Regression Functional Forms: Log Models, Quadratics & Interaction Terms

A simple linear regression assumes that the relationship between X and Y is linear in levels — a one-unit increase in X always produces the same dollar change in Y. In practice, many financial relationships are nonlinear. Firm revenue may grow at a decreasing rate with R&D spending, or the effect of leverage on credit spreads may depend on firm size. Choosing the wrong functional form — known as misspecification — can violate the zero conditional mean assumption, distort coefficient interpretation, and undermine the conclusions drawn from a multiple regression. Understanding log-linear regression models, quadratic specifications, and interaction terms is essential for correctly modeling these relationships.

Why Functional Form Matters

Ordinary least squares (OLS) estimates the conditional mean of Y given X. If you specify that mean incorrectly — for example, fitting a straight line when the true relationship is curved — the estimated coefficients no longer have their intended interpretation. A misspecified model can make a positive effect appear negative, overstate marginal effects at some values of X, and understate them at others.

The RESET test (Regression Equation Specification Error Test) is a widely used diagnostic for detecting functional form misspecification. It works by adding powers of the fitted values to the original regression and testing whether they are jointly significant. A rejection suggests the linear-in-levels specification is inadequate. For more on specification errors and their consequences, see Omitted Variable Bias.

Log-Linear Regression (Log-Level Model)

The log-level model — also called log-linear regression — takes the natural logarithm of the dependent variable while keeping the independent variable in levels. Beyond its interpretive advantages, the log transformation reduces right-skew in the dependent variable and lessens sensitivity to extreme values, making OLS estimates more robust in financial data where variables like revenue and market capitalization span orders of magnitude.

The coefficient β₁ is a semi-elasticity: it measures the approximate percentage change in Y for a one-unit change in X, holding other factors fixed. This interpretation is central to log-linear regression and appears throughout applied finance research.

Level-Log Model (Linear-Log Regression)

The level-log model keeps Y in levels but takes the log of X:

This specification captures diminishing returns: large absolute changes in X produce progressively smaller effects on Y. It is appropriate when proportional changes in the independent variable matter more than absolute changes.

For example, regressing a corporate bond’s credit spread (in basis points) on log(total assets) for a sample of investment-grade issuers yields β₁ = −45.2. A 1% increase in total assets lowers the credit spread by approximately 0.452 basis points — reflecting the diminishing benefit of additional firm size on borrowing costs.

Log-Log Model (Constant Elasticity)

The log-log model takes the natural log of both the dependent and independent variables:

In a log-log regression, the coefficient β₁ is directly interpretable as a constant elasticity. This specification is the standard econometric tool for estimating elasticities in financial and economic research. For the underlying economic theory of elasticity — including demand curves and consumer behavior — see Price Elasticity of Demand.

Quadratic Models

A quadratic specification adds the square of X to capture a relationship that increases (or decreases) at a changing rate:

Unlike a log model where the functional form is fixed, the quadratic allows the direction of the effect to change. When β₁ > 0 and β₂ < 0, Y first increases with X and then decreases — an inverted-U shape representing diminishing and eventually negative returns. The turning point X* is only substantively meaningful if it falls within the observed range of X in the sample; a turning point far outside the data may be an artifact of the functional form rather than a real economic optimum.

Interaction Terms

Continuous × Continuous Interactions

An interaction term allows the effect of one variable to depend on the level of another:

Consider how leverage affects credit spreads differently for firms of different sizes. Regressing credit spread (basis points) on leverage (debt-to-equity ratio), log(total assets in $ millions), and their interaction for 400 corporate bond issuers yields β₁ = 55.0 on leverage, β₂ = −12.1 on log(assets), and β₃ = −8.5 on the interaction. The marginal effect of a one-unit increase in leverage is 55.0 − 8.5 × log(assets). For a mid-size firm with $50 million in assets (log = 3.91), the effect is 55.0 − 8.5(3.91) = 21.8 bps. For a large firm with $500 million in assets (log = 6.21), the effect falls to 55.0 − 8.5(6.21) = 2.2 bps — meaning larger firms face a substantially smaller leverage penalty on borrowing costs.

Dummy × Continuous Interactions

When a binary indicator (0 or 1) is interacted with a continuous variable, the interaction allows different slopes for different groups. In the model Y = β₀ + β₁X + β₂D + β₃(D × X), the slope on X is β₁ when D = 0 and β₁ + β₃ when D = 1, while β₂ captures the intercept shift between groups. For instance, interacting an investment-grade dummy with leverage allows the marginal effect of leverage on credit spread to differ between investment-grade and speculative-grade issuers. For a full treatment of dummy variables in regression, see Dummy Variables in Regression.

Choosing the Right Functional Form

The four basic functional forms each imply a different interpretation of β₁. Choosing among them requires thinking about what kind of relationship makes economic sense for your data.

Common Mistakes

1. Forgetting the percentage interpretation of log models. When the dependent variable is log(Y), the coefficient is not a dollar change. A coefficient of 0.03 means Y increases by approximately 3%, not by 0.03 units. This error leads to dramatically understated effect sizes in financial research.

2. Comparing R-squared across log(Y) and Y models. R-squared from a regression with log(Y) on the left measures explained variation in log-dollars; R-squared from a level model measures explained variation in dollars. A higher R-squared in the log model does not necessarily indicate better predictive accuracy for Y itself.

3. Misinterpreting interaction coefficients. When an interaction term X₁ × X₂ is in the model, reporting β₁ as “the effect of X₁” is incorrect. The marginal effect is β₁ + β₃X₂, which varies with X₂. Evaluating this at the sample mean of X₂ gives the average marginal effect.

4. Reporting a quadratic turning point without checking the sample range. The formula X* = −β₁ / (2β₂) always yields a number, but that number is only economically meaningful if it falls within the range of X observed in the data. A turning point well outside the sample may simply reflect the quadratic capturing mild curvature rather than a true maximum or minimum.

5. Applying log transformations to variables with zeros or negatives. The natural log is undefined for zero and negative values. Variables like net income (which can be negative) or bond default counts (which include zeros) cannot be log-transformed directly. The common workaround log(1 + Y) introduces sensitivity to the units of measurement and should be used cautiously.

Limitations

1. Log models assume multiplicative relationships. If the true relationship between X and Y is additive rather than multiplicative, forcing a log specification introduces bias that cannot be corrected by adding more observations.

2. Quadratic models extrapolate poorly. A quadratic fits well within the sample range but can produce wildly inaccurate predictions outside it. The estimated turning point should always be checked against the actual range of the data — a turning point far outside the observed values may be an artifact of the functional form rather than a real economic phenomenon.

3. Interaction terms increase multicollinearity. Including X₁, X₂, and X₁ × X₂ as regressors can create high correlation among them, inflating standard errors and making individual coefficients difficult to interpret. Centering variables (subtracting their sample means) before constructing the interaction term can reduce this problem.