Markov Chain Model of Rent Burden in the Housing Choice Voucher Program 

Brent D. Mast

U.S. Department of Housing and Urban Development 

Abstract 

This study models rent burden in the Housing Choice Voucher Program as a Markov Chain. The 

model predicts rent burden with program tenure, using longitudinal household data for 2000 

through 2009. Results indicate rent burden increases for many years after admission. Consistent 

with results for unassisted low-income families, the model also predicts considerable mobility 

across burden categories over time. The rent burden formula indicates that HUD policy and 

housing agency policy should not be considered in isolation. Estimates imply that their 

interaction has an effect large in both magnitude and statistical significance. A limitation of the 

Markov Chain model is that it doesn’t estimate variance; I demonstrate a simple method for 

doing so via bootstrapping. 

Acknowledgements 

I thank Chun-Hung Chen, Mark Shroder, and Scott Susin for helpful comments, along with 

participants at a HUD workshop in April 2010. I also thank Mark Perdue for valuable assistance 

constructing the longitudinal data files. 

The contents of this article are the views of the author and do not necessarily reflect the views

or policies of the U.S. Department of Housing and Urban Development or the U.S. government.

1 
 

 

I Introduction 

Since 1937, the United States has made access to affordable housing a national priority. For a 

numerous reasons, assistance has been provided through housing subsidies rather than general 

income transfers. 

Affordable housing assistance was initially provided through public housing projects run by local 

Public Housing Agencies (PHAs). In 1965, PHAs began experimenting with providing 

opportunities in privately owned rental units.1 Private opportunities have been provided through 

a variety of programs. 

There are many arguments for providing assistance in privately owned buildings instead of 

public housing. The primary motivation for increasing private sector housing choices has been 

expanding social and economic opportunities for low-income households receiving housing 

assistance. Another argument is that private owners might have better incentives for operational 

efficiency, thus lowering program costs. 

Since 1995 the largest private options have been Certificates and the Housing Choice Voucher 

Program (HCVP). In 1995 there were 1.2 million households in public housing, compared to 1.3 

million certificates and vouchers.2 

Certificates have been phased out over time, while Housing Choice Vouchers have increased 

dramatically. Today, HUD provides assistance to about 1.1 million households in public housing 

projects, and about 2 million through the Housing Choice Voucher Program. HCVP is the now 

largest U.S. rental assistance program, with an annual budget over $16 billion. 

Voucher recipients choose their own rental units, and rent is paid for partially by the household 

and partially through HUD payments to landlords. The voucher program is administered by 

approximately 2600 local Public Housing Agencies (PHAs). Among the PHAs’ responsibilities 

are selecting applicants to receive vouchers, annual verification of household income, and 

determination of each household’s rent responsibility. 

The Housing Assistance Payment (HAP) under the program is the difference between 1) the 

minimum of gross rent (rent plus utility costs) and the Payment Standard, and 2) the Total 

Tenant Payment (TTP). Payment Standards and TTP are both set by PHAs. TTP is the minimum 

household contribution, which is usually about 30 % of household adjusted income. 

1 See http://www.mphaonline.org/section8.cfm for a concise history of HUD rental assistance programs. 2 http://www.huduser.org/portal/datasets/assthsg/statedata96/index.htm 

2 
If the Payment Standard set by the PHA is in-line with actual rents, households should be able to 

find units meeting HUD quality standards at burden levels around 30 %. Yet in reality, many 

households face burdens considerably greater than 30 %. 

Starting in 1998, when admitted to the program or moving into a new unit, a family’s rent burden 

is required to be between 30 and 40 % of adjusted household income. In other years, rent burden 

has a 30 % floor with no ceiling. 

Data indicate the ceiling for admissions and move-ins is not strictly enforced by all PHAs. In 

2009, 4.216 % of new admissions had rent burdens of at least 41 %, as did 5.480 % of movers. 

Exhibit 1 reports trends in rent burden for new admissions, other households, and total 

households from 1995-2009. After restrictions were placed on burdens at admission or move-in 

in 1998, average rent burdens fell dramatically. Around 2003 burdens increased, and fell again 

starting around 2006. As of 2009, the mean burden was 33.950 % overall, 33.172 % for new 

admissions, and 34.003 % for other households. 

Exhibit 1: Mean Rent Burden Trends, 1995-2009 

33%

35%

37%

39%

41%

43%

45%

199519961997199819992000200120022003200420052006200720082009

New Admissions Other Households Total

N=15,913,767. Source: author’s calculations using MTCS/PIC data 1995-2009. 

While mean burdens in recent years may seem reasonable, many households have burdens 

considerably higher than the mean. Exhibit 2 reports trends for percentages of households with 

burdens of at least 41 %. Since 2000, the fraction of households with burdens above 40 % has 

fluctuated. In 2009, 11.502 % of households fell in this category. 

Rent burdens above 30 % don’t necessarily imply inadequate subsidies. Indeed, the program is 

designed to foster choice. Households may choose burden levels above 30 % for a variety of 

reasons. For instance, households may have a preference for larger homes, newer homes with 

better amenities, or homes in better neighborhoods. 

3 
 

           

       

Exhibit 2: Percent of Households with Burdens >=41 %, 1995-2009 

N=15,913,767. Source: author’s calculations using MTCS/PIC data 1995-2009. 

Between 2000 and 2002, HUD conducted a survey of HCVP households.3 Focusing on housing 

quality, one question asked residents to rate their home on a scale of 1-10. Exhibit 3 reports 

percentages of respondents with burdens of at least 41 % for three categories of home ratings: 1-

3, 4-6, and 7-10. 

Exhibit 3: Percent of Households with Burden>=41 %, by Home Rating 

0%

10%

20%

30%

40%

50%

199519961997199819992000200120022003200420052006200720082009

New Admissions Other Households Total

N=310,314. Source: CSS and MTCS data, 2000-2002 

High burdens are more prevalent for households with higher household ratings. For homes with 

ratings of 1-3, 9.362 % have burdens of at least 41 %. Among families rating their home 7 or 

above, 11.246 % have high burdens. 

3

 Dubbed the Customer Satisfaction Survey (CSS), see Mast (2009) for more information. 

9.25%

9.45%

9.65%

9.85%

10.05%

10.25%

10.45%

10.65%

10.85%

11.05%

11.25%

Rating 1 to 3 Rating 4 to 6 Rating 7‐10

% Households w/ Burden>=41 %

4 
Policy implications depend not only on the number of households with high rent burdens at any 

given time, but also on the tendency of households to have persistent periods of high burden. 

While prior research suggests considerable turnover in the population of households with high 

rent burdens (Martin et al. 2005), prolonged burden may disproportionately affect the most 

disadvantaged households (Susin 2007). 

This study models rent burden in the HCVP as a Markov Chain, using longitudinal data on 

households admitted to the program between 2000 and 2009. The model is a simple and 

powerful tool for examining rent burden patterns with time in program. Estimates imply 

increasing burden with program tenure, and considerable household mobility across rent burden 

categories. The model also estimates large differences in burden due to HUD and PHA policies. 

One limitation of Markov Chain models is that they don’t estimate variance. I demonstrate a 

simple method for doing so via bootstrapping. 

The Markov Chain model predicts burden for households remaining in the program past their 

admission year. Large variation in exit rates with program tenure make exit unsuitable for 

Markov Analysis. Instead, I examine the relationship between burden and program exit with 

survival analysis. Estimates indicate low chances of exit in the admission year. In subsequent 

years, exit rates are larger with moderately burdened households most likely to exit. Households 

admitted with burdens below 32 % are estimated to stay in the HCVP program about .4 years 

longer than other families. 

The next section summarizes the relevant literature. Program guidelines for determining rent 

burden are then discussed, followed by a description of the data. The relationship between 

burden and program tenure is then considered, followed by Markov Chain analysis. Conclusions 

are summarized in the final section. 

II Literature Review 

Rent Burden Studies Many studies have examined rent burden in the general population (i.e., Hill 

2003, HUD 1998, 2000, 2001, 2003). Martin et al. (2005) analyze data from the American 

Housing Survey (AHS) and 2 years of longitudinal data from the Survey of Income of Program 

Participation (SIPP). They find much churn in the population with high rent burdens. 

Susin (2007) analyzes duration of rent burden for low-income families, using 3 years of SIPP 

data. The focus of his study was identifying household characteristics related to prolonged 

periods of high burden. He reports numerous indicators of need, such as non-employment and 

receipt of Supplemental Security Income (SSI), are associated with persistent high burden. 

HUD (2007) compare rent burden estimates from the AHS with SIPP estimates, which were 

mostly adopted from Susin (2007). Compared to the SIPP, the AHS has much better utility cost 

data. Susin (2007) and HUD (2007) use regional AHS data to impute utility costs for SIPP 

participants. 

A few studies have examined rent burden in the HCVP program. McClure (2005) is the most 

recent, using the same data source as this study (HUD’s MTCS data system, described below). 

5 
Analyzing household data for 2000 to 2002, he studies how rent burden varies with a large 

number of household, housing market, and policy variables. McClure finds high rent burdens are 

most prevalent among very low-income families (relative to other low-income HCVP 

households). 

Using only three years of data, McClure’s study doesn’t indicate whether periods of extremely 

low income (and high rent burden) are “chronic or transitory” (McClure 2007, p. 18). This study 

attempts to extend the literature by analyzing rent burden patterns over a 10 year period (2000-

2009). 

Markov Chain Models According to Ross (2007), A Markov Chain model is a discrete time 

stochastic process where the conditional distribution of any future state is independent of past 

states, depending only on the current state. Processes that satisfy this assumption are referred to 

as stationary, because the probability of transitioning to another state is fixed over time. Such 

processes are also referred to as memoryless, because the past has no predictive power. 

HCVP rent contracts are effective for a discrete time period (typically 1 year). And while 

dynamics of HCVP rent burden have yet to be examined, past research on unassisted low-income 

households indicates considerable rent burden churn (Martin et al. 2005, Susin 2007, HUD 

2007). Thus HCVP rent burden may qualify as a discrete time stochastic process. And while rent 

burden in many prior years may influence future burden, the most recent is likely the most 

important. Thus a Markov Chain may be a reasonable model of HCVP rent burden. 

Susin (2007) uses logistic regression to predict rent burden duration. McClure (2005) only 

reports summary statistics. To my knowledge, this is the first study modeling rent burden as a 

Markov Chain. Many researchers have applied Markov Chain models to other housing market 

topics, however. An early example is Clark (1965), who uses a Markov Chain to predict 

geographic movement of households. 

A Markov Chain model is a simple and convenient method to analyze patterns of rent burden 

over time. One limitation of the Markov Chain model is that rent burden must be categorized. 

Yet it is really a category of rent burden that is of policy concern: those above some threshold 

level (traditionally between 40 % and 50 %) considered to cause economic hardship on families. 

The most critical limitation of the Markov Chain model is the stationarity assumption. The 

definition of states in Markov Chains is flexible, however. And Ross (2007) demonstrates that a 

more generalized stochastic process can be transformed into a Markov Chain by defining states 

based on multiple time periods. I follow this strategy by defining rent burden categories based on 

two years of data. 

In general, Markov Chain models are used more for predicting than explaining changes in a 

variable. Regression is more practical for estimating marginal effects for large numbers of 

explanatory variables. Yet Markov Chain states can be defined in ways useful for policy 

analysis. This study demonstrates how a Markov Chain model can predict the impact of HUD 

policy (Fair Market Rent) and PHA policy (the Payment Standard) on rent burden. 

6 
 

While regression may be a better model for exploring causality, many variables correlated with 

rent burden are choice variables potentially influenced by rent burden. For instance, while rent 

burden varies with neighborhood choice, rent burden may also influence location decisions. Thus 

many variables related to rent burden (i.e., neighborhood, unit size, housing quality, moves) are 

potentially endogenous with respect to rent burden. Income as an explanatory variable can also 

be problematic in regression analysis if measured with error, because income is also in the 

denominator of rent burden. 

McClure (2005) thoroughly examined HCVP Rent burden variation across demographic groups. 

While I do examine differences across broad household categories, I do not make estimates for 

particular racial, gender, or geographic subgroups. However the Markov Chain states in this 

study could be defined in ways to make estimates for narrower demographic categories. 

III Rent Burden Formula 

Rent burden (burden) equals the difference between gross rent including utility payments (rent) 

and the Housing Assistance Payment (HAP), divided by household adjusted income (income): 

 burden =[rent – HAP]/income (eq1) 

HAP is the difference between 1) the minimum of gross rent and the Payment Standard (PS), and 

2) the Total Tenant Payment (TTP): 

HAP= min(rent, PS) - TTP (eq2) 

TTP is the minimum household contribution. It is calculated as the maximum of four values: 1) 

30 % of household adjusted income, 2) 10 % of gross income, 3) the welfare rent, and 4) the 

PHA minimum rent. For the vast majority of households, TTP is close to 30 % of household 

adjusted income. Under this assumption, eq2 can be rewritten as: 

 HAP= min(rent, PS) - .3*income (eq3) 

Note that until 1998, households could keep the difference between Payment Standards and rent 

when rent fell below Payment Standards. Thus households had an incentive to search for units 

renting below the Payment Standard in order to reduce their rent burdens. Now, however, the 

minimum rent burden a household can incur is at least 30 %. Thus households have no incentive 

to search for qualifying units renting below the Payment Standard in order to reduce their rent 

burden. 

Yet households may select units with gross rent below the payment standard for a variety of 

other reasons. Indeed, a slight majority of households fall in this category. Accordingly, let H1 

equal the maximum of zero and Payment Standard minus gross rent: 

 H1=max(0, PS-rent) (eq4) 

Thus eq3 can be rewritten as follows, where H1 is nonnegative: 

 HAP= PS - H1 - .3*income (eq5) 

7 
 

 

 

 

HUD attempts to set Fair Market Rent (FMR) at 40 % of the distribution for a unit with a given 

number of bedrooms in each FMR region. In tight markets, FMR is sometimes set at 50 % of 

rent distribution. Let r equal the ratio of the Payment Standard to FMR: 

 r=Payment Standard/FMR (eq6) 

Exhibit 4 depicts the distribution of r. It has a mean of 1.003, a median of 1, and a standard 

deviation of .094. It has a thick lower tail, with a small spike around .9. There are much larger 

spikes around 1 and 1.1, indicating that many PHAs set their payment standard around 100 % or 

110 % of FMR. 

Exhibit 4: Distribution of r=Payment Standard/FMR 

Note: FMR=Fair Market Rent. Source: author’s calculations using MTCS/PIC data for 2000-2009. 

The Payment Standard equals r multiplied by FMR: 

Payment Standard=r*FMR (eq7) 

Substituting eq7 into eq5: 

HAP=r*FMR - H1 - .3*income (eq8) 

Substituting eq8 into eq1: 

burden=.3 + H1 + rent/income - r*FMR/income (eq9) 

Actual rent at the 40th (or sometimes 50th) percentile of the rent distribution in the FMR region 

for a given number of bedrooms (FMR*) equals HUD’s measure (FMR) plus measurement error 

(error): 

8 
 

 

 

 

FMR*=FMR + error (eq10) 

Gross rent is FMR* plus a household preference factor, H2, for a unit with gross rent above or 

below FMR*: 

rent= H2 + FMR* (eq11) 

Substituting eq10 into eq11: 

rent=H2 + FMR + error ` (eq12) 

Substituting eq12 into eq9: 

Burden=.3 + H1/income + H2/income + FMR/income – 

r*FMR/income + error/income (eq13) 

Simplifying: 

Burden=.3 + (H1 + H2)/income + FMR(1-r)/income + error/income (eq14) 

where H1>=0, and (H1 + H2)/income + FMR(1-r)/income + error/income is nonnegative as 

well. That is, rent burden can’t fall below 30 %. 

So rent burden can modeled as function of three main factors: 

A) Household housing preferences. In particular, tendencies to select units with gross rent 1) 

below the Payment Standard, and/or 2) above or below rent at the actual 40th (or sometimes 50th) 

percentile of the rent distribution. Note that the incentive to select units renting below the 

Payment Standard diminished after 1998 when doing so no longer reduced rent burden. 

Rent burden increases significantly with home ratings (see Exhibit 3), suggesting that HCVP 

families are willing to incur higher burdens in order to live in better housing. 

B) FMR(1-r)/income. In words, rent burden increases with FMR interacted with 1 minus the 

ratio of Payment Standard to FMR, relative to household income. 

Exhibit 5 depicts the distribution of f=FMR(1-r)/income, which has a mean and median of 0, and 

standard deviation of .094. For the average household f is 0, thus not affecting rent burden. 

Exhibit 6 reports percentages of families with excess burdens of at least 41 % according to 

f=FMR(1-r)/income in three categories: less than or equal to -.045; -.044 to .013, and .014 or 

above. The cut points are approximately the 33rd and 66th percentiles. The fraction of households 

with excess burdens is slightly higher for the lower third (6.783 %) of the distribution compared 

to the middle third (6.316 %). There is a large increase in burden for families in the upper third 

of the distribution. 12.371 % of households with f>=.014 have burdens of at least 41 %. 

9 
Exhibit 5: Distribution of f=FMR(1-r)/income 

Note: FMR=Fair Market Rent, r=Payment Standard/FMR, and income is household income adjusted per HUD 

guidelines. Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

C) Deviations in FMR from actual 40th or 50th percentile rents, relative to household income. 

While Fair Market Rent may be measured with error, the error is difficult to quantify. It is likely 

that measurement error is much lower in decennial census years, when more data are available 

on rents by bedrooms for small geographic areas. In other years, census estimates are updated via 

random digit dialing. When the American Community Survey is fully implemented, it will 

provide intercensal estimates which may improve FMR measurement. 

FMR regions are fairly large (typically counties), and rents vary within FMR regions. Even if 

FMR is measured well at the FMR region level, this doesn’t necessarily imply that FMR is a 

good measure of neighborhood rents. 

10 
   

       

Exhibit 6: Burdens>=41 %, by Levels of f=FMR(1-r)/income 

5%

6%

7%

8%

9%

10%

11%

12%

13%

f <= ‐.045 ‐.044 <=f<= .013 f >=.014

% Households w/ Burden>=41 %

Note: FMR=Fair Market Rent, r=Payment Standard/FMR, and income is household income adjusted per HUD 

guidelines. Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

IV The Data 

Data are extracted from HUD’s MTCS/PIC data system. The system has quarterly entries for 

each family receiving rental assistance starting in 1995. Data are available on rent burden and a 

large number of other household characteristics. 

For the remainder of the study, I explore data on families admitted to the HCVP program 

between 2000 and 2009. Families admitted before 2000 are excluded due to the rapid decline in 

rent burden from 1995-1999 (see Exhibits 1 and 2). 

I exclude Certificate households because 1) their rent burden regulations differ from HCVP 

regulations, and 2) there are few Certificates today, so the program is of little policy importance. 

I also exclude voucher households in the Moving To Work demonstration program, due to 

discretion in rent burden determination and uneven reporting. Homeownership vouchers are also 

excluded. 

I also drop some outlier observations with suspect data. I exclude households if: 1) adjusted 

annual income is below $250 or above $40,000; or 2) gross rent (including utilities) is above 

$3,000 a month; or 3) rent burden is less than 28 % or more than 90 %. I also exclude households 

with multiple admissions. 

The data system is transaction based. The most common transactions are 1) admissions; 2) 

annual re-exams; 3) interim re-exams due to changes in eligibility factors such as income or 

family size; 4) moves; and 5) exits from the program. The system captures the most recent 

transaction at the end of each quarter. If multiple transactions for a household occur during a 

quarter, only the most recent is available. If there is no transaction during a quarter, the family’s 

entry is a duplicate of the entry for the previous quarter. 

11 
 

   

Rent contracts are effective for one year, and most households have only one transaction per 

year. Therefore most changes in rent burden are annual (not quarterly). Accordingly, for this 

study I employ longitudinal research files that capture the most recent transaction at the end of 

each year for each family. 

The data provide a consistent snapshot for each family on December 31st of each year. Compared 

to the quarterly data, the longitudinal data have been edited somewhat to make them more 

comparable across years. For instance, variable names that change over time are standardized, 

duplicate records are eliminated, and data items from multiple tables are combined into a single 

file. 

V Summary Statistics 

In total, my sample consists of 6,851,746 observations on 1,978,157 HCVP families admitted 

between 2000 and 2009. Exhibit 7 depicts the distribution of observations per family. The 

median number of observations per household is 2, and the mean is 2.974. 27.798 % of 

households have 5 or more observations, and less than 7 % have 7 or more observations. 

Exhibit 8 reports mean burden by years in program. Exhibit 9 reports trends in percentages of 

households with normal rent burdens no more than 31 %; Exhibit 10 reports trends for medium 

burdens between 30 % and 40 %; Exhibit 11 reports trends for excess burdens of at least 41 %. 

Exhibit 7: Observations per Household 

0% 5% 10% 15% 20% 25% 30%

1

2

3

4

5

6

7

8

9

10

% of Households

N=6,851,746 observations on 1,978,157 households. Source: author’s calculations using MTCS/PIC data for 

admissions between 2000 and 2009. 

Trends are reported for five household categories: 1) all households; 2) households with an 

elderly head or spouse (12.609 % of sample); 3) non-elderly households where the head or 

spouse has a disability (25.370 % of sample); 4) other households with children (45.453 % of 

sample); and 5) other households without children (16.568 % of sample). 

12 
       

   

       

   

Exhibit 8: Mean Burden by Years in Program 

N=6,851,746. Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

Exhibit 9: % of Households with Normal Burdens<=31%, by Years in Program 

32.0%

32.5%

33.0%

33.5%

34.0%

34.5%

35.0%

35.5%

0 1 2 3 4 5 6 7 8 9

All Households Elderly

Families w/ disabilities Other w/ children

Other w/o children

N=6,851,746. Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

55%

60%

65%

70%

75%

0 1 2 3 4 5 6 7 8 9

All Households Elderly

Families w/ disabilities Other w/ children

Other w/o children

13 
       

   

       

   

Exhibit 10: % of Households with Medium Burdens between 32% and 40%, by Years in 

Program 

Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

Exhibit 11: % of Households with Excess Burdens>=41%, by Years in Program 

15%

20%

25%

30%

35%

0 1 2 3 4 5 6 7 8 9

All Households Elderly

Families w/ disabilities Other w/ children

Other w/o children

4%

6%

8%

10%

12%

14%

16%

0 1 2 3 4 5 6 7 8 9

All Households Elderly

Families w/ disabilities Other w/ children

Other w/o children

Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

Mean burdens are lowest for elderly families, and highest for the largest category - ‘other 

families with children’. Patterns of burdens are similar for all household categories. In the year 

of admittance (year 0), burdens are relatively low. This is not surprising, given that initially 

burdens are not supposed to exceed 40 %. Note that data for year 0 are for the last record for the 

family in their admission year. For about 26 % of households, the admission year record is not 

for their admission date (see Exhibits 1 and 2 for burden data on admission dates). 

14 
For several years after admission, mean burden increases, along with the percentage of families 

with excess burdens; the share of families with medium burdens decreases. At the end of the 

admission year, the mean burden for all households is 33.752 % of income, and the share of 

households with medium burdens is 31.607 %. The percentage of all households with excess 

burdens of at least 41 % is 7.465 % in the admission year. By year 4, mean burden is 34.251 %, 

the share of households with medium burden is 24.695 %, and the share with excess burdens is 

12.612 %. 

The pattern of normal burdens below 32 % is contrary to the mean for the first several years. 

While the mean increases after admission, so does the share with normal burdens. For the 

admission year, the mean burden for all households is 33.752 % of income, and the share of all 

households with normal burden is 60.876 %. After 2 years, although the mean for all households 

increases to 33.816 %, the share of households with normal burden increases to 64.685 %. In 

year 3, the share of households with normal burdens starts to decrease, reaching 61.940 % by 

year 5. 

While burdens tend to decrease starting around year 5 or 6, this may be an artifact of rapidly 

dwindling observations over time (see Exhibit 7). 

VI Burden and Exit 

Burden and Exit Rates 

 The main focus of my study is examining how rent burden changes with program tenure. Of 

course, rent burden might also influence the likelihood of exit from the program. 

Numerous studies have examined duration in the HCVP program. Olson et al. (2005) and Cortes 

et al. (2009) find a negative relationship between duration and gross rent relative to area housing 

costs. They do not consider the impact of gross rent relative to income, however. 

It’s not possible to predict the sign of the effect of burden on exit. Some families might leave 

sooner if the only rental units available meeting HUD standards are too costly. Yet families with 

rising incomes may exit because they no longer meet income requirements; these families might 

have low burdens when leaving the HCVP program. 

A Markov Chain model isn’t appropriate for examining the relationship between burden and exit 

if exit rates vary with years in program for reasons unrelated to burden. This would violate the 

assumption of fixed transition probabilities over time. 

To explore exit and burden, I estimate logistic regression model yij=αij + eij where y is a binary 

indicator for exit, α is an intercept shift, e is a random error term, i is years in the HCVP, and j is 

the families rent burden category. i equals 0 in the admission year and is truncated at 6. j=1 for 

normal (≤31 %), 2 for medium (32-40 %), and 3 for excess burden (≥41 %). 

For brevity, for the remainder of the study I limit my analysis to non-elderly households at 

admission, with children, where neither the head nor spouse has a disability (the ‘other families 

with children’ category discussed above). I focus on this subset because they are the largest 

family category and tend to have the greatest burdens. 

15 
 

           

   

   

   

   

   

   

   

   

Families with multiple admissions are excluded. I also drop families with two or more 

consecutive years of missing data. Exhibit 12 reports households analyzed by year. Counts are 

reported for new admissions, other households, and all households. In 2000 there are 26,171 

families, all of which are new program participants. By 2009, there are 366,370 households and 

75,022 new admissions. In total, I analyze 2,470,944 observations on 826,080 families. 

Year

New

Admissions

Other

Households

All

Households

Exhibit 12: Households Analyzed by Year 

2000 26171 0 26171
 

2001 91511 2854 94365
 

2002 102776 72169 174945
 

2003 100729 145193 245922
 

2004 77495 192798 270293
 

2005 67635 215766 283401
 

2006 91067 215449 306516
 

2007 100238 244554 344792
 

2008 93436 264733 358169
 

2009 75022 291348 366370
 

Total 826080 1644864 2470944
 

Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

Exhibit 13 reports logistic regression odds ratio estimates, where the reference category is 

families with normal burden in their admission year. All of the effects except for excess burden 

in year 5 have coefficients significant at the .001 level. Mean predicted probabilities of exit are 

plotted in Exhibit 14. 

Years in

program

Burden

category

Odds

ratio

estimate

Lower

95 %

CI

Upper

CI

95 %

Exhibit 13: Logistic Regression Odds Ratio Estimates 

0 Medium 0.971 0.951 0.993
 

0 Excess 0.980 0.946 1.016
 

1 Normal 3.625 3.570 3.681
 

1 Medium 4.474 4.393 4.555
 

1 Excess 2.999 2.924 3.076
 

2 Normal 3.292 3.238 3.348
 

2 Medium 3.573 3.499 3.649
 

2 Excess 3.006 2.923 3.091
 

3 Normal 2.905 2.850 2.961
 

3 Medium 2.965 2.891 3.041
 

3 Excess 2.788 2.697 2.881
 

4 Normal 2.484 2.429 2.540
 

16 
   

   

   

   

   

 

4 Medium 2.827 2.743 2.913

4 Excess 2.499 2.400 2.601

5 Normal 2.109 2.053 2.166

5 Medium 2.464 2.375 2.557

5 Excess 2.263 2.153 2.378

6 Normal 1.646 1.603 1.691

6 Medium 1.917 1.846 1.992

6 Excess 1.789 1.699 1.884

N=2,470,944, -2 log likelihood=1689606.7. Reference category is normal burden in the admission year, and years 

in program is truncated as 6. Source: author’s calculations using MTCS/PIC data for 2000-09 admissions. 

Exhibit 14: Mean Predicted Probabilities of Exit from HCVP Program 

0.05

0.07

0.09

0.11

0.13

0.15

0.17

0.19

0 1 2 3 4 5 6

Normal Burden Medium Burden Excess Burden

N=2,470,944. Years in program is truncated at 6. Source: author’s calculations using MTCS/PIC data for 2000-09 

admissions. 

The probability of exit varies widely with duration regardless of rent burden. Exit hazards are 

lowest in the admission year, and highest in the first year past admission. Exit probabilities tend 

to decline each remaining year. 

Prior research indicates factors such as housing preferences (Olson et al. 2005) and number of 

children (Cortes et al. 2009) may play import roles determining exit. Because of large variation 

in exit probabilities with duration, exit is an unlikely candidate as a Markov Chain. 

In the admission year, there is little difference in exit chances across burden categories. Families 

with normal burden have an estimated .052 exit probability, compared to .051 for families with 

medium or excess burden. 

In each year past admission, households with medium burden are most likely to exit. In year 1, 

exit probabilities are .167 for normal burden, .198 for medium burden, and .142 for excess 

burden families. By year 6, exit chances have declined substantially - .083 for normal, .096 for 

medium, and .090 for excess burden households. 

17 
Exit Rates and the Markov Chain Model 

The Markov Chain model in the next section predicts burden for households remaining in the 

program. Changing exit rates are not a problem for the Markov Chain model if the proportional 

hazard of exiting is constant across burden categories. Yet if the proportion of households exiting 

varies with burden over time, this would lead to changing transition probabilities violating the 

stationarity assumption. 

Odds ratios vary greatly the first couple of years in the program. Starting in year 2, proportional 

exit rates are fairly stable over time. Differences in proportional exit chances between early and 

later years suggest that a Markov chain model needs separate states for families entering the 

program. 

Burden and Program Tenure 

To explore the effect of burden on time in the HCVP program, I estimate a Kaplan-Meier 

survival function stratified by burden category at admission. The survival function estimates the 

probability of remaining in the HCVP program over time. The estimated survival curve is 

depicted in Exhibit 15. 

Households with normal burden in their admission year have the greatest estimated time in 

program, and there is little difference between households admitted with medium or excess 

burden. Median survival time is 5.131 years for households admitted with normal burden, 

compared to 4.715 for those with medium burden and 4.728 for those with excess burden. The 

difference in median tenure between households admitted with normal burden and other 

households is about .4 years. 

Exhibit 15: Survival Probabilities by Burden at Admission 

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9

Normal burden Medium burden Excess burden

Source: author’s calculations using MTCS/PIC data for 2000-09 admissions. 

18 
 

VII Markov Chain Model 

Data 

In this section, I model HCVP rent burden as a discrete time stochastic process. The sample is 

the same group of families with children used in the previous section (see Exhibit 12). While 

household status can change over time, this is just one factor among many possibly affecting rent 

burden. 

Assumptions 

I assume a household’s burden falls into 3 categories, normal, medium, and excess: 

A) Normal burden -- rent burden ≤ 31 % ; 

B) Medium burden -- 32 % ≤ rent burden ≤ 40 %; 

C) Excess burden -- rent burden ≥ 41 %. 

I define 12 states based on rent burden in the prior and current year: 

A -- Normal burden at admission (no burden in the prior year); 

AA -- Normal burden in the prior and current year; 

AB -- Normal burden in the prior year, medium burden in the current year; 

AC -- Normal burden in the prior year, excess burden in the current year; 

B -- Medium burden at admission (no burden in the prior year); 

BA -- Medium burden in the prior year, normal burden in the current year; 

BB -- Medium burden in the prior and current year; 

BC -- Medium burden in the prior year, excess burden in the current year; 

C -- Excess burden at admission (no burden in the prior year); 

CA -- Excess burden in the prior year, normal burden in the current year; 

CB -- Excess burden in the prior year, medium burden in the current year; 

CC -- Excess burden in the prior and current year. 

Current states are known. Burden in subsequent years satisfies the following assumption: the 

conditional distribution of any future state Xn+1 is independent of past states, depending only on 

the current state Xn. That is, when the household is in state i, there is a fixed probability Pij of 

being in state j in the next year. 

Under these conditions, rent burden is a Markov Chain. Let the current year be denoted by year 

0. For households in their admission year, rent burden in year 1 depends only on their current 

burden. For other households, burden in year 1 depends on rent burden in the current and prior 

year. 

Because of differences in burden and proportional exit rates early in the program, the stationarity 

assumption is unlikely to hold making no distinction between burden at admission and in 

subsequent years. Thus the model includes three separate states solely for admission (A, B, and 

C). Modeling rent burden in years 2 and above as dependent on multiple periods also makes the 

stationarity assumption more plausible. Therefore the remaining nine states are based on two 

years of data. 

19 
       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

Transition Probabilities 

Exhibits 16-19 report transition probability matrices Pn for years n=1 to 4, respectively. The 

matrices report the conditional probability Pij of being in state j in n years, given current state i. 

There is a row for each possible current state i, and a column for each possible state j in n years. 

P1

 was calculated by cross tabulating states with states lagged one year. Due to stationarity, the 

transition matrices for years n>1 can be derived by raising P1 to the nth power: P2 =P1

 squared, P3 

=P1 cubed, and P4 =P1 to the 4th power. While the matrices used for calculations are square 

(12x12), 3 columns contain all zeros- households cannot transition to the admission states A, B, 

and C. Thus I report 12x9 matrices excluding nuisance columns for admission states. 

State 

Exhibit 16: P1 Transition Probability Matrix 

AA AB AC BA BB BC CA CB CC 

A 0.853 0.105 0.042 0.000 0.000 0.000 0.000 0.000 0.000
 

AA 0.850 0.110 0.040 0.000 0.000 0.000 0.000 0.000 0.000
 

AB 0.000 0.000 0.000 0.356 0.503 0.140 0.000 0.000 0.000
 

AC 0.000 0.000 0.000 0.000 0.000 0.000 0.313 0.245 0.442
 

B 0.000 0.000 0.000 0.291 0.546 0.163 0.000 0.000 0.000
 

BA 0.760 0.179 0.061 0.000 0.000 0.000 0.000 0.000 0.000
 

BB 0.000 0.000 0.000 0.279 0.577 0.144 0.000 0.000 0.000
 

BC 0.000 0.000 0.000 0.000 0.000 0.000 0.187 0.339 0.474
 

C 0.000 0.000 0.000 0.000 0.000 0.000 0.244 0.255 0.501
 

CA 0.748 0.159 0.093 0.000 0.000 0.000 0.000 0.000 0.000
 

CB 0.000 0.000 0.000 0.269 0.536 0.195 0.000 0.000 0.000
 

CC 0.000 0.000 0.000 0.000 0.000 0.000 0.173 0.255 0.571
 

Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

State 

Exhibit 17: P2 Transition Probability Matrix 

AA AB AC BA BB BC CA CB CC 

A 0.725 0.094 0.034 0.037 0.053 0.015 0.013 0.010 0.019
 

AA 0.722 0.093 0.034 0.039 0.055 0.015 0.013 0.010 0.018
 

AB 0.271 0.064 0.022 0.140 0.290 0.073 0.026 0.048 0.067
 

AC 0.234 0.050 0.029 0.066 0.131 0.048 0.077 0.113 0.252
 

B 0.221 0.052 0.018 0.152 0.315 0.079 0.030 0.055 0.077
 

BA 0.646 0.083 0.031 0.064 0.090 0.025 0.019 0.015 0.027
 

BB 0.212 0.050 0.017 0.161 0.332 0.083 0.027 0.049 0.069
 

BC 0.140 0.030 0.017 0.091 0.181 0.066 0.082 0.121 0.271
 

C 0.183 0.039 0.023 0.069 0.136 0.050 0.087 0.128 0.286
 

CA 0.636 0.082 0.030 0.057 0.080 0.022 0.029 0.023 0.041
 

CB 0.205 0.048 0.016 0.149 0.309 0.077 0.036 0.066 0.093
 

CC 0.130 0.028 0.016 0.069 0.137 0.050 0.099 0.146 0.326
 

Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

20 
       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

State 

Exhibit 18: P3

 Transition Probability Matrix 

AA AB AC BA BB BC CA CB CC 

A 0.654 0.088 0.033 0.051 0.083 0.023 0.017 0.018 0.033
 

AA 0.653 0.088 0.033 0.051 0.084 0.023 0.017 0.018 0.033
 

AB 0.356 0.059 0.022 0.116 0.225 0.060 0.032 0.047 0.082
 

AC 0.306 0.050 0.021 0.085 0.161 0.048 0.062 0.088 0.180
 

B 0.327 0.056 0.021 0.121 0.237 0.064 0.034 0.051 0.089
 

BA 0.612 0.085 0.032 0.059 0.102 0.028 0.019 0.023 0.041
 

BB 0.323 0.056 0.021 0.124 0.243 0.065 0.033 0.050 0.086
 

BC 0.250 0.045 0.019 0.094 0.184 0.054 0.065 0.096 0.194
 

C 0.272 0.046 0.020 0.086 0.167 0.050 0.066 0.096 0.197
 

CA 0.605 0.084 0.032 0.058 0.100 0.028 0.021 0.025 0.047
 

CB 0.315 0.055 0.021 0.121 0.238 0.064 0.036 0.054 0.097
 

CC 0.236 0.042 0.019 0.087 0.171 0.052 0.071 0.104 0.217
 

Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

State 

Exhibit 19: P4 Transition Probability Matrix 

AA AB AC BA BB BC CA CB CC 

A 0.607 0.084 0.031 0.060 0.102 0.028 0.020 0.024 0.044
 

AA 0.607 0.083 0.031 0.060 0.103 0.028 0.020 0.024 0.044
 

AB 0.415 0.065 0.024 0.096 0.185 0.050 0.032 0.047 0.085
 

AC 0.371 0.059 0.023 0.086 0.165 0.047 0.047 0.067 0.135
 

B 0.395 0.063 0.024 0.100 0.192 0.052 0.034 0.050 0.091
 

BA 0.579 0.081 0.030 0.065 0.114 0.031 0.022 0.028 0.051
 

BB 0.393 0.063 0.024 0.101 0.195 0.053 0.034 0.049 0.089
 

BC 0.332 0.054 0.022 0.093 0.180 0.052 0.050 0.072 0.145
 

C 0.346 0.056 0.022 0.089 0.171 0.049 0.050 0.072 0.145
 

CA 0.574 0.080 0.030 0.065 0.114 0.031 0.023 0.029 0.054
 

CB 0.386 0.062 0.023 0.100 0.194 0.053 0.035 0.052 0.095
 

CC 0.320 0.053 0.021 0.091 0.176 0.051 0.053 0.078 0.157
 

Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

Suppose a family is currently in state B: medium burden at admission. P1 indicates a .291 chance 

of normal burden (state BA) next year. The probability of medium burden (state BB) next year is 

.546, and the chance of excess burden (state BC) is .163. 

Burden tends to increase with program tenure. Suppose a family is currently in state A (normal 

burden at admission). Their chance of excess burden increases monotonically with time. In year 

1, their probability of excess burden (state AC) is .042. P2 indicates their probability of excess 

burden in year 2. The family could transition from A to excess burden in year 2 in three possible 

ways (states AC, BC, or CC) depending on their burden in year 1. The probability of AC or BC 

or CC in year 2 equals .034 + .015 + .019=.068. In year 3, their probability of excess burden is 

.089, and in year 4 it increases to .103. 

21 
 

   

       

     

   

   

   

       

     

 

While there is a trend toward increasing burden, there is also considerable mobility across states 

over time. For a family with excess burden at admission (state C), their chance of normal or 

medium burden increases monotonically with time. In year 1, their probability of normal or 

medium burden (state CA or CB) is .244 + .255=.499. Their chance of normal or medium burden 

in year 2 (state AA, AB, BA, BB, CA, or CB) is .641, and in year 3 it increases to .733. By year 

4 their probability of normal or medium burden is .783. 

Multiple Years of Excess Burden 

Suppose a family is in any given current state. What’s their probability of exactly 2 periods of 

excess burden over the next 4 years? There are ൫

ସ

ଶ൯ ൌ6 possible ways this could happen: {B1B2 , 

B1B3, B1B4, B2B3, B2B4, B3B4}, where Bi=excess burden in year i. While the events are not 

independent, the probability can be computed with the inclusion/exclusion formula.4 

Exhibit 20 reports probabilities of 0-4 years of excess burden during the next 4 years for each 

current state. Suppose a family has had excess burden this year and last (state CC). Over the next 

4 years there is a .383 chance of 1 year, a .339 chance of 2 years, and a .120 chance of 3 years of 

excess burden. The probabilities decrease with burden at admission. For families in state AA, the 

corresponding probabilities are .240, .029, and .001, respectively. The probability of 4 years 

excess burden is extremely low for all households. 

Current

state

Probability of

excess

burden

Probability of Probability of

0 years Probability of

1 year excess

burden

2 years

excess

burden

3 years

excess

burden

Probability of

4 years excess

burden

Exhibit 20: Probability of Multiple Years of Excess Burden, Next 4 Years 

A 0.730 0.240 0.029 0.001 0.000
 

AA 0.731 0.239 0.028 0.001 0.000
 

AB 0.506 0.376 0.104 0.013 0.001
 

AC 0.224 0.418 0.275 0.076 0.007
 

B 0.476 0.388 0.119 0.016 0.001
 

BA 0.688 0.270 0.039 0.002 0.000
 

BB 0.491 0.382 0.111 0.014 0.001
 

BC 0.194 0.407 0.298 0.091 0.010
 

C 0.184 0.405 0.306 0.095 0.010
 

CA 0.650 0.296 0.050 0.004 0.000
 

CB 0.444 0.400 0.135 0.020 0.001
 

CC 0.143 0.383 0.339 0.120 0.015
 

Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

4

 According to the inclusion/exclusion formula, the “the probability of the union of n events 

equals the sum of the probabilities ... taken one at time ... minus the sum of the probabilities ... 

taken two at a time plus the sum of the probabilities ... taken three at a time, and so on” (Ross 

2007, p9). 

22 
           

   

           

   

   

   

   

   

   

   

           

           

           

           

Exhibit 21 reports probabilities of 2-4 years of consecutive excess burden over the next four 

years for each current state. For any given number of years of excess burden, the probability of 

consecutive years is less than or equal to the probability for multiple years. For instance, given 

medium burden at admission (state B), the chance of 2 years of excess over the next 4 years is 

.119, compared to a .058 chance of 2 consecutive years. 

Exhibit 21: Probability of Consecutive Years of Excess Burden, Next 4 Years 

Current

state

Probability

of 2

consecutive

years

excess

burden

Probability

of 3

consecutive

years

excess

burden

Probability

of 4

consecutive

years

excess

burden

A 0.014 0.001 0.000

AA 0.014 0.001 0.000

AB 0.051 0.007 0.001

AC 0.152 0.038 0.007

B 0.058 0.008 0.001

BA 0.019 0.001 0.000

BB 0.054 0.008 0.001

BC 0.165 0.046 0.010

C 0.173 0.048 0.010

CA 0.026 0.002 0.000

CB 0.067 0.010 0.001

CC 0.197 0.061 0.015

Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

Predictions 

To gauge the model’s predictive power, Exhibit 22 reports actual and predicted household shares 

with normal, medium, and excess burden for 4 years past admission. Absolute percentage 

differences between actual and predicted outcomes are also reported. Predicted outcomes Xn 

equals X0Pn, where X0 is a row vector of current household shares in each state, and Pn is the year 

n transition probability matrix. 

Exhibit 22: Actual and Predicted Outcomes for 2000-02, 2003-05, and 2000-05 Admissions 

2000‐2002 Admissions

Normal Burden Medium Burden Excess Burden

Absolute % Absolute % Absolute %

Year Actual Predicted difference Actual Predicted difference Actual Predicted difference

0 0.640 0.278 0.082

1 0.678 0.647 4.707% 0.223 0.240 7.033% 0.099 0.114 14.019%

2 0.669 0.636 5.084% 0.222 0.242 9.015% 0.109 0.121 10.657%

3 0.621 0.631 1.469% 0.246 0.242 1.600% 0.132 0.127 4.085%

4 0.574 0.626 8.753% 0.270 0.243 10.453% 0.156 0.130 17.973%

23 
   

           

   

   

   

   

   

   

   

           

           

           

           

   

           

   

   

   

   

   

   

   

           

           

           

           

2003‐2005 Admissions

Normal Burden Medium Burden Excess Burden

Absolute % Absolute % Absolute %

Year Actual Predicted difference Actual Predicted difference Actual Predicted difference

0 0.593 0.320 0.087

1 0.585 0.620 5.890% 0.281 0.259 8.134% 0.135 0.121 10.581%

2 0.573 0.618 7.546% 0.276 0.254 8.004% 0.151 0.127 17.090%

3 0.593 0.619 4.222% 0.260 0.249 4.132% 0.147 0.131 10.847%

4 0.631 0.619 1.908% 0.240 0.248 3.204% 0.129 0.133 3.120%

2000‐2005 Admissions

Normal Burden Medium Burden Excess Burden

Absolute % Absolute % Absolute %

Year Actual Predicted difference Actual Predicted difference Actual Predicted difference

0 0.615 0.300 0.084

1 0.625 0.632 1.188% 0.256 0.250 2.585% 0.119 0.117 1.635%

2 0.619 0.626 1.220% 0.250 0.248 0.638% 0.131 0.124 5.490%

3 0.607 0.624 2.788% 0.253 0.246 3.043% 0.140 0.129 7.866%

4 0.604 0.622 2.955% 0.254 0.246 3.562% 0.142 0.132 7.472%

Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

Three cohorts of admissions are considered: 2000-02, 2003-05, and 2000-05 (2005 is the last 

admission year with 4 years of subsequent data). If the assumption of fixed probabilities is valid, 

the model should predict equally well regardless of program tenure or calendar year. 

In general, predictions are closest for the largest burden category (normal) and worst for the 

smallest category (excess). For 2000-05 for instance, the maximum difference for normal burden 

predictions is 2.955 %, compared to 3.562 % for medium and 7.866 % for excess burden. For 

2000-05 the model slightly over-predicts the share of households with normal burden, and underpredicts shares with medium and excess burden. 

While model accuracy varies with program tenure, the evidence varies by study period. For 

2000-05 admissions, the model predicts better in earlier years. For instance, the excess burden 

prediction for 2000-05 admissions is off 1.635 % in year 1, compared to 7.472 % in year 4. 

For 2003-05 admissions, predictions become better over time. For instance, the medium burden 

prediction is off 8.134 % for year 1, compared to 3.324 % in year 4. 

One possible reason for differences with program tenure may slightly varying proportional exit 

rates between year 1 and later years (see Exhibit 14). Differences by program tenure might be 

mitigated with states based on more years. Of course there is a trade-off between model 

simplicity and accuracy. Defining states based on up to 3 years of data would require 27 more 

states (AAA, AAB, ... ,CCC), and the transition matrices would increase from 12x12 to 39x39. 

Furthermore, there are far fewer households with 3 years of data than 2. Thus the probabilities 

would be more susceptible to outliers. 

24 
 

           

There is also evidence that the transition probabilities vary with calendar year - the model 

predicts better for 2000-05 compared to the two shorter time periods. For example, the year 2 

prediction for medium burden is off .638 % for the combined sample, compared to 9.015 % for 

2000-02, and 8.004 % for 2003-05. 

The model also tends to predict slightly better for 2003-05 compared to 2000-02. For instance, 

the maximum difference in normal burden predictions is 8.753 % for 2000-02, compared to 

7.546 % for 2003-05. 

If the model is used to make predictions for recent cohorts, evidence suggests it might perform 

better excluding the earliest years of data. Another possibility for improving current predictions 

would be calculating transition matrices by Bayesian updating, giving more recent years of data 

greater weight. 

Variance 

One limitation of the Markov Chain model is that it doesn’t estimate variance. In this section I 

demonstrate a method for doing so by re-sampling the data with replacement (bootstrapping). 

Bootstrapping provides a convenient method for computing confidence intervals directly from 

percentiles of the sampling distribution of the mean (Lohr 2007, p. 307). 

I took 1000 100 % random samples with replacement, generating new transition matrices with 

each iteration. Exhibit 23 reports mean predictions for 2009 admissions for the share of 

households with normal burdens in 2010-13, along with 95 % confidence intervals. Confidence 

limits are the 2.5th and 97.5th percentiles. Exhibits 24 and 25 report corresponding projections for 

shares with medium and excess burden, respectively. The admission shares are .606, .308, and 

.087 for normal, medium, and excess burden, respectively. 

Exhibit 23: Predicted Share of Households with Normal Burden, 2009 Admissions 

0.605

0.61

0.615

0.62

0.625

0.63

2009 2010 2011 2012 2013

Lower 95 % CI Mean Upper 95 % CI

Source: author’s calculations using MTCS/PIC data. Transition probabilities are based on admissions between 2000 

and 2009. 

25 
           

           

Exhibit 24: Predicted Share of Households with Medium Burden, 2009 Admissions 

0.24

0.25

0.26

0.27

0.28

0.29

0.3

0.31

2009 2010 2011 2012 2013

Lower 95 % CI Mean Upper 95 % CI

Source: author’s calculations using MTCS/PIC data. Transition probabilities are based on admissions between 2000 

and 2009. 

Exhibit 25: Predicted Share of Households with Excess Burden, 2009 Admissions 

0.085

0.09

0.095

0.1

0.105

0.11

0.115

0.12

0.125

0.13

0.135

2009 2010 2011 2012 2013

Lower 95 % CI Mean Upper 95 % CI

Source: author’s calculations using MTCS/PIC data. Transition probabilities are based on admissions between 2000 

and 2009. 

26 
 

 

 

         

         

         

         

         

         

         

         

The predicted share of families with normal burden increases in year 2010; the confidence 

interval varies from .627 to .629. Predictions decrease each remaining year, while variance 

increases. By 2013, the predicted share with normal burden ranges from .620 to .623. 

The predicted share of households with medium burden decreases each year past admission, 

while the predicted share with excess burden increases. The predicted range with medium 

(excess) burden is .253-.255 (.119-.120) in 2010, and .250-.252 (.125-.127) in 2011. Rates of 

change decrease in 2012. By 2013, the medium burden prediction ranges from .246 to .248, 

while the excess burden prediction varies from .132 to .134. 

Policy Analysis 

In this section I analyze policy effects of f=FMR(1-r)/income discussed in section III. I define 4 

categories based on 2 burden categories and 2 f categories: 

A) Normal/medium burden, low/medium f -- burden ≤ 40 %, f ≤ .006; 

B) Normal/medium burden, high f -- burden ≤ 40 %, f ≥ .007; 

C) Excess burden, low/medium f -- burden ≥ 41 %, f ≤ .006; 

D) Excess burden, high f -- burden ≥41 %, f ≥ .007. 

The f cutpoint is roughly the 66th percentile. I also define 20 states based on the current year for 

admissions, and the current and prior year otherwise: 

A -- Normal/medium burden and low/medium f at admission; 

AA -- Normal/medium burden and low/medium f in the prior and current year; 

AB -- Normal/medium burden and low/medium f in the prior year, normal/medium 

burden and high f in the current year; 

. 

. 

. 

DD – Excess burden and high f in the prior and current year. 

While transition matrices used for computations contain all possible states, the purpose of the 

model is not to predict HUD and PHA policy. For examining the impact of f on burden, 

probabilities conditional on f are more convenient. Exhibit 26 reports the year 1 transition matrix 

conditional on low/medium f in year 1; Exhibit 27 reports corresponding probabilities 

conditional on high f values. 

Exhibit 26: Year 1 Transition Probabilities Conditional on Low/Medium f 

State AA AC BA BC CA CC DA DC
 

A 0.934 0.066 0.000 0.000 0.000 0.000 0.000 0.000
 

AA 0.952 0.048 0.000 0.000 0.000 0.000 0.000 0.000
 

AB 0.000 0.000 0.950 0.050 0.000 0.000 0.000 0.000
 

AC 0.000 0.000 0.000 0.000 0.553 0.447 0.000 0.000
 

AD 0.000 0.000 0.000 0.000 0.000 0.000 0.714 0.286
 

B 0.000 0.000 0.945 0.055 0.000 0.000 0.000 0.000
 

BA 0.955 0.045 0.000 0.000 0.000 0.000 0.000 0.000
 

27 
         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

         

BB 0.000 0.000 0.957 0.043 0.000 0.000 0.000 0.000
 

BC 0.000 0.000 0.000 0.000 0.579 0.421 0.000 0.000
 

BD 0.000 0.000 0.000 0.000 0.000 0.000 0.751 0.249
 

C 0.000 0.000 0.000 0.000 0.501 0.499 0.000 0.000
 

CA 0.861 0.139 0.000 0.000 0.000 0.000 0.000 0.000
 

CB 0.000 0.000 0.889 0.111 0.000 0.000 0.000 0.000
 

CC 0.000 0.000 0.000 0.000 0.438 0.562 0.000 0.000
 

CD 0.000 0.000 0.000 0.000 0.000 0.000 0.587 0.413
 

D 0.000 0.000 0.000 0.000 0.000 0.000 0.705 0.295
 

DA 0.903 0.097 0.000 0.000 0.000 0.000 0.000 0.000
 

DB 0.000 0.000 0.905 0.095 0.000 0.000 0.000 0.000
 

DC 0.000 0.000 0.000 0.000 0.491 0.509 0.000 0.000
 

DD 0.000 0.000 0.000 0.000 0.000 0.000 0.621 0.379
 

Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

Exhibit 27: Year 1 Transition Probabilities Conditional on High f 

State AB AD BB BD CB CD DB DD
 

A 0.872 0.128 0.000 0.000 0.000 0.000 0.000 0.000
 

AA 0.872 0.128 0.000 0.000 0.000 0.000 0.000 0.000
 

AB 0.000 0.000 0.919 0.081 0.000 0.000 0.000 0.000
 

AC 0.000 0.000 0.000 0.000 0.451 0.549 0.000 0.000
 

AD 0.000 0.000 0.000 0.000 0.000 0.000 0.443 0.557
 

B 0.000 0.000 0.915 0.085 0.000 0.000 0.000 0.000
 

BA 0.906 0.094 0.000 0.000 0.000 0.000 0.000 0.000
 

BB 0.000 0.000 0.937 0.063 0.000 0.000 0.000 0.000
 

BC 0.000 0.000 0.000 0.000 0.467 0.533 0.000 0.000
 

BD 0.000 0.000 0.000 0.000 0.000 0.000 0.512 0.488
 

C 0.000 0.000 0.000 0.000 0.416 0.584 0.000 0.000
 

CA 0.775 0.225 0.000 0.000 0.000 0.000 0.000 0.000
 

CB 0.000 0.000 0.834 0.166 0.000 0.000 0.000 0.000
 

CC 0.000 0.000 0.000 0.000 0.346 0.654 0.000 0.000
 

CD 0.000 0.000 0.000 0.000 0.000 0.000 0.356 0.644
 

D 0.000 0.000 0.000 0.000 0.000 0.000 0.422 0.578
 

DA 0.795 0.205 0.000 0.000 0.000 0.000 0.000 0.000
 

DB 0.000 0.000 0.851 0.149 0.000 0.000 0.000 0.000
 

DC 0.000 0.000 0.000 0.000 0.367 0.633 0.000 0.000
 

DD 0.000 0.000 0.000 0.000 0.000 0.000 0.371 0.629
 

Source: author’s calculations using MTCS/PIC data for admissions between 2000 and 2009. 

Suppose a family is admitted with low/medium burden and high f (state B). In year 1 if f is 

low/medium they have a .945 chance of normal/medium burden (state BA), and a .055 chance of 

excess burden (state AB). If f is high they have a .892 chance of normal/medium burden, and a 

.108 chance of excess burden. 

28 
 

         

         

         

         

Exhibit 28 depicts 4 year excess burden projections for 2009 admissions conditional on 

low/medium and high f. 

Exhibit 28: Projected Share of Households with Excess Burden Conditional on f, 2009 

Admissions 

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

2009 2010 2011 2012 2013

Low/medium f lower 95 % CI

Low/medium f upper 95 % CI

High f lower 95 % CI

High f upper 95 % CI

Source: author’s calculations using MTCS/PIC data. Projections are for 2009 admissions with transition 

probabilities based on 2000-09 data. 

Although predicted burdens tend to increase after admission regardless of f, families with high f 

values are much more likely to have excess burden each year. The share of households admitted 

in 2009 with excess burden is .060 for households with low/medium f, compared to .127 for 

those with high f. By 2013, the predicted share of households with excess burden is about 75 % 

higher for those with high f (.160-.164) compared to those with low/medium f (.091-.093). 

Exhibit 29 reports bootstrap conditional densities for the predicted share of households in 2013 

with excess burden. While bootstrap estimates make it possible to conduct likelihood ratio tests 

for differences in burden according f, there is virtually no overlap in the conditional distributions. 

It’s obvious that the interaction of HUD and PHA policy relative to income has an effect very 

large in both magnitude and statistical significance. 

Exhibit 29: Bootstrap Densities for 2013 Excess Burden Predictions Conditional on f, 2009 

Admissions 

29 
Source: author’s calculations using MTCS/PIC data. Projections are for 2009 admissions with transition 

probabilities based on 2000-09 data. 

VIII Conclusion 

This study models rent burden in the Housing Choice Voucher Program as a Markov Chain. The 

model predicts rent burden with program tenure, using longitudinal household data for 2000 

through 2009. 

Results indicate rent burden increases for many years after admission. This is perhaps not 

surprising, given that restrictions on burden at admission don’t apply in following years. 

Consistent with results for unassisted low-income families (Martin et al. 2005, Susin 2007), the 

model also predicts considerable mobility across burden categories over time. For a family with 

burden over 40 % at admission, their estimated chance of burden below 40 % in their next year is 

50 %. Regardless of a family’s current burden, the probability of 4 years of burden above 40 % 

during the next 4 years is very low. 

The rent burden formula indicates that HUD policy and PHA policy should not be considered in 

isolation. Markov Chain estimates imply that the interaction of Fair Market Rent and the 

Payment Standard have effects large in both magnitude and statistical significance. 

One limitation of the Markov Chain model is that it doesn’t estimate variance. I demonstrate a 

simple method for doing so via bootstrapping. 

I also examine the relationship between burden and program tenure using survival analysis. Exit 

probabilities are low in the 1st year in the program, with little differences due to burden. Chances 

of leaving increase after admission and households with moderate burdens tend to have the 

30 
greatest exit rates. Households admitted with burdens below 32 % tend to stay in the program the 

longest. 

References 

31 
 

 

 

 

 

 

 

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33