5 CPI Detailed Report, November 2006

Louise Leonard Campbell

is an economist in the

Office of Consumer Prices

and Price Indexes, U.S.

Bureau of Labor Statistics.

Email:

Campbell.Louise@bls.gov

This paper examines the U.S. Bureau of Labor Statistics (BLS)

past design and results of the age-bias regression model,

as well as its current design. This narrative describes

what has been done and presents current research on areas

where improvement is warranted. The fundamental question is,

“Does this model really capture depreciation for U.S. housing

in the CPI?”

Updating the Housing Age-Bias

Regression Model in the Consumer

Price Index

Louise Leonard

Campbell

Background of Housing Aging Bias

Estimation

The measurement goal of the Consumer Price

Index (CPI), produced by the U.S. Bureau of Labor Statistics (BLS or the Bureau), is to accurately

estimate the price change of a set of constantquality consumer goods and services. The components of the CPI Housing Survey are Rent of

Primary Residence (Rent) and Owners’ Equivalent Rent of Primary Residence (OER). Together,

these two components comprise more than 29

percent of the national CPI; that is, more than 29

percent of all consumption spending is attributed

to renter-occupied and owner-occupied residential dwelling space. The indexes for the housing

components measure changes in the cost of shelter services for renters and homeowners. Rent

estimates are based on contract rents—what tenants pay their landlords for the flow-of-housing

services as provided in the lease. This may include items, such as utilities or furnishings that

are in addition to shelter. OER also uses a flowof-services concept; estimates that owners of

housing units would have to pay to rent equivalent shelter—without additions such as utilities.1

The CPI Housing Survey is the data source for

calculating Rent and OER estimates. The CPI

began using the current housing sample, beginning with the index for January 1999.2

 (This

sample replaced the one used since 1987.) The

current sample is based on the 1990 Census and

is composed entirely of rental units selected to

represent units in both largely renter and largely

owner-occupied neighborhoods. OER estimates

are based on changes in rents of renter units in

the largely owner-occupied neighborhoods.3

With each passing year, the dwellings in the

CPI housing sample deteriorate, losing some

value (depreciate), and deliver less shelter service to their occupants. If this were not taken

into account, the CPI would have a downward

bias. To offset this quality loss, the aging bias

due to depreciation, staff at BLS developed an

age-bias adjustment.4

 This tool, which BLS began using with the CPI for January 1988, makes

a monthly incremental adjustment to each housing unit in the sample to maintain a constant quality index over time.

 The theoretical basis for the hedonic method

used in estimating age bias is found in Randolph

(1988). Randolph estimated that the age bias due

to depreciation was 0.3 to 0.4 percent annually.5

His 1988 regression model, adopted by BLS for

the sample selected from 1980 Census data, accounted for the influence of structure type and
6 CPI Detailed Report, November 2006

the rent control status of units. The age-bias regression model

includes structural characteristics of housing units. Structural

changes examined are changes in the number of bedrooms,

number of bathrooms, number of other rooms (not previously

mentioned), and additions or deletions of central air conditioning units. Subsequently, Bureau researchers realized that

the model missed other dimensions of quality change, although parameter estimates for most housing structural components had reasonable algebraic signs and magnitudes. Since

the age-bias regression provided, as a by-product, estimates

of the value of structural changes, the CPI began using these

estimates in 1989, to quality adjust for structural changes when

changes occurred to sample units.

The CPI used the (old) 1980-Census-based housing sample

from January 1987 through December 1998. In 1999, BLS

replaced that sample with a new one, based on the 1990 Census. Along with the new sample, BLS specified a new regression model to adjust for age bias (depreciation).6

 Staff were

forced to change the model, because some of the variables

used in the pre-1999 model were no longer available. Additionally, this revision was the opportunity to align age-bias

processing with the new sample.

This paper focuses on the change between the pre-1999

hedonic regression model and the model used since then in

producing depreciation estimates for the housing stock.

The Pre-1999 Sample

The hedonic regression model used from 1988 through 1999

follows:

Log(rent) = f [13 structural characteristics variables

(detached, bedrooms, other rooms, complete

kitchen, dishwasher, washer/dryer, oil heat,

electric heat, central air conditioning, extra bathroom, rent control),

various location and survey variables,

13 neighborhood characteristics variables

(renters, race white, large buildings, two or

more autos, without complete plumbing, airconditioned, children age 6 to 18, college students, families below poverty level, elderly

over 65, mobile homes, unemployment, with

college education),

7 dummy variables for services provided

with rent

(gas, electric, parking, furnishings, swimming

pool, other recreation),

6 depreciation variables

(age, age x squared, age x rent control, age x

old, age x detached, age x rooms)]

+ a random error term.

Both models rely on the CPI’s geographic structure. The

CPI geographic sample consists of selected metropolitan areas and urban places (primary sampling units or PSUs) where

the CPI observes rents (and other prices) over time. BLS

generates age-bias estimates for each PSU. In January 1999,

the CPI housing component moved to the revised structure

that the rest of the CPI had adopted one year earlier. Until this

revision of the CPI, there were 88 PSUs–14 in the Northeast,

22 in the Midwest, 33 in the South, and 19 in the West. The

model had dummy variables for these PSUs, and there were

additional dummy variables for structural characteristics for

each PSU within each Census region. Preparing data for the

regressions was quite cumbersome. We weighted rent values

by their relative cost on a unit-by-unit basis, according to the

number of renters they represented within their area and the

estimated owner and renter expenditures of the areas, and

matched the CPI-collected data to 1980 Census data. From

1988 through 1999, we used the specification structure developed by Randolph. This structure generated depreciation,

or age-bias factors, with rent levels of the most recently collected CPI data. For the purpose of this analysis, data for the

year 1997 was chosen, because it was the most logical data to

compare with the most recent annual regression estimates.

This regression model was run for each Census region individually. The model specificity was unique to that region,

and the regression estimates were particular to the geography

of that Census region. Following are the variance analyses

for each region:

Age-Bias

January 1997 Regression Model— NORTHEAST (CP 199612)

Dependent variable: LOGRENT

Analysis of Variance

Degrees of Mean Adjusted

Source freedom square F-Value Prob>F R-Square R-square

Model ..................................... 110 5.10886 75.239 0.0001 0.6945 0.6853

Error ....................................... 3640 0.06790

Total ........................................ 3750
7 CPI Detailed Report, November 2006

Each regression model used at least 110 variables to generate the estimates. Although it is not shown here, the number

of units used in the regressions decreased each year over the

life of the 1987 sample. Because the sample used in age-bias

processing was so closely tied by location to the 1980 Census data, units constructed after 1980 could not be added.

(These units were added to the sample but could not be used

in age-bias regression.) The result was that each year, fewer

units were used in age-bias processing. Dwellings dropped

out for one of four reasons: They were no longer rental units,

they were condemned, they were converted to businesses, or

they were destroyed. Although the regression model was well

specified, the sample used to estimate it through 1999 became

increasingly unrepresentative of the current pool of housing

units.

Focusing on two other descriptive statistics from the tables

above, the adjusted R-square indicates the amount of variation that the model explains with a correction for the degrees

of freedom. The average adjusted R-squared for the old model

structure is quite high at 0.7. Roughly speaking, this implies

that 70 percent of the variation in log rent is explained by

variation in the explanatory variables. The F Value is an indication of confidence in the independent variables in the model,

showing the degree to which this set of variables has statistically significant impact on the dependent variable. The largest F value generated in this model structure is a bit over 100,

at 110.9 for the West region. The fact that it is not larger may

indicate that the models were over-specified.

The model used a complex array manipulation that simulated a maximum likelihood function. Each PSU, and several

of the other independent variables, were arrayed in tandem

with the weighted rent for each housing unit, to generate an

age-bias factor for that specific PSU. The statistical program

(SAS) consistently generated messages indicating co-linearity and warnings that the results were likely biased. Furthermore, staff were not comfortable with the number of zero values that the model created.

The Current Sample: The New Model

When the housing sample drawn from the 1990 Census, the

model was revised. One major decision was to use ZIP Codes

Age-Bias

January 1997 Regression Model— MIDWEST (CP 199612)

Dependent variable: LOGRENT

Analysis of Variance

Degrees of Mean

Source freedom square F-Value Prob>F R-Square Adj R-sq

Model ..................................... 125 3.74195 66.206 0.0001 0.6647 0.6546

Error ....................................... 4175 0.05652

Total........................................ 4300

Age-Bias

January 1997 Regression Model— SOUTH (CP 199612)

Dependent variable: LOGRENT

Analysis of Variance

Degrees of Mean

Source freedom square F-Value Prob>F R-Square Adj R-sq

Model ..................................... 168 5.57731 85.781 0.0001 0.7253 0.7169

Error ....................................... 5457 0.06502

Total ........................................ 5625

Age-Bias

January 1997 Regression Model— WEST (CP 199612)

Dependent variable: LOGRENT

Analysis of Variance

Degrees of Mean

Source freedom square F-Value Prob>F R-Square Adj R-sq

Model ..................................... 110 5.58910 110.900 0.0001 0.7453 0.7386

Error....................................... 4169 0.05040

Total ....................................... 4279
8 CPI Detailed Report, November 2006

to match Census data to CPI data in the set-up phase of the

process. Doing this allowed new construction (built after the

1990 Census) sample units to become part of the analysis. The

other decision was to use unweighted rent data, eliminating

the need for maximum likelihood estimation. The new regression model is:7

Log (rent) = f[10 structural characteristics variables,

various location and survey variables

(detached, bedrooms, bedrooms squared,

other rooms, other rooms squared, oil heat,

electric heat, central air conditioning, window air conditioning, bathrooms),

2 survey variables (A-size, B size),

10 neighborhood characteristics variables

(race white, large buildings, two or more

autos, air-conditioned, children age 6 to 18,

some college, families below poverty level,

elderly 65 & over, mobile homes, unemployment),

3 dummy variables for services provided

with rent

(gas, electric, parking),

5 depreciation variables, and a random

error term

(age, age squared, age x old, age x detached,

age x allrooms)]

+ a random error term.

The CPI Housing Survey, based on the 1990 Census, includes only rental units and asks significantly fewer questions

of respondents about their dwellings than does its predecessor. The number of PSUs was reduced from 88 to 87, and the

various PSUs were not included as dummy variables in the

age-bias model specifications. As a result, the regression

specification became significantly simpler. For the years since

its implementation in 1999, the annual estimate of age bias has

been in the 0.2 to 0.3 range.8

 Results of the current model

design are specified by the analysis of variance as follows:

The current age-bias model has significantly fewer regression variables, significantly more observations, and allows

community variables from Census data to be matched by ZIP

Code to new construction units, as long as the ZIP Codes existed in 1990. The F value of 854.4 is significantly higher than

the one generated through the previous regression model.

However, much of that increase can be explained by the adjustments in the degrees of freedom for the model and the

error. Just as before, the probability of exceeding the F value is

exceedingly small, so it can be stated with confidence that the

fewer number of independent variables, the bigger the impact

on log rent. The independent variables that were deleted did

not contribute that much in explaining the variation in the dependent variable. That is why it was previously stated that the

old model may have been over-specified. Generating the agebias factor no longer requires the complex maximum likelihood

function algorithm.

Naturally, omitted variable bias cannot be ruled out. The

combination of changes may have resulted in the lower Rsquared statistics and the lower values of the annually generated age-bias factor estimates. (See appendix I.) Appendix 1

shows the effect of age-bias values for the Nation annually

since their integration in the production process. The revised

sample, implemented in 1999, consistently yields values less

than 0.3 percent annually.

Data Source Considerations

The most important independent variable in the current model

is age, determined from the year built question. The year the

unit was built is asked of respondents during initiation of the

survey instrument, although we do not need to know the year

built of age of a sample unit to use it to estimate rent change.

We do need it, however, to use the unit in the age-bias regression. All units in the housing survey are rental units, and respondents often do not know the year the dwelling was built.

At initiation, the majority of the units in the current sample, 55

percent, were missing year built. Subsequently, both national

office staff in Washington, D.C., and data collectors in the field

made a strenuous effort to learn the year built from other

sources and BLS has been able to collect year-built data for

many sample units. From 2003 through 2005, the number of

units with year-built data increased to 89 percent.

ZIP Code data for units is important so Census data that

contain neighborhood characteristics can be matched to new

construction built in the 1990s and later. (The database is

updated periodically with appropriate ZIP Codes.)

Future Research

BLS is planning to replace the current housing sample over a

6-year period, beginning in 2009. Up to now, Census informaAge-Bias

January 2006 Regression Model— ALL UNITED STATES (CP 200511)

Dependent variable: LOGRENT

Analysis of Variance

Degrees of Mean

Source freedom square F-Value Prob>F R-Square Adj R-sq

Model ..................................... 30 89.89777 854.37 0.0001 0.5202 0.5196

Error ....................................... 23642 0.10522

Total ........................................ 23672
9 CPI Detailed Report, November 2006

tion has been the basis for the housing survey’s sample frame.

The new sample will be drawn from the 2000 Census. In the

2000 Census, however, detailed neighborhood questions

moved to the long form, so fewer respondents were asked

them. The Census Bureau is implementing a new survey, the

American Community Survey (ACS) to replace much of the

housing data they collect in the Decennial Census. If the ACS

grows as planned, it will be a major data source for the aging

bias model for the Housing CPI.

Current research may eventually lead to changes in the way

the factors are regressed and applied. Age-bias factors might

be processed by regression procedures at index area levels

and applied at the unit level. Preliminary research found that

the weight of units designated as old, units built before 1900,

tends to result in smaller age-bias factors.

Other data sources that BLS researchers believe will enhance regression results have been identified. For example,

researchers will use average income by ZIP Code as one of the

independent variables in the functional form. (This data has

only recently become available from the Internal Revenue Service.9

)

A Small Global Survey

The Ottawa Group is an international organization of individuals responsible for price programs worldwide. In 2003, BLS

staff asked members of the Ottawa Group if their price programs use hedonics of rental units to estimate depreciation for

the housing stock. Over 90 percent of those members surveyed do not estimate depreciation. Just as the current Bureau model is not truly comparable to the previous one, this

lack of comparability is even greater for other countries and

their structure for estimating price change. The Bureau’s regression model is unique; our method of estimating shelter is

also rare.

Conclusion

The Bureau’s two (former and current) age-bias regression

models are not completely comparable. Clearly, each produced, or produces, estimates used in the production of the

Rent of Primary Residence (Rent) and Owners’ Equivalent Rent

of Primary Residence (OER) Indexes. The R-square value for

the current model structure is significantly lower than it had

been previously. However, this is to be expected, since there

are fewer independent variables—and combining all four regions into one—almost certainly increases the variation.

There are still the potential problems of the data source considerations mentioned above and a now-higher likelihood of

omitted variable bias. The CPI program treats the hedonic

regression effort as an area of constant improvement, because

the quality of the index is affected by its being applied correctly.

The first graph in appendix II shows the effect of age-bias

estimates on the Rent and OER Indexes, since 1988. The second graph shows the effect of age-bias adjustments on the published All Items Consumer Price Index for the same period.

Over time, the average increase in the CPI due to age-bias

estimates is approximately 3.0 percent.10 Unit-level estimation

may yield better results than the practice of generating yearly

estimates at the PSU level, and then applying them to individual units on a monthly basis.

Special thanks go to Walter Lane, Frank Ptacek, Randal Verbrugge,

and Ronald Johnson for their helpful and insightful input. 1 Details regarding the estimation of price change for shelter can be

found in the Handbook of Methods, Chapter 17. The Consumer Price

Index, pp. 23-25. 2 See “Revision of the CPI housing sample and estimators,” Monthly

Labor Review by Robert Baskin and Frank Ptacek, pp. 31-39, December

1996, http://stats.bls.gov/opub/mlr/1996/12/art5full.pdf. 3 Current statistical methods and economic theory regarding how

the BLS handles owner-occupied rent can be viewed on line. Also, see

Treatment of Owner-Occupied Housing in the CPI, Robert Poole,

Frank Ptacek and Randal Verbrugge. This paper was presented before

the Federal Economic Statistics Advisory Committee (FESAC),

December 9, 2005, http://stats.bls.gov/bls/fesacp1120905.pdf . 4

 See “Adjusting the CPI shelter index to compensate for the effect

of depreciation,” Walter F. Lane, William C. Randolph and Stephen A.

Berenson, Monthly Labor Review, pp. 34-37, Technical Notes, October 1988— http://stats.bls.gov/opub/mlr/1988/10/rpt1full.pdf. 5

 See Housing “Depreciation and Aging Bias in the Consumer

Price Index,” William C. Randolph, Journal of Business & Economic

Statistics, July 1988, Vol. 6, No. 3, p. 365. 6

 William Thompson, a supervisory economist in the Division of

Consumer Prices and Price Indexes, designed this model in 1999. 7 Ibid.

8 Current statistical methods and economic theory regarding how

the BLS handles owner-occupied rent can be viewed on line. Also, See

Treatment of Owner-Occupied Housing in the CPI, Robert Poole,

Frank Ptacek and Randal Verbrugge. This paper was presented before

the Federal Economic Statistics Advisory Committee (FESAC), December 9, 2005, http://stats.bls.gov/bls/fesacp1120905.pdf, p 29. 9

 Regressions have been performed using 2002 IRS average gross

income (AGI) data by ZIP Codes. Researchers used AGI and log AGI as

independent variables, yielding small improvement in the variance analysis statistics. 10 See “What has happened to price measurement since the Boskin

Report? The U.S. Experience,” David S. Johnson, Stephen B. Reed

and Kenneth J. Stewart, U.S. Bureau of Labor Statistics; “OECD Conference: Inflation Measures: Too High – Too Low – Internationally

Comparable?” Paris 21-22 June 2005, pp. 10-11.

Notes
10 CPI Detailed Report, November 2006

a

 Appendix I

Annual CPI housing age-bias (AB) factors

0.15

0.20

0.25

0.30

0.35

0.40

0.45

1988198919901991199219931994199519961997199819992000200120022003200420052006

Years

Factors
11 CPI Detailed Report, November 2006

 Appendix II

Graph 2. Published CPI & CPI without Housing AB factors

110

150

190

230

1987198819891990199119921993199419951996199719981999200020012002200320042005

Years

Index

Official CPI

CPI w/o AB

110

150

190

230

270

1987198819891990199119921993199419951996199719981999200020012002200320042005

Index

Rent

Rent-AB

OER

OER-AB

 Graph 1. CPI Rent and OER Indexes with and without housing AB factors

Years

aPP

Years

 Graph 2. Published CPI & CPI without housing AB factors