A best-fit line is the straight line used to summarize the average linear relationship between two variables in a scatterplot.
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How it is chosen
In ordinary least squares, the fitted line is written as:
$$
\hat{y} = \hat{\alpha} + \hat{\beta}x
$$
and the coefficients are chosen to minimize the sum of squared residuals:
$$
\min_{\alpha,\beta} \sum_i (y_i - \alpha - \beta x_i)^2
$$
This is why the best-fit line is not just a visual guess. It is an optimization result.
What it tells you
The slope shows the average change in (y) associated with a one-unit increase in (x), and the intercept shows the fitted value when (x=0). The line is useful for prediction and for describing linear association, but it does not by itself prove causation.
Why economists use it
Economists use fitted lines constantly in empirical work because many first-pass questions are about direction, strength, and approximate magnitude of a relationship before moving to richer models.
Knowledge Check
### What does a best-fit line summarize?
- [x] The average linear relationship between two variables
- [ ] The exact value of every data point
- [ ] A legal minimum price
- [ ] A nonlinear equilibrium by definition
> **Explanation:** The line captures the central linear tendency in the data rather than matching every observation perfectly.
### Why are residuals squared in ordinary least squares?
- [x] To penalize larger errors more heavily and avoid positive and negative residuals canceling out
- [ ] To force the line through every point
- [ ] To eliminate the intercept term
- [ ] To prove causality
> **Explanation:** Squaring makes all errors positive and gives larger deviations more weight in the optimization.
### Does a best-fit line prove that x causes y?
- [ ] Yes
- [x] No
- [ ] Yes, whenever the line slopes upward
- [ ] Yes, if the sample is large
> **Explanation:** A fitted line can describe association very well without identifying a causal mechanism.