ECN741: Urban Economics Estimating Housing Demand Professor John Yinger, The Maxwell School, Syracuse University, 2016

Estimating Housing Demand Class Outline  1. Constant Elasticity Case  2. Linear Expenditure System  3. Housing price variable  4. Submarkets and the Demand for Structures  5. Tenure Choice  6. Housing Durability  7. Endogeneity of neighborhood choice  8. Endogeneity of household formation

Estimating Housing Demand Class Outline  1. Constant Elasticity Case  2. Linear Expenditure System  3. Housing price variable  4. Submarkets and the Demand for Structures  5. Tenure Choice  6. Housing Durability  7. Endogeneity of neighborhood choice  8. Endogeneity of household formation

Estimating Housing Demand Constant Elasticity Demand  Let’s begin with a standard formulation of the demand for housing services, H.  The notation ▫ Y = income ▫ P = price of housing services ▫ u = distance from worksite ▫ t = round-trip commuting costs per mile ▫ R = apartment rent = PH ( ≠ land rent, at least not today!!) ▫ V = house value = R/r

Estimating Housing Demand Constant Elasticity Demand, 2  The demand function is where C is a constant  Multiplying both sides by P, we obtain

Estimating Housing Demand Constant Elasticity Demand, 3  Taking logs, we obtain the estimating equation:  In practice, empirical work ignores theory!  Virtually all studies use Y instead of Y- tu; the only exception: Blackley and Follain, Land Economics,  P is usually measured with a metropolitan area construction index; some studies divide R by P to get H;  Some studies use rV instead of R which is fine.

Estimating Housing Demand Constant Elasticity Demand, 4  Example: Zabel, Journal of Housing Economics, March  Uses data from the American Housing Survey (AHS).  The 2001 AHS data, including a few control variables, imply that ▫ θ = ▫ μ =

Estimating Housing Demand Class Outline  1. Constant Elasticity Case  2. Linear Expenditure System  3. Housing price variable  4. Submarkets and the Demand for Structures  5. Tenure Choice  6. Housing Durability  7. Endogeneity of neighborhood choice  8. Endogeneity of household formation

Estimating Housing Demand Linear Expenditure System  Recall this problem with a Stone-Geary utility function:  Recall as well that the resulting demand for H is

Estimating Housing Demand Linear Expenditure System, 2  Now if we multiply both sides by P{u}, we have  This is called a linear expenditure system.  The survival quantities are coefficients to be estimated.  If the price of Z varies in the data, it needs to be included, too.

Estimating Housing Demand Linear Expenditure System, 3  Note that this functional form is quite different from the constant elasticity form.  The linear expenditure system has been widely used in other contexts, but not so much in housing.  An idea for a study: Use standard specification tests to determine which of these specifications is appropriate.  The Davidson/MacKinnon test, for example, calls for including the predicted value from regression A as an explanatory variable in regression B. If it is significant, the specification in regression A adds explanatory power.

Estimating Housing Demand Class Outline  1. Constant Elasticity Case  2. Linear Expenditure System  3. Housing price variable  4. Submarkets and the Demand for Structures  5. Tenure Choice  6. Housing Durability  7. Endogeneity of neighborhood choice  8. Endogeneity of household formation

Estimating Housing Demand The Housing Price Variable  The standard approach allows prices to vary across urban areas, but the basic urban model indicates that P varies within an urban area, too.  Some studies (e.g. Goodman, JUE, May 1988) first regress V on u, A (neighborhood amenities that appear in P ), and X (housing characteristics that appear in H ):  We will examine this type of equation in much more detail later in the course.

Estimating Housing Demand The Housing Price Variable, 2  These studies then predict P based on u and A and use the predicted P in housing demand estimation.  Allowing intra-area variation in P appears to make a big difference:  His price elasticities for owners and renters are and , respectively, much larger (in absolute value) than Zabel’s—as one would expect when correcting the errors in a metropolitan level price variable!  (Zabel knows this but does not have neighborhood variables in his data.)

Estimating Housing Demand Class Outline  1. Constant Elasticity Case  2. Linear Expenditure System  3. Housing price variable  4. Submarkets and the Demand for Structures  5. Tenure Choice  6. Housing Durability  7. Endogeneity of neighborhood choice  8. Endogeneity of household formation

Estimating Housing Demand Submarkets and the Demand for Structures  Zabel’s study with AHS data pools across metropolitan areas, treating each as a submarket.  He allows the coefficients of the X s to vary across metropolitan areas.  Then he defines a “structure price” to be the price of a given housing bundle (set of X s), and “neighborhood price” to be the price of a given neighborhood bundle (set of N s), in each metropolitan area.

Estimating Housing Demand Submarkets and the Demand for Structures, 2  As an aside, some early studies applied the same type of logic to data for a single urban area.  The allow the coefficients of the X ’s to vary across exogenously determined “submarkets” within an urban area.  In my judgment, this approach adds a lot of complexity without much insight.  But the interactions across submarkets are sometimes significant, and some scholars think submarkets are important.

Estimating Housing Demand Submarkets and the Demand for Structures, 3  Back to Zabel: Let N be neighborhood traits (indexed by n ) and X be structural housing traits (indexed by m ). Then Zabel estimates:  The estimated coefficients are allowed to vary across urban areas.

Estimating Housing Demand Submarkets and the Demand for Structures, 4  Zabel’s “structure price index,” P S, is defined by holding the X s at their mean and the λ s at their estimated values  His “neighborhood price index”, P N, is defined by holding the N s at their mean and the η s at their estimated values..  The amount of “structure,” H*, is R/P = R/(P N P S ).  With these terms, he can estimate the constant elasticity demand function using H * as the housing variable and P S and P N as explanatory variables; he finds small, usually significant price elasticities.

Estimating Housing Demand Submarkets and the Demand for Structures, 5  This extension by Zabel goes beyond the single-market setting of an urban model.  In an urban model, H stands for “structure” and P{u, N} is its price.  His approach recognizes that in a multi-market setting, a different notion of “structure price” and “neighborhood price” can be defined.  He is clear about this in his paper. Just remember that his extension only applies when there are multiple markets.  A further extension would be to have a within-market P (a function of u and N ) and across-market structure and neighborhood prices, which is not possible with Zabel’s data.

Estimating Housing Demand Class Outline  1. Constant Elasticity Case  2. Linear Expenditure System  3. Housing price variable  4. Submarkets and the Demand for Structures  5. Tenure Choice  6. Housing Durability  7. Endogeneity of neighborhood choice  8. Endogeneity of household formation

Estimating Housing Demand Adding Tenure Choice  One of the most important behavioral issues in the study of housing markets is tenure choice.  Why do some households decide to buy a house while others choose to rent?  This topic has been widely studied. We will return to it when we study race and ethnicity.  But today it is important because it is a major source of selection bias in estimating housing demand.

Estimating Housing Demand Sample Selection Bias in Estimating Housing Demand  Owner-occupied houses tend to be larger than apartments.  Thus the amount of H (the dependent variable) is correlated—highly—with tenure choice.  This violates the principle that a sample selection rule (a study of just owners or just renters) should not be correlated with the dependent variable.

Estimating Housing Demand Selection Bias in Housing Demand, 2 H Y Owners Renters Error Distribution True Relationship Estimated Line for Owners Estimated Line for Renters H*

Estimating Housing Demand Selection Bias in Housing Demand, 3  One way to handle this is to pool owners and renters.  Use R, apartment rent, as the dependent variable for owners.  Use rV = annualized value as the dependent variable for owners.  But this can be complicated because of differences in tax treatment, depreciation, mortgage interest, maintenance, and risk between owning and renting—all of which affect r.  Owners and renters also may have different elasticities.

Estimating Housing Demand Selection Bias in Housing Demand, 4  Another approach is provided by Goodman (1988 and several more recent articles).  He estimates three equations:  Housing demand for owners  Housing demand for renters  Tenure choice

Estimating Housing Demand Selection Bias in Housing Demand, 5  In his model ψ is the ratio of value to rent, a measure of expected appreciation in owner occupied housing (expected appreciation lowers the real discount rate), which identifies investment incentives.  In addition, λ is the ratio of owner P to renter P, a measure of relative price; f is the probability that a household is an owner; C is a constant, and A is age.  The resulting equations with owner, O, and renter, R, subscripts are:

Estimating Housing Demand Selection Bias in Housing Demand, 6

Estimating Housing Demand Selection Bias in Housing Demand, 7  Now overall housing demand is  Differentiating with respect to income yields the overall income elasticity, θ:

Estimating Housing Demand Selection Bias in Housing Demand, 7  Goodman results, are as follows:  Recall that Goodman also accounts for intra- urban variation in P. If you are interested in housing demand, his work is well worth studying!

Estimating Housing Demand Class Outline  1. Constant Elasticity Case  2. Linear Expenditure System  3. Housing price variable  4. Submarkets and the Demand for Structures  5. Tenure Choice  6. Housing Durability  7. Endogeneity of neighborhood choice  8. Endogeneity of household formation

Housing Demand Theory Accounting for the Long Lifetime of Housing  The long lifetime of housing makes housing decisions different from many other decisions.  One implication is that housing demand is more closely linked to permanent income than to temporary income.  Households may not immediately adjust their housing consumption in response to temporary income shocks.  Thus, temporary shocks are like measurement error, and the income elasticity of demand for housing is higher when permanent income is used instead of current income.

Housing Demand Theory The Long Lifetime of Housing, 2  Olsen (1988 Handbook chapter) emphasizes the long time frame of housing decisions.  Because housing decisions are forward-looking, models of housing demand need to consider wealth, age, and expectations.  At the very least, studies should try to have controls for wealth, education, and age (which help to predict permanent income).

Housing Demand Theory The Long Lifetime of Housing, 3  The Olsen chapter is valuable because it presents all these issues in fairly straightforward models.  He shows, for example, that one might also want to interact age with other parameters in the demand model, because age changes expectations and time horizons!  Moreover, the formulations given above implicitly assume static expectations, and a formal treatment of expectations might alter the estimating equation.

Estimating Housing Demand Class Outline  1. Constant Elasticity Case  2. Linear Expenditure System  3. Housing price variable  4. Submarkets and the Demand for Structures  5. Tenure Choice  6. Housing Durability  7. Endogeneity of neighborhood choice  8. Endogeneity of household formation

Housing Demand Theory The Endogeneity of Neighborhood Choice  Households must decide where to live as well as how much housing to consume.  These decisions are related, so failure to consider neighborhood choice might bias estimates of housing demand.  Papers that make this argument include Rapaport (JUE, September 1997) and Iaonnides and Zabel (J. of Applied Econometrics, Sept./Oct. 2003).

Housing Demand Theory The Endogeneity of Neighborhood Choice, 2  Rapaport’s methodology is complicated.  She estimates a logit model of neighborhood choice and then includes the predicted probabilities in a constant- elasticity housing demand equation.  In effect, she adds to the standard model an additional variable for each neighborhood.  She finds that accounting for neighborhood choice increases the estimated (absolute value of) price elasticities considerably.

Housing Demand Theory Endogeneity of Neighborhood Choice, 3  A similar endogeneity will show up later in the class when we study “hedonics,” which is the name economists give to a regression of house value or rent on neighborhood and housing attributes.  Because households compete for entry into desirable neighborhoods, prices rise with neighborhood quality.  As a result, households simultaneously select housing price and neighborhood quality.  As we will see, this form of endogeneity makes it difficult to study the demand for neighborhood amenities.

Estimating Housing Demand Class Outline  1. Constant Elasticity Case  2. Linear Expenditure System  3. Housing price variable  4. Submarkets and the Demand for Structures  5. Tenure Choice  6. Housing Durability  7. Endogeneity of neighborhood choice  8. Endogeneity of household formation

Housing Demand Theory The Endogeneity of Household Formation  Individuals must decide how to form themselves into households when they make housing decisions.  Single adults may decide to move in with other single adults or with their parents when the price of housing is high.  Elderly parents may move in with their adult children when the price of housing is high.

Housing Demand Theory Endogeneity of Household Formation, 2  A few studies have shown that accounting for household formation decisions is important in estimating housing demand.  See, for example, Boersch-Supan and Pitkin (JUE, September 1988).  This study uses nested logit analysis; in level 1, adults decide on household formation; in level 2 they decide on tenure; and in level 3 they decide on housing type.

Housing Demand Theory