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Power Transformations for Consumer 

Expenditure Data

Taylor J. Wilson

Branch of Information and Analysis

2018 Microdata Users Workshop

18-20 July 2018
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Roadmap

CE data distributions

Defining the power transformation

Picking the optimal lambda

Income-Expenditure relationship
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Data Distributions

 Most expenditure and income data exhibit a right skew. 

 This results from a fixed lower bound and extreme values in the 

right tail. 

 May need to correct the distribution into a normal shape.
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Source: Public Use Microdata 2016 – First quarter 2017

Income Distribution – First Quarter 2017
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Defining the Power Transformation

yyttrrrrrrrr =

yyλ − 1

λ λ ≠ 0

ln yy λ = 0

 Power Transformations can be 

expressed as a simple power 

in a linear context. 

 We care about finding a λ that 

works for our purposes. 

 Log transformation is a special 

case of power transforms. 
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Picking the Optimal Lambda

 Choose lambda such that skew is minimized. 

 Statistical programs estimate this with maximum likelihood 

estimation. 

 Visualize skew by plotting it against the parameter. For 

convenience select parameters with function names. 
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-60

-50

-40

-30

-20

-10

0

10

20

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Skew by Selected Lambda Parameters

λ

Skew
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-60

-50

-40

-30

-20

-10

0

10

20

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Skew by Selected Lambda Parameters

λ

Skew

λ = 0.25
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Source: Public Use Microdata 2016 – First quarter 2017

Transformed Income Distribution – First Quarter 2017
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Income-Expenditure Relationship

 Bivariate incomeexpenditure relationship 

is heteroskedastic. 

 Explore two strategies

Fix heteroskedasticity 

with a log transformation

Fix heteroskedasticity 

with a power 

transformation. 
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Income-Expenditure Relationship

 Log transformation 

improves the relationship 

between the variables. 

 Outliers still seem to exist 

in the left tail of the 

distribution. 

 Regression line is still 

biased. 
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Income-Expenditure Relationship

 Optimal lambda creates a 

much better shape to the 

data. 

 Outliers are treated such 

that the bias is lessened 

or eliminated. 

 Regression line describes 

the data more accurately.
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Down side of power transformation

 Log transformations allow for 

direct interpretability of the 

coefficient as a constant elasticity. 

 Power transformations require 

some additional calculus to 

‘correct’ the beta and turn it into 

an elasticity at a point (x, y).

εyy,xx =

dy

dx

xx

yy

= ββ1

εyy,xx =

dy

dx

xx

yy

= ββ1

xxmm−1

yy −1

mm

nn

xx

yy
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Questions?

The paper accompanying this presentation can be found 

at the following address: 

https://www.bls.gov/cex/ce_methodology.htm#reports
Contact Information

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Taylor J. Wilson

(202) 691-6550

Wilson.Taylor@bls.gov