Applications of Simple Linear Regression in Finance

A regression line is the best straight-line summary of how one variable moves with another. On the exam, you will fit it, test it, interpret it, and use it (especially as capital asset pricing model (CAPM)).

To describe and explain a simple linear regression, start with the population model $Y_i=b_0+b_1{X}_i+{\epsilon }_i$. Y is the dependent variable, X is the independent variable, and epsilon is the random error. Ordinary least squares (OLS) handles coefficient estimation by picking the line that minimizes the sum of squared residuals $\sum (Y_i-{\hat{Y}}_i{)}^2$. Squaring penalizes large misses and produces unique closed-form estimates.

$${\hat{b}}_1=\frac{\sum (X_i-\bar{X})(Y_i-\bar{Y})}{\sum (X_i-\bar{X}{)}^2},{\hat{b}}_0=\bar{Y}-{\hat{b}}_1\bar{X}$$

KEY: Slope tells you the change in Y for a one-unit change in X. Intercept is the predicted Y when X = 0 (often economically meaningless, but mathematically required to anchor the line).

You identify violations through residual diagnostics, with each assumption tied to its own residual plot.

Common mistakes

• Confusing SEE with the prediction interval half-width. SEE measures average residual size from the fitted line. The forecast standard error $s_f$ is larger because it adds parameter uncertainty. Trap: using SEE × 1.96 as the interval.
• Treating heteroskedasticity as a coefficient-bias problem. Wrong. It corrupts standard errors only. Coefficient estimates remain unbiased and consistent.
• Reading the intercept as economically meaningful. Often $X=0$ is outside the sample range. Intercept anchors the line; do not over-interpret unless X = 0 is plausible (CAPM's alpha is the exception, it has direct economic meaning).

Bottom line

• OLS picks the slope and intercept that minimize the sum of squared residuals. Slope $b_1=\text{Cov}(X,Y)/\text{Var}(X)$; intercept $b_0=\bar{Y}-b_1\bar{X}$.
• Four assumptions: linearity, independence (no autocorrelation), homoskedasticity, normally distributed errors. Each has a residual-plot diagnostic.
• $R^2=\text{RSS}/\text{SST}$; F = MSR/MSE; t-stat on slope = ${\hat{b}}_1/s_{{\hat{b}}_1}$. In simple regression, $F=t^2$.
• SEE is the typical residual measured in the units of Y, capturing average distance from the fitted line.

Exam shortcut

For ANOVA: in simple regression, regression df = 1, residual df = $n-2$, and $F=t^2$. For prediction intervals: forecast standard error is always larger than SEE, so any answer choice equal to SEE × t-critical is a distractor. For CAPM: dependent variable is excess return, slope is beta, intercept is alpha. If a problem hands you raw returns, subtract $R_f$ first.

The full lesson (about 2,210 words, 15 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• applications of simple linear regression in finance

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