CFA Level I Exam · Applications of Simple Linear Regression in Finance

Log-Lin, Lin-Log and Log-Log Regression Models Explained

Updated 6 October 2026 · Fact-checked

These are regression forms where you take the natural log of the dependent variable, the independent variable, or both, to fit a curved relationship with a straight-line model. Log-lin: a 1-unit change in X changes Y by about 100×b1 percent. Lin-log: a 1% change in X changes Y by b1÷100 units. Log-log: b1 is an elasticity.

Understand Functional Forms: Log-Lin, Lin-Log and Log-Log Models

Simple linear regression fits a straight line: Y = b0 + b1X + ε. But many financial relationships are not straight. A company's revenue may grow at a steady percentage rate, so it rises along a curve. Fitting a straight line to a curve gives poor fit and residuals that show a pattern.

A fix is to transform one or both variables with the natural logarithm (ln). After the transformation the relationship becomes linear in the transformed variables, so you can still use ordinary least squares and all the usual tests. The model stays linear in the parameters b0 and b1.

There are three common forms. In the log-lin model only Y is logged: ln(Y) = b0 + b1X. In the lin-log model only X is logged: Y = b0 + b1 ln(X). In the log-log model both are logged: ln(Y) = b0 + b1 ln(X). The name reads dependent variable first, then independent variable. Log means that variable is logged; lin means it is left in levels.

The key skill is interpretation, because the slope no longer means 'change in Y per one-unit change in X'. In log-lin, the slope is a relative change in Y per unit of X. In lin-log, it is an absolute change in Y per relative change in X. In log-log, it is a relative change in Y per relative change in X, which is an elasticity.

The log-lin model is the one used for constant growth. If Y_t = Y_0 × e^(b1 t), then ln(Y_t) = ln(Y_0) + b1 t. Regressing ln(Y) on time gives b1 as the continuously compounded growth rate per period. You choose a form by looking at the scatter plot, the residual plot and fit measures such as R² and standard error of estimate. Only compare R² across models if the dependent variable is the same.

Key formulas to remember

Log-lin model: ln(Y) = b0 + b1X + ε; Y is logged, X is not. A 1-unit rise in X changes Y by about 100 × b1 percent (relative change). Used for constant growth with X as time.
Lin-log model: Y = b0 + b1 ln(X) + ε; X is logged, Y is not. A 1% change in X changes Y by about b1 ÷ 100 units. Slope is an absolute change in Y.
Log-log model: ln(Y) = b0 + b1 ln(X) + ε; Both logged. A 1% change in X changes Y by about b1 percent. b1 is the elasticity of Y with respect to X.
Predicting Y from a log-dependent model: Ŷ = e^(predicted ln Y); If the dependent variable is ln(Y), compute the fitted ln(Y) first, then exponentiate to get Y in original units.
Constant growth as log-lin: ln(Y_t) = b0 + b1 t, growth rate ≈ b1 per period (continuously compounded); The equivalent periodic growth rate is e^b1 − 1.
Choosing a form: Pick the form whose residuals show no pattern and that fits well; Check the scatter plot, residual plot, and R² or standard error only among models with the same dependent variable.

How to solve Functional Forms: Log-Lin, Lin-Log and Log-Log Models questions

Use this method for any question on functional forms. Most items ask you to name the form, interpret a coefficient or compute a prediction.

1Look at which variables carry ln. Name the form: ln on Y only is log-lin, on X only is lin-log, on both is log-log.
2Identify what the question asks: interpret the slope, predict Y, or choose a form.
3For interpretation, use the rule for that form: log-lin gives a percent change in Y per 1-unit change in X; lin-log gives units of Y per 1% change in X; log-log gives percent change in Y per 1% change in X.
4Apply the scaling. Log-lin: multiply b1 by 100 to get percent. Lin-log: divide b1 by 100 to get units of Y per 1% change in X.
5For prediction, substitute X (or ln X) into the equation. If the dependent variable is ln(Y), exponentiate the result to get Y.
6For form choice, match the pattern: constant percentage growth over time suggests log-lin; diminishing effect of X on Y suggests lin-log; proportional relationships or elasticity suggest log-log.
7Check units and sign. Make sure your answer states percent versus units and the right direction of change.

Quickest way: Read the name, apply the rule

When to use it: Use it for interpretation questions in the exam, where you have about 90 seconds per item.

Say the form aloud: first word is Y, second word is X.
If Y is log, the answer is a percent change in Y; if Y is lin, it is a unit change in Y.
If X is log, the trigger is a percent change in X; if X is lin, it is a one-unit change in X.
Scale: a log-lin answer is b1 × 100 percent; a lin-log answer is b1 ÷ 100 units.
If asked for Y and the dependent variable is ln(Y), compute e^x before choosing an option (TI BA II Plus: 2ND then LN gives e^x; HP 12C: press g then LN, because e^x is the blue g function on the LN key).
Eliminate options that give the wrong unit (percent versus units), then confirm the scale.

Common mistakes in Functional Forms: Log-Lin, Lin-Log and Log-Log Models

Reading the log-lin slope as a unit change in Y.
Students carry over the ordinary regression meaning of the slope.
Fix: If Y is logged, the slope is a relative change. Multiply b1 by 100 to get the percent change in Y per 1-unit change in X.
Forgetting to divide by 100 in a lin-log model.
The slope looks like a plain effect, but it relates to a 1-unit change in ln(X), which is a 100% change.
Fix: Lin-log: a 1% change in X changes Y by b1 ÷ 100 units.
Reporting ln(Y) as the predicted value.
The equation gives the log of Y and the student stops there.
Fix: Exponentiate the fitted value with e^x to get Y in original units.
Mixing up the order of the names.
The names are easy to reverse under time pressure.
Fix: The dependent variable is named first. Log-lin means ln(Y) on X.
Comparing R² between models with different dependent variables.
R² looks like a universal scorecard.
Fix: Compare R² or standard error only when the dependent variable is the same. Otherwise use residual plots and the economic logic of the relationship.
Treating b1 as an exact percent change for large moves.
The 100 × b1 reading is an approximation that works for small changes.
Fix: Use it as the standard exam reading, and use e^b1 − 1 only if the question asks for the exact growth rate.

Worked examples

Example 1

An analyst regresses ln(sales) on time in years for a global retailer and gets ln(sales) = 4.00 + 0.06t. What is the continuously compounded annual growth rate, and what is predicted sales (in millions of USD, to the nearest 0.1) at t = 10? (e^4.6 ≈ 99.484)

Show the solution

The form is log-lin: Y is logged and X is time.
The slope 0.06 means about 100 × 0.06 = 6% continuously compounded growth per year.
At t = 10: ln(sales) = 4.00 + 0.06 × 10 = 4.60.
Exponentiate: sales = e^4.60 ≈ 99.484.
Rounded to one decimal: 99.5 million USD.

Answer: Growth is about 6% per year, and predicted sales at t = 10 are about USD 99.5 million.

Example 2

An economist estimates a log-log demand model: ln(Q) = 8.0 − 1.5 ln(P), where Q is quantity demanded and P is price. For a 1% rise in price, quantity demanded changes by about: A) −150% B) −1.5% C) +1.5%

Show the solution

Both variables are logged, so the model is log-log.
In a log-log model the slope is an elasticity: a 1% change in X changes Y by about b1 percent.
Here b1 = −1.5, so a 1% rise in price changes quantity demanded by about −1.5%.
Option A scales the slope by 100 for no reason. Option C has the wrong sign.

Answer: B

Example 3

A separate analyst estimates a lin-log model of fund inflows: Y = 20 + 12 ln(X), where Y is inflows in EUR millions and X is assets under management. For a 1% rise in X, inflows change by about: A) EUR 0.12 million B) EUR 1.2 million C) EUR 12 million

Show the solution

Only X is logged, so the model is lin-log and Y stays in levels (EUR millions).
In a lin-log model a 1% change in X changes Y by about b1 ÷ 100 units.
b1 ÷ 100 = 12 ÷ 100 = 0.12.
Inflows rise by about EUR 0.12 million.
Option C forgets to divide by 100. Option B divides by the wrong factor.

Answer: A

Exam tips

Identify the form from the placement of ln before doing anything else; many items are solved by interpretation alone.
Watch the units in the options: percent versus units of Y is often the whole trap.
If the dependent variable is ln(Y), check whether the question wants Y or ln(Y) before you pick a number.
Constant percentage growth over time points to log-lin; an elasticity wording points to log-log.
With three options and no penalty for wrong answers, always answer. Eliminating one option raises your chance of a correct guess from one in three to one in two.

Practice questions from Applications of Simple Linear Regression in Finance

Functional Forms: Log-Lin, Lin-Log and Log-Log Models: frequently asked questions

When should I use a log transformation in regression?

Use it when the scatter plot is curved, when the effect is proportional rather than constant, or when residuals fan out or show a pattern. A log form can straighten the relationship so OLS fits better. Variables must be positive to take a log.

How do I interpret a coefficient in a log-log regression?

The slope is an elasticity. A 1% change in X is associated with about a b1 percent change in Y. For example, a slope of −1.5 means a 1% rise in X goes with a 1.5% fall in Y.

What is the difference between log-lin and lin-log?

In log-lin only Y is logged, so the slope gives a relative change in Y per one-unit change in X. In lin-log only X is logged, so the slope divided by 100 gives the unit change in Y per 1% change in X.

Can I compare R² across log-lin and linear models?

Not directly. R² measures variation in the dependent variable, and ln(Y) and Y are different variables. Compare only models with the same dependent variable, and use residual plots to judge fit.