JOHN L. GOODMAN, JR.

C A U S E S A N D I N D I C A T O R S O F H O U S I N G Q U A L I T Y

(Received 25 May, 1977)

ABSTRACT. This paper examines the feasibility and validity of one method for com- bining measures of different dimensions of the quality of a household's housing situation into a summary index value. Housing quality is treated as an unobservable variable for which there are multiple observable causes and indicators. Alternative mathematical models are specified, and their parameters are estimated using data from a sample of low-income renter households in a major U.S. metropolitan area.

1. INTRODUCTION

The Congxess hereby declares that the general welfare and security of the Nation and the health and living standards of its people require ... the realization as soon as feasible of the goal of a decent home and a suitable living environment for every American family ....

Section 2 Housing Act of 1949

One problem that has hampered federal efforts to pursue and achieve the goal

established by Congress 27 years ago is the absence of any consensus defini-

tion of a 'decent home and a suitable living environment ' .

I t is far easier to enumerate dwelling units than to evaluate them and

their suitability for their occupants. Yet, quality is no less important than

quanti ty in the pursuit of the Congressionally-mandated goal. There have

been repeated calls for more "meaningful concepts and indicators of housing

deprivation", and for "definit ions that more accurately reflect housing

problems as perceived by households ' ' t . The task is not simple, since there

are a "mult ipl ici ty of housing objectives" and "difficulties which surround

efforts to cast them in measurable terms" 2.

The purpose of the paper is to examine one potential method for combining

measures of different dimensions o f the quality of a family's housing situation

into a summary index value. The usefulness o f such an index for charting the

Social Indicators Research $ (1978) 195-210. All Rights Reserved Copyright �9 1978 by D. Reidei Publishing Company, Dordrecht, Holland

196 JOHN L. GOODMAN, JR.

nation's housing progress over time, or for comparing the housing quality of areas or individuals at a point in time, is apparent.

A current example of the need for good measures of housing quality is provided by the Experimental Housing Allowance Program (EHAP). EHAP is a federally-sponsored, large-scale social experiment that is attempting to

establish the feasibility and desirability of providing subsidies directly (i.e., housing allowances) to low-income households, a Outcome measures are required in order to evaluate the subsidy approach. The evaluation effort would be facilitated if improvements in the quality of a family's housing situation could be expressed by a single index.

Creating a housing quality index is easy. There are an infmite number

of ways of combining any selected set of components into a summary measure. However, creating a housing quality index that is meaningful for use in framing and carrying out housing policy is more difficult. The index needs to be validated according to some reasonable criterion. This paper examines the possibility of allowing the calibration of the index to be determined by the actions of housing consumers. The logic is that if there is some entity called housing quality that is the object of households' housing demand, then one sensible strategy is to let the composition of a housing quality index be determined by consumer's preferences as revealed by their actions in the housing market.

Most existing approaches to housing quality measurement emphasize the characteristics of the dwelling unit and sometimes the neighborhood. But what if the notion of housing quality is broadened to something more like 'quality of a family's housing situation' or 'household housing circumstances'? The object of housing policy is not, after all, housing units. The object is rather the welfare of citizens. This realization calls for a broader definition of housing quality, a definition which considers not only the features of a dwelling unit and neighborhood, but rather those features in relation to the housing needs and financial capabilities of the unit's inhabitants. Has a family achieved a 'suitable living environment' if it occupies a safe and sanitary dwelling unit only by spending half of its income for rent and by crowding two or three family members into each room of the unit? It is clear that if housing quality is construed to be the inverse of housing deprivation, then both housing and occupant characteristics are pertinent to an assessment of housing quality.

Some past research has acknowledged that housing quality should be

HOUSING QUALITY 197

given this broader interpretation and has attempted to estimate the incidence

of housing deprivation not only according to unit and neighborhood charac- teristics, but also in the dimensions of overcrowding and financial burden. 4 This approach compares dwelling unit with household characteristics and

defines housing deprivation as a mismatch between unit and occupants. While the direction of the association of unit and neighborhood charac-

teristics, rent burden, and overcrowding to quality of the housing situation

is clear, far less clear are the exact calibrations of the relationships between the indicators and housing quality. The problem is that housing quality (or its inverse, housing deprivation) is an abstract, theoretical entity with no

straightforward operational definition. One opinion is that housing quality is a subjective concept that is determined according to the eyes of the beholder, so that there is no one combination of indicators to which all observers will agree. Even if this is true, the possibility remains that there is

some weighting scheme that averages diverse opinions in establishing a calibra- tion scheme and resulting housing quality index, s

In this paper, the calibration scheme for combining the indicators or components into a single measure of housing quality is determined by the consumers' preferences as revealed by their behavior in the housing market. The approach is similar to a standard analysis of the demand for housing. The twist is that the dependent variable is an unobservable variable - housing quality. Recently developed statistical methodology is used to solve for the parameters of a model that relates both the causes of housing quality and its indicators to the underlying unobservable variable.

The approach adopted in this paper offers at least the possibility of providing a meaningful method for combining different dimensions of a family's housing situation into one continuous index. Estimation of the model will reveal whether rent burden, crowding, and unit and neighbor- hood characteristics are merely indicators of a single variable called housing quality that is pursued (demanded) by households. If such a variable is found to exist, the model will tell us how the three indicators are to be added up to determine the index value for the quality of a family's housing situation. 6

The next section of the paper describes a statistical model for analyzing both the causes and the indicators of housing quality and also discusses the data set and variables used in this study. The results from estimation of the basic model and an'alternative model are described in the subsequent section. The last section summarizes the findings and draws some conclusions.

198 J O H N L. G O O D M A N , JR .

II. T H E M O D E L A N D D A T A

The analysis in this paper is restricted to one individual housing market. This restriction simplifies our task considerably, since market or supply-

side forces - the price of housing in particular - can be assumed invariant 7 across households and thus do not need to appear as variables in the model. The housing market on which attention is focused is the low-income rental market of a large urbanized area.

Given that the housing supply-type variables are controlled by design, what we are left with is an analysis of how demand-type variables influence housing quality. (In the remainder of the paper, 'housing quality' is used in the broad sense of 'quality of a household's housing situation'.) The first equation in our model specifies the level of housing quality attainment to be a linear additive function of a number of variables:

(1) Y* = ~ l X l +...+akXk+...+r +e,

where Y* is housing quality (unobservable), the X's comprise a set of ob- servable variables that determine a household's housing quality attainment, and e is an unobservable error term.

Housing quality is not directly measurable, but there are multiple ob- servable indicators of housing quality. Specifically, assume that the multiple indicators are linear functions of Y*, as follows:

Y1 = BIY* + Ut

(2) rm =Bm Y* + w,,,

YM = BM Y* + UM

Yl through Ym are observable indicators of the unobservable Y*, and U l through Um am unobservable error terms. The M indicators of I"* are assumed to comprise an exhaustive list. In other words, Y* can be defined in terms of the M indicators. The error terms in the M + 1 equation linear model defined by Equations (1) and (2) are assumed to be independent of the X's.

The model, then, has two components. Equation (1) is the causal com- ponent, and Equations (2) comprise the indicator component. Estimation of the parameters of Equations (1) will establish the sensitivity of housing

HOUSING QUALITY 199

quality to each of the proposed determinants. Estimation of the parameters of Equations (2) will establish the strength of the association of the proposed indicators to the underlying housing quality variable.

The model is specified in its reduced form by substituting Equation (1) into Equations (2). In the reduced form, the indicators are specified as functions of the causes and error terms:

where

Yrn = BmalX l + .... + BrnatcXtc +Brae + Urn,

= f fmiX l + ... + f f m K X K + Vrn,

~rrnk= Brnak (m = 1, .-.,M; k = 1,--.,K)

and the reduced form disturbances are

V m = Brne+U m ( m = 1,.. . ,M).

Data for this study are from a random sample of low-income rental house- holds in Pittsburgh and surrounding Allegheny County, Pennsylvania, in 1973. The data were collected in connection with the Demand Experiment of the Experimental Housing Allowance Program. 8

Consider now the specific variables appearing in the models that will be estimated. In Equation (1), the four variables (K = 4) hypothesized to deter- mine the quality of a family's housing are income, family size, head's educa- tion, and race. These variables appear frequently in the literature as deter- minants of housing consumption?

Annual household income (Xl) . Household income should be positively related to housing quality.

Family size (X~). The larger the family, the greater the amount of space required. Greater space requirements are often met by reductions in other dimensions of housing such as neighborhood quality. As family size increases, the requirements for non-housing expenditures increase as well. Thus, the direction of the independent effect of family size on housing quality attain- ment is probably negative.

Education (.1"3). Education can influence housing quality through at least two channels. First, there may be different tastes for housing consumption relative to other goods and services at different education levels. Second, educational level might be correlated with elements of long run or permanent income not reflected in current annual income. Education is measured as a

200 JOHN L. GOODMAN, JR.

dummy variable taking the value 1 if the household head is at least a high school graduate.

Race (X4). This variable could conceivably show differential tastes for

housing across racial groups, but there is no prior evidence that would sup- port the existence of such differentials. The main purpose of race as a predictor of housing quality attainment in our models is to control for intramarket differences in housing prices and availability that are attributable to racial discrimination and housing market segregation. Households headed by whites are coded 1 on X4 and all minorities (mostly blacks) are coded zero.

TABLE I

Descriptive statistics for input variables (n = 1256)

Variable Description Mean Standard deviation

Yt Rent burden (percent of annual 0.324 0.135 household income spent on rent)

]:2 Crowding (persons per hundred square 0.448 0.221 feet of interior living space)

I:3 Unit & neighborhood quality 85.7 20.6 (estimated hedonic index score)

]:3 a Unit & neighborhood quality 112.4 34.3 (rent & utilities minus adjustment for furnishings)

XI Annual household income (net of 4708 2025 income taxes and alimony payments)

X 2 Family size (number of occupants 3.17 1.78 exclusive of roomers and boarders)

X3 Education (1 = head a high school 0.437 0.496 graduate, 0 = head did not finish high school)

X4 Race (I = white 0.774 0.418 0 = minority, mostly blacks)

SIMPLE CORRELATIONS

YI ]:2 Y3 Y3a XI X2 X3 X4

Yt 1.000 ]:2 - 0.367 1,000 Y3 0.218 -0 .098 1.000 Y3a 0.334 0.096 0.650

Xt -0.652 0.409 0.259 X2 -0 .328 0.805 0.057 X3 -0.083 0.056 0.250 X4 0.023 -0.141 0.171

1.000

0.375 1.000 0.302 0.516 1.000 0.252 0.264 0.066 1.000 0.063 0.034 -0.180 0.031 1.000

HOUSING QUALITY 201

The expectation is that minority households will attain a lower level of hous-

ing quality than whites of the same income, family size, and education. 1~

Having run through the list of predictors of housing quality attainment, we turn now to the dependent variables. The preceding section discussed the nature of housing quality as an unobservable theoretical construct for which there are multiple observable indicators. The model estimated in this paper will consider three (M--3) indicators of housing quality: physical characteristics of dwelling unit and neighborhood; overcrowding or the popu- lation density level within the unit; and the financial burden that the rental of the unit imposes upon the occupant family. The housing quality indicators are defined as follows:

Financial burden (YI). A crude measure is provided by the ratio of rent (including utilities but not furnishings) to household income. A reasonable argument can be made that paying 25 percent of income for rent imposes a greater financial burden on a family with a $5000 annual income than on a family with an annual income of $20 000. But this problem is not acute in our sample, since almost all households are of low income. (Seventy-four percent of the sample households have a gross annual family income of less than $6000. Only about 5 percent of the households have an income higher than $10 000.)

Crowding (Y2). Measured as number of occupants per hundred square feet of interior floor space in the dwelling unit. This is a more termed measure than the standard crowding variable, persons per room.

Unit and neighborhood quality score (Y3). A hedonic index score for each dwelling unit is used here. The hedonic index used in this analysis is an improved version of an index developed and estimated previously, zl The index value reflects the summed market value of dwelling unit and neighbor-

hood characteristics which were found to bear a statistically significant relationship to market rent.

The correlation matrix along with the means and standard deviations of the variables appear in Table I. 12

Ill. SOLVING FOR THE PARAMETERS

The model presented in the preceding section is similar in structure to the MIMIC (Multiple Indicators and Multiple Causes) model discussed by J6reskog and GoldbergerJ 3 The MIMIC model, in turn, is a relatively simple member

202 JOHN L. GOODMAN, JR.

of the very general class of models for which J6reskog has recently developed maximum likelihood solution algorithms. 14 Heuristically, the maximum

likelihood solution algorithm involves choosing values for the parameters that will result in the closest match between the actual variance/covariance matrix of the variables in the system with the variance/covariance matrix

derived from the model's estimates of endogenous variables. The algorithm is discussed in more detail by J6reskog Is and by J6reskog and Van Thillo. ~ The computer program described in the latter reference was used to estimate the models in this paper.

The coefficients that will be displayed in the following diagrams are the path coefficients. Path coefficients are standardized regression coefficients which more readily show the relative importance of the independent variables in the system, since all variables are transformed to have a mean of zero and a standard deviation of one. ~ 7

Figure I presents the estimated path model. Is Looking first at the causes

of housing quality, it can be seen the family size has the strongest effect of the four X's. The path coefficient from X2 to Y* implies that a change of one standard deviation in family size causes an average change of 0.702 standard deviations in Y*. Income has the second strongest impact of the four X's on Y*, and, as with family size, the relationship is positive. Educa- tion and race have minimal impacts on Y*.

Turning to the indicator portion of the model, the crowding variable is the best indicator of housing quality under this specification, as determined by the magnitude of the path from Y* to ]'2. By this criterion, rent burden (Yl) serves less well as an indicator of deprivation, and the hedonic index score (Y3) is the poorest of the three indicators.

The model of Figure 1 restricts the elements of ~r, the reduced-form para- meter matrix defined in the previous section. The restrictions on ~r amount to forcing a proportionality onto the impacts of the four X variables upon the indicators. For example, the influence of income on any one of the indicators is always 53 percent (i.e., ~l/a2) as strong as, and in the same direction as, the influence of family size. This set of restrictions is necessary if all of the causality is to be funnelled through Y*. The funnelling, in turn, is necessary for the derivation of estimated values of Y* for individual households.

Following the principles of path analysis, the B coefficients in the model of Figure 1 are also the values of the (unobserved) correlations between Y* and the indicators. This fact allows the B's to serve as weights to be attached

HOUSING QUALITY 203

x3.. .~~ ~ V 2

v3 . - ~ g a M ~ - u3

Fig. 1. MIMIC model of causes and indicators of housing quality. Note: All coefficients standardized (i:e., beta coefficients). • _- 937 with 15 degrees of freedom, which calls for rejection of the null hypothesis that this model 'fits' the data at the 0.001 level

of significance. Variables def'med in Table I.

to each of the indicators in adding them up into a single measure, Y*. (The analogy is to the use of factor loadings in combining several valuables into a

factor score in factor analysis.) For example, the estimated model in Figure l implies that the package of housing attributes signified by F* is not that closely related to dwelling unit and neighborhood quality (Fs , as measured by the hedonic index score). Thus, F 3 should not be given much weight in adding the three indicators into a composite such as y..19

When the indicator error terms are allowed to be freely correlated, as in Figure l, the MIMIC model is very similar to canonical correlation analysis. 2~ The X coefficients are identical to the factor loadings of the X's onto the first canonical X variable, and the B coefficients are transformations of the factor loadings of the Y's onto the first canonical Y variable. One way of interpreting the results of the model of Figure l is to view the solution as showing the linear combination of the causes of housing quality and the linear combina- tion of the indicators that are most highly correlated with each other. This is essentially what canonical correlation does. 21 A less mechanical, more sub- stantive interpretation is that the a values show the sensitivity of housing quality to its theoretically plausible determinants. The B values show the 'package' (i.e., the linear additive combination) of housing quality attributes that are purchased.

Note that the signs of the paths from Y* to Y2 and }'3 are positive, while

204 JOHN L. GOODMAN, JR.

the sign to Y l is negative. This combination of signs causes problems for the proposed summary measure of housing quality; it follows from these signs that Y* cannot be strictly interpreted as either housing quality or housing deprivation. This is because increases in F* are not unambiguously associated either with improvement or wi[h deterioration in the specific measureable indicators of the housing condition of the household. As Y* increases, rent burden decreases and the hedonic score increases. These represent improve- ments in quality. But at the same time, crowding increases, implying a decline in the family's housing quality. As u increases, quality increases according to two indicators and decreases according to one. 22

Even if F* cannot be interpreted as housing quality, the possibility remains that it is a valid single-dimensional measure of the family's housing situation, reflecting the package of attributes that families buy and the tradeoffs involved in that package. However, a X 2 test calls for rejection of the hypothesis that the model of Figure 1 fits the data. 23 In addition, the generally weak associations between the indicators and F* are causes for concern. Only 26 percent (0.515 2 x lO0 percent) of the variance in rent burden is attributable to variation in F*, and only 3 percent (0.170 2 x 100 percent) of the variance in the hedonic index value is captured by the un- observable F*. The message appears to be that the three indicators cannot be reduced to a single measure of the family's housing situation.

Waiving the restrictions on the reduced-fonn coefficients of ~r is equiva- lent to regressing each of the three indicators directly on all four of the causes. The path diagram for this totally unrestricted model is shown in Figure 2. This procedure allows the observed correlation matrix of all the variables to be perfectly fitted, reducing the X 2 to zero. The unrestricted model of Figure 2 forfeits much of the simplicity of the restricted model of Figure 1. Specifically, the effects of the causes on the indicators of housing quality are no longer constrained to funnel through a single unidi- mensional measure (F*). To compare the model of Figure l with the un- restricted model of Figure 2, look at the coefficient matrices of Table II. The top matrix gives the derived reduced-form estimates, based on the restriction that ~r -- aB'. The bottom matrix gives the path coefficients from regressing each of the indicators on the four Xs directly (as depicted in Figure 2), imposing no restrictions on the values of 7r. The comparison shows that the derived and direct coefficients are generally consistent in displaying only weak dependence of the indicators on education and race of head. But the

HOUSING QUALITY 205

X 1

Y1

X 2

Y2

X3 V3

Fig. 2. Path diagram for totally unrestricted model.

coefficients for income and family size vary considerably between the two approaches. In particular, compared to the direct estimates, the derived

coefficients understate the influence of income on rent burden and the

influence of family size on crowding. On the other hand, the derived

coefficients overstate the dependence of rent burden on family size

and the dependence of crowding on income) 4 In fact, the derived estimate

implies a counter-intuitive positive association between income and crowding.

TABLE 1| Derived and direct reduced-form parameter estimates derived

(&8' from Figure 1)

Xl X2 X, X4 R 2

Y] -0.228 -0.362 -0.021 -0.023 0.27 F2 0,326 0.518 0.030 0.033 0.54 Y3 0.075 0.119 0.007 0.008 0.03

DIRECT [(X~X) -1 X'Yasin Figure 2]

XI X2 Xa X4 R 2

Yl -0.697 0.034 0.096 0.051 0.44 Y2 - 0.011 0.811 0.005 0.005 0.65 Y3 0.228 -0.046 0.188 0.149 0.13

206 I O H N L. G O O D M A N , JR.

Another way of assessing the validity of the model of Figure 1 is by com- paring the explanatory power of that model with the totally unrestricted model of Figure 2. The R 2 statistics in the last column of Table II show the percent of variance in the indicator that is explained when the influence of the causes is 'funnelled' through Y* (derived estimates) and when the causes are allowed to determine the indicator value directly (direct estimates). This comparison shows the unrestricted model to have substantially more

explanatory power.

IV. S U M M A R Y AND C O N C L U S I O N S

This paper has explored the possibility of reducing the dimensionality of the measurement of a family's housing situation. Specifically, we have tested a model in which housing quality is assumed to be a single, continuously measured, but inherently unobservable variable for which rent burden, crowding, and unit and neighborhood features are observable indicators. The determinants of housing quality have been assumed to be income, family size, education, and race.

The analysis leads to the conclusion that there is no single variable called 'housing quality', or perhaps more precisely 'housing situation', that is reasonable in terms of its relationship both to the proposed causes of housing quality and to the examined indicators of housing quality. What is shown is that there is no linear additive combination of the three indicators of housing quality that can be expressed as a linear additive function of the four postulated causes of housing quality and still be consistent with the observed correla- tions among the variables. The presumed causes of housing quality do not affect the indicators equally or even proportionately. The influence of the causes on the indicators cannot be funnelled through a single summary measure. Such funnelling is necessary for the calibration of the proposed index.

If the model had shown that the three indicators were in fact indicators of the same underlying variable called housing quality that households demand, then a convenient population-determined set of weights would have been provided for combining the indicators into an overall quality measure. Any attempt to simplify reality through a model will result in some distortion of reality. It was never expected that the MIMIC model would fit the data exactly. But the model fits the data so poorly that too much reality is forfeited

HOUSING QUALITY 207

for the sake of simplicity - simplicity in this case being the expression of

housing quality as a single variable. This analysis has used data on one county's low-income renter population.

Only one set of determinants of housing quality and only one set of indicators have been examined) s The relationships among all of the variables have been constrained to be linear and additive. All of these restrictions work to limit

the scope of the conclusions that can be drawn from this study. Nonetheless, the rejection of the MIMIC model is so definitive that the basic conclusions should be unchanged under a number of reasonable alternations of the variables and sample. Even if the model had not been rejected, the issue of the 'portability' of the results to other housing markets and time periods would remain.

As stated at the outset, it is not difficult to construct an index for measuring housing quality. It is difficult to construct an index for that can be validated or justified according to some reasonable criterion. In this paper, an attempt has been made to construct an index that acquires validity through its role as the unobserved object of families' housing demand. The conclusion is that the three indicators used in this analysis cannot be combined into a single variable called housing quality that households pursue in the market. Housing depriva- tion - by one interpretation the inverse of housing quality - is the creation, of policy makers and policy analists. Within the framework imposed by the model estimated in this paper, the concept of housing quality has no single counterpart in the preferences of households.

The Urban Insa'tute, Washington, D. C.

ACKNOWLEDGEMENTS

A number of colleagues have reviewed this paper, and it has benefited from their comments and criticisms. Special acknowledgement is due to David Haleru, who provided statistical and data processing assistance for this research.

This research was funded by the U.S. Department of Housing and Urban Development. Opinions expressed are those of the author and do not necessarily represent the views of the Urban Institute or its sponsors.

208 JOHN L. GOODMAN, JR.

NOTES

Arthur P. Solomon and Reilly Atkinson, 'The Nation's Housing Needs: 1975-1980', in Committee on Banking, Housing and Urban Affairs, United States Senate, Estimates o f Housing Needs 19 75-1980, Committee Print (Government Printing Office, Washington D.C., 1975), p. 75. 2 William G. Grigsby and Louis Roscnburg, Urban Housing Policy (APS Publications, Inc., New York, 1975), p. 57. 3 U.S. Department of Housing and Urban Development, Office of Policy Development and Research, Housing Allowances: The 1976 Report to Congress (Government Printing Office, Washington, D.C., 1976). 4 See for example, Arthur P. Solomon, 'Housing Deprivation in the United States', Chapter 2 of Analysis o f Selected Census and Welfare Program Data to Determine the R elation o f Household Characteristics, Housing Market Characteristics, and Administrative Welfare Policies to a Direct Housing Assistance Program, Draft Final Report (Joint Center for Urban Studies of the Massachusetts Institute of Technology and Harvard University, Cambridge, 1974). See also John C. Weicher, 'Policy and Economic Dimen- sions of a "Decent Home and a Suitable Living Environment" ', paper presented at the meetings of the American Real Estate and Urban Economics Association, Washington, D.C., May 21, 1976. A historical review and discussion of these measures of housing deprivation can be found in William C. Baer, 'The Evaluation of Housing Indicators and Housing Stadards: Some Lessons for the Future', Public Policy 24 (1976), 361-393. 5 Hedonic index construction involves this type of weighting, since each feature of the dwelling unit is allocated a 'shadow price' or weight signifying the market value of that feature as determined by the relationship between the feature and the market value of the entire dwelling unit. Market prices, in turn, are determined in part by the aggrega- tion of the diverse preferences of housing consumers. For a discussion of hedonic indices as applied to housing, see Cynthia Thomas and Thomas King, 'Housing Allow~hce Household Experiment Design: Part 3, Response Measures and Scaling Approaches', Working Paper 205-3 (The Urban Institute, Washington, D.C., 1972). 6 Freedom from crowding and dwelling unit/neighborhood quality are more readily interpretable as objects of housing demand than is rent burden, which is more a result of demand for housing and for other goods. A strict interpretation of the model as examining the demand for (freedom from) rent burder[ could cause problems. However, using rent burden as one of several indicators of the quality of a housing situation, which in turn is the object of housing demand, seems legitimate.

This assumption is not entirely correct, since there is some evidence of intra-market variation in the price per unit of housing services. See, for example, Ann B. Schnare and Raymond J. Struyk, 'Segmentation in Urban Housing Markets', Journal o f Urban Economics 3 (1976), 146-166. But since the sample is restricted to the low-income rental submarket of one county, price variations are mitigated and those that remain are in all probability not large enough to influence our conclusions. 8 For a description of the data used in this analysis, consult Abt Associates, Inc., Working Paper on Early Findings (Cambridge, Mass., 1975).

The prior research is reviewed and the determinants of housing consumption are discussed in more detail in Abt Associates, Inc., Working Paper on Early Findings, pp. 92-112. Io This expectation is consistent with the conclusions of other analysis of these same data, as reported in Chapter 4 of Abt Associates, Inc., Early Findings Working Paper. 11 See Jeanne E. Goedert, Larry, J. Ozanne, and Robert W. Tinney, 'Development of Hedonic Regressions for Measuring Housing Quality', Working Paper 216-20 (The Urban Institute, Washington, D.C., 1975). The actual index value used in this analysis is based on neighborhood (Census tract) characteristics as well as the dwelling unit

HOUSING Q U A L ITY 209

characteristics utilized in the referenced report. The hedonic index provides a method for purging reported rent of price differences associated with, for example, race of occupants and length of tenure. The computed hedonic index value is then used to indicate the market-determined value - or level of housing services - provided by a given structure and location. Note that the hedonic index in this analysis is itself an indicator of neighborhood and unit quality, which in turn is one indicator of the quality of a family's housing situation. i~ The positive correlations between income and both family size and crowding could be partially attributable to the sample disign, since larger families had higher income eligibility limits for selection into the sample. However, relatively few of the households had incomes near the eligibility limit, and controlling for the spurious correlation attributable to the sample design causes only slight adjustment to the correlations listed in Table I. ~ Karl G. J~reskog and Arthur S. Goldberger, 'Estimation of a Model with Multiple Indicators and Multiple Causes of a Single Latent Variable', Journal o f the American Statistical Association 70 (1975), 631-639 . J6reskog and Goldberger's MIMIC model imposes the additional constraint that the error terms for the indicator equations be uncorrelated with each other. Preliminary efforts at fitting this basic MIMIC model to our data revealed the assumption of independent error terms among the indicator equations to be untenable. ,4 Karl G. J~reskog, 'A General Method for Estimating a Linear Structural Equation System', in A. S. Goldberger and O.D. Duncan (eds.), Structural Equation Models in the Social Sciences (Seminar Press, New York, 1973). The particular advantages of the J6reskog techniques for estimating our models are the facility of the technique for handling (1) overidentified models and (2) unobservable variables. The maximum likeli- hood solution is the best possible way to handle the 'excess' information of an overident- ified system, and using the excess information to estimate inherently unobservable parameters is straightforward within the JSreskog framework. Is Ibid. ~s Karl G. J6reskog and Marielle van Thillo, LISREL: A General Computer Program for Estimating a Linear Structural Equation System Involving Multiple Indicators o f Un- measured Variables, Educational Testing Service Research Bulletin 72 -56 (Princeton, N.J., December 1972). a~ Path analysis is a method of presenting results from regression analysis that makes the model's assumptions explicit and can facilitate the interpretation of results. Single- headed arrows indicate the assumed direction of causality. The path coefficients give the average change (measured in standard deviations) in the dependent variable that is caused by a change of one standard deviation in the independent variable. Double-headed arrows represent correlations between variables to which no causal association is assigned. Path analysis allocates all of the variance in dependent variables among the causually prior variables. The precent of variance of a dependent variable explained by a model is given by I minus the square of the path from the error term to the dependent variable. For a more complete description of path analysis, see Otis Dudley Duncan, 'Path Analysis: Sociological Examples', American Journal o f Sociology 72 (1966), 1-16. ts To simplify the diagram, the correlations among the four Xs are not shown. Since the indicator error terms are allowed to be freely correlated, the common disturbance, e, is empirically indistinguishable from the correlations among the three indicator errors Ul, U2, and U3. Thus, the allocation of the error between the two components is arbitrary. In the model portrayed in Figure l , �9 has been constrained to zero. For a discussion of this point, see Robert M. Hauser and Arthur S. Goldberger, 'The Treatment of Unobservable Variables in Path Analysis', in Herbert L. Costner (ed.), Sociological Methodology 1971 (Jossey-Boss Inc., San Francisco, 1971), p. 100. as As mentioned previously, the assumption in this paper is that there are no indicators

210 JOHN L. GOODMAN, JR.

of Y* other than the three included in the model. In other words, Y* is defined in terms of the three Y's. If there were unincluded indicators, then the three Y's could not validly be added up into a summary measure. 20 Hauser and Goldberger, 'The Treatment of Unohservable Variables', pp. I 1 2 - I 14. 2J The correlation between the first canonical variate for the vector X and the first canonical varlate for the vector Y is 0.84. The first canonical variate for Y extracts 40 percent of the variance in the vector of indicators, following the formulae in William W. Cooley and Paul R. Lohnes, Multivariate Data Analysis (Wiley, New York, 1971), Chapter 6. 2~ The problem with the direction of the relationships cannot be rectified by transforming one or more of the indicators, since this would cause changes to the input correlation matrix and just another manifestation of the same problem in the resulting estimated parameters.. 23 A likelihood-ratio technique can be used to test the null hypothesis - that the restric- tions imposed by the model are valid - against the alternative that all of the elements of the reduced form matrix are unconstrained. The null hypothesis is that the estimated model fits the data in the sense that the observed correlation matrix can be matched by the correlation matrix as estimated subject to the constraints imposed by the model. The • for the model in Figure 1 calls for rejection of the model at the 0.001 level of sig- nificance. For more on the statistical test, consult J6reskog and Goldberger, 'Estimation of a Model', p. 633. The likelihood-ratio test should not, however, be construed as the def'mitive or exclus/ve measure of the validity of the model. The computed value, as with any chi-square, is a direct function of the sample size. Thus, a model for which the null hypothesis was not rejected when the sample size was 1000 might well be rejected with otherwise identical data, if only the sample size were increased to 2000, Another draw- back of the likelihood-ratio chi-square is that it does not straightforwardly indicate the extent of the deviation of the computed from the observed correlation matrix, but only whether the deviation is great enough to reject the null hypothesis of no deviation other than that attributable to sampling error. 24 A number of studies of housing demand support the direct estimate of a low income elasticity of demand for housing space. One of the earlier and better known studies reaching this conclusion is Margaret G. Reid, Housing and Income (University of Chicago Press, Chicago, 1962). 2s The exception to this statement is that adjusted gross rent (variable Yaa in Table I) was substituted for the hedonic index score (F3) in the models discussed in this paper. This substitution resulted in somewhat different parameter estimates, but does not alter the conclusions drawn.