Lesson 6 — Simple & Multiple Regression (interpretation for business/economics)

Why this matters (motivation)¶

Regression is one of the most widely used tools in business and economics because it helps quantify relationships:

• How is sales associated with marketing spend?

• Are higher prices associated with lower demand?

• Which factors are correlated with churn or customer satisfaction?

But regression is also easy to misuse, especially when people confuse association with causation.

The regression mindset: from question to model¶

Today’s scope¶

We focus on:

• fitting regressions,

• interpreting outputs,

• and communicating limitations.

We will not claim causality unless the design supports it.

Simple regression (one predictor)¶

Example¶

$Y$ = monthly sales
$X$ = marketing spend

Interpretation: if $\beta_1 = 2.5$, then an extra unit of spend is associated with +2.5 units of sales (on average), in this dataset.

Multiple regression (controls)¶

Why add controls?¶

To reduce confounding (informally):

• If both marketing spend and season affect sales, a simple regression might mix the two effects.

Example:

• $Y$ = sales

• $X_1$ = marketing spend

• $X_2$ = season indicator (e.g., Q4 = 1)

• $X_3$ = price

Now $\beta_1$ is the association between sales and marketing, after accounting for season and price.

How to read regression output (what we focus on)¶

1) Coefficients¶

• sign: positive/negative association

• magnitude: “how much change in $Y$ per unit of $X$”

• units matter (e.g., dollars vs thousands)

2) Uncertainty (standard errors / confidence intervals)¶

We care about whether the estimate is precise enough to be actionable.

3) Fit (R-squared)¶

• R² tells you how much variation in $Y$ the model explains in-sample.

• A high R² does not guarantee good decision-making, and a low R² does not mean the model is useless.

Regression and causality (a careful note)¶

Regression describes a relationship. Causal interpretation requires stronger assumptions/design.

Common threats:

• Omitted variables: a third factor affects both X and Y

• Reverse causality: Y influences X (e.g., firms spend more on marketing when sales rise)

• Measurement error: X or Y is noisy or mismeasured

• Selection bias: data is not representative

Mini case: sales, marketing, and seasonality¶

Question: Is marketing spend associated with sales after accounting for seasonality and price?

Steps:

1. EDA reminder: plot sales over time and by category/region

2. Fit simple regression: sales ~ marketing

3. Fit multiple regression: sales ~ marketing + price + season + category/region dummies (as appropriate)

4. Compare coefficient on marketing and discuss how the interpretation changes

5. Write a short manager memo with one caveat

Mini-lab (Google Colab)¶

In-class checkpoints

1. Fit a simple regression and interpret $\beta_1$ in units.

2. Fit a multiple regression that adds at least 2 controls.

3. Compare the coefficient on your main variable (e.g., marketing) across the two models:

   • Did it change sign or magnitude?

   • Why might that happen?

4. Create one diagnostic plot (residuals vs fitted OR histogram of residuals).

5. Write a short “manager memo” (5–7 lines) in the notebook.

Submission (after class)

• Share the Colab link (view permission) or export to PDF.

• Include a short memo + one caveat as Markdown.

AI check (responsible use for regression)¶

Good prompt examples

• “Generate statsmodels code to run OLS of sales on marketing, price, and a season dummy, with a clear summary output.”

• “How should I interpret the coefficient on marketing when price and season are included?”

• “Suggest one diagnostic plot and explain what it checks.”

Bad prompt example

• “Write my conclusion that marketing causes sales to rise” (without a causal design)

Review questions (quiz / reflection)¶

1. What changes about the interpretation of $\beta_1$ when you move from simple to multiple regression?

2. Give one example of an omitted variable that could bias your regression.

3. Why is “high R²” not the same as “good model”?