Lesson 8 of 9 · All lessons in this module
  1. The regression model and least squares
  2. Interpreting coefficients, fitted values and residuals
  3. The four assumptions and residual plotsVideo lesson
  4. Sums of squares, \(R^2\), the ANOVA table and the SEEVideo lesson
  5. Testing the slope: t-test, correlation test and F-testVideo lesson
  6. Intercept tests, indicator variables and p-values
  7. Forecasts and prediction intervals
  8. Functional forms: log-lin, lin-log and log-logVideo lesson
  9. Integrated cases: reading a full regression printout

Lesson 8 of 9 · 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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