Logarithmic specifications
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Logarithmic specifications. Jane E. Miller, PhD. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Overview. Types of logarithmic specifications Prose interpretation of coefficients from logarithmic specifications

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Logarithmic specifications

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Logarithmic specifications

Logarithmic specifications

Jane E. Miller, PhD

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


Overview

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 specifications1

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

Types of logarithmic specifications

  • Lin-lin

  • Lin-log

  • Log-lin

  • Log-log

    • Also known as “double log”


Lin lin specifications

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 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 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 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

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

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

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.


Interpreting coefficients from logarithmic specifications

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.


Summary

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.


Suggested resources

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.


Supplemental online resources

Supplemental online resources

  • Podcasts on

    • Defining the Goldilocks problem

    • Resolving the Goldilocks problem – model specification

  • Online appendix on interpreting coefficients from logarithmic specifications.


Suggested practice exercises

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.


Contact information

Contact information

Jane E. Miller, PhD

[email protected]

Online materials available at

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


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