01:220:322 Chap.7 Functional Form: Logs, Quadratics and Interactions
Functional Form: Logs, Quadratics and Interactions
Logs revisited
Logarithms appear throughout the course. A log outcome makes a slope an approximate percentage change; a log regressor means the slope over one hundred is the effect of a one percent change; logs on both sides give an elasticity. The approximation is best for small changes.
Quadratics
Adding the square of a regressor lets its effect rise or fall with its level.
The slope at a chosen value is the linear coefficient plus twice the quadratic coefficient times that value. When the two coefficients have opposite signs there is a turning point, a maximum for a hump and a minimum for a U. Always ask whether the turning point falls inside the observed data.
Testing curvature
A t test on the squared term asks whether the relationship is linear.
A claim that the variable has no effect restricts both terms and needs a joint F test.
Interactions
Multiplying two regressors lets the effect of one depend on the level of the other. The coefficient on the first variable alone is then its effect when the second equals zero, which may be an impossible value.
Evaluating the effect at the sample mean, or centring the variables before multiplying, gives a meaningful main-effect coefficient with its own standard error.
What exams ask
Typical items ask for a marginal effect at a given value, a turning point, and the correct joint null when a variable enters more than once.
Logs with a quadratic
When the outcome is in logs and a regressor enters with its square, the marginal effect formula gives an approximate proportionate change; multiply by one hundred to report a percentage.
Evaluating the fitted equation at two neighbouring values and subtracting gives the exact change, which is slightly different for larger steps.
Choosing where to evaluate
Report effects at values that exist in the data, typically the mean, the median or a round number inside the range.
For interactions, show the effect at two contrasting values of the moderating variable, which communicates the finding better than any single coefficient.
What this chapter covers
- 01
Log and elasticity interpretations
- 02
Quadratic marginal effects
- 03
Turning points
- 04
Testing linearity versus curvature
- 05
Interaction terms and effects at the mean
- 06
Centring
Worked example · free
Marginal effect and turning point
- 2Slope at 20: 0.052 - 2(0.0008)(20) = 0.020, about 2.0% more output.
- 2Turning point: 0.052/(2 x 0.0008) = 32.5 hours, after which fitted output declines.
Key terms
- Marginal Effect
- The change in the predicted outcome from a small change in one regressor, evaluated at a chosen value.
- Turning Point
- The value of a regressor where a quadratic relationship changes direction.
- Interaction Term
- The product of two regressors, which lets the effect of one depend on the other.
- Centring
- Subtracting a variable's mean before building an interaction so that main effects are read at the mean.
- Quadratic Term
- The square of a regressor added to a model so that its effect can change with its level.
Functional Form: Logs, Quadratics and Interactions FAQ
Why is the coefficient on x alone not the effect of x in a quadratic?
With a squared term the slope changes with x. The linear coefficient is the slope only at x equal to zero, so the effect must be computed at a value that matters, such as the mean.
How do I test that a variable with an interaction has no effect?
Set the coefficient on the variable and on every interaction containing it to zero, and test them jointly with an F statistic from the test command.
What if the turning point lies outside the data?
Then the curve never actually turns within the observed range, and the quadratic is simply capturing increasing or decreasing returns. Do not interpret the turning point as a real feature.
How do I tell whether a quadratic shape is real?
Check the t statistic on the squared term, then locate the turning point relative to the data. A significant square with a turning point far outside the observed range describes diminishing returns rather than a genuine reversal.
Does centring change the interaction coefficient?
No. Subtracting means before multiplying leaves the interaction coefficient and the fit unchanged. It changes only the main-effect coefficients, which become effects at the means with directly printed standard errors.
Exam move
Sketch the fitted curve for every quadratic you meet and mark the turning point, then compute one interaction effect at two different values of the other variable.
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