Logarithmic specifications

1. Logarithmic specifications Jane E. Miller, PhD The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.

2. Overview • Types of logarithmic specifications • Prose interpretation of coefficients from logarithmic specifications • Considerations for contrast size for logarithmic specifications • Descriptive statistics for multivariate models with logarithmic specifications The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.

3. Logarithmic specifications • Another approach to comparing βs across variables with different ranges and scales is to take logarithms of the • dependent variable (Y), • independent variable(s) (Xis), • or both. • The βs on the transformed variable(s) lend themselves to straightforward interpretations such as percentage change.

4. Types of logarithmic specifications • Lin-lin • Lin-log • Log-lin • Log-log • Also known as “double log”

5. Lin-lin specifications • Review: For OLS models in which neither the IV nor the DV is logged, βmeasures the change in Y for a 1-unit increase in X1, • the changes are measured in the respective units of the IV and DV. • In the lingo of logarithmic specifications, these models are termed “lin-lin” models because they are linear in both the IV and DV Y = β0 + β1X1

6. Lin-log specifications • Lin-log models are of the form Y = β0 + β1 lnX1. Where lnX1 is the natural log (base e) of X1 • For such models, β1 ÷ 100 gives the change in the original units of the DV for a 1 percent increase in the IV. • E.g., in a model of earnings, βlog(hours worked) = 5,905.3: • “Each 1 percent increase in monthly hours worked is associated with a NT$ 59 increase in monthly earnings.”

7. Log-lin specifications • Log-lin models are of the form lnY = β0 + β1X1. • For such models, 100  (eβ – 1) gives the percentage change in Y for a 1-unit increase in X1, • Where the increase in X1 is in its original units. • E.g., “For each additional child a woman has, her monthly earnings are reduced by 3.6 percent.”

8. Log-log specifications • Log-log models are of the form lnY = β0 + β1lnX1 • For such models, β1 estimates the percentage change in the Y for a one percent increase in X1. • This measure is known in economics as the elasticity (Gujarati 2002). • E.g., “A 1 percent increase in monthly hours worked is associated with a 0.6% increase in monthly earnings.”

9. Choice of contrast size for logarithmic models • Caveat: The scale of the logged variable must be taken into account when choosing an appropriate-sized contrast. • E.g., a 1-unit increase in ln(monthly hours worked) from 5.3 to 6.3 is equivalent to an increase from 200 to 544 hours per month. • That contrast is nearly a 2.5 fold increase in hours. • Implies working three-quarters of all day and night-time hours, 7 days a week.

10. Review: Assess whether a 1-unit increase in the variable is the right sized contrast • Always consider whether a 1-unit increase in the variable as specified in the model makes sense in its real world context! • Topic • Distribution in the data • If not, use theoretical and empirical criteria for choosing a fitting sized contrast. • See podcast on measurement and variables approaches to resolving the Goldilocks problem

11. Descriptive statistics to report if you use a logarithmic specification • In a table of descriptive statistics, report the mean and range both • In the original, untransformed units, such as income in dollars, which are • more intuitively understandable • easier than the logged version to compare with values from other samples. • In the logged units, so readers know the range and scale of values to apply to the estimated coefficients.

12. Interpreting coefficients from logarithmic specifications • Taking logs of the IV(s) and/or DV affects interpretation of the estimated coefficients. • If your models include any logged variables, report the pertinent units as you write about the βs, especially if • your specifications include a mixture of logged and non-logged variables; • you are testing the sensitivity of your findings to different logarithmic specifications.

13. Summary • Consider whether a logarithmic specification fits your: • Topic, • Data, • Field. • Report descriptive statistics for each variable in original and transformed units. • Convey the pertinent units for each coefficient as you interpret it. The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.

14. Suggested resources • Chapter 10 of Miller, J. E., 2013. The Chicago Guide to Writing about Multivariate Analysis, 2nd edition. • Gujarati, Damodar N. 2002. Basic Econometrics. 4th ed. New York: McGraw-Hill/Irwin. • Miller, J. E. and Y. V. Rodgers, 2008. “Economic Importance and Statistical Significance: Guidelines for Communicating Empirical Research.” Feminist Economics 14 (2): 117–49.

15. Supplemental online resources • Podcasts on • Defining the Goldilocks problem • Resolving the Goldilocks problem – model specification • Online appendix on interpreting coefficients from logarithmic specifications.

16. Suggested practice exercises • Study guide to The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. • Questions #9 and 10 in the problem set for chapter 10 • Suggested course extensions for chapter 10 • “Reviewing” exercise #4. • “Applying statistics and writing” questions #5 and 6. • “Revising” questions #7 and 9.

17. Contact information Jane E. Miller, PhD jmiller@ifh.rutgers.edu Online materials available at http://press.uchicago.edu/books/miller/multivariate/index.html