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

Why this matters¶

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

• Are countries with higher income associated with higher (or lower) divorce outcomes?

• Are higher prices associated with lower demand?

• Which factors are correlated with churn or customer satisfaction?

But regression is easy to misuse—especially when we confuse association with causation.

Where regression sits in the AI/ML/DS map¶

The regression mindset: from question to model¶

Today’s running example (new dataset)¶

We use:

• divorce-raw-1992.csv — divorce data

• Unit: country (2019 cross-section)

• Outcome $Y$: divorce–marriage ratio
  d_rate = divorce_rate / marriage_rate

• Main predictor $X$: income (rescaled GDP per capita)
  income_pc = gdp_pc / 1000

• Control candidate: GDP growth (gdp_gr)

Step 0: Start with a picture (scatter plot)¶

Before any equation, draw the relationship.

• If the scatter is basically a cloud: the relationship may be weak or non-linear.

• If there is a visible upward/downward pattern: regression helps summarize it.

Step 1: Correlation (a warm-up)¶

Correlation is a standardized measure of linear association.

• It ranges from -1 to +1.

• It is symmetric: $\mathrm{corr}(X,Y)=\mathrm{corr}(Y,X)$.

• It does not tell you “how much $Y$ changes when $X$ changes” in units.

Step 2: Simple regression (one predictor)¶

A simple regression estimates a line:

$Y = \beta_0 + \beta_1 X + \varepsilon$

• $\beta_1$ tells you how $Y$ changes (on average) with a one-unit change in $X$.

• $\varepsilon$ captures everything else not in the model.

Our example (simple regression)¶

• $Y$: d_rate

• $X$: income_pc

Interpretation template:

“A one-unit increase in income_pc (i.e., $1{,}000$ dollars of GDP per capita) is associated with a $\beta_1$ change in d_rate, on average, in this sample.”

Step 3: Multiple regression (adding a control)¶

Multiple regression adds additional predictors:

$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \varepsilon$

Now $\beta_1$ is interpreted as the association between $Y$ and $X_1$ holding other included variables constant.

Why add controls?¶

To reduce confounding (informally):

• If both income level and macroeconomic conditions are related to divorce/marriage patterns, a simple regression may mix these relationships.

Today we add:

• gdp_gr (GDP growth) as a simple control.

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

2. Uncertainty (standard errors / confidence intervals)

• How precise is the estimate?

3. Fit (R-squared)

• How much of the variation in $Y$ is explained (in-sample)?

• Do not treat $R^2$ as “truth” or “usefulness” by itself.

Regression and causality (a careful note)¶

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

Common threats:

• Omitted variables

• Reverse causality

• Measurement error

• Selection bias

Mini-lab (Google Colab)¶

In-class checkpoints¶

1. Load divorce_raw.csv and filter to year = 2019.

2. Create:

   • income_pc = gdp_pc / 1000

   • d_rate = divorce_rate / marriage_rate * 1000

3. Make a scatter plot: d_rate vs income_pc.

4. Compute correlation between d_rate and income_pc.

5. Fit simple regression: d_rate ~ income_pc and interpret $\beta_1$.

6. Fit multiple regression: d_rate ~ income_pc + gdp_gr and interpret how $\beta_1$ changes (if it does).

7. Make one diagnostic plot (residuals vs fitted OR histogram of residuals).

8. Write a short “manager memo” (5–7 lines): headline + evidence + caveat.

Submission (after class)¶

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

• Include your memo + one caveat as Markdown.

AI check (responsible use)¶