Fitting a regression model yields numbers: $\hat{\beta}_0 = 23.5$, $\hat{\beta}_1 = 4.2$. But what do these numbers mean? How do we communicate them to stakeholders? What claims can we legitimately make?

Coefficient interpretation is where regression meets the real world. A technically correct model is worthless if we interpret it incorrectly or overclaim its implications. This page develops the skills to interpret coefficients rigorously, honestly, and usefully.

The slope $\hat{\beta}_1$ is the heart of simple linear regression. It quantifies the relationship between $x$ and $y$.

Marginal Effect Interpretation

The slope represents the expected change in $y$ for a one-unit increase in $x$:

$$\hat{\beta}_1 = \frac{\Delta \hat{y}}{\Delta x}$$

If $\hat{\beta}_1 = 4.2$, then increasing $x$ by 1 unit is associated with an expected increase of 4.2 units in $y$.

Example: In a regression of house price ($y$, in thousands of dollars) on square footage ($x$, in hundreds of square feet):

$$\widehat{\text{Price}} = 50 + 4.2 \times \text{SqFt}$$

Interpretation: "For each additional 100 square feet, the expected price increases by $4,200."

Sign of the Slope

• $\hat{\beta}_1 > 0$: Positive relationship. As $x$ increases, $\hat{y}$ increases.
• $\hat{\beta}_1 < 0$: Negative relationship. As $x$ increases, $\hat{y}$ decreases.
• $\hat{\beta}_1 = 0$: No linear relationship (but could be nonlinear).

Magnitude of the Slope

The magnitude $|\hat{\beta}_1|$ indicates the steepness of the relationship. But be careful: a "large" slope in one context (prices) may be "small" in another (probabilities). Practical significance depends on the domain.

The intercept $\hat{\beta}_0$ is often overlooked, but understanding it correctly prevents common errors.

Mathematical Definition

The intercept is the predicted value of $y$ when $x = 0$:

$$\hat{y}|_{x=0} = \hat{\beta}_0$$

It's where the regression line crosses the vertical axis.

When the Intercept is Meaningful

The intercept has a natural interpretation only when $x = 0$ is:

1. A possible value in the data range
2. Scientifically meaningful

Example of meaningful intercept: In a regression of test score on hours studied, $\hat{\beta}_0$ represents the expected score with zero study time (the baseline before any studying).

The Intercept's Role in Prediction

Even when not directly interpretable, the intercept is essential for accurate predictions. It positions the line vertically so that predictions are correct on average. Removing or constraining the intercept to zero (when inappropriate) biases predictions.

Centering to Create Meaningful Intercepts

A useful technique: center $x$ by subtracting its mean. If we regress $y$ on $(x - \bar{x})$:

$$\hat{y} = \hat{\beta}_0^* + \hat{\beta}_1(x - \bar{x})$$

Now $\hat{\beta}_0^* = \bar{y}$ (the mean of $y$), and the intercept represents the predicted value at the average $x$. This is often more interpretable.

The raw slope $\hat{\beta}_1$ depends on the scales of $x$ and $y$. To compare effects across different variables or studies, we standardize.

Standardized (Beta) Coefficients

If we standardize both variables to have mean 0 and standard deviation 1:

$$z_x = \frac{x - \bar{x}}{s_x}, \quad z_y = \frac{y - \bar{y}}{s_y}$$

And regress $z_y$ on $z_x$, the slope is:

$$\hat{\beta}1^{\text{std}} = r{xy}$$

The standardized slope equals the correlation coefficient! This is unit-free.

When to Use Standardized Coefficients

Example: In predicting salary from years of education and years of experience:

• Raw: "Each year of education → $5,000 more salary"
• Standardized: "Education has twice the effect of experience (β̂ = 0.4 vs 0.2)"

Both are correct; they answer different questions.

Perhaps no topic in regression is more important—or more frequently misunderstood—than the distinction between association and causation.

What Regression Measures

Ordinary regression measures association: the pattern of covariation between $x$ and $y$ in observed data. The slope $\hat{\beta}_1$ tells us how $y$ tends to differ when $x$ differs.

What Regression Does NOT Automatically Measure

Regression does not, by itself, tell us what would happen if we intervened to change $x$. Intervention is a causal concept; covariation is an associational concept.

Why Might Association ≠ Causation?

Three main threats to causal interpretation:

1. Confounding: A third variable $z$ causes both $x$ and $y$. Even with no direct $x → y$ effect, we observe an $x$-$y$ association.

2. Reverse causation: $y$ causes $x$, not vice versa. Example: Successful companies advertise more (success → advertising), not advertising → success.

3. Selection bias: The sample itself is selected based on $x$ or $y$ in ways that distort relationships.

When Can We Make Causal Claims?

Causal interpretation is justified when:

1. Randomized experiment: $x$ is randomly assigned, breaking confounding. (Gold standard)

2. Natural experiment: Some external event randomly varies $x$. (Approximates randomization)

3. Careful observational design: All confounders measured and controlled for, with strong theory. (Requires strong assumptions)

Safe Language for Observational Data

Use associational language:

• ✅ "X is associated with higher Y"
• ✅ "Higher X predicts higher Y"
• ✅ "For each unit difference in X, we observe 5 units difference in Y"

Avoid causal language:

• ❌ "X causes Y"
• ❌ "Increasing X will increase Y"
• ❌ "To raise Y, raise X"

Regression coefficients are estimated from observed data. Using them to predict far outside the observed range—extrapolation—is treacherous.

What is Extrapolation?

Given data with $x$ ranging from 10 to 50, predicting at $x = 45$ is interpolation (within range). Predicting at $x = 100$ is extrapolation (outside range).

Why is Extrapolation Dangerous?

The linear relationship is only confirmed within the observed range. Beyond it:

1. The true relationship may become nonlinear
2. The underlying mechanism may change
3. No data exists to detect or correct errors
4. Predictions can become absurd

Quantifying Extrapolation Risk

Recall the prediction variance at a new $x_0$:

$$\text{Var}(\hat{y}0) = \sigma^2 \left[ \frac{1}{n} + \frac{(x_0 - \bar{x})^2}{S{xx}} \right]$$

The term $(x_0 - \bar{x})^2/S_{xx}$ grows as $x_0$ moves away from the mean. Prediction intervals widen rapidly during extrapolation—the model honestly admits uncertainty.

Best Practices

1. Report the range of $x$ in your data
2. Be explicit when predictions are extrapolations
3. Use wide prediction intervals for extrapolated values
4. Consider domain knowledge about boundary behavior

Regression results must be communicated to various audiences: fellow data scientists, business stakeholders, policymakers, and the general public. Tailoring your interpretation is essential.

For Technical Audiences

Provide complete details:

"The OLS estimate is $\hat{\beta}_1 = 4.23$ (SE = 0.45, p < 0.001, 95% CI: 3.35–5.11). For each additional 100 sqft, expected price increases by $4,230, holding other factors constant. The model explains 78% of variance (R² = 0.78)."

For Business Stakeholders

Focus on actionable insights with uncertainty:

"Larger homes sell for more—roughly $4,000 more per 100 additional square feet. This relationship is statistically reliable. However, this doesn't mean adding 100 sqft will increase any particular home's value by exactly $4,000."

For General Audiences

Simplify without distorting:

"We found that bigger homes tend to sell for higher prices. On average, a home 100 square feet larger sells for about $4,000 more."

Even experienced practitioners make interpretation mistakes. Here are the most common and how to avoid them:

Defensive Practices

1. Write out interpretations in words before presenting—errors become obvious
2. Include units in every equation you present
3. Have a non-statistician review your plain-language summary
4. Ask yourself: "What would have to be true for my causal interpretation to be valid?"
5. Report effect sizes alongside p-values
6. Visualize predictions to check for absurdities

Interpreting regression coefficients correctly is where technical analysis meets real-world communication. Let's consolidate:

What's Next

We've covered model formulation, derivation, geometry, and interpretation. The final piece is understanding the assumptions that underpin all of this theory. Under what conditions are OLS estimates unbiased, consistent, and efficient? When do our confidence intervals and p-values mean what we think they mean? The next page addresses these critical questions.