{\beta}}=(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{Y}\) when \(\mathbf{X}'\mathbf{X}\) is invertible.

✅ Adding regressors changes coefficient interpretation to a ceteris paribus interpretation.

✅ Matrix OLS and statsmodels produce the same coefficient estimates.

✅ Nested models help us see how omitted factors can alter estimated slopes.

Next Lecture

Lecture 2 — Interpretation of Coefficients

We will cover: - partial effects in MLR - ceteris paribus language - standardized coefficients - R² versus adjusted R²