Line of Best Fit

What it is

A line of best fit (best-fit line) is a straight line that summarizes the relationship between two variables in a scatter plot by minimizing the distance between the line and the observed data points. It is the primary output of linear regression and is used to identify trends, quantify relationships, and make simple predictions.

Why it matters

• Shows the overall trend in noisy data.
• Quantifies the direction and strength of a linear relationship.
• Provides a simple predictive model (extrapolation or interpolation).
• Used across fields: finance, economics, science, engineering, and social sciences.

How it’s determined

The most common method is ordinary least squares (OLS). OLS fits a line that minimizes the sum of the squared vertical distances (residuals) between observed values and the predicted values on the line.

For simple linear regression (one independent variable):
– Model: y = a + b x
– Slope (b) measures change in y per unit change in x.
– Intercept (a) is the predicted value of y when x = 0.

Closed-form formulas:
– b = Σ[(xi − x̄)(yi − ȳ)] / Σ[(xi − x̄)²]
– a = ȳ − b x̄

In multiple regression (multiple independent variables), the model generalizes to:
– y = c + b1 x1 + b2 x2 + … + bn xn
Each coefficient bj represents the partial effect of xj on y, holding other variables constant. Coefficients are typically estimated by solving the normal equations or using matrix-based OLS in statistical software.

Residuals and model fit

• Residual = observed y − predicted y.
• Sum of squared residuals (SSR) is minimized by OLS.
• Goodness of fit is often assessed with R² (proportion of variance in y explained by the model) and by inspecting residual plots for patterns (nonrandom patterns suggest model misspecification).

Straight line vs best-fit curve

Although the phrase “line of best fit” implies linearity, data may be better described by a curve. Common alternatives:
– Polynomial (quadratic, cubic)
– Logarithmic or exponential
– Power or root functions
Selecting a more complex form can improve fit but risks overfitting; prefer the simplest model that adequately captures the pattern.

Practical use in finance

Financial analysts use best-fit lines to:
– Estimate relationships (e.g., stock price vs. earnings per share).
– Build regression-based factor models (e.g., explaining a stock’s returns with market indices and macro variables).
– Make short-term projections by extrapolating trends, with the caveat that relationships can change over time.

How to compute one

• Visual approximation: sketch a line through the middle of points (quick but imprecise).
• Analytical OLS: compute using the formulas above (suitable for small datasets).
• Software: use Excel, R, Python (statsmodels, scikit-learn), or statistical packages that return coefficients, diagnostics, and plots.

Limitations and cautions

• Correlation ≠ causation: a best-fit line describes association, not causation.
• Outliers can strongly influence the fitted line.
• Extrapolation beyond observed data can be misleading.
• Linear models may be inappropriate for nonlinear relationships.

Key takeaways

• The line of best fit summarizes a linear relationship by minimizing squared residuals.
• OLS is the standard method; coefficients quantify the effect of predictors.
• Curves can replace lines for nonlinear patterns, but simplicity and interpretability are important.
• Use diagnostics and domain knowledge to check model validity before using the line for prediction.