Using a Two-Part Markov Latent 

Class Model to

Examine Expenditure

Report Quality

Brian Meekins

Bureau of Labor Statistics

N. Clyde Tucker

American Institutes for Research

Any opinions expressed in this paper are those 

of the authors and do not constitute policy of 

the Bureau of Labor Statistics
Latent Class Analysis

 Uses repeated measurements from panel survey 

data to estimate classification error

 Does not require external validation data; 

estimates of error directly from panel data

 LCA used to study measurement or response 

error (VandePol and deLeeuw 1986; Tucker 

1992; Van de Pol and Langeheine 1997; Bassi et 

al. 2000; Biemer and Bushery 2000; Tucker, et 

al. 2002, 2003, 2004, 2005, 2006, and 2008); 

Meekins et al. (2011)
3

U.S. Consumer Expenditure 

Interview Survey (CEIS)

 ~ 6,000 CU’s/year

 CU’s interviewed every 3 months about 

prior 3 months expenditures

 4 consecutive interviews on each CU

 15 years of CEIS: 1996-2010

 Unweighted analysis

 31 commodity categories analyzed
Dental Computer games Childcare

Prescription drugs Computer equipment Pets and pet supplies

Eye care Books Major Vehicle Repairs

Clothing Cable Minor Vehicle Repairs

Infant clothing Music License/registration

Clothing accessories Internet (2001+) HH electricity

Clothing services Sports equipment HH gas

Sewing Major appliances HH trash service

Shoes Minor appliances Phone

Jewelry Electronics HH services

Events (e.g. sporting/theatre)

Commodity Categories

4
2nd Order Markov

5

A B C D

W X Y Z

= = = =
Mover-Stayer

6

M

A B C D

W X Y Z

= = = =

1, P(W),P(X), P(Y), and P(Z) are unconstrained.

M= 2, P(W=1) = P(X =1) = P(Y =1) = P(Z = 1) = 1

3, P(W=1) = P( X =1) = P(Y =1) = P(Z = 1) = 0
 Markov or Mover-Stayer model assumptions

 Equal measurement error across all interviews

 No False Positives

Model Assumptions

( 2 | 2) 1

( 1| 2) 0

  

  

i i

i i

P a w

P a w

i i i i jk

i i i i

P c j y k P d j z K q

P a j w k P b j x k

      

    

( | ) ( | )

( | ) ( | )
LCA MOVER-STAYER OVER TIME 

(1 YEAR POOLED COHORTS)

8




Summary of Previous Findings

 Assessment of MLCA for the detection 

of change in measurement error/time

 Accuracy rates/all estimates noisy

 Estimates are reasonable

 Useful: Sensitive to survey changes

13
Add Expenditure

 Adds information to the model

 Allows for simultaneous (as opposed to 

two-stage estimation of unreported 

expenditure)

 If auto-correlative effects are large –

stabilization of estimates should result

14
Expenditure data raw

15
Expenditure Data (ln)

16
Autoregressive (Observed)

17

Cable Clothing

Drugs Music
Two-part Model (Olsen 

and Schaeffer 2001)

18

A B C D

iy

sc ic

sy

e1 e2 e3 e4
Modified MS

19

M =1

A B C D

W X Y Z

= = = =

Mover
Modified MS

20

M =2

A B C D

W X Y Z

= = = =

e1 e2 e3 e4

Stayer

Purchaser
Modified MS

21

M =3

A B C D

W X Y Z

= = = =

Stayer

NonPurchaser
Objective Diagnostics

 Fit Statistics

L-square

Entropy

BIC
Results Model Fit

23

Model log L BIC Entropy

2

nd order -34769.864 69579.396 .961

Mover-Stayer -30532.432 61183.868 .845

Two part -63092.649 126294.386 na

Modified MS -148951.831 297949.663 .983

Cable
Results Model Fit

24

Model log L BIC Entropy

2

nd order -50629.582 101298.83 .801

Mover-Stayer -49623.976 99366.956 .793

Two part -121605.028 243319.143 na

Modified MS -227619.635 455285.270 .935

Clothing
Results Model Fit

25

Model log L BIC Entropy

2

nd order -48549.757 97139.181 .768

Mover-Stayer -45002.673 90124.351 .766

Two part -106225.730 212560.548 na

Modified MS -214846.321 429920.735 .924

Drugs
Results Model Fit

26

Model log L BIC Entropy

2

nd order -16041.502 32122.673 .200

Mover-Stayer -15990.318 32099.640 .576

Two part -22532.573 45174.233 na

Modified MS -123067.662 246363.416 .759

Major Appliances
Results Model Fit

27

Model log L BIC Entropy

2

nd order -32777.309 65594.285 .503

Mover-Stayer -31134.973 62388.951 .769

Two part -46334.975 92779.037 na

Modified MS -135818.829 271865.751 .823

Music
Results, 

Reports/Expenditure

28

Model P(A=1|W=1)

Missing 

Expenditure

/QTR

Missing 

Expenditure

/CU QTR

2

nd order .984 $1,751.96* $1.17*

Mover-Stayer .984 $1,751.96* $1.17*

Two part na $18,358.56 $12.24

Mod MS .984 $996.16 $.66

*Estimated from P(A|W)

Cable
Results, 

Reports/Expenditure

29

Model P(A=1|W=1)

Missing 

Expenditure

/QTR

Missing 

Expenditure

/CU QTR

2

nd order .910 $25,059.97* $16.71*

Mover-Stayer .846 $46,124.29* $30.75*

Two part na -$26,799.50 -$17.87

Mod MS .742 $33,992.79 $22.66

*Estimated from P(A|W)

Clothing
Results, 

Reports/Expenditure

30

Model P(A=1|W=1)

Missing 

Expenditure

/QTR

Missing 

Expenditure

/CU QTR

2

nd order .817 $60,399.74* $40.27*

Mover-Stayer .903 $28,966.10* $19.31*

Two part na -$33,316.50 -$22.21

Mod MS .877 $17,222.49 $11.48

*Estimated from P(A|W)

Drugs
Results, 

Reports/Expenditure

31

Model P(A=1|W=1)

Missing 

Expenditure

/QTR

Missing 

Expenditure

/CU QTR

2

nd order ? ? ?

Mover-Stayer ? ? ?

Two part na $5,182.59 $3.46

Mod MS < 0 -$27,784.80 $-18.52

*Estimated from P(A|W)

Major Appliances
Results, 

Reports/Expenditure

32

Model P(A=1|W=1)

Missing 

Expenditure

/QTR

Missing 

Expenditure

/CU QTR

2

nd order .445 $58,551.30* $39.34*

Mover-Stayer .741 $16,409.11* $10.94*

Two part na -$1,840.63 -$1.23

Mod MS .187 $31,017.09 $20.68

*Estimated from P(A|W)

Music
Are Models Worth It?1

 Generally more information is better

 Time consuming – estimation is slow

 Model fit does suffer more than expected

 Are estimates of missing expenditure 

superior?

LCA Mover-Stayer, 2nd order, are vetted, 

stable over time, estimates make sense, 

internal validity, validation with external 

sources

33
Are Models Worth It?2

 Estimates are no more believable for difficult 

expenditure categories (e.g. major 

appliances)

 Two part latent growth produces very 

different estimates

 Some support for modified mover-stayer

 Much more testing is needed

Grouping variables

Examine estimates over time

Validation with external sources 34
Contact Information

Brian Meekins

Office of Survey Methods Research

202-691-7594

meekins.brian@bls.gov
Estimating Magnitude of Underreported 

Expenditures for

False Negative: Notation

Total reported expenditures for persons with 

characteristics, c

Accuracy rate for persons with characteristics, c, 

estimated from M-S model; i.e. P(A=1|W=1)

True total expenditures persons with 

characteristics, c

True total expenditures persons with 

characteristics, c for true positives

True total expenditures persons with 

characteristics, c for false negatives
Assumptions

 No false positive reports of expenditures

 Reported expenditures are accurate; i.e., 

 Mean expenditures for reporters and mean 

expenditures for nonreporters are equal
Estimate of Underreports Due to 

False Negatives

Under these assumption, an estimate of is 

Thus, an estimate of is 
Mover-Stayer Model

Assumptions

Population can be divided into:

 Persons who purchase the item in each quarter 

(“purchase-stayers”)

 Persons who do not purchase the item in any 

quarter (“nonpurchase-stayers”)

 Persons whose purchase behavior is not 

consistent across the quarters (“movers”)

Additional Assumption

 No false positive reports. Persons who report a 

purchase are assumed to have actually made 

that purchase.
Definition of Latent Variables

Where,

1, if one or more purchases of an item during the 

W= quarter (“purchaser”)

2, if no purchase (“non-purchaser”)

with similar definition for X, Y, Z for 2nd, 3rd, and 

4th interview
41

Definition of Indicator Variables

Define for Interview 1,

A = 1 if reported as a purchaser for the quarter

2 if reported as non-purchaser

with similar definition for B, C, D

for 2nd, 3rd, and 4th interviews
42

Grouping Variables

1. Family size

2. Refusal to answer income question

3. Derived variable combining records use 

and interview length

4. Income class