3.3 Percent change interpretation

Understand interpretations on the log scale, why log transforms result in percentage change interpretations.

Key facts: The log of a product is the sum of the logs.
$ln(1 + \delta) \sim \delta$ for small $\delta$.

Take a log (natural) regression.

$ln(y) = \beta_0 + \beta_1 ln(x).$

Increase x by $\delta$ percent, how does ln(y) shift?

$ln(y^\ast) = \beta_0 + \beta_1 ln(x\times(1 + \delta)).$

$ln(y^\ast) = \beta_0 + \beta_1 ln(x) + \beta_1 ln(1 + \delta).$

$ln(y^\ast) \sim \beta_0 + \beta_1 ln(x) + \beta_1\delta.$

How much did ln(y) shift?

$ln(y^\ast) - ln(y) \sim \beta_1\delta$.

$ln(y^\ast/y) \sim \beta_1\delta$.

$ln(y/y + (y^\ast-y)/y) \sim \beta_1\delta$.

$ln(1 + (y^\ast-y)/y) \sim \beta_1\delta$.

$(y^\ast-y)/y \sim \beta_1\delta$.

Finally: percentage change in y is $\beta_1\delta$.