Lesson 8 of 8 · 13 min

Functional forms: log-lin, lin-log and log-log

When the relationship is curved, transform Y, X or both with natural logs so that a straight line fits, then read the slope as a relative change and convert forecasts back.

In short

  • Log-lin: ln⁡Y=b0+b1X\ln Y = b_0 + b_1X. Slope = relative (%) change in Y for an absolute change in X. Suits constant growth rates.
  • Lin-log: Y=b0+b1ln⁡XY = b_0 + b_1\ln X. Slope = absolute change in Y for a relative change in X.
  • Log-log (double-log): ln⁡Y=b0+b1ln⁡X\ln Y = b_0 + b_1\ln X. Slope = relative change in Y for a relative change in X, an elasticity.
  • Log-lin forecasts must be converted back: Y^=eln⁡Y^\hat Y = e^{\ln \hat Y}.
  • Choose the form with the best R2R^2, F and SEE and random residuals, but compare those measures only across models with the same dependent variable.

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Functional forms: log-lin, lin-log and log-log · Simple Linear Regression