Urban Studies Research

Volume 2015 (2015), Article ID 121978, 13 pages

http://dx.doi.org/10.1155/2015/121978

## Testing Estimates of Housing Cost Differences among US Metropolitan Areas

Robert B. Pamplin, Jr. School of Business Administration, University of Portland, 5000 N. Willamette Boulevard, Portland, OR 97203, USA

Received 11 June 2014; Revised 16 October 2014; Accepted 23 October 2014

Academic Editor: Eric Koomen

Copyright © 2015 Todd Easton. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper investigates the accuracy of six measures of housing cost differences among US metropolitan areas. Using Census data from 177 metropolitan areas, it tests the measures in two ways. First, it tests the ability of changes in the measures to predict changes in the shelter component of the metropolitan CPI from 1990 to 2000. Second, it tests the ability of the measures themselves to predict a proxy in 2000. A measure based on Fair Market Rents calculated by HUD placed second on the first test but did badly on the second. The housing component of the ACCRA index, a living cost measure frequently used by researchers, performed poorly on both tests. The top performer on both tests was a measure based on the average rent per room for a metropolitan area’s dwellings. Researchers wishing to control for living cost differences among places should consider including it in their living cost index.

#### 1. Introduction

Researchers would like to investigate real earnings differences among urban areas in the US, but there is no good, official measure of living cost differences among places^{1}. For example, the Consumer Price Index (CPI) measures changes in costs over time in a place, not differences in costs among places. Researchers have responded to this problem with a variety of strategies. The most frequent of these is to use the ACCRA Cost of Living Index, an index compiled by a nonprofit organization. The suitability of this response is uncertain. While the ACCRA Index is available for a large set of metropolitan areas, data to calculate it are collected by volunteers and its accuracy has seldom been compared to practical alternatives.

This research contributes to a solution to this problem, in two ways. First, it evaluates alternative measures of* housing* cost differences among US metropolitan areas.^{2} That evaluation is central to deciding how to best measure living costs, because the cost of housing is, by far, the largest source of variation in living costs among metropolitan areas.^{3} Second, because the housing cost portion of the ACCRA Index is one of the measures it evaluates, this research offers evidence regarding the wisdom of using the full ACCRA Index as a measure of living cost differences.

This study evaluates six housing cost measures against two benchmarks. Initially, I test the ability of* changes* in each of the measures to predict changes in the shelter portion of the metropolitan CPI in 25 large metropolitan areas. Then, I test the ability of each measure to predict a proxy for a metropolitan area’s housing costs: the average size of its dwellings. Using this proxy allows me to test performance of housing cost measures in 176 metropolitan areas.

This research builds on a prior paper by Easton, one which evaluated the same six measures of housing cost differences [1]. He tested them against a different benchmark: a housing cost index taken from work by Aten [2]. She created an experimental measure of living costs in 26 large metropolitan areas, using Bureau of Labor Statistics (BLS) data collected to calculate the CPI. By utilizing two new benchmarks and, in one test, a much larger set of metropolitan areas, this research provides additional evidence regarding the best way to measure housing cost differences among areas.

In addition to Easton’s paper, two other articles provide some evaluation of metropolitan living cost measures. Koo et al. create an index to measure metropolitan price levels between July 1988 and June 1989 [3]. They estimate price levels in 22 CPI metropolitan areas using BLS data and calculate a mean absolute difference of 7.8% between their new index and the ACCRA Index; they conclude the ACCRA Index has substantial errors, errors that result mostly from sampling and aggregation bias. Curran et al. provide a theoretical overview of alternative cost of living measures, concentrating on those calculated by the Council for Community and Economic Research (also known as C2ER), the National Research Council (NRC), and the Economic Research Institute [4]. They conclude C2ER’s measure, the ACCRA Index, is best, because it includes prices of a broad set of goods (not just housing) and carefully specifies the goods to be priced. They fault NRC’s measure for ignoring nonhousing prices and for using a biased measure of housing costs (Fair Market Rents calculated by the Department of Housing and Urban Development). They fault the Economic Research Institute’s measure for inconsistencies in the goods priced and for implausible component weights for low income families.

Between 1991 and 2008, at least six articles relied solely on the ACCRA Index to measure living costs: Browne and Trieschmann’s study of the compensation of full professors at research universities [5], Cutler and Glaeser’s examination of residential segregation’s impact on real earnings differentials between African American workers and other workers [6], Gisser and Dávila’s analysis of earnings differences between unskilled farm workers and unskilled urban workers [7], Olson et al.’s study of the influence of local wage levels and living costs on pay for federal government jobs [8], Easton’s study of immigration’s impact on the wages of native workers [9], and Glaeser and Tobio’s examination of the influence of climate and Southern location on real wage income [10]. Not one of these papers included an evaluation of the ACCRA Index’s accuracy or an examination of the sensitivity of results to measuring living costs using alternative methods.

Three papers improve on those mentioned above by using multiple measures of living costs. Dumond et al. study metropolitan variation in living costs, amenities, and wages [11]. They argue that estimates of real wages should rely on partial adjustment, that is, on models that only include living cost measures as independent variables. They estimate a log living costs coefficient of .526 in a model predicting 1989 log wages with controls for amenities. When they replace the ACCRA Index living cost measure with an index they create from the BLS comparative cost index, the coefficient falls to .366.^{4} Winters’ 2009 article also predicts nominal wages at the metropolitan level [12]. With the ACCRA Index, he calculates a log living cost coefficient of .314, a value which rises to .760 when he replaces the ACCRA Index with his rent-based index. He calculates the rent-based index by replacing the housing cost portion of the ACCRA Index with a hedonic measure of rents for tenant-occupied dwellings.^{5} The rent-based index is a weighted average of the hedonic measure and the nonhousing cost portion of the ACCRA Index. Moretti studies the earnings advantage of high-education workers, by comparing the earnings of workers with a college degree or more to the earnings of workers with only a high school degree [13]. For example, he calculates that the nominal earnings advantage of the first group was 60 log points in 2000. He then constructs five living cost indexes to calculate real wages at the metropolitan level. One index measures housing costs with the average rent of 2- and 3-bedroom apartments and nonhousing costs with the ACCRA Index. Using it, the real earnings advantage is 54 log points in 2000. Another index measures housing costs with the average rent of 2- and 3-bedroom apartments and imputes nonhousing costs using the relationship between housing and nonhousing costs in CPI data. Using that index, the real earnings advantage is 51 log points in 2000.

#### 2. Six Measures of Housing Costs

This section presents the six housing cost measures tested.^{6} Measure A, Measure B, and Measure C embody hedonic approaches to valuing housing services and require extensive calculations using Census data. Measure D also uses Census data, but only for a simple calculation. Measure E and Measure F are similarly straightforward but use different datasets. Measure E is calculated directly from HUD Fair Market Rents and Measure F is calculated directly from the ACCRA Index.

Measure A estimates rents on tenant-occupied units in each metropolitan area. By ignoring owner-occupied dwellings, it implicitly assumes that rents and owners’ equivalent rents move in tandem. The approach is analogous to the one the CPI uses to value housing services.^{7} I estimate the following relationship in each metropolitan area in the sample:
where is the rent, net of utilities, for the th housing unit in the metropolitan area’s sample; is a vector of attribute rents, with one element for each dwelling attribute; is a vector of the dwelling attributes; and is a random disturbance term.^{8}

Once attribute rents have been estimated, I use them to predict the average tenant-occupied unit’s rent in each metropolitan area and then calculate the housing cost index by dividing each metropolitan area’s predicted rent by the average predicted rent in all metropolitan areas.^{9}

Measure B uses both rented and owner-occupied units to estimate housing costs. Within each metropolitan area, it estimates a dwelling’s cost, which is either what the household pays in rent or the value of the dwelling:
where is the th dwelling’s “cost” (the monthly rent, net of utilities, if the unit is rented, or the market value, if the unit is owner-occupied) and is a dummy variable identifying whether the th unit is an owner-occupied dwelling.^{10}

This approach, developed by Crone et al. [14], assumes the attributes of a rented dwelling affect its rent by the same proportion as the attributes of an owner-occupied dwelling affect its market value. While this assumption is restrictive, it allows owner-occupied housing to directly influence estimated attribute rents. Once attribute rents are estimated, the Measure B housing cost index is calculated with them in the same manner as the Measure A index.^{11}

Measure C is like Measure B, in the fact that it includes both rented and owned units, but it pools all metropolitan areas to estimate where is the cost for dwelling in metropolitan area ; is a vector of coefficients, one for each metropolitan area; and is a vector of dummy variables, one for each metropolitan area in the sample.

Two aspects of this relationship should be noted. First, since it is estimated for all metropolitan areas simultaneously, rather than for each one individually, it is much easier to estimate than (1) or (2). Second, since it constrains to be the same across metropolitan areas, it will improve estimates of owner’s equivalent rents if speculation pushes house values above what fundamentals warrant in some areas.

Measure C housing cost index is calculated from the coefficients estimated for the dummy variables in vector . The antilog of each coefficient is taken, to get an estimate of each metropolitan area’s average rent. Then, each area’s average is divided by the across-metropolitan area average rent.

As mentioned above, the fourth, fifth, and sixth housing cost measures require little calculation. The fourth and fifth are based on tenant-occupied dwellings, while the sixth is based mostly on owner-occupied housing. To calculate the fourth, Measure D, I calculate the average rent per room for each metropolitan area and then create the housing cost index by dividing those averages by the mean of all the metropolitan averages. Rather than relying on Census data, Measure E uses Fair Market Rents (FMRs) calculated by the Department of Housing and Urban Development (HUD). A metropolitan area’s FMR generally estimates the 40th percentile of the rent distribution for tenant-occupied units three years or older.^{12} To calculate the housing cost index for Measure E, each metropolitan area’s FMR for two-bedroom apartments is divided by the mean of the two-bedroom FMRs for all metropolitan areas in the sample. Measure F is simply the ACCRA Index’s housing component divided by 100, so that its scaling matches that of the other measures. Dwellings selected for C2ER’s sample are meant to be typical of those lived in by managers and professionals in the top quintile of the income distribution. C2ER calculates the housing component using a weighted average of the average monthly mortgage payment on a new 2,400-square-foot home and the average rent for a two-bedroom apartment. The weights are based on Consumer Expenditure Survey results for top quintile households. For example, in 2012 the weights were about 82% and 18%, respectively.^{13}

#### 3. Data

There are four sources for the data used in the study. Data to calculate most of the housing cost measures and the housing cost proxy come from the 5% Public Use Microdata Set (PUMS) of the 1990 and 2000 Censuses [15]. Additional data to compute housing cost measures come from Fair Market Rents calculated by HUD.^{14} C2ER, also known as the Council for Community and Economic Research, provided the ACCRA Index and its components; the variable used here is the housing cost component. The metropolitan CPI comes from the BLS.

I use Census data for 177 metropolitan areas, the ones that had a population greater than 200,000 in both 1990 and 2000. For metropolitan areas with populations under 400,000, all the PUMS records are included. However, sampling rates are reduced as metropolitan populations grow, to keep dataset sizes manageable. For example, for areas with populations over 2 million, the sampling rate is 10%. The dataset includes 580,000 dwellings in 1990 and 730,000 in 2000.

#### 4. Methodology

The six housing cost measures described above are plausible measures of differences in the level rents and owner’s equivalent rents among metropolitan areas. Two tests evaluate their accuracy.

##### 4.1. Test One

The first test is based on a simple idea: even though the housing cost measures gauge price differences among places, not changes over time, a good housing cost measure should accurately track changes in price. If it does not, a measure that is accurate in one year will not be accurate subsequently. Since the shelter portion of the metropolitan CPI should accurately track changes in housing costs in major metropolitan areas, I test the accuracy of each of the six measures by seeing how well it predicts changes in the metropolitan CPI by estimating
where is the change in the shelter portion of the CPI in metropolitan area , from 1990 to 2000, and is the change in a particular housing cost measure (e.g., Measure A) in metropolitan area , from 1990 to 2000.^{15}

After predicting the change in the shelter CPI with each housing cost measure, I use -squared to compare the accuracy of the predictions.^{16}

##### 4.2. Test Two

The second test compares the ability of the housing cost measures to predict a proxy: the average size of a metropolitan area’s dwellings. The justification of the proxy is this: changes in the opportunity cost of space should change the amount of space households consume. Since the Census lacks more precise measures, I use the number of rooms in a dwelling to measure its size. The second test estimates where is the average number of rooms per dwelling, across all households in metropolitan area ; is the value of a particular housing cost measure for metropolitan area ; is a vector of six controls for metropolitan area : average household size, average household income, and the proportions of the population that are black, Hispanic, Asian, and recent immigrants; and is a random disturbance term.

Controls are included to evaluate the unique information each housing cost measure contains. Rises in household size are expected to increase the number of rooms a household dwells in, by increasing its demand for space. Space is expected to be a normal good, so that rises in household income are associated with rises in the number of rooms a household occupies. Measures of metropolitan ethnicity control for differences in taste and wealth among ethnic groups. The proportion of the population that is recent immigrants also controls for differences in taste and wealth, but this time between immigrants who arrived recently and others.

I implement the second accuracy test in two ways. First, I gauge the marginal contribution made by each of the six measures of housing costs, by predicting the proxy with just the controls and then adding each measure in turn. Second, I gauge the total contribution by predicting the proxy with each housing cost measure alone. In combination, the rise in -squared in the first exercise and the actual -squared in the second exercise provide a minimum and a maximum magnitude for each measure’s contribution to explaining the variation in the proxy.

#### 5. Results

This section presents the results of the two accuracy tests described above. It first presents the results of estimating (4), the equation predicting the change in an area’s metropolitan CPI, using the first five housing cost measures (Measure F cannot be included, since C2ER recenters its index annually). Then, it presents the results of using all six measurements to estimate (5), the equation predicting the proxy.

##### 5.1. Test One

Shelter CPI changes are predicted for 25 large metropolitan areas for which the BLS calculated metropolitan CPIs in 1990 and 2000. Table 1 reports the results. Table 2 reports means and standard deviations for variables in Table 1, as well as for each variable appearing in subsequent tables. Comparing the -squared values in Table 1, Measure D (based on average rent per room) is, by a substantial margin, the best predictor of the change in the shelter CPI. The 1990 to 2000 change in Measure D predicts 69% of the variation in the CPI change. Measure E (based on Fair Market Rents) is second best and Measure C (the pooled hedonic measure of tenant and owners’ equivalent rents) is third best.