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 fit it, test it, read its analysis of variance table, forecast with it, and apply it to the capital asset pricing model (CAPM).

Picture a scatter plot of six years of data. GDP growth sits on the horizontal axis and a retailer's sales growth on the vertical axis. The points drift upward, but they do not sit on a line. Simple linear regression draws the one straight line that best summarizes that drift.

$$Y_i = b_0 + b_1 X_i + \varepsilon_i$$

Y is the dependent variable, the one you are trying to explain. X is the independent variable, the explanatory variable. The intercept $b_0$ and slope $b_1$ are the regression coefficients. The error term $\varepsilon_i$ is everything the line misses. Once you estimate the line, each observation's miss is its residual: the actual Y minus the fitted $\hat{Y}$.

Ordinary least squares (OLS) picks the intercept and slope that make the sum of squared residuals as small as possible. Squaring does two jobs.

Common mistakes

• Using SEE as the forecast standard error. In Example 2, 2.8 ± 2.00 × 4.55 gives a half-width of 9.10. The correct $s_f$ is 4.59, for a half-width of 9.18. Trap: the interval built from SEE is always slightly too narrow, and it appears as a distractor.
• Dividing SSE by n − 1. Two coefficients were estimated, so the error degrees of freedom are n − 2. In Example 1, $\sqrt{3.2/5}$ = 0.80 is the trap; the correct SEE is 0.894.
• Computing R² as SSE / SST. That is the unexplained share. In Example 1, 3.2 / 37.5 = 0.085 is the trap; $R^2$ is 34.3 / 37.5 = 0.915.

Bottom line

• OLS minimizes squared residuals. Slope = Cov(X,Y) / Var(X); intercept = $\bar{Y}$ − slope × $\bar{X}$, so the line passes through the means.
• Four assumptions: linearity, homoskedasticity, independence, normality. Residual plots reveal violations.
• SST = SSR + SSE. $R^2$ = SSR / SST. MSE = SSE / (n − 2). SEE = $\sqrt{\text{MSE}}$. F = MSR / MSE = $t^2$ for the zero-slope t.
• Slope standard error = SEE / $\sqrt{\sum (X - \bar{X})^2}$. Test any hypothesized slope with n − 2 degrees of freedom.

Exam shortcut

When a question hands you an ANOVA table with a blank, rebuild it from the two anchors: degrees of freedom are 1 and n − 2, and every mean square is its sum of squares over its degrees of freedom. When a question gives only the variance of X, the sum of squared X deviations is (n − 1) times that variance.

The full lesson (about 4,328 words, 29 min read) adds 5 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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