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Logarithmic specifications

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### Logarithmic specifications

Jane E. Miller, PhD

The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.

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.

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.

Types of logarithmic specifications

- Lin-lin
- Lin-log
- Log-lin
- Log-log
- Also known as “double log”

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

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.”

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.”

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.”

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.

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

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.

- In the original, untransformed units, such as income in dollars, which are

Interpreting coefficients from logarithmic specifications specification

- 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.

Summary specification

- 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.

Suggested resources specification

- 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.

Supplemental online resources specification

- Podcasts on
- Defining the Goldilocks problem
- Resolving the Goldilocks problem – model specification

- Online appendix on interpreting coefficients from logarithmic specifications.

Suggested practice exercises specification

- 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.

Contact information specification

Jane E. Miller, PhD

Online materials available at

http://press.uchicago.edu/books/miller/multivariate/index.html

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