1. Logarithmic specifications Jane E. Miller, PhD The Chicago Guide to Writing about Multivariate Analysis, 2 nd 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, 2 nd 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) (X i s), – 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, β measures the change in Y for a 1-unit increase in X 1, – 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 = β 0 + β 1 X 1

6. Lin-log specifications Lin-log models are of the form Y = β 0 + β 1 lnX 1. Where lnX 1 is the natural log (base e) of X 1 For such models, β 1 ÷ 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, β log(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 = β 0 + β 1 X 1. For such models, 100  (e β – 1) gives the percentage change in Y for a 1-unit increase in X 1, – Where the increase in X 1 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 = β 0 + β 1 lnX 1 For such models, β 1 estimates the percentage change in the Y for a one percent increase in X 1. – 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, 2 nd edition.

14. Suggested resources Chapter 10 of Miller, J. E., 2013. The Chicago Guide to Writing about Multivariate Analysis, 2 nd 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