Dynamics of the Affordable Housing Stock: Microdata ... · Braid (1981) studies filtering in rental housing markets.Bond and Coulson (1989) analyze the process of neighborhood change

Dynamics of the Affordable Housing Stock:Microdata Analysis of Filtering

C. Tsuriel Somerville and Cynthia Holmes*

Abstract

This article identifies the factors that change the stock of market housing affordable to low-incomehouseholds. During the past 15 years policy makers and academics have concentrated on the shortageof good-quality affordable housing for low-income households. Among the triggers of that interest arethe increase in homelessness, the disappearance of rental units affordable to the least well off membersof society, and the implications of growing income inequality. We take advantage of the panel natureof the metropolitan surveys of the American Housing Survey to model the movement of individual hous-ing units in and out of the stock of units affordable to low-income households.

We use a multinomial logit methodology to estimate the effects of unit, neighborhood, and market char-acteristics and conditions on the status of a unit over time. We compare the alternative outcomes for anaffordable unit with the outcomes for unaffordable rental stock. One objective of this comparison is todetermine whether these factors have symmetric effects across different segments of the housing mar-ket. Our empirical results suggest that movements are more sensitive to variation in neighborhood con-ditions than to unit characteristics or movements in market rents or prices.

Keywords: Affordability; Rental housing

Introduction

During the past 15 years policy makers and academics have focused on the shortage of good-quality affordable housing for low-income households. Among the triggers for that interestare the increase in homelessness, disappearance of rental units affordable to the least welloff members of society, and falling real incomes at the lower end of the income distribution.Using data sources including the U.S. Bureau of the Census, researchers have examined themismatch between the aggregate housing needs of low-income households and the availablesupply of affordable rental units. One aspect of this relationship that has not been studiedis the process and course of the movement over time of units in and out of the affordablehousing stock. In this article we identify the unit, neighborhood, and market characteristicsassociated with a higher probability that a unit will stay in the stock of rental units affordableto low-income households or move out because of an increase in rent, conversion to owner-occupancy, or demolition. We apply the same empirical model to identify the factors associ-ated with whether a rental unit currently unaffordable to low-income renters will filter

Journal of Housing Research · Volume 12, Issue 1 115© Fannie Mae Foundation 2001. All Rights Reserved. 115

* C. Tsuriel Somerville is Associate Professor and Cynthia Holmes is a doctoral student in the Faculty of Commerceand Business Administration at the University of British Columbia (UBC).

The authors would like to thank participants at the American Real Estate and Urban Economics AssociationMidyear Meetings, May 2000; Amy Bogdon; and two unnamed referees for comments and suggestions. We thankthe University of British Columbia for providing financial support for this research through a UBC Humanities andSocial Sciences grant. All errors are the responsibility of the authors.

down into the affordable housing stock. Using data on individual housing units, this articleis thus a study of filtering, the process by which units move from one housing submarket toanother.

Reports such as Rental Housing Assistance at a Crossroads (U.S. Department of Housing andUrban Development [HUD] Office of Policy Development and Research 1996) and The Stateof the Nation’s Housing (Harvard University Joint Center for Housing Studies, annual) havehelped to identify the problems facing America’s poorest households in obtaining suitable andaffordable shelter. To develop policies that can create affordable housing opportunities, it isimportant to know what governs the transition of units in and out of the affordable stock.We present a set of findings that can help policy makers identify housing market and neigh-borhood conditions that worsen the housing problems facing low-income renters. Ourapproach is to look at individual units in successive waves of the American Housing Survey(AHS) metropolitan area sample. Our work differs from much of the literature on affordablehousing in that we focus exclusively on market housing, excluding both public housing andunits in which the household receives a subsidy.1 Our emphasis is not on identifying thematch between this stock and low-income households, but on identifying the determinantsof the movement of individual units in and out of this stock.

Controlling for the ratio of a unit’s rent to the affordable cutoff, we find that the probabilityof a unit filtering up (becoming unaffordable) during a four-year period is most sensitive toneighborhood characteristics. That is true even for higher-quality units and those in marketswith rising rents, factors that do increase the probability of filtering up. Factors of particularimportance in affecting whether units filter up are the satisfaction of residents with theirneighborhood and the concentration of affordable units in a neighborhood. The affordableunits in neighborhoods with a relatively lower concentration of affordable units are more like-ly to filter up, so that units in a pocket of affordable units in an otherwise unaffordable neigh-borhood are more likely to filter up than are similar units located elsewhere. In contrast, unitcharacteristics matter more for the probability that an affordable unit will be demolished;inadequate and older units are more likely to exit the affordable stock via demolition orabandonment.

Although we identify determinants of the movement of units out of the affordable stock, weare less successful for those that filter down into the affordable stock. Controlling for the dif-ference between a unit’s rent and the affordability cutoff, the probability that during a four-year period a unit currently in the stock of units unaffordable to low-income households willfilter down appears to be somewhat random, though filtering down is less likely to occur fornewer units in better neighborhoods. These results show that neighborhoods move towardhomogeneity. In a neighborhood with a larger affordable stock, individual unaffordable unitshave a higher probability of filtering down than if they were located elsewhere. The sym-metrical effect holds for affordable units in a neighborhood that contains a relatively largerstock of unaffordable units. The policy implication is that neighborhood characteristics mustbe taken into consideration for designing policies favorable to helping preserve the afford-

116 C. Tsuriel Somerville and Cynthia Holmes

1 Strictly speaking, private units in which the occupying household receives a subsidy are part of the market stock.Because of data problems in the AHS, we cannot determine whether the reported rent is pre- or postsubsidy. Toaddress this problem we exclude units that initially have subsidies and treat all units that subsequently obtainthem as being affordable.

able stock. Although heterogeneous neighborhoods have benefits, it is harder to retain theaffordable stock in those locations.

Our research draws on existing studies of the stock of low-income housing. There is a largebody of work on low-income housing policy and measures of the affordable stock. Among themany publications in this literature are those of Bogdon, Silver, and Turner (1994), whichexamines housing conditions and needs and emphasizes that affordability is the most seri-ous housing problem in most parts of the United States; Nelson (1994), which discusses thematch between the affordable stock and low-income households; and O’Flaherty (1996), whichinvestigates the economics of homelessness. Nelson and Vandenbroucke’s (1996) seminalwork charting the size of and change in the aggregate low-income housing stock is of par-ticular relevance to our analysis.

Nelson and Vandenbroucke use the panel nature of the AHS metropolitan area survey datato chart the movements of individual units in and out of various segments of the low-incomehousing stock. Paying careful attention to the AHS questions and the weights assigned indi-vidual observations, they document the dynamics in the aggregate size of the affordable stockby metropolitan statistical area (MSA).They also provide a comparison of aggregates of thesemovements with measures of the overall MSA housing supply and some neighborhood char-acteristics. Our article builds on their work by presenting an empirical model to identify thecontributions of unit, neighborhood, and housing market conditions in determining the move-ment of individual rental units, both into and out of the low-income housing stock. In devel-oping the model we rely on the insights from the theoretical literature on filtering.

This literature is quite old, with Ratcliff (1949) credited with providing the first explicitdefinition of filtering. The older empirical treatments are well surveyed by Brzeski (1977).Sweeney (1974) is credited with being the first to provide a thorough theoretical treatmentof filtering, in which the level of maintenance affects the rate of depreciation. The theoreticalliterature includes publications that expand his model to include other issues. For instance,Arnott, Davidson, and Pines (ADP) (1983) allow for maintenance and rehabilitation, andBraid (1981) studies filtering in rental housing markets. Bond and Coulson (1989) analyze theprocess of neighborhood change in a model in which the value of housing is related to neigh-borhood characteristics.

The recent empirical filtering literature does not directly examine individual units, but looksfor outcomes consistent with filtering. Phillips (1981) uses cross-sectional data to comparemean neighborhood income with descriptive statistics of the neighborhood housing stock;Weicher and Thibodeau (1988) test for the effect of new construction on the low-income hous-ing stock. A more targeted study is Susin’s (1999) examination of the effect of Section 8 hous-ing vouchers on rents for the least expensive third of units. He uses the AHS neighborhoodsample and finds a fairly inelastic supply curve, that is, little filtering, as rents are clearlyhigher in the presence of vouchers. Our work differs from the work reported in this literature;instead of using aggregate measures, we conduct our analysis at the level of the individualunit.

The remainder of the article is organized as follows. In the next section, we present the the-oretical and empirical framework for the article. The following section includes a descriptionof the data. Our empirical results are presented in the subsequent section, and we concludewith a discussion on policy implications and an outline of additional avenues for research.

Dynamics of the Affordable Housing Stock: Microdata Analysis of Filtering 117

Theoretical Framework

The landlord of any rental unit faces an optimization problem for the current level and futurepath of net rents. Borrowing the general framework from ADP (1983), we define rent per unitof structure quantity Rt at time t as a function of unit quality st, neighborhood quality Yt, andexogenous market conditions Zt:

Rt = R(st , Yt , Zt). (1)

Rents are rising in both unit quality and neighborhood quality. Unit quality evolves as a func-tion of quality in the previous period and the current level of effective maintenance m*

t:

st = S(st–1, m*t). (2)

There is an underlying rate of depreciation such that for m*t < α, we have falling quality

st < st–1. The effect of maintenance expenditures will vary with the age of the unit, so thateffective maintenance is the expenditure on maintenance mt, adjusted to reflect the build-ing’s age (t – t0), where t0 is the construction date:

m*t = M[mt, (t – t0)]. (3)

It is possible for m*t < 0 even if mt > 0, because the rate of unit depreciation is rising with

building age; that is, for quality to be maintained, expenditures must be higher in older units,∂m*/∂t < 0. The partial derivatives for quality (equation 2) are thus:

∂S/∂s > 0, ∂S/∂m > 0, ∂S/∂t < 0. (4)

Combining equations 1 through 4, we have the landlord’s optimization process. We abstractaway from the redevelopment decision addressed in ADP (1983). Instead, landlords choose toabandon the property at time T. Landlords choose a path of maintenance expenditures andan abandonment date to maximize profits:

(5)

The dynamic programming solution to equation 5 is analogous to an equation that includesredevelopment in ADP (1983).

The aspect that is important to us is the return to maintenance, because we understand ren-ovation and rehabilitation expenditures to be a form of maintenance. We are concerned withdetermining under what conditions a unit will move out of the affordable stock. This movecan occur because of an increase in the price of the unit’s neighborhood quality Y or marketconditions Z larger than the growth in household income for the low-income segment.2 Alter-natively, a landlord can invest enough in maintenance to increase the quality level of theunit, raising rent. Although the optimal expenditure on maintenance will depend on these

( )[ ],,),(,maxT

0t0

,dt .emZY)tmMS(sR –rt

ttttTm

∫=

−=π


2 Neighborhood and market conditions will interact with maintenance returns.

values, the maximum loss of quality is constrained by the exogenous rate of depreciation.3

Units will also drop out of the low-income stock if the landlord decides to abandon them, thatis, reach time T. The condition for abandonment is that the holding costs of the unit exceedthe present discounted value of some future positive rental flow, so that there is no reasonto incur short-term losses (R < m). There is a symmetric outcome for units currently not inthe affordable stock. They can filter down because of exogenous changes in neighborhoodprices and market factors or because unit quality declines. We combine all the factors weexpect to affect the likelihood that a unit will filter up or down into a linear reduced formXβ, where X is composed of the vectors [x, Y, Z], x being the unit-specific characteristics thathelp determine s.

Data Description

We use the AHS metropolitan surveys to create a panel of individual rental units in 44MSAs covering the years 1984 to 1994. Each MSA is surveyed every three or four years inwaves of approximately 11 MSAs per survey, so that we have three years of observations for23 of the MSAs and two years for the remaining 21. Our panel size is constrained by the intro-duction of a new survey questionnaire in 1984 and a new sample in 1995.4

Advocacy groups and many reports decry the loss of affordable housing, yet there is no singleaccepted definition of this stock. One approach is to use HUD’s fair market rents (FMRs) asthe cutoff for affordability.5 A second is to identify those units affordable to renters whosehousehold income is a given percentage of the MSA median household income (Nelson 1994;Nelson and Vandenbroucke 1996). A third alternative is to use one of the consumption bun-dle-based measures, such as Stone’s (1993) shelter-poverty methodology. Ultimately, thisreport is not about identifying the size of the affordable stock but modeling its evolution.Although FMRs are easily obtained (by definition they set the affordable stock at 40 percentof the rental stock occupied by recent movers), using them would limit our ability to study theevolution of the affordable stock because there would be almost no variation in the affordablestock’s share of the total stock both across MSAs and over time within an MSA. Methodolo-gies such as Stone’s add complexity and complication, without adding much to an under-standing of the dynamics of the stock, which is our goal. Consequently, we take the secondapproach.


3 For a given current rent R, a unit is more likely to filter (up and down) when R is closer to the cutoff level that delin-eates affordability.

4 DiPasquale and Somerville (1995) demonstrate how to merge the 1974–83 AHS data with data from 1984–94, butthe earlier period does not report precise rents. Combining the two sets would introduce bias in our results becausewe must set a precise cutoff for affordability.

5 According to HUD (see <http://www.huduser.org/datasets/fmr.html>), the methodology for calculating FMRs fromcensus or AHS surveys is as follows: “The level at which FMRs are set is expressed as a percentile point within therent distribution of standard-quality rental housing units. The current definition used is the 40th percentile rent,the dollar amount below which 40 percent of the standard-quality rental housing units are rented. The 40th per-centile rent is drawn from the distribution of rents of all units occupied by recent movers (renter households whomoved to their present residence within the past 15 months). Public housing units and units less than 2 years oldare excluded.”

We follow the conventional definition that a household is low-income if it is at or below the35 percent of median household income distribution in the surveys of the AHS.6 We make thiscalculation for each MSA–year combination, using the distribution for four-person families.Assuming that 30 percent of household income should go to gross rent, we obtain the afford-able gross rent cutoff.7 We map this cutoff to different unit sizes using the same ratios HUDfollows in defining FMRs by unit size.8 We do not calculate distributions by household size andthen designate the appropriate unit size for each household size; our objective is to trackhousing units, not define the appropriate level of housing consumption.

This study analyzes the evolution of the affordable stock. Like a repeat sales estimator, a unitmust appear in at least two surveys to be included in our sample. As a result we exclude unitsthat for whatever reason appear in only one survey. We define the “evolution of the afford-able stock” as the set of discrete outcomes that a unit identified as affordable in the first sur-vey year can have in the subsequent survey year. The baseline is that it is still classified asaffordable. Alternatively, the unit can become unaffordable because its rent now exceeds theaffordability cutoff, it becomes owner-occupied, or it is demolished or converted.9 For rentalunits identified as unaffordable in the first survey year, we have a similar set of possible out-comes with the status quo remaining unaffordable as the baseline.

In defining our data set we have to exclude certain affordable units. We are interested in themarket affordable stock, so we exclude public housing from the data set. Units in which theoccupying households receive a government subsidy are problematic. We would prefer to in-clude them; however, the unreliability of the rents reported for these units means that includ-ing them could bias our results. Unpublished work by McArdle (n.d.) at the Joint Center forHousing Studies indicates that for many households that receive a government voucher orother subsidy, the actual gross rent cannot be distinguished from the gross rent paid (net ofthe subsidy). We choose to exclude units in which the occupant receives a subsidy in the firstsurvey year. However, a unit whose occupying household did not receive a subsidy in the firstsurvey year, but did in the second, is considered to be affordable in the second survey. Thatapproach does not cause any loss; treating subsidized units as a separate category into whichunits can move does not qualitatively change our results.

Table 1 shows the frequency of each outcome for movements out of both the affordable andunaffordable stocks. The period is between any two AHS metropolitan surveys. The three-


6 We recognize that the AHS is not the preferred source for identifying income distributions, but as long as any biasin this is uncorrelated with housing characteristics our analysis will not be affected.

7 Throughout we use rent to refer to gross rents. One complication is that in 1989, the survey question about utili-ty costs was changed, resulting in a shift in responses. To correct for this, we follow Nelson and Vandenbroucke(1996) and adjust reported utility costs for 1989 and later years.

8 These ratios are a percentage of the four-person family 30 percent cutoff as follows: 0 bedrooms, 70 percent; 1 bed-room, 75 percent; 2 bedrooms, 90 percent; 3 bedrooms, 104 percent; 4 bedrooms, 116 percent; and increasing by 12percentage points for each additional bedroom up to 14 bedrooms. Any unit with more than 14 bedrooms is auto-matically classified as unaffordable.

9 Demolished or converted units include units that were converted to business use, eliminated in a conversion,abandoned, destroyed by disaster, demolished, or condemned. The category also includes units with an interior nowexposed to the elements and mobile home sites that no longer have a home on them.

survey cycle is only for units that appear in the AHS for all three surveys.10 Consistent withthe results of Nelson and Vandenbroucke (1996), there is considerable churning in and outof the affordable stock because of changes in rent, though tenure conversions and demolitionare small by comparison. Not surprisingly, units in the unaffordable stock are less likely tobecome government-subsidized or to be demolished, but are more likely to convert to owner-occupancy than are units initially classified as affordable. The unaffordable stock is muchlarger; therefore, a larger absolute number of these units is demolished or has occupants whoreceive government subsidies. Hidden in those numbers is an absolute increase in the num-ber of affordable units: a net gain of approximately 1,700 units. If a subsidy makes those for-merly unaffordable units affordable, that increase rises by another 184 percent. This result,however, may be misleading because AHS will tend to exclude units with a change in occu-pant, which are the units most likely to experience rent increases.11 The lower part of table 1


Table 1. Aggregate Dynamics of Rental Stock

Number Percentage

Units beginning as affordableRemain affordable 4,171 45.3Become unaffordable because of rent increase 2,928 31.8Become subsidized 760 8.3Become owner-occupied 506 5.5Are demolished or converted 837 9.1

Total 9,202

Units beginning as unaffordableRemain unaffordable 54,298 78.1Become affordable because of rent decrease 6,007 8.6Become subsidized 3,185 4.6Become owner-occupied 4,703 6.8Are demolished or converted 1,369 2.0

Total 69,562

Units beginning as affordable—three-survey cycleRemain affordable 830 35.9Become unaffordable because of rent increase 660 28.6Become subsidized 300 13.0Become owner-occupied 248 10.7Are demolished or converted 273 11.8

Total 2,311

Note: For the top two data sets, only units that had observations for two consecutive years are included.For the bottom data set, only units that had observations for three consecutive years are included. Ifa unit dropped out of the sample, it is excluded. We also exclude units that are initially governmentsubsidized or classified as public housing. A unit is defined to be affordable if the sum of rent and util-ities is less than 30 percent of household income for a household at 35 percent of the median incomefor four-person families for that year in that city. To reflect different unit sizes, an adjustment is madebased on the number of bedrooms.

10 If an MSA had three survey years, we were able to examine the movement separately between the first and sec-ond and second and third surveys. We treat these as two separate observations.

11 We expect that new occupants are less likely to respond to the AHS survey than are occupants who have respond-ed in the past. Rents for a unit tend to increase more with unit turnover. Thus, we are likely to undercount unitswhose rents rise, which means an undercount of units that move out of the affordable stock because the new rentexceeds the affordability cutoff.

shows that when the period of analysis is extended to cover three surveys, the percentage ofunits that remains affordable is similar, though obviously lower.

The affordable stock varies by MSA. Table 2 shows the percentage of rental units that are af-fordable by MSA by survey year. The share varies from a low of 1.8 percent in Anaheim to ahigh of 31.4 percent in Cleveland. These percentage changes can vary substantially from sur-vey to survey with differences in growth rates of MSA incomes and rent. For instance, in Mil-waukee the affordable share rises from 9.5 to 20.1 percent, whereas in Dallas it rises from6.7 to 16.5 percent, before falling to 8.4 percent.

There are four classes of variables that enter into the optimal maintenance and renovationdecision: unit characteristics, neighborhood characteristics, market variables, and control vari-ables. All observations in our sample appear in two successive AHS surveys, but to ensureexogeneity we take the values from the year of the first survey. Exceptions are the variablesthat measure the percentage change in rents and prices. The first category of variables, unitcharacteristics, includes unit adequacy (a dummy variable coded 1 if the unit is defined asadequate by AHS standards);12 unit age, a dummy for multi-unit buildings; and the numberof units in the structure (coded 1 for single-family houses).

To provide more detailed geography, the AHS divides each MSA into zones, which are socioe-conomically homogeneous areas of approximately 100,000 people. To identify the second cat-egory of variables, we create a set of neighborhood characteristics defining neighborhood asthe zone. These include various ratios describing the housing stock in the zone: the ratio ofrental units to all units, of affordable units to all rental units, of public housing units to allrental units, and of subsidized units to all rental units in the zone. Other neighborhood vari-ables include the average age of the rental stock, percentage of households headed by anAfrican American, and median household income in the zone.

The third category of independent variables is market characteristics. We use three separatespecifications to measure market conditions. The first specification uses MSA fixed effects.The second specification uses an adjusted Herfindahl index to measure the concentration ofaffordable units in an MSA.13 The final specification uses the percentage changes in MSAhedonic prices and rents. For the last two we cannot include MSA fixed effects because welack enough within-MSA variation or year-observations per MSA for them to be sufficientlyuncorrelated with the MSA dummies. We use DiPasquale and Somerville’s (1995) method-ology for generating hedonic price and rent series from the AHS. The price and rent series dif-fer in that the first uses mean MSA values for characteristics, whereas the second uses meanvalues of affordable stock. In constructing both series, we included zone dummies in thehedonic equations to capture differences across zones in mean price and rent changes.


12 Adequacy is an AHS-coded summary variable based on responses to questions about physical problems in theunit. The lack of hot piped water or a flush toilet would classify a unit as severely inadequate; multiple leaks andholes in the floor and walls would result in the unit being classified as moderately inadequate.

13 An MSA’s Herfindahl score is calculated by treating the ratio of a zone’s affordable stock to its total rental stockas its market share, with the values normalized to sum to one. Larger MSAs will have more zones, ensuring thatthey will have lower scores; therefore, we multiply the Herfindahl score by the number of zones in the MSA to cor-rect for that bias.


Table 2. Affordable Stock

Percentage ofRental Units Classified as Affordable

MSA Survey Year 1 Survey Year 2 Survey Year 3

Anaheim 4.3 1.8 2.3Atlanta 12.6 12.4 NABaltimore 19.4 20.6 NABirmingham 24.6 26.7 24.3Boston 9.8 7.3 8.8Buffalo 17.4 29.2 26.0Chicago 17.3 11.9 NACincinnati 23.5 20.8 NACleveland 21.3 30.7 31.4Columbus 14.4 17.6 NADallas 6.7 16.5 8.4Denver 5.1 21.1 NADetroit 24.9 28.7 29.0Fort Worth 8.3 11.4 10.4Hartford 13.2 10.6 NAHouston 23.8 20.2 NAIndianapolis 16.2 16.6 15.7Kansas City 26.2 19.6 NALos Angeles 5.5 3.7 NAMemphis 17.5 25.7 21.5Miami 3.5 4.5 NAMilwaukee 9.5 11.0 20.1Minneapolis 8.9 11.2 12.3New Orleans 8.5 9.1 NANew York 14.0 12.6 NANewark 12.6 11.3 NANorfolk 7.3 4.5 5.3Oklahoma City 8.7 20.6 17.9Philadelphia 10.7 19.1 NAPhoenix 7.0 5.9 7.7Pittsburgh 13.7 15.8 NAPortland 9.9 7.0 NAProvidence 20.7 12.7 9.8Riverside 8.3 3.9 3.6Rochester 10.4 12.3 NASt. Louis 24.0 26.3 NASalt Lake City 5.7 16.9 12.6San Antonio 6.1 6.1 NASan Diego 2.5 2.2 2.9San Francisco 12.4 9.3 7.9San Jose 5.9 4.6 4.0Seattle 9.9 8.3 NATampa 6.5 6.9 4.9Washington, DC 15.0 14.0 10.7

Note: NA means there were only two years of survey data for that MSA. A unit is defined to beaffordable if the sum of rent and utilities is less than 30 percent of household income for a house-hold at 35 percent of the median income for four-person families for that year in that city. To reflectdifferent unit sizes, an adjustment is made based on the number of bedrooms. Subsidized units andpublic housing units are not included in the count of affordable units.

Table 3 shows the Herfindahl values where a zone is analogous to a firm, and share of anMSA’s affordable units is analogous to market share. We pool observations across surveys forthe concentration calculation. If the unadjusted Herfindahl score is 1.0 (scaled from the stan-dard 10,000), all the affordable units are concentrated in one zone. Before adjustment, theindex almost directly reflects an inverse ordering based on size (number of zones). Multiply-ing by the number of zones adjusts for this market-size effect. Phoenix has the highest degreeof concentration of affordable units; Denver has the most dispersed pattern of affordability.The ranking of scores in table 3, however, does not exhibit any obvious geographic or demo-graphic pattern.

In table 4 we present descriptive statistics for the variables for affordable and unaffordableunits separately. Comparing these two sets, we find that difference of means t-tests rejectequality of means for nearly all variables. Qualitatively, affordable units are in poorer con-dition and are located in older and smaller buildings. Tenants have a notably longer meanstay in the affordable units, 6.3 versus 2.9 years. Affordable units are both more concen-trated in space than are rental units in total and much more likely to be in areas with ahigher proportion of African Americans. Although other differences are statistically signifi-cant, they are not meaningful. Nominal house price growth for the affordable bundle wasslightly lower, but the growth in rents was surprisingly higher. The difference in rentchanges (which are calculated at the zone rather than the unit level) are a result of the factthat affordable and unaffordable units do not have the same distribution across space, andprice and rent changes vary by area.

Empirical Results

We estimate the model using a multinomial logit specification in which any observation i = 1to n can fall into one of k groups. For a unit currently in the low-income stock, these groupsare as follows: (1) remaining in the low-income stock, (2) filtering up (defined as having arent that surpasses the affordability threshold), (3) converting to owner-occupied, or (4)being demolished. For each observation we have the following probability for all k groups:

. (6)

Equation 6 is unidentified unless we set eXβ1 = 1.The standard procedure is to present the oddsratio, the ratio of the probability that i ∈ k (k ≠ 1) relative to the probability that i ∈ 1. Forinstance,

. (7)

The multinomial regression results are presented as appendix tables.

Pr( )Pr( )

ii

e

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X

X

j

kX

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1

1

2

2

2 2

β

β β

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1


Multinomial logit regression output can be difficult to interpret because it measures effectsrelative to a given outcome. Therefore, we present a set of tables that show the sensitivity ofrelative probabilities to given changes in the values of right-hand-side variables. One way to


Table 3. Concentration of Affordable Housing

Herfindahl Number AdjustedMSA Score of Zones Score Sorted by Adjusted Score

Anaheim 0.0833 12 1.000157 Phoenix 1.1889 Atlanta 0.0614 17 1.043453 Providence 1.1468 Baltimore 0.0609 17 1.034901 Detroit 1.1267 Birmingham 0.1764 6 1.058648 Houston 1.1129 Boston 0.0324 31 1.003236 Riverside 1.1115 Buffalo 0.1026 10 1.026487 Dallas 1.0761 Chicago 0.0183 56 1.025338 Cleveland 1.0717 Cincinnati 0.0849 12 1.018861 Birmingham 1.0586 Cleveland 0.0670 16 1.071745 Memphis 1.0545 Columbus 0.1135 9 1.021467 San Francisco 1.0496 Dallas 0.0673 16 1.076092 Atlanta 1.0435 Denver 0.1012 9 0.910906 St. Louis 1.0411 Detroit 0.0417 27 1.126739 Baltimore 1.0349 Fort Worth 0.1691 6 1.014802 Kansas City 1.0311 Hartford 0.1433 7 1.003189 Philadelphia 1.0306 Houston 0.0696 16 1.112890 Buffalo 1.0265 Indianapolis 0.1266 8 1.012531 Chicago 1.0253 Kansas City 0.1146 9 1.031134 Washington, DC 1.0236 Los Angeles 0.0228 44 1.002142 Oklahoma City 1.0233 Memphis 0.1757 6 1.054472 Columbus 1.0215 Miami 0.0716 14 1.001830 New York 1.0198 Milwaukee 0.1016 10 1.015826 Cincinnati 1.0189 Minneapolis 0.0723 14 1.011996 Milwaukee 1.0158 New Orleans 0.1116 9 1.004355 Fort Worth 1.0148 New York 0.0123 83 1.019809 Pittsburgh 1.0130 Newark 0.0252 40 1.009084 Indianapolis 1.0125 Norfolk 0.1674 6 1.004155 Minneapolis 1.0120 Oklahoma City 0.1706 6 1.023330 Rochester 1.0106 Philadelphia 0.0294 35 1.030618 San Antonio 1.0097 Phoenix 0.0915 13 1.188906 Newark 1.0091 Pittsburgh 0.0724 14 1.012961 Seattle 1.0081 Portland 0.1430 7 1.001151 Salt Lake City 1.0061 Providence 0.1433 8 1.146762 New Orleans 1.0044 Riverside 0.1111 10 1.111459 Norfolk 1.0042 Rochester 0.1684 6 1.010551 Boston 1.0032 St. Louis 0.0612 17 1.041136 Hartford 1.0032 Salt Lake City 0.2012 5 1.006108 Tampa 1.0027 San Antonio 0.1122 9 1.009688 Los Angeles 1.0021 San Diego 0.1000 10 1.000271 Miami 1.0018 San Francisco 0.0456 23 1.049601 San Jose 1.0012 San Jose 0.1001 10 1.001156 Portland 1.0012 Seattle 0.0630 16 1.008073 San Diego 1.0003 Tampa 0.0836 12 1.002666 Anaheim 1.0002 Washington, DC 0.0445 23 1.023581 Denver 0.9109

Note: The Herfindahl score is calculated in the usual way, but using the percentage affordable in a zone as the mar-ket share. The interpretation is that if the unadjusted Herfindahl score = 1, all the affordable units are concentrat-ed in one zone. The adjusted score is the Herfindahl value multiplied by the number of zones in the MSA to controlfor differences in MSA size. Higher adjusted scores mean that affordable units are more concentrated.

126C

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esTable 4. Descriptive Statistics of Variables Influencing Maintenance and Renovation Decisions

t-testAffordable Units Unaffordable Units on Mean

Variables Mean SD Min. Max. Mean SD Min. Max. Diff.

UnitAdequacy of unit (1 if adequate, 0 otherwise) 0.72 0.45 0.00 1.00 0.90 0.30 0.00 1.00 37.44 Age of unit 46.56 19.58 0.00 76.00 27.91 20.64 0.00 76.00 85.33 Unit is part of multi-unit building (1 if yes, 0 if no) 0.70 0.46 0.00 1.00 0.76 0.43 0.00 1.00 12.43 Number of units in building 8.35 19.00 1.00 303.00 13.63 29.19 1.00 317.00 23.25

Neighborhood (at zone level)Ratio of subsidized units to all rental units 0.11 0.06 0.00 0.49 0.10 0.06 0.00 0.80 19.52 Average age of rental units 37.15 13.67 7.77 64.89 28.28 12.92 4.00 64.89 58.85 Ratio of public housing units to rental units 0.06 0.07 0.00 0.51 0.04 0.05 0.00 0.51 31.33 Ratio of rental units to all units 0.46 0.17 0.09 1.00 0.43 0.15 0.04 1.00 19.09 Ratio of affordable units to rental units 0.27 0.19 0.00 0.84 0.11 0.11 0.00 0.84 81.07 Percentage African-American head of household 0.27 0.30 0.00 0.98 0.13 0.18 0.00 0.98 44.30 Median household income 21,487 8,665 6,060 70,500 27,650 8,998 6,060 70,500 63.83

MarketAdjusted Herfindahl score 1.04 0.04 0.91 1.19 1.03 0.05 0.91 1.19 19.01 Hedonic price change in MSA (all units) 0.11 0.29 –0.35 1.44 0.10 0.27 –0.35 1.44 0.63 Hedonic rent change in MSA (all units) 0.21 0.14 –0.16 0.53 0.17 0.16 –0.16 0.53 23.62 Hedonic price change in MSA (affordable units) 0.07 0.38 –0.48 1.91 0.08 0.34 –0.48 1.91 1.93 Hedonic rent change in MSA (affordable units) 0.23 0.11 –0.11 0.49 0.21 0.12 –0.11 0.49 19.38

ControlNumber of years current resident has occupied unit 6.33 8.60 0.00 82.00 2.92 4.96 0.00 80.00 34.39 Ratio of rent to cutoff of affordability 0.76 0.20 0.00 0.99 1.62 0.46 0.00 5.37 319.24

Note: Only the units that were included in two consecutive surveys are included. Units that drop out of the sample are excluded. All price and rent changes aremeasured in nominal dollars. The mean values in the affordable units column and the unaffordable units column for the hedonic price and rent changes differbecause these two categories of units are not distributed identically across MSAs. SD = Standard Deviation. Min. = Minimum. Max. = Maximum. Mean Diff. =Mean Difference.

think of these measures is as pseudoelasticities, in that they give a sense of the sensitivity ofrelative rather than absolute changes in the probability of an outcome occurring. Tables 5through 7 present the results for the probability that a unit will move out of the stock of af-fordable units relative to the probability of remaining in the affordable stock. Moving out ofthe stock of affordable units can occur in three ways: unit rent increases (table 5), unit becomesowner-occupied (table 6), or unit is demolished or converted (table 7). For the given changein the variable, the tables report the percentage point change in the odds ratio, which is therelative marginal probability between moving into the category and remaining affordable.For both continuous and dummy variables, we evaluate these marginal changes in proba-bility for 10 percent and for one standard deviation increases above the mean values in theright-hand-side variables.

Table 5 reports the sensitivity of the relative probability of filtering up to our set of unit, zone,and market characteristics. Looking at the effects of a 10 percent increase in values of right-hand-side variables, it is not surprising that the relative probability is most sensitive to in-creases in the control variable—the ratio of the initial period rent to the affordable cutoff—because that variable indicated how close a unit is to the affordability “border.” Signs and ef-fects are robust across specifications. Older, less adequate units and those in larger buildingsare less likely to filter up, relative to staying affordable. In general, the relative probabilityof filtering up is more sensitive to changes in zone values. Units are more likely to filter upif they are located in an area with many rental units but relatively fewer affordable units. Thereason could be that the current rent level in the units incorrectly prices the zone, or that thereis a much higher return for improving unit quality in these zones than in poorer-quality zones.Controlling for these ratios and unit characteristics, the presence of public and subsidizedunits and the mean age of a zone’s rental stock do not affect the probability of filtering up,relative to that of staying affordable. In contrast, units in areas with higher concentrationsof African-American-headed households are less likely to filter up.

In specifications 2 and 3, we replace the MSA fixed effects with variables aggregated at theMSA level. The degree of concentration of affordable units in space does not matter. In addi-tion, movements in the general level of rents do not affect the probability of filtering up, rel-ative to remaining affordable. Movements in both rents and prices for the affordable bundledo increase that probability. However, the relative probability that a unit will filter up is high-er in MSAs in which house prices are rising faster, even controlling for changes in rents. Weinterpret this difference between the effects by hedonic bundle as supporting Susin’s (1999)result that the affordable and unaffordable submarkets are not well integrated.

Unit characteristics are a more important factor for determining whether a unit is more like-ly to leave the affordable rental stock because it becomes owner-occupied relative to stayingaffordable and a rental unit. Results in table 6 indicate that the newer, higher-quality unitsare most likely to convert. The difficulty in converting multifamily rental buildings to condo-minium status and selling those units when other units are occupied by renters means thatunits in larger multifamily buildings are much less likely to make that transition. Relativeto table 5, zone characteristics are far less important, though relative to a unit staying afford-able, this transition is less likely to occur in zones with a greater presence of African Amer-icans, which is consistent with the lower rates of homeownership among African Americans.Comparing specifications 1 with 2 and 3, we find that units are more likely to convert inMSAs with an older rental stock and in MSAs where, controlling for price changes, rents arerising.


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esTable 5. Change in the Relative Probability that Affordable Rental Units

Are Lost through Higher Rents

Specification 1 Specification 2 Specification 3

10% 1 SD 10% 1 SD 10% 1 SDIncrease Increase Increase Increase Increase Increase

Variables (%) (%) (%) (%) (%) (%)

UnitAdequacy of unit 2.71 18.20 2.67 17.89 2.28 15.14Age of unit –4.30 –16.87 –4.75 –18.52 –5.03 –19.51Unit is part of multi-unit building 1.25 8.52 1.37 9.39 1.24 8.43Number of units in building –0.43 –9.42 –0.70 –14.70 –0.68 –14.33

Neighborhood (at zone level)Ratio of subsidized units to all rental units NS NS NS NS NS NSAverage age of rental units NS NS NS NS NS NSRatio of public housing units to rental units NS NS NS NS NS NSRatio of rental units to all units 4.49 17.55 6.74 27.13 6.89 27.77Ratio of affordable units to rental units –3.72 –23.06 –4.74 –28.51 –4.62 –27.89Percentage African-American head of household –0.89 –9.39 –1.20 –12.53 –0.96 –10.11Median household income NS NS 0.00 0.00 0.00 0.00

MarketMSA fixed effects present? .yes yes no no no noAdjusted Herfindahl score — — NS NS NS NSHedonic price change in MSA (all units) — — 0.40 11.39 — —Hedonic rent change in MSA (all units) — — NS NS — —Hedonic price change in MSA (affordable units) — — — — 0.13 6.93Hedonic rent change in MSA (affordable units) — — — — 4.89 24.60

ControlNumber of years current resident has occupied unit –0.97 –12.35 –0.88 –11.34 –0.90 –11.57Ratio of rent to cutoff of affordability 5.27 14.13 5.28 14.14 5.29 14.16

Note: We report percentage point change in the odds ratios that are due to a 10 percent increase from the mean, and due to an increase equal to one standarddeviation from the mean. The odds ratios are relative to the outcome with the unit remaining affordable or becoming subsidized. The MSA dummies are used inspecification 1 but are not reported. SD stands for Standard Deviation. NS indicates the variable was not significant at the 5 percent level. A dash indicates thevariable was not used in this specification.

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Table 6. Change in the Relative Probability that Affordable Rental UnitsAre Lost by Becoming Owner-Occupied



Variables (%) (%) (%) (%) (%) (%)

UnitAdequacy of unit 2.28 15.15 1.99 13.11 NS NSAge of unit –6.91 –26.00 –6.42 –24.34 –6.38 –24.21Unit is part of multi-unit building –11.56 –55.48 –10.77 –52.77 –10.82 –52.94Number of units in building –0.85 –17.70 NS NS NS NS

Neighborhood (at zone level)Ratio of subsidized units to all rental units NS NS NS NS NS NSAverage age of rental units NS NS 7.33 29.71 5.98 23.83Ratio of public housing units to rental units NS NS NS NS NS NSRatio of rental units to all units NS NS NS NS NS NSRatio of affordable units to rental units NS NS NS NS NS NSPercentage African-American head of household –2.60 –25.34 –2.17 –21.58 –2.17 –21.56Median household income NS NS 0.00 0.00 0.00 0.00

MarketMSA fixed effects present? yes yes no no no noAdjusted Herfindahl score — — NS NS NS NSHedonic price change in MSA (all units) — — NS NS — —Hedonic rent change in MSA (all units) — — 2.07 14.82 — —Hedonic price change in MSA (affordable units) — — — — NS NSHedonic rent change in MSA (affordable units) — — — — 2.64 12.77

ControlNumber of years current resident has occupied unit NS NS NS NS NS NSRatio of rent to cutoff of affordability NS NS NS NS NS NS


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esTable 7. Change in the Relative Probability that Affordable Rental Units

Are Lost as a Result of Demolition or Conversion



Variables (%) (%) (%) (%) (%) (%)

UnitAdequacy of unit –5.52 –29.91 –5.37 –29.17 –5.26 –28.66Age of unit 8.08 38.65 8.14 38.97 8.35 40.08Unit is part of multi-unit building –2.47 –15.18 –2.45 –15.05 –2.37 –14.63Number of units in building NS NS NS NS NS NS

Neighborhood (at zone level)Ratio of subsidized units to all rental units NS NS NS NS NS NSAverage age of rental units –6.54 –22.04 –11.03 –34.96 –10.38 –33.17Ratio of public housing units to rental units NS NS NS NS NS NSRatio of rental units to all units 5.01 19.70 NS NS NS NSRatio of affordable units to rental units NS NS NS NS NS NSPercentage African-American head of household NS NS NS NS NS NSMedian income NS NS NS NS NS NS

MarketMSA fixed effects present? yes yes no no no noAdjusted Herfindahl score — — NS NS NS NSHedonic price change in MSA (all units) — — NS NS — —Hedonic rent change in MSA (all units) — — NS NS — —Hedonic price change in MSA (affordable units) — — — — NS NSHedonic rent change in MSA (affordable units) — — — — NS NS

ControlNumber of years current resident has occupied unit –0.96 –12.33 –1.07 –13.56 –1.10 –13.92Ratio of rent to cutoff of affordability –6.29 –15.40 –6.30 –15.41 –6.27 –15.33


The sensitivities of the probability a unit will be demolished or converted to nonresidentialuse relative to remaining in the affordable rental stock are presented in table 7. This prob-ability is also primarily a function of unit characteristics. The relative likelihood is lower fornewer, higher-quality units. The marginal absolute value effects of these variables are greaterfor the demolition probability than for the conversion to owner-occupied status. Controllingfor unit age, the demolition/conversion probability is higher in older zones. In comparison withtables 5 and 6, zone effects are less important.

In tables 8 through 10 we present the percentile changes in the relative probability that a unitwill move out of the stock of rental units unaffordable to low-income households. In all casesthe change is relative to the probability of remaining an unaffordable rental unit.

Table 8 presents the determinants of the relative probability that a unit will filter down be-cause the unit’s rent falls below the affordability cutoff relative to remaining an unaffordablerental unit. Relative to filtering up as shown in table 5, most zone and unit characteristicshave smaller but similar effects on the relative probability that a unit will filter down. Still,there are some interesting differences. First, zone income matters for filtering down. The rel-ative probability of doing so falls with increases in median zone income, controlling for othercharacteristics and MSA fixed effects. Second, an individual unit is more likely to filter down,although contingent on being in a multifamily building, the relative probability rises withbuilding size. Third, downward filtering is more likely in MSAs in which the affordable stockis less concentrated, controlling for the share of affordable public, subsidized, and rental unitsin the zone in question. As might be expected, downward filtering is less likely when rents arerising, for both individual units and affordable bundles. Downward filtering is more sensitiveto mean MSA rent changes than is upward filtering.

In table 9 we present the factors that affect whether an unaffordable unit will become owneroccupied. These factors do differ in several clear ways from the results for the affordable stockpresented in table 6. Most important, zone characteristics are more significant in this case.If a unit is unaffordable to low-income renters, its relative probability of becoming an owner-occupied unit rises as share of affordable rental units in the zone rises. The true effect mayactually be negative, but with a multinomial logit specification all we know is the effect rela-tive to remaining an unaffordable rental unit.

The relative probability that a rental unit unaffordable to low-income households will be de-molished or converted to nonresidential use is described in table 10. Older, less adequate unitsare more likely to be demolished than to remain unaffordable. However, as with the compar-ison between tables 6 and 9, in table 10 the relative probability of demolition or conversionto nonresidential use for unaffordable rental units is more sensitive to zone characteristicsthan is the case for the affordable rental units in table 7. The relative probability of demoli-tion rises with the share of units in the zone that are affordable and the percentage of house-holds in the zone that have an African-American head of household, and it falls with medianzone income. As in table 7, as the age of the unit relative to the mean age of the zone’s rentalstock falls (i.e., controlling for age, the mean age of the zone’s rental stock rises), demolition isless likely to occur.


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esTable 8. Change in the Relative Probability that Unaffordable Rental Units

Join Affordable Stock through Lower Rents



Variables (%) (%) (%) (%) (%) (%)

UnitAdequacy of unit –2.10 –6.82 –1.77 –5.78 NS NSAge of unit 1.65 12.84 1.88 14.75 1.90 14.94Unit is part of multi-unit building –3.27 –17.03 –2.91 –15.32 –2.91 –15.27Number of units in building 0.24 5.33 0.25 5.44 0.26 5.67

Neighborhood (at zone level)Ratio of subsidized units to all units 1.28 7.50 2.08 12.38 2.32 13.88Average age of rental units –1.39 –6.19 NS NS 1.00 4.67Ratio of public housing units to rental units NS NS NS NS 0.25 3.14Ratio of rental units to all units NS NS –1.65 –5.77 –1.22 –4.27Ratio of affordable units to rental units 0.84 9.22 0.81 8.88 0.89 9.77Percentage African-American head of household 0.72 10.97 0.47 7.00 0.45 6.74Median household income –24.16 –59.34 –24.16 –59.34 –24.16 –59.34

MarketMSA fixed effects present? yes yes no no no noAdjusted Herfindahl score — — NS NS –0.59 –14.51Hedonic price change in MSA (all units) — — NS NS — —Hedonic rent change in the MSA (all units) — — –0.42 –16.53 — —Hedonic price change in MSA (affordable units) — — — — 0.35 2.01Hedonic rent change in MSA (affordable units) — — — — –39.01 –99.98

ControlNumber of years current resident has occupied unit 0.17 0.48 0.16 0.45 0.17 0.50Ratio of rent to cutoff of affordability 0.00 0.00 0.00 0.00 0.00 0.00


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Table 9. Change in the Relative Probability that Unaffordable Rental UnitsBecome Owner Occupied



Variables (%) (%) (%) (%) (%) (%)

UnitAdequacy of unit NS NS NS NS NS NSAge of unit NS NS NS NS NS NSUnit is part of multi-unit building –14.93 –59.70 –14.55 –58.66 –14.55 –58.69Number of units in building NS NS NS NS NS NS

Neighborhood (at zone level)Ratio of subsidized units to all units NS NS NS NS NS NSAverage age of rental units NS NS 1.50 7.02 NS NSRatio of public housing units to rental units NS NS 0.38 4.71 0.37 4.56Ratio of rental units to all units –2.08 –7.23 –2.12 –7.36 –1.97 –6.86Ratio of affordable units to rental units 1.88 21.70 1.47 16.63 1.48 16.67Percentage African-American head of household –0.77 –10.68 –0.81 –11.16 –0.80 –10.99Median household income –24.16 –59.34 NS NS NS NS

MarketMSA fixed effects present? yes yes no no no noAdjusted Herfindahl score — — 0.83 24.17 0.75 21.61Hedonic price change in MSA (all units) — — NS NS — —Hedonic rent change in MSA (all units) — — NS NS — —Hedonic price change in MSA (affordable units) — — — — NS NSHedonic rent change in MSA (affordable units) — — — — 13.62 771.83

ControlNumber of years current resident has occupied unit 0.25 0.70 0.28 0.78 0.28 0.79Ratio of rent to cutoff of affordability 0.00 0.00 0.00 0.00 0.00 0.00


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esTable 10. Change in the Relative Probability that Unaffordable Rental Units

Are Lost as a Result of Demolition or Conversion



Variables (%) (%) (%) (%) (%) (%)

UnitAdequacy of unit –6.79 –20.87 –7.43 –22.64 –7.36 –22.45Age of unit 9.76 99.06 9.59 96.87 9.63 97.35Unit is part of multi-unit building –5.54 –27.39 –5.47 –27.10 –5.46 –27.06Number of units in building 0.45 10.09 NS NS NS NS

Neighborhood (at zone level)Ratio of subsidized units to all units NS NS NS NS NS NSAverage age of rental units –4.58 –19.29 –8.38 –32.97 –7.66 –30.50Ratio of public housing units to rental units NS NS NS NS NS NSRatio of rental units to all units NS NS NS NS NS NSRatio of affordable units to rental units 1.07 11.83 1.00 11.07 1.03 11.43Percentage African-American head of household 0.90 13.83 0.94 14.53 0.94 14.64Median household income –24.16 –59.34 –24.16 –59.34 –24.16 –59.34

MarketMSA fixed effects present? yes yes no no no noAdjusted Herfindahl score — — NS NS NS NSHedonic price change in MSA (all units) — — NS NS — —Hedonic rent change in MSA (all units) — — –0.50 –19.02 — —Hedonic price change in MSA (affordable units) — — — — NS NSHedonic rent change in MSA (affordable units) — — — — –26.17 –99.42

ControlNumber of years current resident has occupied unit –0.54 –1.52 –0.56 –1.58 –0.54 –1.52Ratio of rent to cutoff of affordability 0.00 0.00 0.00 0.00 0.00 0.00


Conclusion

By studying the affordable stock, we seek to add to the understanding of housing marketconditions facing low-income households. In contrast to the existing literature, our objectiveis not to measure the aggregate stock or to determine how well it matches the needs of low-income households. Instead, we seek to identify the relative importance of unit, neighborhood,and market characteristics in affecting the dynamics of the low-income stock, of both unitsmoving out of the stock and units filtering down into the stock. We find that, although unitcharacteristics matter, the probabilities of units filtering up or down relative to remainingeither affordable or unaffordable are more sensitive to neighborhood conditions. These condi-tions include the share of rental units in the neighborhood, affordable share of the total rentalstock, and neighborhood incomes. Surprisingly, the share of units that are public housing orsubsidized provides little information about the relative probability of filtering.Although over-all movements in market house prices and rents are less important, the movements in rentsfor units with the structure and location characteristics of the affordable stock are quiteimportant. This suggests that the affordable housing submarket is not strongly integratedwith the larger housing market.

Our results have important implications for policy makers. Most critical is that policies de-signed to preserve the affordable stock and those that encourage neighborhood diversity maywork at cross-purposes. It is the affordable units in better neighborhoods that are most at riskof filtering up. Although it is possible for filtering up and down to affect each other in thesemixed neighborhoods, our evidence shows that the probability that a unit will filter up and outof the affordable stock is more sensitive to neighborhood quality and the mix of units than isthe probability that it will filter down.

The results we present are provocative and suggest a number of avenues for future research.Among them are testing for robustness across alternative definitions of affordability and de-veloping more varied measures of market conditions, at both MSA and neighborhood levels.In addition, it should be possible to tie this work more explicitly to the filtering literature bycontrolling for the variation over time in unit quality and quantity.


Appen

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esTable A.1. Multinominal Logit Estimation Results: Affordable Rental Units

Specification 1 Specification 2 Specification 3Pseudo R2 = 9.1% Pseudo R2 = 7.4% Pseudo R2 = 7.7%

Rent Owner- Demolished/ Rent Owner- Demolished/ Rent Owner- Demolished/Variables Rises Occupied Converted Rises Occupied Converted Rises Occupied Converted

Unit Adequacy of unit 1.4501 1.3681 0.4540 1.4416 1.3149 0.4646 1.3680 1.3009 0.4721

(1 if adequate, 0 otherwise) (5.268) (2.232) (8.183) (5.301) (2.045) (8.226) (4.503) (1.947) (7.990)Age of unit 0.9906 0.9847 1.0168 0.9896 0.9859 1.0170 0.9890 0.9859 1.0174

(5.305) (4.409) (5.444) (5.957) (4.243) (5.571) (6.295) (4.217) (5.702)Unit is part of multi-unit building 1.1951 0.1715 0.6986 1.2160 0.1950 0.7008 1.1929 0.1935 0.7085

(1 if yes, 0 if no) (2.662) (13.361) (3.399) (3.005) (12.880) (3.473) (2.698) (12.950) (3.364)Number of units in building 0.9948 0.9898 0.9965 0.9917 0.9904 0.9972 0.9919 0.9903 0.9968

(3.349) (2.003) (1.076) (4.962) (1.813) (0.875) (4.845) (1.840) (0.985)

Neighborhood (at zone level)Ratio of subsidized units to all 1.0971 0.7308 4.0858 1.8478 1.7787 1.9490 1.4478 1.1141 1.6813

rental units (0.141) (0.255) (1.207) (1.148) (0.535) (0.731) (0.686) (0.102) (0.582)Average age of rental units 0.9991 0.9992 0.9820 1.0047 1.0192 0.9690 0.9974 1.0158 0.9709

(0.189) (0.081) (2.094) (1.375) (2.796) (5.225) (0.712) (2.215) (4.735)Ratio of public housing units to 0.3321 0.9154 4.6587 0.5798 0.5024 2.9630 0.4105 0.4225 2.9839

rental units (1.848) (0.074) (1.774) (1.067) (0.641) (1.539) (1.722) (0.802) (1.547)Ratio of rental units to all units 2.5965 1.6180 2.8903 4.1243 1.9619 1.8112 4.2477 1.4596 1.6446

(3.421) (0.853) (2.169) (6.427) (1.458) (1.603) (6.909) (0.860) (1.405)Ratio of affordable units to 0.2426 0.7867 0.8632 0.1632 0.6205 0.9067 0.1710 0.6054 0.8732

rental units (4.978) (0.450) (0.319) (7.681) (1.060) (0.259) (7.396) (1.114) (0.331)Percentage African-American 0.7167 0.3726 1.1224 0.6362 0.4398 1.0175 0.6975 0.4402 1.0054

head of household (1.994) (2.970) (0.420) (3.167) (2.791) (0.075) (2.507) (2.800) (0.233)Median household income 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 1.0000 1.0000 0.9999

(1.687) (0.864) (1.577) (4.369) (2.986) (0.511) (3.380) (2.055) (0.586)

Market MSA dummies Used in estimation, .— .— .— .— .— .—

(1 for each of 44 MSAs) but excluded for .— .— .— .— .— .—presentation clarity.

Adjusted Herfindahl score .— .— .— 2.4885 3.9276 0.7434 1.6570 7.4110 1.3043 .— .— .— (1.380) (1.101) (0.249) (0.781) (1.675) (1.514)

Hedonic price change in MSA .— .— .— 1.4602 1.0066 0.7969 .— .— .—(all units) .— .— .— (3.911) (0.032) (1.138) .— .— .—

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Table A.1. Multinominal Logit Estimation Results: Affordable Rental Units (continued)


Rent Owner- Demolished/ Rent Owner- Demolished/ Rent Owner- Demolished/Variables Rises Occupied Converted Rises Occupied Converted Rises Occupied Converted

Hedonic rent change in MSA .— .— .— 1.1883 2.6853 1.5157 .— .— .—(all units) .— .— .— (0.814) (2.335) (1.151) .— .— .—

Hedonic price change in MSA .— .— .— .— .— .— 1.1955 0.9767 0.7930 (affordable units) .— .— .— .— .— .— (2.399) (0.149) (1.469)

Hedonic rent change in MSA .— .— .— .— .— .— 7.9769 3.1113 0.7179 (affordable units) .— .— .— .— .— .— (6.805) (2.014) (0.677)

Control Number of years current resident 0.9848 1.0046 0.9848 0.9861 1.0051 0.9832 0.9858 1.0044 0.9827

has occupied unit (4.546) (0.745) (2.609) (4.171) (0.849) (2.904) (4.260) (0.729) (2.984)Ratio of rent to cutoff 1.9627 0.8588 0.4261 1.9635 0.9205 0.4257 1.9658 0.9320 0.4278

of affordability (4.444) (0.518) (3.598) (4.464) (0.284) (3.627) (4.462) (0.242) (3.602)

Note: The dependent variable has four possible values. An affordable unit can become unaffordable as a result of a rent increase, becoming owner-occupied, orbeing demolished or converted. The excluded outcome is to remain affordable. The first number reported is the unit odds ratio eb, and the number in parenthe-ses is the Z statistic. The odds ratio is the probability of outcome i divided by the probability of the null (or excluded) outcome, and is equal to eXB. The unit oddsratio is the odds ratio for a 1-unit increase to the independent variable. So it is not b that is reported in the table, but eb. The Z statistic is based on the nullhypothesis that b = 0, which is equivalent to the unit odds ratio eb = 1.

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esTable A.2. Multinominal Logit Estimation Results: Unaffordable Rental Units


Rent Falls/ Owner- Demolished/ Rent Falls/ Owner- Demolished/ Rent Falls/ Owner- Demolished/Variables Subsidized Occupied Converted Subsidized Occupied Converted Subsidized Occupied Converted

Unit Adequacy of unit 0.7900 1.0251 0.4578 0.8197 1.0130 0.4244 0.8453 1.0038 0.4279

(1 if adequate, 0 otherwise) (5.575) (0.372) (9.258) (4.762) (0.195) (10.508) (1.008) (0.057) (10.376)Age of unit 1.0059 1.0009 1.0339 1.0067 1.0011 1.0334 1.0068 1.0011 1.0335

(7.052) (0.800) (15.759) (7.989) (1.006) (15.733) (8.058) (1.014) (15.777)Unit is part of multi-unit building 0.6458 0.1190 0.4726 0.6775 0.1263 0.4771 0.6783 0.1262 0.4776

(1 if yes, 0 if no) (13.253) (50.898) (9.768) (12.026) (50.145) (9.910) (11.957) (50.147) (9.891)Number of units in building 1.0018 0.9986 1.0033 1.0018 0.9988 1.0017 1.0019 0.9988 1.0018

(3.322) (1.637) (2.172) (3.381) (1.229) (1.074) (3.523) (1.304) (1.152)

Neighborhood (at zone level)Ratio of subsidized units to all 3.6294 0.7196 1.0645 8.0080 0.7513 0.2760 10.1420 0.7548 0.3524

rental units (4.383) (0.847) (0.075) (8.161) (0.794) (1.827) (9.087) (0.784) (1.480)Average age of rental units 0.9951 0.9967 0.9835 0.9984 1.0053 0.9695 1.0035 1.0040 0.9722

(2.104) (1.041) (2.752) (0.954) (2.304) (7.110) (2.002) (1.711) (6.281)Ratio of public housing units to 1.7626 1.3302 0.7111 1.3858 2.5872 0.2865 1.8936 2.5131 0.3513

rental units (1.801) (0.575) (0.433) (1.153) (2.030) (1.773) (2.245) (1.964) (1.476)Ratio of rental units to all units 0.7671 0.6101 0.6576 0.6759 0.6045 0.7668 0.7501 0.6261 0.8620

(1.943) (2.611) (1.203) (3.586) (3.209) (0.973) (2.756) (3.085) (0.562)Ratio of affordable units to 2.1792 5.6693 2.6862 2.1207 3.8914 2.5287 2.2782 3.9055 2.6020

rental units (4.715) (6.811) (2.306) (5.305) (6.015) (2.609) (5.775) (6.022) (2.678)Percentage African-American 1.7594 0.5419 2.0190 1.4438 0.5261 2.0876 1.4248 0.5316 2.0989

head of household (6.119) (3.793) (3.087) (4.647) (4.428) (3.836) (4.462) (4.356) (3.864)Median household income ($) 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

(6.878) (3.122) (3.006) (9.927) (0.766) (6.512) (6.602) (0.985) (5.682)

Market MSA dummies Used in estimation, .— .— .— .— .— .—

(1 for each of 44 MSAs) but excluded for .— .— .— .— .— .—presentation clarity.

Adjusted Herfindahl score .— .— .— 0.6933 2.2242 2.4312 0.5605 2.0599 1.8250 .— .— .— (1.212) (2.135) (1.180) (1.972) (2.018) (0.830)

Hedonic price change in MSA .— .— .— 0.9330 1.1186 0.8129 .— .— .—(all units) .— .— .— (1.373) (1.727) (1.529) .— .— .—

Dynam


ousing Stock:M


iltering139

Table A.2. Multinominal Logit Estimation Results: Unaffordable Rental Units (continued)


Rent Falls/ Owner- Demolished/ Rent Falls/ Owner- Demolished/ Rent Falls/ Owner- Demolished/Variables Subsidized Occupied Converted Subsidized Occupied Converted Subsidized Occupied Converted

Hedonic rent change in MSA .— .— .— 0.5883 1.0306 0.5383 .— .— .—(all units) .— .— .— (5.261) (0.228) (2.421) .— .— .—

Hedonic price change in MSA .— .— .— .— .— .— 1.1818 1.0163 0.9552 (affordable units) .— .— .— .— .— .— (4.236) (0.301) (0.426)

Hedonic rent change in MSA .— .— .— .— .— .— 0.1842 1.5477 0.3542 (affordable units) .— .— .— .— .— .— (12.881) (2.702) (3.271)

Control Number of years current resident 1.0104 1.0153 0.9672 1.0098 1.0171 0.9660 1.0108 1.0172 0.9673

has occupied unit (4.447) (4.373) (4.029) (4.237) (4.884) (4.155) (4.662) (4.903) (4.016)Ratio of rent to cutoff 0.1054 1.9828 0.5496 0.1077 1.9385 0.5636 0.1049 1.9501 0.5645

of affordability (51.744) (16.902) (6.256) (52.143) (17.186) (6.149) (52.449) (17.389) (6.126)

Note: The dependent variable has four possible values. An affordable unit can become unaffordable as a result of a rent increase, becoming owner-occupied, orbeing demolished or converted. The excluded outcome is to remain affordable. The first number reported is the unit odds ratio eb, and the number in parenthe-ses is the Z statistic. The odds ratio is the probability of outcome i divided by the probability of the null (or excluded) outcome, and is equal to eXB. The unit oddsratio is the odds ratio for a 1-unit increase to the independent variable. So it is not b that is reported in the table, but eb. The Z statistic is based on the nullhypothesis that b = 0, which is equivalent to the unit odds ratio eb = 1.

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Documents

Dynamics of the Affordable Housing Stock: Microdata ... · Braid (1981) studies filtering in rental housing markets.Bond and Coulson (1989) analyze the process of neighborhood change