Logs, quadratics and units: reading a coefficient

Economics deals in proportions: a 10% rise in price, a 2% growth rate, a share of income. Natural logarithms turn proportional changes into additions: a 1% rise in x raises ln x by about 0.01, and in general the change in ln x is approximately the proportional change in x. Putting a log on one side or the other of a regression gives four forms, each reading the slope differently:

Two notes on precision. First, the percentage reading of a log-level coefficient is an approximation that is good only for small b: the exact change in y for one more unit of x is 100(e^b − 1)%, so b = 0.08 means 8.33%, not 8%. Second, “log” in econometrics means the natural log, ln. A paper that uses logs to base 10 will say so, and the coefficients then differ by the factor ln 10.

Here are two relationships, one real and one invented, under all five forms (the quadratic is treated below). On the World Bank data, pick “level-log”: life expectancy rises by 6.03 years for each unit rise in ln of GDP per head, which is 0.06 years for a 1% rise, or 0.57 years for a 10% rise. Then compare the straight line (R² = 0.518) with the level-log curve (R² = 0.715): rich countries gain less from extra income than poor ones, and the log captures it with a single slope.

The survey data show the opposite lesson: we wrote spending as a straight line in income, so the level-level fit is the right one there, with R² = 0.606. The log forms still run, but their R² values (0.539 for level-log, 0.538 for log-level and 0.546 for log-log) are not all comparable. The R² of a regression with ln y as the dependent variable measures the share of the variation in ln y that is explained, a different quantity from the share of the variation in y. Only the level-level and level-log fits (both with y as the dependent variable) can be compared by R².

The elasticity of y with respect to x is the percentage change in y per 1% change in x. For a smooth relationship it is the slope times x over y:

Two traps. The slope dy/dx and the elasticity are different things: a table that lists −b ÷ x² as “the elasticity” of a reciprocal model has listed the slope. And the elasticity of the semi-log model ln y = a + bx is b·x, not b; b alone is the semi-elasticity, the proportional change in y for one more unit of x.

In the survey, the level model gives spending a slope of 0.534 per unit of income, and at the means (income 15.44, spending 12.63) the elasticity is 0.534 × 15.44 ÷ 12.63 = 0.652. The log-log fit, which has a constant elasticity, gives 0.597. The two agree to within a few hundredths, as they should: both are measuring how spending responds, in percentages, to income.

Reading an estimated elasticity. If a log-log demand equation has a price coefficient of −0.03 (significant), a 1% rise in price lowers quantity demanded by 0.03%. The absolute value is below 1, so demand is inelastic (above 1 is elastic, equal to 1 is unit elastic). The coefficient is the elasticity directly, with no further information needed: that is what the opening question tests. A significant coefficient says the elasticity is not zero, not that it is large.

A series growing at a constant rate g follows Y_t = Y₀(1 + g)^t. Take logs: ln Y_t = ln Y₀ + t ln(1 + g). So regressing ln Y on a time trend t gives a slope b = ln(1 + g), and the growth rate is

Real data. Regressing the natural log of India’s real GDP on a time trend for 1960 to 2025 (World Bank, constant rupees) gives a slope of 0.0521 with R² = 0.987, so the trend growth rate is about 5.2% a year (exactly 5.3%). From 1991 onwards the slope is 0.0613, a growth of about 6.1% a year. Two things to remember. The R² of a trend regression is high for almost any trending series, which is no evidence of anything (lesson 15). And a single trend slope averages over the whole period, so it hides acceleration within it.

To let the effect of x change with x itself, add its square:

If b > 0 and c < 0 the curve rises at a falling rate (diminishing returns) and peaks at x*. After x* the marginal effect is negative. For a wage equation estimated as predicted wage = 2.5 + 0.40 exper − 0.008 exper² (standard error of the squared term 0.0011), the marginal effect of experience is 0.40 − 0.016 exper: it falls steadily as experience grows (diminishing returns throughout), it is 0.24 at 10 years, and it reaches zero at 0.40 ÷ (2 × 0.008) = 25 years. The t statistic of the squared term shows whether the curvature is real: here 0.008 ÷ 0.0011 = 7.3, far beyond 2.

The World Bank data again: life expectancy on GDP per head in $ thousand and its square gives 63.42 + 0.523x − 0.0032x², with R² = 0.653 against 0.518 for the straight line. The turning point is at x* = 81.7 thousand dollars, which only 6 of the 186 economies exceed. That is a warning, not a finding: the quadratic bends down because it must eventually (a parabola does), and the data hardly reach that region. Extrapolating a quadratic past the data produces nonsense, such as longer lives falling with income, which no theory predicts. The level-log curve (R² = 0.715) fits better with fewer parameters and never turns down.

Change the unit of a variable and the regression says exactly the same thing in different numbers. Take the survey regression of spending on income, both in ₹ thousand, and rescale:

The rules, which follow from the algebra and can be tested with the table:

• Multiply y by c (rupees for thousands, c = 1,000): every coefficient and standard error is multiplied by c. The t statistics, the F statistic and R² do not change.
• Multiply x by c: its coefficient and its standard error are divided by c, because one new unit of x is c times smaller. The intercept, t statistics and R² do not change.
• In log forms, rescaling shifts the logs by a constant: ln(cy) = ln c + ln y. Only the intercept changes. The slope of a log-log model, an elasticity, is unit-free.
• A dummy variable is not affected by rescaling x, and a coefficient in percentages (a semi-elasticity) is not affected by the unit of y.

One of the Delhi School 2019 questions gives a home-sales regression of price (in ₹ thousand) on, among other things, the house’s age in years, with a coefficient of 0.090. It asks what the coefficient would be if age were measured in decades. A decade is ten years, so one new unit of age is ten times larger and the coefficient is multiplied by ten: 0.090 × 10 = 0.900.

The key printed with the paper marks 0.009, which is what dividing by ten gives. Dividing by ten would be the right adjustment if age were measured in units ten times smaller than a year, not larger. That is an error in the official key. The check that settles it is the one in the table above: when a variable is expressed in units that are k times bigger, its coefficient is k times bigger (income in rupees rather than thousands made the slope a thousand times smaller; age in decades rather than years makes it ten times larger). The question bank stores 0.900 and says so; the question is served in the Delhi School set at the end of the module with 0.900 as the right answer, and flagged as disagreeing with the paper’s printed key.

The Delhi School’s questions here (a quadratic wage equation, the units of a coefficient in a home-sales regression) are in the hard-problems lesson at the end of the module. For logs and growth in simple regressions see the regression primer.