Applications of Simple Linear Regression in Finance

Regression is the workhorse for measuring how one financial variable responds to another: a stock to the market, a bond spread to growth, sales to advertising. This module covers how a simple linear regression is estimated, the assumptions behind it and how to check them, how to judge the fit and significance of the result, how to forecast with a prediction interval and alternative functional forms, and how regression is used to estimate beta and alpha in the capital asset pricing model.

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What you will learn

1. Describe the simple linear regression model and calculate and interpret its intercept and slope estimated by ordinary least squares.

Model: Y = b0 + b1 X + error; Y is the dependent variable, X the independent variable.
OLS minimizes the sum of squared residuals.
Slope = Cov(X, Y) / Var(X); intercept = mean of Y - slope x mean of X.
The fitted line passes through the means; slope = r x s_Y / s_X.
Slope = change in Y per one-unit change in X; the intercept is Y when X = 0.

2. Describe the assumptions of simple linear regression and use residual analysis to identify their violations.

Assumptions: linearity, homoskedasticity, independence of errors, normally distributed errors.
Curved residual pattern = non-linearity; fanning residuals = heteroskedasticity.
Runs of same-signed residuals over time = serial correlation, which understates standard errors.
Heteroskedasticity and serial correlation leave coefficients unbiased but make significance tests unreliable.
Check for outliers and influential observations that can drive the fitted line.

3. Evaluate the goodness of fit of a simple linear regression and the significance of its coefficients using the coefficient of determination, the standard error of estimate, ANOVA and t-tests.

SST = SSR + SSE; R^2 = SSR / SST (= r^2 in simple regression).
SEE = square root of [SSE / (n - 2)] = square root of MSE.
F = MSR / MSE with 1 and n - 2 df; in simple regression F = t^2 for the slope.
t = (estimated slope - hypothesized slope) / standard error of the slope, with n - 2 df.
Test against the economically relevant value (e.g., beta = 1), not only against zero.

4. Calculate and interpret predicted values and prediction intervals, and describe log-lin, lin-log and log-log functional forms.

Predicted value = b0 + b1 x X_f.
Prediction interval = Y-hat +/- t_c x s_f, with n - 2 df.
s_f includes the SEE and parameter uncertainty; the interval widens away from the mean of X.
Log-lin: slope = relative change in Y per unit of X (growth); log-log: slope = elasticity.
Convert ln Y forecasts back with the exponential; compare R^2 only for the same dependent variable.

5. Estimate and interpret the beta and alpha of the capital asset pricing model using a simple linear regression of excess returns.

CAPM: E(R_i) = R_f + beta_i [E(R_M) - R_f].
Estimate beta as the slope of a regression of the asset's excess returns on the market's excess returns.
Beta = Cov(R_i, R_M) / Var(R_M); alpha (intercept) = mean excess return - beta x mean market excess return.
R^2 = share of variance that is systematic; 1 - R^2 = unsystematic share.
Estimates depend on the market proxy, data frequency and window; test alpha = 0 and beta = 1.

Module at a glance

OLS slope = Cov(X, Y) / Var(X); intercept = mean Y - slope x mean X; the line passes through the means.
Check residuals for non-linearity, heteroskedasticity, serial correlation and non-normality.
R^2 = SSR / SST; SEE = square root of [SSE / (n - 2)]; F = MSR / MSE = t^2 in simple regression.
Test coefficients with t = (estimate - hypothesized value) / standard error, df = n - 2.
Prediction interval = Y-hat +/- t_c x s_f, wider away from the mean of X.
Log-lin slope = growth per unit of X; log-log slope = elasticity; exponentiate ln Y forecasts.
CAPM regression: slope = beta, intercept = Jensen's alpha; R^2 = systematic share of risk.

This free module has 5 lessons, 13 practice questions with full explanations.

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