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Project 8/9: User’s Guide to 

Income Imputation in the CE
2 — U.S. BUREAU OF LABOR STATISTICS • bls.gov

NOTE:

The content of this training presentation 

is derived from the “User’s Guide to 

Income Imputation in the CE” 

(https://www.bls.gov/cex/csxguide.pdf)
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In this project, you will learn:

◼ How income data are collected in the 

Consumer Expenditure Surveys (CE)

◼ Why and how these data are imputed when 

missing

◼ How imputation affects “you-sers”
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Income Collection in the CE:

◼ Interview Survey

1

st and 4th interview (2015 onward; 2nd and 5th

previously)

◼ Diary Survey

One time only (1st or 2nd week, at interviewer’s 

discretion)
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In both surveys, components are:

◼ Collected for:

The consumer unit as a whole (e.g., INTRDVX),

or

Each member 14 or older (e.g., SEMPFRMX)

◼ Summed to consumer unit total (FINCBTAX, 

Interview; FINCBEFX, Diary)

◼ Subject to nonresponse. This leads to biased 

statistics (means, standard errors, etc.).
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First, respondents are asked for each component: “Did you or 

any member of your household receive [type of income]?”

◼ If yes, then asked: “What was the amount?” 

To which respondent reports:

Actual value; If unknown or refused:

– Bracket value; if unknown or refused:

• No information (“invalid blank”)

◼ If no:

Next source is collected.

But if all “no,” the respondent is an “All Valid 

Blank” (AVB) reporter.
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How are missing data handled?

◼ Historical Data:

1972-73, 1980-2003:

– “Complete Reporter” definition is in effect:

• Complicated: “Reference person”-based, but not always;

• Does not mean “all valid” reporters of income.

2001: Brackets introduced to Interview Survey.

2004: Brackets introduced to Diary Survey.

◼ Current Data:

2004-present: Missing incomes are imputed.
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Income Imputation Highlights

◼ Enables CE to fill in blanks due to 

nonresponse;

◼ Particular methodology is called “multiple 

imputation,” because there is more than one 

imputed value for each income source not 

reported.
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Why “multiple” imputation?

◼ Technical reasons, related to variance.

From the User’s Guide:

– Multiple imputation “yields variance estimates that 

take into account the uncertainty built into the data 

from the fact that some observations are imputed, 

rather than reported.” (P. 1, section I.A.)

In other words: Multiply imputed data are 

designed to have larger variances than “singly” 

imputed data because, by definition, imputed data 

are “best guesses,” not actual values.
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How are data multiply imputed?

◼ If respondent reports actual value:

Five “imputations” appear in the dataset, 

replicating the amount reported.

– Example: Respondent reports value of INTRDVX to be 

$100. INTRDVXm = $100 (where m is number of 

imputations, and 1<=m<=5 in CE)
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How are data multiply imputed? 

(Continued)

◼ Bracket Reports:

Through an algorithm, a random value within the 

bracket range is drawn, and serves as the first 

imputation.

Process is repeated four times.

Example:

– Respondent reports $0<INTRDVX<=$999.

– Values as small as $1 and as large as $999 are plausible 

(e.g., $10; $494; $384; $875; and $132 is a plausible 

string of imputed values for INTRDVX1-5)
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How are data multiply imputed?

(We’re nearly done...)

◼ Regression-based, when respondent reports no 

information beyond receipt

1. Income reported by similar consumer units is regressed on 

independent variables.

2. Coefficients are “shocked” (i.e., random noise is added to 

each).

3. Predicted values are produced using the “shocked” model 

coefficients.

4. Predicted values from first “shocked” model are each 

“shocked”; The resulting values are used to fill in invalid blanks 

where they occur.

5. This process is repeated four times, starting at step 2.
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How are data multiply imputed?

(Exciting Conclusion!)

◼ In case of AVB:

1. Impute receipt (or lack thereof) for each source of income.

2. If receipt is imputed, treat observation as a standard “modelbased” case.
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Some Key Points:

◼ Reiteration:

Each income variable has not one, but five 

imputed values;

When reported, each imputed value equals 

reported value;

When bracket range is reported, imputed values 

differ, but all fall within the bracket range.

◼ New point:

When model-based imputation is used, values have no 

preset bound, but are always positive (except for variables 

like SEMPFRMX, for which losses are possible).
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Some variables include summations 

of imputed values:

◼ FSMPFRX1:

Equals sum of SMPFRMX1 for each member of 

the consumer unit (CU); i.e., first “F” of 

“FSMPFRX1” is for “Family”

Some SMPFRMX1 within the CU may be valid 

reports, others imputed (bracket or model-based)

◼ FINCBTX1:

Sum of all components (including FSMPFRX1), 

imputed or not
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NOTE: This applies to ALL 

“INCOMEm” variables:

◼ FSMPFRXm, m=imputation number, 1<=m<=5:

Equals sum of SMPFRMXm for each member;

Some SMPFRMXm within the consumer unit may 

be valid reports, others imputed (bracket or 

model-based)

◼ FINCBTXm, 1<=m<=5:

Sum of all components (including FSMPFRXm), 

imputed or not
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To identify quantity of, and reason for, 

imputation, “flag variables” are available.

◼ Naming convention: End in “I”. Examples:

INTRDVXI

SEMPFRMI

FSMPFRMI
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Possible Values:

◼ 100: No imputation on variable or subcomponents 

(i.e., variable or subcomponents are validly reported)

◼ 2nn: Only model-based imputation is performed on 

variable or subcomponents

◼ 3nn: Only bracket imputation is performed on 

variable or subcomponents

◼ 4nn: At least one model-based and at least one 

bracket imputation are performed (summary 

variables only)

◼ 5nn: “AVB” case
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What does “nn” mean?

◼ Number of subcomponents imputed.

Always equal to “01” for non-summary variables 

(e.g., INTRDVX or SEMPFRMX).

Minimum of “01” for summary variables.

– Example: A consumer unit has three members 

reporting receipt of SEMPFRMX.

• One reports the value (SEMPFRMI=100).

• Another provides no information (SEMPFRMI=201).

• The third reports a bracket (SEMPFRMI=301).

• FSMPFRMI is 402. 
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Important for CE Microdata Users:

◼ Once again, each income variable has not one, 

but five imputed values.

◼ Microdata users must use all five values to 

obtain valid results.
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Computing Means

◼ Unweighted (i.e., Sample) Means

◼ X is the value of income from consumer unit i 

for imputation j (where 1 <= j <= 5 in CE)

◼ n is the number of rows (varies by data set)

◼ m is the number of columns (always 5 in CE)

( )/( )

1 1

X n m

m

j

n

i

 ij 

= =
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Computing Unweighted Means

◼ Applied Example:

 INTRDVX is the value reported (or not).

 INTRDVX1,...,INTRDVX5 are the values imputed.

1. Sum all imputed values, i.e., INTRDVX1,...,INTRDVX5 

(100 + 100 + ... 100 + 50 + ... + 20 + 80);

2. Divide total (1,200) by total number of observations 

(n*m=2*5=10);

3. Mean = 120

INTRDVX INTRDVX1 INTRDVX2 INTRDVX3 INTRDVX4 INTRDVX5

100 100 100 100 100 100

D 50 250 300 20 80
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Computing Unweighted Means

◼ Alternatively, use INTRDVXM:

Find the mean of each row (INTRDVXM).

Add the row means: 100+140=240.

Dividing this by the number of rows (2).

Mean=120, the same value obtained by finding 

the mean of all 10 observations.

INTRDVX INTRDVX1 INTRDVX2 INTRDVX3 INTRDVX4 INTRDVX5 INTRDVXM

100 100 100 100 100 100 100

D 50 250 300 20 80 140
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BONUS:

◼ INTRDVXM is already computed for you on the 

PUMD files!

◼ This makes it easy to compute either a sample 

mean (as just demonstrated), or a weighted 

mean (as you shall soon see...).
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Computing Weighted Means: 

Estimating Population Means

◼ Consider the following data:

◼ Based on FINLWT21:

The first CU represents 5,000 units in U.S.;

The second CU represents 7,500.

◼ Using INTRDVXM and FINLWT21, the weighted mean is: 

[(100*5,000) + (140*7,500)]/(5,000 + 7,500) = 124.

INTRDVX INTRDVX1 INTRDVX2 INTRDVX3 INTRDVX4 INTRDVX5 INTRDVXM FINLWT21

100 100 100 100 100 100 100 5,000

D 50 250 300 20 80 140 7,500
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Computing Variances:

Unweighted (Sample) Data

◼ Five steps for computing variances for unweighted means:

1. Compute the mean of each column of completed data 

(INTRDVX1 through INTRDVX5)

2. Calculate the average of the five means

3. Calculate the variance of the MEAN of each column of data

4. Calculate the average of these variances (of means)

5. Calculate the variance (actual, not variance of mean) between

(or among) the five complete data mean estimates

INTRDVX INTRDVX1 INTRDVX2 INTRDVX3 INTRDVX4 INTRDVX5

100 100 100 100 100 100

D 50 250 300 20 80
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Computing Variances

◼ Then, the final step is to insert them into the formula 

for total variance:

◼ Tm is total variance; SQRT(Tm) is the standard error

◼ Um is the variance of the MEAN of the mth column; 

and Uഥ

m is the average of these variances (of means)

◼ m is the number of columns (5 columns in this case)

◼ Bmmeasures the variance of the five means (one for 

each column) ( ) Tm

Umm Bm

1

1

−

= + +
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Computing Variances

◼ So, in this example:

1. Compute the mean of each column of completed data (INTRDVX1 through 

INTRDVX5): 75; 175; 200; 60; 90

2. Calculate the average of the five complete data estimates: 

(75+175+200+60+90)/5 = 600/5 = 120

3. Calculate the variance of each column of data: 1,250; 11,250; 20,000; 3,200; 

200; divide each by 2 (because n=2) to get each variance of mean: U1=625, ..., 

U5=100

4. Calculate the average of these variances of means (U1+...+U5)/5=3,590

5. Calculate the variance between (or among) the five complete data mean 

estimates: Var (75,...,90)=[(75-120)2+ ... +(90-120) 2]/m-1, where m=5; Var

(75,...,90)=3,987.5

INTRDVX INTRDVX1 INTRDVX2 INTRDVX3 INTRDVX4 INTRDVX5

100 100 100 100 100 100

D 50 250 300 20 80

MEAN 75 175 200 60 90
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Computing Variances

◼ Then, in the final step is to insert them into the 

formula for total variance:

◼ Um is the variance of the MEAN of the mth column; 

and Uഥ

m is the average of these variances: 3,590 

◼ m is the number of columns (5 columns in this case)

◼ Bmmeasures the variance of the means of each of 

the five columns: 3,987.5

◼ T is the total variance:

3,590 + (1+0.2)*3,987.5 = 8,375 ( ) Tm

Umm Bm

1

1

−

= + +
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Computing Variances: 

IMPORTANT NOTICE

◼ Proper computation of variances using multiply imputed data 

is more complicated than computing means.

 You MUST use all five columns of imputed data do obtain the correct 

variance.

 You MUST NOT compute the variance of any old column (INTRDVX1 

only; INTRDVX2 only), or even of the “M” column (i.e., INTRDVXM) 

and call it a day.

◼ All right, don’t listen to me.

 But your variances will be biased, possibly quite seriously. 

 And the direction of the bias (“too large” or “too small”) is not 

predictable!
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Computing Regression Results

◼ To compute regression coefficients and standard errors, 

use repeated-imputation inference (RII).

The proper estimation uses all five implicates for income by 

estimating the regression model once with each implicate. 

Estimating coefficients with RII is similar to mean estimation.

Estimating standard error with RII is similar to variance 

estimation.

◼ RII applies to both weighted and unweighted regression 

analysis. However, for simplicity, only unweighted 

regressions are described herein.
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Computing Regression Results: 

Coefficients

◼ Objective: Compute y =  + I+ X +  using imputed income.

◼ To obtain estimates of the , , and , the regression model is 

estimated five times, once for each implicate:

 y = a1 + b1(FINCBTX1) + g1X,

 y = a2 + b2(FINCBTX2) + g2X,

 y = a3 + b3(FINCBTX3) + g3X,

 y = a4 + b4(FINCBTX4) + g4X, and

 y = a5 + b5(FINCBTX5) + g5X.

◼ Average a1 to 5 to get ; b1 to 5 to get ; and g1 to 5 to get . 
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Computing Regression Results: 

Standard Error (SE) of a Coefficient

◼ Objective: Compute SEs for , , and .

◼ Same steps as computing variance of income, except the 

coefficient is treated as the column mean. For example, to 

compute SE():

 Compute the VARIANCE of a1, then a2,..., then a5. (Your computer 

software may do this. If not, it should provide the SE of each. Square 

SE for each coefficient a1,..., a5 to obtain the variance of a1,..., a5.)

 Compute the average of the variances of a1,..., a5; call the result Uഥ

m.

 Compute the variance of the VARIANCES of a1,..., a5; call the result Bm.

 Compute 

 The square root of Tmis the SE().

( ) Tm

Umm Bm

1

1

−

= + +
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This concludes the “basics” of 

using imputed CE income data. 

For more applications, see:

“User’s Guide to Income Imputation in the 

CE” 

(https://www.bls.gov/cex/csxguide.pdf)
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ATTENTION SAS USERS:

You have a “macro” available to 

compute the following exercises. 

Details in “special topics” after 

feedback session.
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Project 8

1. Create data set containing data for a collection year 

from 4 quarterly FMLI files.

2. Find the unweighted mean for income using 

FINCBTXM for all CUs by region of residence:

a. Northeast (REGION=“1");

b. Midwest (REGION=“2");

c. South (REGION=“3);

d. West (REGION=“4");

e. Suppressed (REGION=“ “)—affects selected combinations 

of PSU and STATE, done to maintain confidentiality 

3. See next slide to compute standard error
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Project 8

3. Same steps as computing variance of income, except the 

coefficient is treated as the column mean. For example, to 

compute SE():

 Compute the VARIANCE of a1, then a2,..., then a5. (Your computer 

software may do this. If not, it should provide the SE of each. Square 

SE for each coefficient a1,..., a5 to obtain the variance of a1,..., a5.)

 Compute the average of the variances of a1,..., a5; call the result Uഥ

m.

 Compute the variance of the VARIANCES of a1,..., a5; call the result Bm.

 The square root of Tmis the SE().
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Unweighted Mean & Standard Error

Project 8 Results

(FMLI211-FMLI214)

REGION Mean Income Before Tax Total Standard error

of Income

Suppressed $66,779.98 $2,778.30

Northeast (1) $98,551.01 $1,669.53

Midwest (2) $84,908.21 $1,284.43

South (3) $78,894.41 $1,081.29

West (4) $93,865.96 $1,285.97
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Project 9

1. Create data set containing data for a collection year 

from 4 quarterly FMLI files.

2. Find the income coefficient (i.e., “β”) and its SE for 

Food at Home: 

a) Add FDHOMEPQ and FDHOMECQ, and annualize 

the resulting variable FDHOME: 

FDHOME=4*(FDHOMEPQ+FDHOMECQ)

b) Regress FDHOME on FINCBTX1, ..., FINCBTX5

c) Compute mean and standard error of “β”
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Regressions using unweighted multiply imputed data

Project 9 Results

(FMLI211-FMLI214)

Type Estimate Total Variance Total Standard Error

INTERCEPT 4,897.59 1923.53100 43.85808

Income Coeff. (MPC) 0.019 1.250216E-7 0.00035358