Updated for the 2026-2027 CFA® Level I curriculum.
The functional form of a regression determines whether each variable enters the model in its original units or as a natural logarithm. The coefficients are still estimated using least squares, but the logged variables change how you interpret the slope.
For CFA Level I, the main task is to identify which variables are logged and translate the slope coefficient into a clear statement about units or percentages.
Quick Answer
Simple linear regression can take four functional forms: level-level, log-level, level-log, and log-log. The position of the logarithm determines how you interpret the slope. A variable in levels changes in units, while a logged variable changes in percentage terms. A log-log slope measures elasticity because it connects a percentage change in with a percentage change in .
Key Takeaways About Functional Forms of Simple Linear Regression
A level-level model relates a unit change in to a unit change in .
A log-level model relates a one-unit change in to an approximate percentage change in .
A level-log model relates a 1% change in to a unit change in .
A log-log model relates a 1% change in to a percentage change in .
The slope in a log-log regression is an elasticity.
Log-based percentage interpretations are usually approximations for small changes.
Logging a variable may help represent proportional relationships, diminishing marginal effects, or highly skewed data.
A transformation should still be evaluated using model fit and residual diagnostics.
The most common exam mistake is mixing up whether the dependent or independent variable is logged.
What You Need to Know for CFA Level I
For CFA Level I, focus on:
Recognizing the four functional forms from their equations.
Identifying whether the dependent variable, independent variable, both variables, or neither variable is logged.
Interpreting correctly in units or percentages.
Applying the required multiplication or division by 100.
Explaining why the log-log slope is an elasticity.
Distinguishing an approximate percentage interpretation from an exact change.
Translating an estimated regression equation into a practical investment statement.
The official learning outcome asks candidates to describe the different functional forms of simple linear regressions.
What Is a Functional Form in Regression?
A functional form describes how the dependent and independent variables appear in a regression equation.
A variable in levels remains in its original measurement units. Depending on the variable, those units might be dollars, percentage points, years, shares, or another numerical measure.
A logged variable is expressed using its natural logarithm, written as . Logging changes the scale of the variable and allows the coefficient to describe a relative or percentage change.
The model remains linear in its coefficients. For example, the following equation contains a logged independent variable, but and still enter the model linearly:
The functional form changes the interpretation of , rather than the basic least-squares process used to estimate it.
What Are the Four Functional Forms of Simple Linear Regression?
Level-Level Model
In a level-level model, neither variable is logged.
A one-unit increase in is associated with a -unit change in .
For example, if , a one-unit increase in is associated with a three-unit increase in .
Log-Level Model
In a log-level model, the dependent variable is logged and the independent variable remains in levels.
A one-unit increase in is associated with an approximate change in .
For example, if , a one-unit increase in is associated with an approximate 4% increase in .
This form is also sometimes described as a log-linear regression.
Level-Log Model
In a level-log model, the dependent variable remains in levels and the independent variable is logged.
A 1% increase in is associated with an approximate -unit change in .
For example, if , a 1% increase in is associated with an approximate 0.25-unit increase in .
This form may also be called a linear-log regression.
Log-Log Model
In a log-log model, both variables are logged.
A 1% increase in is associated with an approximate change in .
For example, if , a 1% increase in is associated with an approximate 0.65% increase in .
The slope is an elasticity because it measures the percentage change in one variable relative to a percentage change in another.
Where:
= dependent variable for observation
= independent variable for observation
= estimated intercept
= estimated slope coefficient
= natural logarithm
= error term
How Do You Interpret the Slope in Each Functional Form?
Functional Form | Equation | Change in | Change in | Interpretation of |
|---|---|---|---|---|
Level-level | One unit | Units | units | |
Log-level | One unit | Percentage | Approximately | |
Level-log | 1% | Units | Approximately units | |
Log-log | 1% | Percentage | Approximately |
A useful exam shortcut is to read the equation from left to right:
Look at the dependent variable to determine whether the result for should be stated in units or percentages.
Look at the independent variable to determine whether the change in should be stated in units or percentages.
How Can You Identify the Functional Form in an Exam Question?
Use the following three-step process.
1. Check the Dependent Variable
The dependent variable appears on the left side of the equation.
If appears in levels, describe its change in units.
If appears, describe its change as a percentage.
2. Check the Independent Variable
The independent variable appears on the right side of the equation.
If appears in levels, begin with a one-unit change.
If appears, begin with a 1% change.
3. Apply the Correct Scaling
Use the placement of the logarithms to determine whether the coefficient needs to be multiplied or divided by 100.
Log-level: multiply by 100.
Level-log: divide by 100.
Log-log: read directly as an elasticity.
Level-level: read directly in units.
This process is usually faster and safer than memorizing the names alone. Similar names such as log-level and level-log are easy to reverse under exam pressure.
Why Do Analysts Use Logarithms in Regression?
To Model Proportional Relationships
Many financial relationships are easier to understand in percentage terms.
A $10 million increase in sales may be significant for a small company and relatively minor for a large company. A percentage increase provides more context because it measures the change relative to the company’s starting size.
To Represent Diminishing Marginal Effects
A level-log model allows the effect of on to become smaller as grows.
The marginal effect in a level-log model is:
As becomes larger, becomes smaller.
For example, an additional $1 million of advertising may have a larger effect when the existing advertising budget is $2 million than when it is $200 million.
To Compress Large Differences in Scale
Financial data can contain observations that differ greatly in size. Company sales, asset values, and market capitalizations often span several orders of magnitude.
Taking logarithms reduces the distance between very large and very small values. This can make proportional patterns easier to model and may reduce skewness in the data.
To Support Constant-Elasticity Models
A log-log regression assumes that the elasticity between and remains constant.
If the estimated slope equals 0.65, the model associates any small percentage change in with a change in that is approximately 65% as large in percentage terms.
A logarithmic transformation does not automatically improve a model. Analysts should still examine its residuals, assumptions, fit, and economic logic.
When Are the Percentage Interpretations Approximate?
The common interpretations are based on small percentage changes.
Log-Level Exact Interpretation
For a log-level model, the exact percentage change in for a change of is:
For a one-unit increase in :
When is small:
For example, if , the approximation is 4%, while the exact increase is:
Level-Log Exact Interpretation
For a level-log model, the exact change in when moves from to is:
For a 1% increase in :
Log-Log Exact Relationship
For a log-log model:
For small changes, the familiar approximation is:
CFA Level I questions usually emphasize the approximate interpretations. The exact formulas help explain why the approximations work and when they may become less precise.
Worked Example: Four Functional Forms for One Company
An analyst studies a consumer goods company. Sales, market value, and advertising expenses are measured in millions of dollars. Earnings growth is measured in percentage points.
Level-Level: Sales and Advertising
The analyst estimates:
A $1 million increase in advertising is associated with a $3 million increase in predicted sales.
At an advertising expense of $10 million:
Predicted sales equal $70 million.
At an advertising expense of $11 million:
The $1 million increase in advertising raises predicted sales by $3 million.
Log-Level: P/E Ratio and Earnings Growth
The analyst estimates:
A one-percentage-point increase in expected earnings growth is associated with an approximate 4% increase in the predicted P/E ratio.
The approximate interpretation is:
The exact percentage change is:
Because the coefficient is small, the approximation is close to the exact result.
Level-Log: Sales and Advertising
The analyst estimates:
A 1% increase in advertising is associated with an approximate $0.25 million increase in predicted sales.
The exact change for a 1% increase is:
This form also implies diminishing marginal effects. The same dollar increase in advertising has a smaller effect when the existing advertising budget is already large.
Log-Log: Market Value and Sales
The analyst estimates:
A 1% increase in sales is associated with an approximate 0.65% increase in predicted market value.
The slope of 0.65 is the elasticity of market value with respect to sales.
Because the elasticity is below 1, market value changes less than proportionally to sales. For example, a 10% increase in sales produces an exact predicted increase in market value of approximately:
Each equation uses a different functional form, so the same-looking slope coefficient can carry a very different meaning.
Common Exam Traps
Reversing log-level and level-log. The first term in the name refers to the dependent variable.
Treating the level-log slope as an elasticity. The level-log slope gives a unit change in , not a percentage change.
Reading a log-level coefficient as a unit change. A logged dependent variable changes in percentage terms.
Forgetting to multiply a log-level coefficient by 100. A coefficient of 0.04 corresponds to approximately 4%, not 0.04%.
Forgetting to divide a level-log coefficient by 100. A coefficient of 25 corresponds to approximately 0.25 units for a 1% increase in .
Multiplying the log-log coefficient by 100. A slope of 0.65 already means a 1% increase in is associated with an approximate 0.65% increase in .
Treating the approximate interpretation as exact. The approximation becomes less precise as the coefficient or change becomes larger.
Assuming a logged model automatically fits better. Functional form should be supported by the data, residual diagnostics, and economic reasoning.
Ignoring the measurement units. A one-unit change may mean one dollar, one million dollars, one year, or one percentage point.
Practice Question
An analyst estimates the following regression, where the dependent variable is a company’s price-to-earnings ratio and expected earnings growth is measured in percentage points:
If expected earnings growth rises by one percentage point, the predicted price-to-earnings ratio increases by approximately:
0.04 units
4%
0.04%
Correct Answer: B
The dependent variable is logged, while the independent variable remains in levels. This is a log-level model.
In a log-level model, a one-unit increase in is associated with an approximate change in .
The predicted P/E ratio therefore increases by approximately 4%.
The exact increase would be:
Option A. 0.04 units treats the model as level-level and ignores the logarithm applied to the dependent variable.
Option C. 0.04% reads the coefficient directly as a percentage without multiplying by 100.
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FAQs About Functional Forms of Simple Linear Regression
What Are the Four Functional Forms of Simple Linear Regression?
The four functional forms are level-level, log-level, level-log, and log-log.
They differ according to whether the dependent variable, independent variable, both variables, or neither variable is expressed as a natural logarithm.
What Is the Difference Between Log-Level and Level-Log Regression?
A log-level regression logs the dependent variable. A one-unit increase in is therefore associated with an approximate percentage change in .
A level-log regression logs the independent variable. A 1% increase in is associated with a unit change in .
How Can I Remember Which Variable Comes First in the Name?
The dependent variable comes first.
Log-level: logged dependent variable, level independent variable.
Level-log: level dependent variable, logged independent variable.
Looking at the left side of the equation before interpreting the coefficient can prevent the two forms from being reversed.
Why Is the Log-Log Slope an Elasticity?
Elasticity measures the percentage change in one variable associated with a 1% change in another variable.
Because both variables are logged, the slope in a log-log model directly links a percentage change in with a percentage change in .
Why Are Logarithms Used in Regression?
Logarithms help analysts model proportional relationships, express effects in percentage terms, compress large differences in scale, and represent diminishing marginal effects.
A log transformation should still be supported by the economic relationship and the model’s diagnostic results.
Are the Percentage Interpretations Exact?
The common interpretations are approximations for small changes.
For a log-level model, the exact percentage change is calculated using . For a log-log model, the exact relationship depends on the ratio of the new and original values of . The approximations used in CFA questions are generally accurate when the changes are small.