Does Collateral Value Affect Asset Prices? Evidence from a Natural Experiment in Texas
DDoes Collateral Value Affect Asset Prices?Evidence from a Natural Experiment in
Texas ∗ Albert Alex Zevelev
Baruch College, City University of New York
Abstract
Does the ability to pledge an asset as collateral, after purchase, affect its price?This paper identifies the impact of collateral service flows on house prices, exploitinga plausibly exogenous constitutional amendment in Texas that legalized home equityloans in 1998. The law change increased Texas house prices 4%; this is price-basedevidence that households are credit-constrained and value home equity loans to fa-cilitate consumption smoothing. Prices rose more in locations with inelastic supply,higher prelaw house prices, higher income, and lower unemployment. These estimatesreveal that richer households value the option to pledge their home as collateral morestrongly. (JEL R0, R3, G0, E21, E44, G2) ∗ I thank my discussants Marc Francke, Sanket Korgaonkar, Adam Nowak, and Johannes Stroebel fordetailed comments. I thank Sumit Agarwal, Brent Ambrose, Zahi Ben-David, Sean Chu, Anthony DeFusco,Gilles Duranton, Vadim Elenev, Alex Gelber, Todd Gormley, Ben Keys, Olivia Mitchell, Jonathan Parker,Tomasz Piskorski, Nikolai Roussanov, Todd Sinai, Boris Vabson, Stijn Van Nieuwerburgh, Jessica Wachter,Susan Wachter, Vincent Yao, and Yildiray Yildirim, as well as seminar participants at AMES, AREUEA,ASSA, Baruch, Brandeis, Dallas Fed, EFA, EMES, Federal Reserve Board, Houston Finance, IAAE, JohnsHopkins, Maryland, MMM, NASMES, NYCREC, Penn State, PFMC, Philadelphia Fed, Rice, ReCapNet,SEM, UT Dallas, Wharton, and XXVI Finance Forum for helpful comments. I thank Albert Saiz for sharinghis data. I thank Guido Imbens for sharing his synthetic control code. I thank Cheng Chen and Sung Son forexcellent research assistance. I gratefully acknowledge funding from the PSC CUNY Research Foundationunder grant 60202-00-48. All errors are my own.Send correspondence toAlbert Alex Zevelev, 1 Bernard Baruch Way, New York, NY 10010.Email: [email protected]. a r X i v : . [ ec on . GN ] F e b Introduction
Real estate is the largest source of collateral used by American households (FRBNY, 2016).A home can be pledged as collateral for a loan at the time of purchase or after purchasevia a home equity loan (HEL). Theory predicts that an asset’s price should reflect all ofits benefits including options to pledge it as collateral. This paper seeks to quantify if theability to pledge an asset as collateral, after purchase, affects the price.Prior to 1998, a home in Texas could only be pledged as collateral for a purchase mort-gage or home improvement loan. Beginning in 1998, Proposition 8 (a Texas constitutionalamendment inspired by federal tax reform and a circuit court ruling) greatly expanded theset of mortgages available to Texans. Texas homeowners gained access to HELs, cash-outrefinance loans, and reverse mortgages; however, the total value of all new liens on thehome after purchase could not exceed 80% of its appraised price. Because Texas was theonly state with such restrictions, this law change provides a unique source of exogenousvariation; it expanded future debt capacity (allowing homeowners to extract home equityvia HELs) without affecting purchase debt capacity (the amount homebuyers could borrowat purchase was unaffected).While many papers study the impact of home equity borrowing on consumption andinvestment (Section 1.1), the current paper examines the impact on house prices. There arethree related questions. First, what was the total impact of the law change on house prices?Second, which locations were more affected? Third, what was the mechanism throughwhich the law change affected house prices? The law change could have raised house pricesdirectly through demand for collateral service flows or indirectly by affecting other variablesrelated to house prices. The identifying assumption behind the first question, the causaleffect, is that the law change was exogenous conditional on fixed effects and controls. Theidentifying assumption behind the direct mechanism is that the law change did not affectother variables related to house prices. This paper does not estimate willingness to pay forthe embedded option to pledge a home as collateral. This paper exploits this plausibly exogenous law change to identify the impact of col-lateral service flows on house prices. The estimation requires detailed Texas house price Other products that allow future home equity extraction, including cash-out refinance loans and reversemortgages, will be referred to collectively as “HELs” for ease of exposition. See Appendix E for a timeline of relevant laws. Kuminoff and Pope (2014) show that capitalization effects are not equal to willingness to pay. Furthermore, zip codes with higher prelaw house prices, higher income, andlower unemployment rates saw a greater rise in prices, indicating that households in richerlocations valued this option more. While it has been shown that an expansion in purchasemortgage debt capacity has a greater impact on ex ante lower-priced properties (Landvoigt,Piazzesi, and Schneider, 2015), this paper shows that an expansion in future HEL debtcapacity has a greater impact on ex ante higher-priced properties. This can be explained by(1) the greater tax shield available to rich households, (2) the tendency of rich householdsto be more financially literate and aware of these financial products (Lusardi and Mitchell,2014), and (3) the tendency of rich households to be more likely to qualify for HELs as theytend to have better credit and more stable income (DeFusco and Mondragon, 2018).The heterogeneity analysis refines the interpretation of the main result. The impact ofHEL legalization on house prices is not just a measure of the extent to which householdsexpect to be credit-constrained, but also a measure of the extent to which householdsexpect to qualify for HELs to smooth consumption. For example, a recently unemployedhomeowner may want to borrow but not qualify for a HEL. On the other hand, a homeownerwith high stable income (hence a high marginal tax rate) and good credit is more likely tobe approved for a bigger loan at a lower effective interest rate. The mechanism through which HEL legalization increased house prices could haveoccurred through two broad types of channels: This paper uses supply elasticity as a source of heterogeneity, not for identification. The values of these variables are averaged for each zip code before the law change. Stolper (2015) found that richer homeowners are more likely to use HELs for their children’s tuition.
3. Direct Channel: the law caused a rise in demand for owner-occupied housing due tothe new option allowing homes to be pledged as collateral after purchase.2. Indirect Channel: the law affected other variables that affect house prices. For ex-ample, if the law increased investment enough to stimulate the local economy, thisincrease could have raised demand for housing, consequently raising the price.The mechanism is investigated in two ways. First, falsification tests show that the lawchange did not affect observed variables related to prices including rent, population, andincome. There was a small (0.3%) but statistically significant rise in unemployment, whichworks against the indirect channel. Second, estimates in border cities (such as Texarkana)are similar to the bigger samples, providing further support for the direct channel. Bordercities are often viewed as one economy. Hence, if the law change stimulated the economy,the indirect channel would affect both the Texas and control sides of the border city, whilethe direct channel would only affect the Texas side. Together, these results provide evidencethat most of the effect was through the direct channel.Finally, in partial equilibrium the law change should have increased demand for owner-occupied housing relative to renting (in Texas) because owners can pledge their home ascollateral but renters cannot. On the other hand, in general equilibrium, the rise in houseprices from this law change should have caused a reduction in demand. The net effect ofthis law change on homeownership is theoretically ambiguous. Estimates show that Texashomeownership rates, single-family building permits, and population were unaffected by thelaw. These results indicate that house prices rose enough to keep the marginal homebuyerindifferent between owning and renting.Inference about external validity to the collateral value of other assets, in other loca-tions, at other times should be made with caution. This paper finds that after-purchasecollateral service flows had a positive impact on the price of owner-occupied housing inTexas after 1998. The conceptual framework predicts that there should be a positive effectfor other assets, in other locations, at other times. In particular, the effect is likely higherfor housing in other states, because Texas is the only state with an 80% limit on futurehome equity extraction. Unfortunately, it is hard to quantify the collateral value of otherassets because it is rare to find large discontinuous changes in lending laws.This paper makes three contributions:1. It empirically identifies if, and to what extent, the option to pledge a home as collateral,4fter purchase, affects its price.2. It provides price-based evidence that households are borrowing-constrained. The treat-ment effect is a measure of the extent to which households value HELs to facilitatefuture consumption smoothing.3. It provides new evidence on the distribution of credit supply. Richer households thatget a greater mortgage interest tax shield, are more financially literate, and are morelikely to qualify value HELs more strongly.
Many strands of literature study borrowing constraints and collateral. Theoretical literaturehas shown that frictions such as adverse selection, moral hazard, and the inability to fullypledge future labor income restrict borrowing. Borrowers often pledge collateral to mitigatethese frictions.A second strand of literature studies the relationship between housing, credit con-straints, and consumption. Households that are, or fear they will be, credit-constrainedshould have a stronger demand for assets that facilitate their future ability to borrow. Ifprices reveal information, the magnitude of the treatment effect estimated in this paper canbe interpreted as a measure of the extent to which households value HELs as a tool to relaxcredit constraints.A third strand of literature studies the relationship between property prices, firm in-vestment, and entrepreneurship. There is evidence that rises and declines in propertyvalues, which affect debt capacity, amplify firm investment in the United States and Japanbut not in China. While theory predicts that firms prefer to own assets that serve as bet-ter collateral, none of these papers empirically identifies if and to what extent asset pricesreflect this. Barro (1976); Hart and Moore (1994); Lustig and Van Nieuwerburgh (2005). Agarwal, Hadzic, and Yildirim (2015); Agarwal and Qian (2017); Bhutta and Keys (2016); Carroll,Otsuka, and Slacalek (2011); Chen, Michaux, and Roussanov (2016); DeFusco (2018); Hurst and Stafford(2004); Leth-Petersen (2010); Mian and Sufi (2011); Sodini, Van Nieuwerburgh, Vestman, and von Lilienfeld-Toal (2017). Chaney, Sraer, and Thesmar (2012); Kerr, Kerr, and Nanda (2019); Schmalz, Sraer, and Thesmar(2017); Wu, Gyourko, and Deng (2015). In particular, Huang and Kilic (2018) argued that gold is a bettersource of collateral than platinum for historical and institutional reasons. For example, goldis formally recognized as collateral by the Basel Accords and is accepted as collateral bybroker dealers, while platinum is not. They contended that, in times when the probabilityof a consumption disaster is high, agents prefer gold for its collateral benefits, which isreflected in the price. The results in this paper provide support for the “Flight to Collateral”phenomenon (Fostel and Geanakoplos, 2008) as distinct from “Flight to Liquidity”, ashomeowners value the collateral privileges of real estate despite its lack of liquidity.A fifth strand of literature studies the impact of total mortgage leverage (includingpurchase mortgages) on house prices. In particular, Favilukis et al. (2017) showed that(total) mortgage leverage played a quantitatively important role in explaining the housingboom-bust cycle.It is worth emphasizing that this paper focuses on nonpurchase mortgage leverage asopposed to total or purchase leverage. A purchase mortgage can be used to buy a house Argyle, Nadauld, Palmer, and Pratt (2018); Benmelech and Bergman (2009, 2011); Huang and Kilic(2018); Mann (2017); McAndrew and Thompson (2007); Chen, Chen, He, Liu, and Xie (2019). Adelino, Schoar, and Severino (2012); An and Yao (2016); Anenberg, Hizmo, Kung, and Molloy (2017);Arslan, Guler, Kuruscu, et al. (2020); Di Maggio and Kermani (2017); Favara and Imbs (2015); Favilukis,Ludvigson, and Van Nieuwerburgh (2017); Greenwald and Guren (2019); Orlando (2018); Bednarek, te Kaat,Ma, and Rebucci (2019). This was not a relevant factor in Texas because the law includedan 80% LTV limit for all liens after purchase. Note that there is different terminology in the literature for how the ability to pledgean asset as collateral affects its price. Fostel and Geanakoplos (2008) call this the collateralvalue, Brumm, Grill, Kubler, and Schmedders (2015) call this the collateral premium, andHe, Wright, and Zhu (2015) call this the liquidity premium.
Texas lending laws have been studied widely in the history, legal, and recently economicliterature. Restrictions on mortgage lending existed in Texas before it became a U.S.state. These laws, protecting homes from creditors, carried over to the state’s original con-stitution in 1845 and to subsequent versions (Internet Appendix E). Before 1998, the Texasconstitution protected homes from foreclosure except for nonpayment of property taxes, thepurchase mortgage, mechanics’ liens for home improvement, and refinance loans. UnderTexas law, refinance loans were only allowed up to the balance, permitting homeowners toget a lower payment (if interest rates fell) but not to borrow an additional amount. Sup-pose a Texan borrowed $ $ $ Conforming loans with LTV >
80% require private mortgage insurance (PMI). For example, a borrowerwho wants a 90% LTV loan can avoid PMI by getting a first lien mortgage with 80% LTV and a secondlien mortgage with 10% LTV. The 80% limit is based on the appraised value of the property at the time of future equity extraction. Abdallah and Lastrapes (2012); Forrester (2002); Kumar (2018); McKnight (1983); Texas Legisla-tive Council (2016). Home improvements require building permits and raise property taxes based on the reassessed value ofthe home; hence borrowers were unlikely to use mechanics’ liens to extract equity for consumption. Hence cash-out refinance loans were illegal before 1998.
7t al. (2016) find that 61% of all refinance mortgages in the United States are cash-outloans. The fraction of cash-out loans was over 80% in years when the interest rate incentivewas high. Before 1998, the only way for a Texas homeowner to extract home equity was tosell the home at substantial transaction costs. The literature (Internet Appendix E) has linked the movement to amend Texas lendinglaws to two main factors: federal tax reform and a circuit court ruling. First, the Tax ReformAct of 1986 made mortgage interest the only form of interest on consumer credit that is taxdeductible. This tax shield made HELs more attractive than other forms of debt such as autoloans, personal loans, and credit card debt. While HELs expanded throughout the UnitedStates after this tax reform, Texans did not have access to this additional tax shield. Second,the Fifth Circuit (with jurisdiction over Louisiana, Mississippi, and Texas) ruled in 1994 thatfederal regulations superseded the Texas constitution. This temporarily overturned Texasrestrictions on HELs. Even though subsequent actions by lawmakers quickly reestablishedthe restrictions, this ruling brought attention to amending Texas mortgage laws.Since these lending restrictions were in the Texas constitution, they could only beoverturned by a constitutional amendment. This requires a joint resolution from both thestate Senate and House. If the resolution is approved with a two-thirds vote in each, itbecomes a proposition, which must then be passed by a majority of the state’s voters in areferendum.Joint Resolution 31, to amend the state constitution and legalize home equity lending,was passed by the Texas House and Senate in May 1997. This resulted in Proposition 8,which was approved by Texas voters in November 1997. The amendment became effectiveJanuary 1, 1998, legalizing after-purchase mortgages with a combined balance of up to 80%of the appraised market value of the home. Suppose that after 1998, a Texan’s home wasappraised to be worth $ $ × $ $ $ $ The transaction cost of selling a home is ≈ This section constructs a model to help interpret the results below. The model has threemain goals: (1) to illustrate how pledgeability affects the standard asset pricing equation, (2)to study the heterogeneity of collateral service flows, (3) to clarify the different mechanismsthrough which collateral service flows affect house prices. Consider a household that valuesnondurable consumption c t and durable housing h t , which depreciates at rate δ t . Thehousehold can borrow up to κ t ≡ min { κ t , LT V t } of its house value, where κ t is the legallimit and LT V t is the most lenders will lend at time t . It solvesmax E ∞ (cid:88) t =0 β t u ( c t , h t ) s.t. c t + p t h t +1 + a t +1 ≤ y t + p t h t (1 − δ t ) + (1 + r t ) a t (DBC λ t ) − a t +1 ≤ κ t p t h t (CC µ t )where u ( c t , h t ) is the household’s flow utility from consumption and housing, assumed twicecontinuously differentiable and strictly concave in each argument. λ t is the multiplier onthe dynamic budget constraint (DBC) and µ t is the multiplier on the collateral constraint(CC). c t is the num´eraire, and p t is the price per unit of housing. a t is the amount savedor borrowed at interest rate r t , and y t is income.The solution to the household’s problem (derived in Appendix F) reveals that houseprices reflect service flows from shelter and collateral: p t (cid:124)(cid:123)(cid:122)(cid:125) price = E t β u c ( t + 1) u c ( t ) (cid:124) (cid:123)(cid:122) (cid:125) discount factor × u h ( t + 1) u c ( t + 1) (cid:124) (cid:123)(cid:122) (cid:125) housingservice flow (rent) + µ t +1 κ t +1 p t +1 u c ( t + 1) (cid:124) (cid:123)(cid:122) (cid:125) collateralservice flow + (1 − δ t +1 ) p t +1 (cid:124) (cid:123)(cid:122) (cid:125) resale price (1a) µ t +1 = u c ( t + 1) − E t +1 [ β (1 + r t +2 ) u c ( t + 2)] (1b) The collateral constraint can be written in different ways. See Appendix J for a comparison. µ t +1 >
0, and its FOCs reflect the collateralservice flow. Any model with a collateral constraint has this type of collateral term (denoted
CSF t ). Even though it is ubiquitous, it is hard to quantify, as it is rare to find a settingwith a large exogenous shock to the pledgeability of housing κ t .In equilibrium, the only way the collateral service flow can be positive is if there is atleast one unconstrained household that is lending with µ t +1 = 0. Hence a general equilibriummodel requires heterogeneity for CSF t +1 >
0. A convenient way to model this heterogeneityis with an impatient borrower and a patient lender (Iacoviello, 2005; Kiyotaki and Moore,1997). An alternative way to model positive collateral service flow is to allow interestrates to be exogenous and to assume the representative agent borrows from a deep-pocket,risk-neutral, international lender (Bianchi et al., 2012).The results above can help illustrate heterogeneity in the collateral service flow. First,index each household with superscript j . Second, rewrite the effective interest rate forhousehold j as r jt = (1 − τ jt ) i jt , where τ jt is household j ’s marginal income tax rate and i jt is the contract rate charged to household j . Finally, index household j ’s credit constraint κ jt ≡ min (cid:8) κ t , LT V jt (cid:9) . Note that (1) households with higher marginal income tax rates τ jt enjoy a bigger mortgage interest deduction and (2) different households are chargeddifferent interest rates and credit limits depending on their risk (measured by their creditscore and stability of income). The collateral service flow for household j can be decomposedfrom Equations (1a) and (1b): CSF jt (cid:124) (cid:123)(cid:122) (cid:125) collateralservice flow = 1 u c ( t ) × µ jt (cid:124)(cid:123)(cid:122)(cid:125) desire to borrow × κ jt (cid:124)(cid:123)(cid:122)(cid:125) fraction you can borrow × p t (cid:124)(cid:123)(cid:122)(cid:125) value of collateral (2a)= 1 u c ( t ) × (cid:0) u c ( t ) − E t [ β (cid:0) − τ jt ) i jt (cid:1) u c ( t + 1)] (cid:1) × κ jt × p t (2b)This decomposition shows that the value of being able to pledge housing as collateral de-pends on the desire to borrow µ jt and the debt capacity κ jt p t (the amount a homeowner canborrow), which depends on j (cid:48) s access to credit.The component µ jt (the multiplier on the collateral constraint) is a measure of the Bianchi, Boz, and Mendoza (2012); Greenwald (2016); He et al. (2015); Iacoviello (2005). A progressive income tax implies a regressive mortgage interest deduction. µ jt = 0,making CSF jt = 0. The collateral service flow can still affect property prices today if thereis a desire for HELs in the future. Observe from equation 2b that the desire to borrowdepends negatively on the effective interest rate that household j faces. Households thatface lower effective interest rates (because they have better credit scores or more stabledocumented income) should have a greater demand for HELs. The estimates in Section 5.3are consistent with this prediction.The component κ jt ≡ min (cid:8) κ t , LT V jt (cid:9) depends on (a) the amount households can legallyborrow ( κ t ) and (b) the amount lenders are willing to lend ( LT V jt ). If HELs are illegal( κ t = 0), equation 2a implies CSF t = 0, as in Texas before 1998. Abdallah and Lastrapes(2012) and Kumar (2018) showed that HELs and cash-out refinance loans were indeedutilized by Texans after 1998. It is worth noting that the amount lenders are willing tolend,
LT V jt , depends on the borrower’s characteristics: borrowers with better credit scoresand stable documented income are more likely to qualify for bigger loans (Brueggeman andFisher, 2011). These underwriting requirements help explain the heterogeneity results inSection 5.3.The model above does not distinguish between the ability to pledge an asset as collateralat the time of purchase and in the future. Appendix G considers a three-period setting wherepurchase mortgage collateral service flows are disentangled from HEL collateral service flows.In this setting, the borrower’s HEL debt capacity is equal to κ t p t h t − B pmt , where B pmt is theremaining balance on the purchase mortgage used to buy the home. For example, considera household that owns a home appraised to be worth p t h t = $100 k with a remaining balanceon its purchase mortgage of B pmt = $50 k . If this household can borrow up to κ t = 80% ofthe price, then its HEL debt capacity would be 80% × $100 k − $50 k = $30 k . This paper estimates the impact of the HEL legalization on various outcome variables witha generalized difference-in-differences (DID) methodology. The main analysis uses fourgeographically nested samples (United States, Border-State, Border, and CBCP), explained The Texas constitution set κ t = 0 for after-purchase mortgages prior to 1998, and κ t = 0 .
80 after. Only zip codes that include both Texas and non-Texas locations within a 50-mileradius are kept. The fourth, and most local, sample is the Contiguous Border County Pair(CBCP) sample. This sample consists of all zip codes that belong to a contiguous countyon either side of the Texas border. Since a given Texas county may border more thanone non-Texas county, data in this sample are stacked as in Dube et al. (2010), making itpossible to identify county pair by year fixed effects. The local samples can help control for unobserved heterogeneity to the extent thathouses near each other are more likely to be affected by the same local shocks such as hurri-canes and factory shutdowns. In particular, the border city Texarkana, the only Metropoli-tan Statistical Area (MSA) in Texas that includes another state, can be viewed as oneeconomy. So if the law change affected the economy on the Texas side of Texarkana, itshould have also affected the economy on the Arkansas side.This paper estimates: y i,s,t = α i + θ t + β DID
T exas s × P ost t + Γ X i,s,t + ε i,s,t (static DID) y i,s,t = α i + θ t + (cid:88) k (cid:54) =1997 η k T exas s × k + Γ X i,s,t + ε i,s,t , (dynamic DID)where y i,s,t is the outcome variable. In the main regressions, it is the log real house priceindex. In further regressions, real house price growth, log real rent, log population, log realincome per capita, unemployment rate, homeownership rate, and log single-family buildingpermits are also considered. The index i corresponds to the most local level of the outcome Dube, Lester, and Reich (2010); Heider and Ljungqvist (2015); Severino and Brown (2017). This stacking procedure creates more observations in the CBCP sample than in the Border sample,despite the fact that the CBCP sample is nested in the Border sample. i is the zip code. For population, income, employment,and permit regressions, i is the county Federal Information Processing Standard (FIPS)code. For rent and homeownership rate regressions, i is the MSA. The index s correspondsto the level of treatment, which is the state in all regressions. T exas s is the treatment groupindicator variable, which is equal to 1 if s = T exas . P ost t is the treatment period indicatorvariable, which is equal to 1 for t ≥ k is a year indicator variable, which is equal to1 for t = k . α i and θ t are the location and time fixed effects. ε i,s,t is an error term assumedto be conditionally uncorrelated with the treatment.Specifications in the main analysis include location fixed effects, time fixed effects,time trends, and national oil prices interacted with MSA dummies. The CBCP sample alsoincludes county pair by year fixed effects. Robustness tests also consider national interestrates interacted with state dummies as well as real income per capita and population.This paper also investigates heterogeneity in the treatment effect – that is, whetherthe law change had a different impact in different locations. To study the sensitivity of theeffect to various observable measures of heterogeneity H i , this paper estimates: y i,s,t = α i + θ t + β H, T exas s × P ost t + β T exas s × H i + β P ost t × H i + β H T exas s × P ost t × H i + Γ X i,s,t + ε i,s,t (DDD)In this specification, the average treatment effect (ATE) is an affine function of H i :ATE ( H i ) = β H, + β H H i The coefficient β H, is the estimated average treatment effect if H i = 0, and β H = ∂ ATE( H i ) ∂H i is the sensitivity of the average treatment effect to a rise in H i .For example, theory predicts that a rise in demand should have a smaller impact onhouse prices in elastically supplied locations where it is easier to build real estate (Figure 4).This corresponds to the hypothesis β elasticity <
0. The coefficient β elasticity, is the estimatedimpact of the law change on prices in a hypothetical location where the asset (housing) isin perfectly inelastic supply.This paper investigates treatment effect heterogeneity in elasticity, prelaw median houseprices, income, and unemployment. Prelaw variables are set equal to their average valuebefore 1998 to ensure they are unaffected by the treatment.13 .2 Identification The main goal of this paper is to identify the impact of the HEL legalization on houseprices. The identifying assumption is that the error term is conditionally uncorrelated withthe treatment. The identifying assumption can be defended because the constitutionalamendment, approved by the state House, Senate, and ultimately by Texas voters in areferendum, was motivated by federal tax reform and a circuit court ruling (Section 2).These factors are not linked to Texas house prices and other outcome variables studied in thispaper. Forrester (2002) reviews the extant literature detailing the adoption of Proposition8, and none of the legislative histories suggest that the law was passed with any intent tostimulate the economy.Several further steps are taken to defend the identification of the main effect: theimpact of the law change on house prices.1. Various levels of fixed effects control for different types of unobserved heterogeneity.Zip code fixed effects remove time-invariant zip code-specific differences.Time fixed effects remove location-invariant time-specific differences.County pair by year fixed effects (in the CBCP sample) remove border county pair byyear differences.Estimates remain positive and statistically significant across specifications.2. Four geographically nested samples and four border city samples control for localunobserved heterogeneity such as hurricanes and factory shutdowns.Estimates remain positive and statistically significant across sample restrictions.3. Various covariates at national and local levels are included to control for observedtime-varying sources of heterogeneity.The estimates remain positive and statistically significant.4. This paper re-estimates the main effect with the ZHVI, which is constructed differ-ently from the FHFA index, to test if the results are robust to the method used formeasuring house prices.The estimates remain positive, statistically significant, and similar to the main esti-mates using the FHFA data. .The main analysis does not use ZHVI data because it begins too late in April 1996.
14. Dynamic estimates are analyzed to rule out the risk of upward sloping pretrends.Pretrends are parallel across samples conditional on fixed effects and controls (nationaloil prices interacted with MSA dummies). For two of the samples, Border State andCBCP, the oil price interactions in the baseline specification are required for parallelpretrends.6. Synthetic control estimates are analyzed to further investigate the pretrends, andplacebo tests are conducted in untreated zip codes.The synthetic control estimates are similar to the main results and there is no placeboeffect in untreated zip codes.7. A heterogeneity analysis investigates whether zip codes located in relatively inelasticMSAs (where it is harder to build) experienced a bigger treatment effect. This is asanity check to see if the results are consistent with standard theory (Figure 4). Zip codes in relatively inelastic MSAs had a bigger treatment effect.8. Falsification tests investigate if other outcome variables (log real rents, real income percapita, and unemployment rates) were affected. If the law change was correlated withpolicies designed to stimulate the economy, these other variables would be positivelyaffected as well.There was a small (0 . E [ ε i,s,t | Texas s × Post t , α i , θ t , X i,s,t ] = 0.While the fixed effects and border samples help control for several levels of local unobservedheterogeneity, they alone cannot eliminate state specific time-varying differences. Within-state time-varying fixed effects, such as county by year fixed effects, cannot be included While some authors criticize using this measure as an instrument for house prices (Davidoff, 2016), theestimates here do not use supply elasticity for identification but as a source of heterogeneity. s × Post t . For example, supposethere was a tax cut passed in Texas but not in other states around the same time as theHEL legalization. This tax cut could affect Texas house prices and would not be absorbedby the fixed effects. In this case, it is not possible to separately identify the impact of theHEL legalization from the tax cut. If this type of Texas-specific shock existed at the sametime as the treatment, we could not interpret the coefficients β DID and η k as the causaleffect of the HEL legalization on house prices.This paper argues that this type of contemporaneous Texas-specific shock is not likelybased on analysis of the law change and falsification tests. The paper explores the in-stitutional setting in the legal, history, and economic literature in Section 2 and InternetAppendix E. Federal tax reform and a circuit court ruling motivated the HEL legalization.None of the legislative histories suggest that the law was passed with any intent to stimulatethe economy. Moreover, if such a contemporaneous Texas-specific shock existed, it wouldhave likely affected variables related to house prices. The falsification tests discussed above,both in this paper and in other papers (Kumar (2018); Kumar and Liang (2019)), find thatthese variables were not affected in a way that would stimulate house prices.A secondary goal is to investigate the mechanism behind the main effect. The mecha-nism is investigated in two ways. First, falsification tests are conducted on observed localeconomic outcome variables. If the law change had a positive effect on these variables, thatwould provide evidence that the indirect mechanism played an important role. Estimatespresented in Table 6 show that the HEL legalization did not lead to economically or statis-tically significant changes in real rents, population, or real income per capita. There wasa small (0.3%) but statistically significant rise in unemployment, which works against theindirect channel. Second, the indirect effect should at least partially cancel out in the bordercity samples, to the extent that they can be viewed as one economy. Estimates in bordercities remain positive, providing additional evidence that the effect was mostly direct.Under additional (stronger) assumptions, the regression coefficients can be linked tothe model (Appendix F). If the law had no indirect impact on prices, housing supply wasperfectly inelastic and households were fully aware of and understood the new home equityextraction products, and if the functional forms for the representative Texas homebuyer areas specified in the model, then the first coefficient in the dynamic difference-in-differences16egression can be tied to the model: η = (cid:18) P DV ( CSF ) p (cid:19) Texas = E (cid:104)(cid:80) ∞ j =1 (1 − δ ) j − M , j × CSF j (cid:105) p Under these assumptions, the regression coefficient η is equal to the present value ofHEL collateral service flows (which themselves depend on future house prices) divided bythe prelaw price.The model teaches us that house prices are the present value of housing service flows(rent) and collateral service flows. The HEL legalization had an indirect effect if it affectedhousing service flows. Housing service flows can be directly measured by rents, or indirectlymeasured by variables that determine housing demand and housing supply. The falsificationtests discussed above show that the law change did not affect rents directly, nor did it affecteconomic outcome variables that affect housing demand. Further estimates show that thelaw change did not affect housing supply, as single-family building permits were not affected.In summary, there are several nested levels of interpretation of the treatment effectcoefficients β DID and η k . If we assume that the law change was exogenous conditionalon fixed effects and controls, β DID is the causal effect of the HEL legalization on Texashouse prices. If we also assume that the law change did not have a significant effect onother variables that affect house prices, β DID is the causal effect of the HEL legalization onTexas house prices through the direct, collateral service flow, channel. If we also assume thestructure of the model described above and in Appendix F, η is equal to the present valueof HEL collateral service flows (which themselves depend on future house prices) dividedby the prelaw price. The data used in this paper are summarized in Table 7. The main outcome variable used inthis study is the log real house price index. It is notoriously hard to access detailed Texashouse price data because Texas is a nondisclosure state. Household-level data sets such asCoreLogic (and DataQuick) do not have good Texas data for the relevant time periods. TheZillow Home Value Index (ZHVI) is not used in the main analysis because the data begin Except for a small rise in the unemployment rate, which if anything would reduce housing demand andthus the indirect effect. Zip codesin sparsely populated locations tend to be larger than zip codes in densely populated areas.While parts of the Texas border are sparsely populated, the Border sample has 110 zip codes(within 50 miles of the Texas border) and the CBCP sample has 73 zip codes (in contiguousborder counties).Data used for controls, heterogeneity analysis, and falsification tests come from severaldifferent sources. Supply elasticity data are available at the MSA level from Saiz (2010).Employment data at the county level are from the BLS. Median house price data at thezip code level and rent data at the MSA level are from Zillow. Income data at the countylevel are from the BEA. Population data at the county level, single-family building permitdata at the county level, and homeownership rate data at the MSA level are from theCensus. Homeowner survey data at the household level are from the American HousingSurvey (AHS). One must be careful in merging the data sets since the same zip code canbe located in more than one county. Each zip code is assigned to the county with themaximum allocation factor (e.g., if 75% of zip code z is in county A and 25% is in county B , then zip code z is assigned to county A ). U.S. oil price data are from the EIA. U.S.interest rates are constructed as in Himmelberg, Mayer, and Sinai (2005) by correcting the10-year Treasury bond rate for inflation with the Livingston Survey of inflation expectations.Nominal variables are deflated using the CPI for all urban consumers from the BLS as inGlaeser, Gottlieb, and Gyourko (2012). For details about this new data set, see Bogin, Doerner, and Larson (2019). Estimates
This section presents and discusses the empirical analysis. First, Section 5.1 examinessummary statistics and raw house price growth in the treatment and control groups. Next,Section 5.2 studies the impact of the HEL legalization on Texas house prices, the mainquestion of the paper. Section 5.3 investigates heterogeneity of the treatment effect. Section5.4 conducts falsification tests and investigates the mechanism by analyzing the law’s impacton other outcome variables. Section 5.5 studies whether the HEL legalization affectedthe marginal homebuyer, examining the impact on homeownership, population, and singlefamily building permits. Finally, Section 5.6 discusses external validity and compares themain results to the quantitative macroeconomic literature.
Table 1 presents summary statistics, comparing variables in Texas and border states beforeand after the law change. Before the law change, the average LTV for primary purchasemortgages was 83.17% in Texas and 79.29% in border states. After the law change, TexasLTV fell 5.6% to 77.57% while LTV in the border states rose slightly from 2.23% to 81.52%.The drop in LTV makes intuitive sense. Before 1998, Texas homeowners had one shot topledge their home as collateral, so they naturally borrowed more for their home purchase.After 1998, Texas homeowners gained future opportunities to extract equity, so they didnot need to borrow as much at the time of purchase.Mortgage interest rates were similar in Texas and border states both before and afterthe law change. Texas had higher median real house price levels than border states bothbefore and after the law change. Real rents in Texas were similar to rents in border states,both before and after the law change. Texas had slightly higher population growth thanborder states before and after the law change. Both Texas and border states had a smalldrop in population growth after the law change. Real income per capita in Texas was slightlyhigher than in border states, both before and after the law change. Texas and border stateshad similar unemployment rates before and after the law change. The unemployment ratefell slightly less in Texas than in border states after the law change. Texas had a lowerhomeownership rate than border states, both before and after the law change. Texas had The only way to extract home equity after purchase was to sell the home at significant transactioncosts.
This section presents the main results of this paper, the impact of HEL legalization onTexas house prices. Estimates are presented for four geographically nested samples and thedata are weighted by the inverse of the number of zip codes in each state. The baselinespecification contains year fixed effects, zip code fixed effects, a time trend interacted withan MSA dummy, and national log real oil prices interacted with an MSA dummy. Thespecification in the CBCP sample also includes contiguous border county pair by year fixedeffects as in Dube et al. (2010). In the main analysis, robust standard errors are computedthree ways: (1) clustered by state, (2) clustered by zip code, and (3) spatially correlated(Conley (1999)). This strategy is conservative in the border samples because there arerelatively few clusters to precisely estimate standard errors clustered at the state level. All samples have enough zip codes to precisely estimate standard errors clustered at the zipcode level. The largest standard error of the three is reported. Estimates from the baselinespecification will be presented first, followed by an exploration of robustness.Estimates from static regressions across the four samples (Table 2, columns 1–4) showthat the HEL legalization raised Texas house prices 3 . − . In the CBCP sample, standard errors are (1) double clustered by state and border county pair, (2)double clustered by zip code and border county pair, and (3) spatially correlated (Conley (1999)). This follows Cameron and Miller (2015): “The consensus is to be conservative and avoid bias and to usebigger and more aggregate clusters when possible, up to and including the point at which there is concernabout having too few clusters.” .
5% in the U.S. sample, 6 .
16% in the Border State sam-ple, 4 .
76% in the Border sample, and 4 .
13% in the CBCP sample. There is no systematicpattern in the size of the effect as the sample becomes more refined. The preferred estimate,ˆ β DID = 4 . η k > This section investigates whether the main results in Table 2 are robust to (1) the methodused for estimating standard errors, (2) the inclusion of covariates and an alternative spec-ification of the outcome variable, (3) the restriction to border city samples, and (4) themethod used to construct the house price index. Estimates in Table 8 explore whether thetreatment effect remains significant when standard errors are conventional (OLS), robust(EHW), clustered by five-digit zip code, three-digit zip code, county FIPS code, MSA, state,or spatially correlated (SHAC). For the CBCP sample, standard errors are also clustered by21ounty pair, double clustered by county pair and zip code, and double clustered by countypair and state. All estimates are statistically significant at least at the 5% significancelevel. The only clear pattern is that the main treatment effect is significant at the 1% levelin all samples when standard errors are spatially correlated as in Conley (1999) or clusteredat the three-digit zip code or smaller level. There does not seem to be an obvious pattern asstandard errors are clustered at larger levels. In the U.S. and CBCP samples, the standarderror falls as we move from MSA to state clusters, whereas the standard error estimates riseslightly in the Border State and Border samples.Estimates in Table 3 investigate whether the treatment effect in Table 2 is robust tothe inclusion of covariates including national interest rates interacted with state dummies,real income per capita, and population. The table also explores an alternative specificationof the outcome variable using annual real house price growth (as opposed to log real houseprices). The treatment effect remains positive and statistically significant across samplesand specifications. In specifications using log real house prices as the outcome variable,including covariates slightly reduces the estimates in the U.S. and Border State samplesand slightly raises the estimates in the Border and CBCP samples. In specifications usingreal house price growth as the outcome variable, the estimates are larger in the U.S. sampleand smaller in the other three samples.Estimates in Table 4 study the impact of HEL legalization on house prices in four bordercity samples. The advantage of these smaller samples is they can potentially provide bettercontrol for local unobserved heterogeneity. In addition, these estimates can help alleviateconcerns about the indirect mechanism to the extent that border cities can be viewed asone economy. The samples are (1) Texarkana, the only MSA in Texas with zip codes inanother state, (2) the Dallas-Fort Worth (DFW) Combined Statistical Area (CSA), whichincludes zip codes in Bryan County, Oklahoma, (3) the El Paso-Las Cruces CSA, whichincludes zip codes in Do˜na Ana County, New Mexico, and (4) the Texoma area, whichincludes counties in Oklahoma near Lake Texoma. The estimates in these border samplesare between 3 .
75% and 5 .
97% and statistically significant. Estimates in Table 9 investigate whether the treatment effect in Table 2 is robust to the I thank an anonymous referee for this suggestion. Not to be confused with the DFW MSA, which is entirely in Texas. El Paso-Las Cruces was delineated as a CSA by the Office of Management and Budget in 2013. Standard errors are clustered by zip code in these smaller border city samples as they only contain twostates. from the FHFA index used in the mainanalysis. These estimates remain positive, statistically significant, and similar to the mainestimates. Another advantage of the ZHVI index is that it can be directly interpreted asthe median dollar house value in a zip code year. Table 9 also presents estimates of theHEL legalization on real ZHVI levels, finding an inflation-adjusted effect between $2 , . , . This section investigates the impact of HEL legalization on Texas house prices using thesynthetic control method (Abadie, Diamond, and Hainmueller, 2010). Athey and Imbens(2017) called this “the most important innovation in the policy evaluation literature inthe last 15 years. This method builds on difference-in-differences estimation, but usessystematically more attractive comparisons.” Cavallo, Galiani, Noy, and Pantano (2013)extend the method to allow for multiple units to experience treatment. This is useful becauseeach Texas zip code received the treatment. Doudchenko and Imbens (2016) generalize thesynthetic control method to allow nonconvex weights and a permanent additive differencebetween the treated and control units. They show that this powerful generalization nestsmany existing approaches as special cases including classical difference-in-differences andmatching methods.Let Y z,t denote the outcome variable for treated zip code z in year t and Y j,t the outcomefor untreated zip code j . Let Y z,pre and Y j,pre be vectors of the outcome variables in thepretreatment years. Let Y C,pre be a matrix of predictors whose columns consist of Y j,pre (outcome variables for all control zip codes in the pretreatment years) as well as other controlvariables (national oil prices). The Doudchenko and Imbens (2016) estimator minimizes thedistance between the treated outcome and an affine combination of the untreated outcomefor the pretreatment period, regularized by the elastic-net (en) penalty (Zou and Hastie,2005): (ˆ µ en , ˆ ω en ) = argmin µ,ω (cid:107) Y z,pre − µ − Y C,pre · ω (cid:107) + λ · ( α (cid:107) ω (cid:107) + (1 − α ) (cid:107) ω (cid:107) ) .The main analysis does not use ZHVI data because it begin too late in April 1996. λ determines the amount of regularization, and α determines the type. Thecase α = 1 corresponds to a LASSO penalty function, which captures a preference forparsimony via a small number of nonzero weights. The case α = 0 corresponds to a Ridgepenalty function, which captures a preference for smaller weights. Doudchenko and Imbens(2016) propose a cross-validation procedure to select the regularization parameters λ and α that minimize the average mean squared prediction error for all untreated units.These estimates give the counterfactual outcome for treated zip code z if it did notreceive the treatment as a function of the control zip codes:ˆ Y z,t (0) = ˆ µ en + Y C,t ˆ ω en , where Y C,t is a row vector consisting of outcomes for the control zip codes and national oilprices in year t . The identifying assumption is that the relationship between the treated andcontrol outcome variables, given by ˆ µ en and ˆ ω en , would have remained the same in absenceof the treatment. While this is defended in the same way as in Section 4.2, the advantageof the synthetic control estimator is a more attractive control group with tighter pretrends.The estimated treatment effect for zip code z is the gap (i.e., difference) between theobserved and counterfactual outcome ˆ η z,t = Y z,t − ˆ Y z,t (0). The extension by Cavallo et al.(2013) allows for more than one zip code to experience treatment. Let z ∈ { , . . . , Z } bethe index for all treated zip codes; then the average treatment effect in Texas (across allTexas zip codes) in year t is given by ˆ η T X,t = Z (cid:80) Zz =1 ˆ η z,t . The average post-treatment effectis defined similarly ˆ η T X = { t ≥ } (cid:80) t ≥ ˆ η T X,t .The synthetic control estimates using the Border State sample are presented in Figure3. Panel A plots the treatment effect for each Texas zip code ˆ η z,t in gray and the treatmenteffect averaged across all Texas zip codes ˆ η T X,t in blue. Panel B plots the correspondingplacebo estimates for each untreated zip code in gray as well as the average across untreatedzip codes in blue.Observe that before the treatment year, 1998, the treatment effect in both the Texasand control zip codes is approximately zero as intended. After 1998, the treatment effectin Texas zip codes is mostly positive, whereas the placebo treatment effects in control zipcodes are equally positive and negative.The treatment effect averaged over all post-treatment years in Texas is ˆ η T X = 4 . η C = − . This section explores if and in what ways the effect differed across treated zip codes. Figure6 presents histograms and summary statistics of the treatment effect for each treated zipcode in the four geographically nested samples. These estimates are from regressions in thebaseline specification (Table 2, columns 1–4), except the term Texas × Post is interactedwith an indicator for each zip code. The histograms reveal that there is heterogeneity inthe effect across zip codes.Table 5 presents estimates from triple-difference regressions to investigate treatmenteffect heterogeneity along four dimensions: housing supply elasticity, income per capita, theunemployment rate, and median house price level. Estimates using the Saiz (2010) measureof supply elasticity (Table 5, column 1) show that zip codes in more elastic MSAs saw asmaller rise in prices. This is consistent with predictions from a partial equilibrium model;a rise in housing demand should have a bigger impact on prices in locations where it is rela-tively hard to build (Figure 4). The rise in house prices was 0 .
9% lower per unit of elasticity.If housing supply was perfectly inelastic, the average treatment effect would be the intercept6 . (cid:16) ˆATE ( H i ) = ˆ β H, + ˆ β H H i (cid:17) for each Texas MSA in Figure 5. The treatment effect varies considerably from 5 .
25% inthe most inelastic MSA (Galveston) to 2% in the most elastic MSA (Sherman).Table 5, column 2, investigates heterogeneity by prelaw log real income per capita. Zipcodes in higher-income counties saw larger treatment effects. A 1% higher prelaw real incomeper capita corresponds to a 0 . .
7% smaller treatment effect.Table 5, column 4, investigates heterogeneity by each zip code’s prelaw log real medianhouse price level. Zillow estimated median house prices are used because the level of the While this measure of elasticity is widely used as an instrumental variable for house prices (Mian andSufi, 2011), not all authors agree it is ideal (Davidoff, 2016). This paper does not use elasticity as aninstrument, but as a source of heterogeneity. Ex ante pricier zip codes sawa bigger treatment effect. A 1% higher prelaw real median house price level corresponds toa 0 . × Post is not interesting in thisspecification because there were no observations with zero house price levels.These results complement Landvoigt et al. (2015). While Landvoigt et al. (2015) findthat the credit expansion during the housing boom had a bigger impact on ex ante lower-priced homes in San Diego, this paper finds that the (exogenous) HEL legalization had abigger impact on ex ante higher-priced homes in Texas. The credit expansion in Landvoigtet al. (2015) increased access to borrowers seeking all loans secured by housing; in particularpurchase mortgages became available to many households that previously did not qualify.These previously purchase-constrained households tended to buy lower-priced homes. Incontrast, the Texas law change did not affect access to purchase mortgages, but ratherthe ability of existing homeowners to extract equity by borrowing after they were alreadyhomeowners.Together, the estimates in Table 5 (columns 2–4) provide evidence that households inmore prosperous zip codes (with ex ante higher house price levels, higher income, and lowerunemployment) value the option to pledge their home as collateral more strongly. There areseveral possible explanations. First, richer households typically enjoy a bigger tax shield onmortgage interest because (a) they have higher marginal tax rates and (b) they are morelikely to itemize deductions. Second, these households tend to be more financially literateand are more likely to be aware of these equity extraction products (Lusardi and Mitchell,2014). Third, rich households are more likely to qualify for these new loans as they tend tohave better credit and more stable income (DeFusco and Mondragon, 2018).To be clear, these results do not show that poorer households do not value HELs.The treatment effect was still positive (but smaller) in poorer zip codes. These resultsreveal that households expect to borrow more when they are not under distress (e.g., foreducation or entrepreneurship). These results highlight the asymmetric benefits of homeequity borrowing due to mortgage underwriting requirements: households are less likely toqualify for HELs when they are experiencing economic distress. Zillow data are not used in the main analysis because the sample begins too late in April 1996. .4 Channels This section investigates the mechanism behind the treatment effect on house prices bystudying the impact of HEL legalization on various other outcome variables. The treatmenteffect could have occurred through two channels:1. Direct Channel: the law caused a rise in demand for owner-occupied housing due tothe new option allowing homes to be pledged as collateral.2. Indirect Channel: the law affected other variables that affect house prices. For exam-ple, if the law increased investment enough to stimulate the local economy, this couldhave raised demand for housing, thus raising the price.If variables known to affect house prices such as rent, population, income, and unemploymentwere not affected by the law change, that would provide evidence that the direct channeldrove the treatment effect.Estimates presented in Table 6 show that the law did not lead to economically orstatistically significant changes in real rents, population, or real income per capita. Therent regressions are particularly informative as classical economic theory predicts that theprice of housing should have been equal to the present discounted value of rents before thelegalization of HELs: p t = E t (cid:80) ∞ j =1 Rent t + j (1+ r t +1 ) ×···× (1+ r t + j ) . After the HEL legalization, the priceshould reflect both the housing service flow (rent) and the additional collateral service flow: p t = E t (cid:80) ∞ j =1 Rent t + j + CSF t+j (1+ r t +1 ) ×···× (1+ r t + j ) . The rent regression helps reassure us that the treatmenteffect is not due to indirect effects on rent. There was a small but statistically significant risein the unemployment rate of 0 .
3% (Table 6, column 4), but this works against the indirectchannel as higher unemployment is associated with lower house price growth. In addition,Kumar and Liang (2019) find that Texas GDP was unaffected by this law change.While the falsification tests address concerns about observed economic impact of thelaw change, estimates in border cities help control for unobserved economic impact to theextent that a border city (such as Texarkana) is one economy. Estimates in the border citysamples (Table 4) are positive and statistically significant, providing additional support infavor of the direct channel.Various other concerns regarding the mechanism behind the treatment effect are con-sidered and addressed below: 27. Home Improvement Loans: if Texans used HELs to improve their homes, the rise inhouse prices might be due to the higher quality of the properties and not due to thedemand for HELs.A: Home improvement loans were available before 1998 (Section 2).2. Piggy-Back Loans: HELs could have increased purchase mortgage debt capacity forhouseholds that used second liens (“piggy-back mortgages”) to avoid mortgage insur-ance and obtain bigger loans (Lee et al., 2013).A: This was not a relevant factor in Texas because of the 80% LTV limit for all liensobtained after purchase.Together, the falsification tests and border city estimates suggest that the law changedid not have a significant impact on variables related to house prices, providing evidencethat the treatment effect occurred mainly through the direct channel.
This section investigates whether the law change affected homeownership in Texas. Thelaw change created a new benefit for homeownership: the option to extract equity withoutselling the home. Hence, if households value this option, and if house prices were heldconstant, demand for owner-occupied housing would be expected to rise. In equilibrium,the rise in demand should have raised house prices until the marginal buyer was indifferentbetween owning and renting. The same logic applies to potential owners deciding whetherto live in Texas (or a nearby state).Estimates presented in Table 6, columns 2 and 5–7, show the law did not lead toeconomically or statistically significant changes in population, MSA-level homeownershiprates, household-level homeownership (using AHS survey data), or single-family buildingpermits. In addition, Kumar (2018) finds no evidence of migration using IRS tax returndata. These estimates provide evidence that the rise in house prices was sufficient to offsetthe rise in demand for ownership, keeping the marginal buyer indifferent between rentingand owning. 28 .6 External validity and discussion
This paper finds that after-purchase collateral service flows had a positive impact on theprice of owner-occupied housing in Texas (over 500 treated zip codes) after 1998. Theconceptual framework predicts that collateral service flows should have a positive impact onthe prices of other assets, in other locations, at other times. In particular, the effect is likelyhigher for housing in other states, because Texas is the only state with an 80% limit on homeequity extraction. More generally, collateral service flows for purchase loans are expectedto have a positive effect on asset prices as well. Unfortunately, it is notoriously difficultto empirically identify the impact of purchase mortgage leverage on house prices due tosimultaneity: on the one hand, a rise in purchase mortgage credit supply can raise housingdemand and thus house prices; on the other hand, higher house prices require potentialhomebuyers to get bigger purchase mortgages.There is little empirical work to compare the estimate to, because it is unusual to finda setting where it was illegal to pledge an asset as collateral after purchase. However, thereis a growing literature in quantitative macroeconomics that seeks to understand whetherhouseholds are liquidity-constrained and the role of illiquid housing wealth. In particular,Gorea and Midrigan (2018) observe that housing is an important component of wealth forAmerican households and seek to quantify the extent to which housing wealth is illiquid.They study this question using a life-cycle model with uninsurable idiosyncratic risks inwhich they explicitly model key institutional details including LTV constraints, payment-to-income (PTI) constraints, long-term amortizing mortgages, and transaction costs of refi-nancing. Their model predicts that three-quarters of homeowners are liquidity-constrained,in that they would be better off if they could convert housing equity into liquid wealth.These homeowners are willing to pay five cents, on average, for every additional dollar ofliquidity extracted from their homes. Liquidity constraints increase the average marginalpropensity to consume out of a transitory income windfall by about 40%. Their modelpredicts that frictions that prevent homeowners from tapping home equity are sizable.Gorea and Midrigan (2018) proceed to simulate the Texas HEL legalization in theirmodel. They find that the addition of the option to extract home equity raises equilibriumhouse prices 5.5%. This is within the range of the main estimates (Table 2, columns 1–4) and slightly higher than the preferred estimate in the CBCP sample of 4.13%. This In the Survey of Consumer Finances, about 70% of U.S. households own a home, and housing equityaccounts for about 80% of the median homeowner’s wealth.
A large body of literature studies the impact of credit constraints on borrowing, consump-tion, and investment. This paper finds that there is also an impact on the prices of assetsthat can be pledged as collateral. Estimates using zip code data show that the HEL legal-ization raised Texas house prices 4.13%. If households fear they will be credit-constrained,they should value assets that facilitate their future ability to borrow. Hence, the treatmenteffect estimated in this paper can be interpreted as price-based evidence that householdsare credit-constrained and value HELs to facilitate future consumption smoothing.Prices rose more in locations with inelastic housing supply, higher prelaw income, lowerunemployment, and higher house price levels. This reveals that richer households valuethe option to pledge their home as collateral more strongly. This heterogeneity can beexplained by (1) the greater tax shield available to rich households, (2) the tendency ofrich households to be more financially literate and aware of these financial products, and(3) the tendency of rich households to be more likely to qualify for HELs as they tendto have better credit and more stable income. The law change did not affect variablesknown to be related to house prices such as rent, population, and income. There was asmall but statistically significant rise in the unemployment rate, which works against theindirect channel. Moreover, the border and border city samples, which help control forlocal unobserved heterogeneity, provide further evidence that the effect was direct. Theseresults indicate that the treatment effect was mainly driven by the direct channel. Finally,the law change did not affect Texas population, homeownership, or building permits. Thisoffers evidence that the rise in house prices was sufficient to keep the marginal homebuyerindifferent between owning and renting.There are several avenues for future work. It would be interesting to estimate howcollateral service flows affect the price of other assets such as stocks and Treasury bonds.Between 1934 and 1974, the Federal Reserve changed the initial margin requirement (reg-ulation T) for the U.S. stock market 22 times (Jylh¨a, 2018). A good experiment would30ompare the same stock (possibly traded on different exchanges) but where certain sharesof the stock are not affected by the margin requirements. It would also be interesting todisentangle the components of collateral service flows. Loans secured by housing have twobenefits: a lower interest rate and a higher debt capacity. It would be helpful to separatelyidentify the fraction of collateral service flow that is due to interest rate savings comparedto debt capacity.In conclusion, owner-occupied housing comes with a valuable option to pledge the homeas collateral in the future. The legalization of HELs in Texas provides evidence that houseprices reflect this. 31 eferences
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Appendix A. Figures
A.1 Percent Change in Real House Prices
Figure 1:
Annual Percent Change in Real House Prices (demeaned) in theUnited States, Texas, and Border States -4-2024 D e m eaned pe r c en t c hange This figure plots the annual demeaned percent change in real house prices in the UnitedStates, Texas, and its four border states. There is a vertical red line in 1998, the yearhome equity loans were legalized in Texas. The house price data are from the FHFAAT Index. House prices are deflated by the CPI-U, as explained in the paper. Datasources can be found in Table 7. .2 Pretrends Figure 2:
Impact of HEL Legalization on Texas House Prices in FourGeographically Nested Samples -.050.05.1.15 E s t i m a t ed c oe ff i c i en t s Sample: United States -.050.05.1.15 1995 1996 1997 1998 1999 2000 2001 2002
Sample: Border State -.050.05.1.15 E s t i m a t ed c oe ff i c i en t s Point estimates 95% CI
Sample: Border -.050.05.1.15 1995 1996 1997 1998 1999 2000 2001 2002
Sample: CBCP
This figure plots point estimates ˆ η k and 95% confidence intervals from the dynamicregression in Table 2. There is a vertical red line in 1998, the year of the law change.Data sources can be found in Table 7. .3 Synthetic Controls Figure 3:
Impact of HEL Legalization on Texas House Prices UsingSynthetic Controls
Panel A: Zip Codes in Texas − . − . . . . year G ap i n r ea l hou s e p r i c e g r o w t h Gap per Texas zip codeAverage gap, across Texas zip codesAverage post treatment gap = 4.58%
Panel B: Zip Codes in Border States − . − . . . . year G ap i n r ea l hou s e p r i c e g r o w t h Gap per control zip codeAverage gap, across control zip codesAverage post treatment gap = −0.24%
Panel A presents the gap (treatment effect) in real house price growth foreach treated zip code ˆ η z,t (gray) estimated using a synthetic control sampleconsisting of zip codes in border states, as explained in Section 5.2.2. Theblue curve is the average gap across all treated zip codes each year ˆ η T X,t .The average post-treatment effect in Texas is ˆ η T X = 4 . η C = − . .4 Supply Elasticity Theory Figure 4:
The impact of a rise in demand on house prices in cities withdifferent supply elasticities Q supplyInelastic Q supplyElastic Q demandOld Q demandNew This figure compares the impact of a rise in housing demand on houseprices in two cities with different supply elasticities. Price (P) is on thevertical axis, and quantity (Q) is on the horizontal axis. Initially, the priceof housing is the same in both cities. A rise in demand causes prices torise more in the relatively inelastic city. This figure is from a discussion byLoretta Mester at the Federal Reserve Bank of Philadelphia. .5 Elasticity Fitted Values Figure 5:
Impact of HEL Legalization on Texas House Prices by MSAElasticity
This figure plots the fitted average treatment effect (cid:16) ˆATE ( H i ) = ˆ β H, + ˆ β H H i (cid:17) for each MSAagainst housing supply elasticity. The estimates are from triple-difference regressions presentedin Table 5. This figure includes all 21 Texas MSAs. Galveston, the most inelastic MSA, had a5.25% effect, whereas Sherman, the most elastic MSA, had a 2% effect. The measure of housingsupply elasticity is from Saiz (2010). Data sources can be found in Table 7. Appendix B. Tables
B.1 Summary Statistics
Table 1:
Summary Statistics
Sample: Pre-1998 Post-1998Variables Texas Control Difference Texas Control DifferenceLTV (%) 83.17 79.29 3.88* 77.57 81.52 –3.95***(1.46) (0.73) (1.50) (0.99) (0.49) (0.40)Interest Rate (%) 7.58 7.58 0.00 6.80 6.77 0.03(0.17) (0.08) (0.16) (0.28) (0.14) (0.04)Real House Price ( $ $ ) 482.67 488.56 –5.88 502.77 502.96 –0.19(5.16) (1.38) (22.78) (4.56) (1.22) (20.65)Population Growth (%) 1.74 1.08 0.66** 1.62 0.93 0.68***(0.07) (0.07) (0.22) (0.09) (0.08) (0.09)Real Income Per Cap ( $ ) 12,585.55 11,885.64 699.91** 14,115.58 13,312.92 802.66***(93.08) (86.11) (159.86) (105.47) (97.57) (113.15)Unemployment Rate (%) 6.30 6.38 –0.08 5.41 5.22 0.19(0.12) (0.11) (0.60) (0.07) (0.07) (0.41)Homeownership Rate (%) 56.96 60.88 –3.91 60.80 64.52 –3.72*(0.80) (0.90) (2.92) (0.71) (0.79) (1.43)Single-Family Permits 798.46 313.24 485.22*** 1,304.91 400.52 904.39***(55.69) (52.80) (42.80) (88.26) (83.68) (53.88)Saiz Supply Elasticity 3.06 3.60 –0.54 3.06 3.60 –0.54(0.43) (0.47) (0.36) (0.43) (0.47) (0.36)This table presents summary statistics for the treatment group (Texas) and control group (borderstates) six years before and after the law change. The standard deviation of each variable is inparentheses below the average. Tests for differences between treatment and control groups usingrobust standard errors as described in the paper are also reported. Nominal variables are adjustedfor inflation, as explained in Section 4. Supply elasticity, as estimated by Saiz (2010), does notvary over time in the sample. Data sources can be found in Table 7. * p < .1; ** p < .05; ***p < .01. .2 Main Table 2:
Impact of HEL Legalization on Texas House Prices (1) (2) (3) (4) (5) (6) (7) (8)Variables United States Border State Border CBCP United States Border State Border CBCPTexasPost 0.0350*** 0.0616** 0.0476** 0.0413*** (0.0099) (0.0221) (0.0151) (0.0126)
Texas95 -0.0027 0.0155 0.0096 0.0159 (0.0117) (0.0131) (0.0108) (0.0107)
Texas96 0.0009 -0.0029 0.0027 0.0101 (0.0129) (0.0142) (0.0110) (0.0085)
Texas98 0.0126 0.0498*** 0.0322** 0.0260 (0.0147) (0.0163) (0.0111) (0.0166)
Texas99 0.0298*** 0.0462*** 0.0448** 0.0283* (0.0103) (0.0110) (0.0140) (0.0146)
Texas00 0.0421*** 0.0388** 0.0429* 0.0462*** (0.0134) (0.0136) (0.0155) (0.0128)
Texas01 0.0468*** 0.0749*** 0.0571** 0.0533** (0.0122) (0.0139) (0.0133) (0.0127)
Texas02 0.0407*** 0.0871*** 0.0587*** 0.0583*** (0.0130) (0.0143) (0.0117) (0.0104) N R This table reports estimates of the effect of a law change (which legalized HELs in Texas) on house pricesin geographically nested samples. Each column reports a separate regression estimated at the zip code yearlevel where the dependent variable is the log of the real house price index. In columns 1–4, coefficientsare reported for the interaction of the Texas dummy with an indicator for whether the year of observationfalls on or after 1998. In columns 5–8, coefficients are reported for interactions of the Texas dummy withyear indicators. All specifications include zip code and year fixed effects, state time trends, and nationaloil prices interacted with MSA dummies. Specifications using the CBCP sample also include contiguousborder county pair by year fixed effects. Standard errors, computed using methods described in Section 5.2,are reported in parentheses. Data sources can be found in Table 7. * p < .1; ** p < .05; *** p < .01. .3 Specification Robustness Table 3:
Impact of HEL Legalization on Texas House Prices, SpecificationRobustness (1) (2) (3) (4)Variables United States Border State Border CBCP A. Outcome: Log real house price indexBaseline 0.0350*** 0.0616** 0.0476** 0.0413*** (0.0099) (0.0221) (0.0151) (0.0126)
Covariates 0.0330*** 0.0558** 0.0480** 0.0499*** (0.0086) (0.0196) (0.0166) (0.0159) N B. Outcome: Real house price growthBaseline 0.0389*** 0.0563*** 0.0317** 0.0409*** (0.0071) (0.0079) (0.0110) (0.0124)
Covariates 0.0374*** 0.0539*** 0.0288** 0.0375*** (0.0055) (0.0065) (0.0100) (0.0125) N This table reports estimates of the effect of a law change (which legalized HELs in Texas) on houseprices in geographically nested samples. Each entry reports a separate regression estimated at thezip code year level where the dependent variable is the log of the real house price index in panel Aor real house price growth in panel B. All specifications include zip code and year fixed effects, statetime trends, and national oil prices interacted with MSA dummies. Specifications using the CBCPsample also include contiguous border county pair by year fixed effects. Specifications in rows labeled“Covariates” include U.S. interest rates interacted with state dummies, real income per capita, andpopulation. Standard errors, computed using the methods described in Section 5.2, are reported inparentheses. Data sources can be found in Table 7. * p < .1; ** p < .05; *** p < .01. .4 Border City Samples Table 4:
Impact of HEL Legalization on Texas House Prices in BorderCity Samples (1) (2) (3) (4)Variables Texarkana DFWBorder ElPasoLasCruces TexomaTreated post1998 0.0588** 0.0405** 0.0597*** 0.0375**(0.0091) (0.0131) (0.0133) (0.0126) N
39 65 195 78 R This table reports estimates of the effect of a law change (which legalized HELs in Texas) on houseprices in various border city samples. Column 1 uses data from the Texarkana MSA, column 2 usesdata from the Dallas-Fort Worth (DFW) CSA, column 3 uses data from the El Paso Las Cruces CSA,and column 4 uses data from the Texoma area. Each column reports a separate regression estimatedat the zip code year level where the dependent variable is the log of the real house price index. Incolumns 1–4, coefficients are reported for the interaction of the Texas dummy with an indicator forwhether the year of observation falls on or after 1998. All specifications include zip code and year fixedeffects, state time trends, and national oil prices interacted with MSA dummies. Specifications usingthe CBCP sample also include contiguous border county pair by year fixed effects. Standard errorsare reported in parentheses. Data sources can be found in Table 7. * p < .1; ** p < .05; *** p < .01. .5 Treatment Effect Heterogeneity Table 5:
Impact of HEL Legalization on Texas House Prices, TreatmentEffect Heterogeneity (1) (2) (3) (4)VariablesTexas × Post 0.063*** -1.002*** 0.080*** -0.367***(0.011) (0.246) (0.015) (0.122)Texas × Post × Elasticity -0.009**(0.004)Texas × Post × LogRealIncomePre 0.109***(0.026)Texas × Post × UnemploymentRatePre -0.007***(0.002)Texas × Post × LogRealMedianHousePricesPre 0.036***(0.011) N R H i H i H i .0201 .0803 -.0619 .131TE Min H i .0525 -.03 .066 .000534 This table reports estimates of the effect of a law change (which legalized HELs in Texas) onhouse prices. Each column reports a separate regression in which the treatment effect is allowedto vary based on four measures of heterogeneity: supply elasticity, prelaw log real income percapita, unemployment, and log real median house prices. Prelaw variables are set equal to theiraverage before 1998. The dependent variable is the log of the real house price index. The firstrow reports coefficients for the interaction of the Texas dummy with an indicator for whether theyear of observation falls on or after 1998. This represents the treatment effect if the measure ofheterogeneity is equal to zero. The remaining rows report the coefficient on the triple interactionbetween the Texas dummy, an indicator for the treatment period, and one of four measures ofheterogeneity. All specifications include zip code and year fixed effects, state time trends, andnational oil prices interacted with MSA dummies. Specifications using the CBCP sample alsoinclude contiguous border county pair by year fixed effects. Standard errors are clustered at thestate level and are reported in parentheses. When interpreting the parameters in this table, takeinto account that some measures of heterogeneity are in logs while others are not. Data sourcescan be found in Table 7. * p < .1; ** p < .05; *** p < .01. .6 Other Outcome Variables: Falsification, Channels, MarginalBuyer Table 6:
Impact of HEL Legalization on Other Outcome Variables (1) (2) (3) (4) (5) (6) (7)Log Home Home LogLog Log Real Income Unemployment Ownership Owner Single FamilyVariables Real Rent Population Per Capita Rate Rate AHS PermitsTexas × Post 0.003 0.000 -0.001 0.003** 0.003 0.004 0.048(0.005) (0.007) (0.005) (0.001) (0.004) (0.004) (0.033) N R This table reports estimates of the effect of a law change (which legalized HELs in Texas) onsix outcome variables: Log Real Rent, Log Population, Log Real Income Per Capita, the Unem-ployment Rate, the Homeownership Rate, and Log Single-Family Building Permits. Coefficientsare reported for the interaction of the Texas dummy with an indicator for whether the year ofobservation falls on or after 1998. All specifications include year fixed effects along with locationfixed effects at the most local level possible. Standard errors are clustered at the state level andare reported in parentheses. Data sources can be found in Table 7. * p < .1; ** p < .05; *** p < .01. For Online Publication: Online Appendix Figures
C.1 Treatment Effect Heterogeneity by Zip Code
Figure 6:
Impact of HEL Legalization on Texas House Prices,Heterogeneity Across Zip Codes
Obs: 520 Avg: .035 StD: .085 Min: -.13 Max: .34 . . . F r a c t i on -.1 0 .1 .2 .3Point Estimate per Zip Code US Obs: 520 Avg: .062 StD: .085 Min: -.11 Max: .37 . . . F r a c t i on -.1 0 .1 .2 .3 .4Point Estimate per Zip Code Border-State
Obs: 63 Avg: .048 StD: .042 Min: -.034 Max: .16 . . . . . F r a c t i on -.05 0 .05 .1 .15Point Estimate per Zip Code Border
Obs: 40 Avg: .05 StD: .04 Min: -.0075 Max: .16 . . . F r a c t i on CBCP
Note . This figure presents histograms and summary statistics of the treatment effectfor each zip code in the geographically nested samples. There is a vertical red line atthe mean. Data sources can be found in Table 7. For Online Publication: Online Appendix Tables
D.1 Datasets
Table 7:
Datasets
Variable Level Source .
665 zips 576 zipsMedian Rent (historic) MSA Zillow 313 MSAs 21 MSAsSupply Elasticity MSA Saiz (2010) 269 MSAs 21 MSAsEmployment county BLS 2,581 fips 153 fipsIncome county BEA 2,581 fips 153 fipsPopulation county Census 2,581 fips 153 fipsBuilding permits county Census 3,072 fips 229 fipsHomeownership Rate MSA Census 75 MSAs 5 MSAsHomeownership Household Survey AHS 123 MSAs 9 MSAsOil Prices US EIA - -10 year Treasury US Treasury - -Inflation Expectations US Livingston survey - -CPI-U US BLS - -
Note . This table lists sources for the different variables used in this paper. All data are annualexcept for the FHFA state house price index, Zillow rent index, and the Zillow Home ValueIndex. The Metropolitan Statistical Areas (MSAs) are defined by the United States Office ofManagement and Budget. The Federal Information Processing Standards (fips) codes uniquelyidentify counties. The AHS survey data is at the household level, but the most local geographiclevel available is the MSA. .2 Standard Error Robustness Table 8:
Impact of HEL Legalization on Texas House Prices, StandardError Robustness (1) (2) (3) (4)METHOD US Border-State Border CBCPTexasPost 0.0350 0.0616 0.0476 0.0413OLS (0.0049)*** (0.0045)*** (0.0098)*** (0.0116)***
EHW (0.0029)*** (0.0037)*** (0.0087)*** (0.0127)***
Zip5 (0.0030)*** (0.0045)*** (0.0097)*** (0.0115)***
Zip3 (0.0129)*** (0.0164)*** (0.0148)*** (0.0120)***
Fips (0.0149)** (0.0179)*** (0.0143)*** (0.0120)***
Msa (0.0167)** (0.0210)*** (0.0139)*** (0.0116)***
State (0.0075)*** (0.0221)** (0.0151)** (0.0039)***
SHAC (0.0099)*** (0.0121)*** (0.0098)*** (0.0111)***
Pair (0.0128)***
Pair Zip5 (0.0126)***
Pair State (0.0083)*** N R Note . This table repeats the estimates in Table 2: columns 1-4, using a variety of different methods toestimate standard errors. Each column corresponds to the sample and each row (after the first) correspondsto the method for estimating standard errors. All specifications include zip code and year fixed effects, statetime trends, and national oil prices interacted with MSA dummies. Specifications using the CBCP samplealso include contiguous border county pair by year fixed effects. Data sources can be found in Table 7. *p < .1; ** p < .05; *** p < .01. .3 House Price Index Robustness Table 9:
Impact of HEL Legalization on Texas House Prices, House PriceIndex Robustness (1) (2) (3) (4) (5) (6) (7) (8)SAMPLES US US Border-State Border-State Border Border CBCP CBCPVARIABLES (lnY) ( $ Y) (lnY) ( $ Y) (lnY) ( $ Y) (lnY) ( $ Y)Texas × Post 0.046*** 5,363.34*** 0.031*** 2,714.41*** 0.042*** 4,020.57*** 0.050*** 3,027.09***(0.008) (971.54) (0.003) (136.47) (0.002) (100.10) (0.007) (380.86)N 1,277,495 1,277,495 126,806 126,806 14,662 14,662 14,948 14,948 R Note . This table repeats the estimates in Table 2: columns 1-4, using data from Zillow. Each sampleestimates the treatment effect two ways: using the log house real price index and using the real ZHVI(Zillow’s measure of median home value in a zip-code). All specifications include zip code and year fixedeffects, state time trends, and national oil prices interacted with MSA dummies. Specifications using theCBCP sample also include contiguous border county pair by year fixed effects. Data sources can be foundin Table 7. * p < .1; ** p < .05; *** p < .01. For Online Publication: Texas Legal Timeline
This appendix provides a brief timeline of relevant laws regarding home equity borrowingin Texas. • Article XVI, Section 50 of Texas Constitution of 1876 protects homesteads from fore-closure for failure to pay all debts except the purchase loan, property taxes, or amechanic’s lien. • This section has only been amended twice before 1997; the first amendment extendedprotections to single adults in 1973 and the second amendment of 1995 related todivorce proceedings. • December 1994, Texas Senate Interim Committee on Home Equity Lending comes outin strong support of easing restrictions on lending with limits on home equity lendingto protect consumers, and called for the amendment proposal to be put on the ballotfor voters to decide. The proposal did not gather the two-thirds majority in the Houseof Representatives, but it did pass the Senate, which was a first for such a proposal.The committee’s report was incorporated into House Joint Resolution 31 which passedthe Texas House and Senate in May 1997. • Voters approved Proposition 8 on 11-4-1997 with almost 60% voting yes of the 1.17million votes cast, begins , legalized Home Equity Loans without restrictionson how the money could be used, however the total value of all liens on the home cannot exceed 80% of the fair market value . Reverse mortgages and cash-outrefinance loans were also legalized. • Several problems with Proposition 8 in the first year.HELs not allowed for those who live on more than one acre of land.Reverse mortgage rules made the loans ineligible for purchase by Fannie Mae. • Proposition 2 on the November 2, 1999 ballot corrected the shortcomings.Article XVI, Section 50 was amended to clear up the issues with reverse mortgages.Article XVI, Section 51 was amended by increasing the acreage limit used to defineurban households to 10 acres. Abdallah and Lastrapes (2012); Forrester (2002); Kumar (2018); McKnight (1983); Stolper (2015);Texas Legislative Council (2016) For Online Publication: Household’s Problem
This appendix solves the household’s problem presented in Section 3.max E ∞ (cid:88) t =0 β t u ( c t , h t )s.t. c t + p t h t +1 + a t +1 ≤ y t + p t h t (1 − δ t ) + (1 + r t ) a t (DBC λ t ) − a t +1 ≤ κ t p t h t (CC µ t )Savings: a t +1 at time t . If a t +1 >
0, save, if a t +1 <
0, borrow.The multiplier on the dynamic budget constraint (DBC): λ ( s t ) β t π ( s t ) > µ ( s t ) β t π ( s t ) ≥ µ ( s t ) β t π ( s t ) [ κ ( s t ) p ( s t ) h t ( s t − ) + a t +1 ( s t )] = 0State variables in s t : y ( s t ), h t ( s t − ), a t ( s t − ), r t ( s t − ), δ ( s t ), κ ( s t ).Choice variables in s t : c ( s t ), h t +1 ( s t ), a t +1 ( s t ). L = ∞ (cid:88) t =0 (cid:88) s t ∈ S t β t π ( s t ) u ( c ( s t ) , h ( s t ))+ λ ( s t ) β t π ( s t ) (cid:2) y ( s t ) + p ( s t ) h t ( s t − )(1 − δ ( s t )) + (cid:0) r t ( s t − ) (cid:1) a t ( s t − ) (cid:3) − λ ( s t ) β t π ( s t ) (cid:2) c ( s t ) + p ( s t ) h t +1 ( s t ) + a t +1 ( s t ) (cid:3) + µ ( s t ) β t π ( s t ) (cid:2) κ ( s t ) p ( s t ) h t ( s t − ) + a t +1 ( s t ) (cid:3) In s t we have: y ( s t ), h t ( s t − ), a t ( s t − ), δ ( s t ), κ ( s t ).In s t we choose: c ( s t ), h t +1 ( s t ), a t +1 ( s t ). (Any two will pin down the third.) L c ( s t ) = β t π ( s t ) u ( s t ) + λ ( s t ) β t π ( s t ) [ − ⇔ λ ( s t ) = u ( s t ) 55 h t +1 ( s t ) = (cid:88) s t +1 | s t β t +1 π ( s t +1 ) u ( s t +1 )+ λ ( s t ) β t π ( s t ) (cid:2) − p ( s t ) (cid:3) + (cid:88) s t +1 | s t λ ( s t +1 ) β t +1 π ( s t +1 ) (cid:2) p ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + (cid:88) s t +1 | s t µ ( s t +1 ) β t +1 π ( s t +1 ) (cid:2) κ ( s t +1 ) p ( s t +1 ) (cid:3) = 0 L a t +1 ( s t ) = λ ( s t ) β t π ( s t ) [ − µ ( s t ) β t π ( s t ) [1]+ (cid:88) s t +1 | s t λ ( s t +1 ) β t +1 π ( s t +1 ) (cid:0) r t +1 ( s t ) (cid:1) = 0 ⇔ λ ( s t ) β t π ( s t ) = µ ( s t ) β t π ( s t ) + (cid:88) s t +1 | s t λ ( s t +1 ) β t +1 π ( s t +1 ) (cid:0) r t +1 ( s t ) (cid:1) ⇔ λ ( s t ) = µ ( s t ) + β (cid:0) r t +1 ( s t ) (cid:1) E t [ λ ( s t +1 )]recall λ ( s t ) = u ( s t ) (consumption foc) ⇔ u ( s t ) = µ ( s t ) + β (cid:0) r t +1 ( s t ) (cid:1) E t [ u ( s t +1 )]This is the famous liquidity-constrained Euler equation! If the collateral constraint is notbinding ( µ = 0) or doesn’t exist, then this collapses to the usual frictionless Euler equation.56sing λ ( s t ) = u ( s t ), we combine the consumption and housing FOCs. u ( s t ) β t π ( s t ) (cid:2) p ( s t ) (cid:3) = (cid:88) s t +1 | s t β t +1 π ( s t +1 ) u ( s t +1 )+ (cid:88) s t +1 | s t u ( s t +1 ) β t +1 π ( s t +1 ) (cid:2) p ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + (cid:88) s t +1 | s t µ ( s t +1 ) β t +1 π ( s t +1 ) (cid:2) p ( s t +1 ) κ ( s t +1 ) (cid:3) u ( s t ) (cid:2) p ( s t ) (cid:3) = E t βu ( s t +1 )+ E t u ( s t +1 ) β (cid:2) p ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + E t µ ( s t +1 ) β (cid:2) p ( s t +1 ) κ ( s t +1 ) (cid:3) p ( s t ) = E t β u ( s t +1 ) u ( s t )+ E t β u ( s t +1 ) u ( s t ) (cid:2) p ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + E t β µ ( s t +1 ) u ( s t ) (cid:2) p ( s t +1 ) κ ( s t +1 ) (cid:3) p ( s t ) = E t β u ( s t +1 ) u ( s t ) u ( s t +1 ) u ( s t +1 )+ E t β u ( s t +1 ) u ( s t ) (cid:2) p ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + E t β u ( s t +1 ) u ( s t ) µ ( s t +1 ) u ( s t +1 ) (cid:2) p ( s t +1 ) κ ( s t +1 ) (cid:3) p ( s t ) = E t (cid:20) β u ( s t +1 ) u ( s t ) (cid:18) u ( s t +1 ) u ( s t +1 ) + (cid:2) p ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + µ ( s t +1 ) u ( s t +1 ) (cid:2) p ( s t +1 ) κ ( s t +1 ) (cid:3)(cid:19)(cid:21) t (cid:124)(cid:123)(cid:122)(cid:125) price = E t β u ( t + 1) u ( t ) (cid:124) (cid:123)(cid:122) (cid:125) sdf × u ( t + 1) u ( t + 1) (cid:124) (cid:123)(cid:122) (cid:125) housing service flow + µ ( t + 1) u ( t + 1) κ t +1 p t +1 (cid:124) (cid:123)(cid:122) (cid:125) collateral service flow + (1 − δ t +1 ) p t +1 (cid:124) (cid:123)(cid:122) (cid:125) resale price p t = E t [ M t +1 × ( s t +1 + CSF t +1 + (1 − δ t +1 ) p t +1 )]We can use the equilibrium value of the multiplier: µ ( t + 1) = u ( t + 1) − β (1 + r t +2 ( s t +1 )) E t +1 [ u ( t + 2)] µ t = u c ( t ) − β (1 + r t +1 ) E t [ u c ( t + 1)]If there is no collateral constraint or the collateral constraint never binds ( µ ( s t ) = 0), thenhousing has no collateral value ( CSF t = 0). 58 .1 Price Decomposition Define M t,j ≡ M t × M t +1 · · · M j − × M j Define the total dividend d t ≡ s t + CSF t p t = E t [ M t +1 × ( d t +1 + (1 − δ ) p t +1 )]= E t [ M t +1 × ( d t +1 )] + (1 − δ ) E t [ M t +1 × p t +1 ]= E t [ M t +1 × ( d t +1 )] + (1 − δ ) E t [ M t +1 × E t +1 [ M t +2 × ( d t +2 + (1 − δ ) p t +2 )]]= E t [ M t +1 × ( d t +1 )] + (1 − δ ) E t [ M t +1 × M t +2 × ( d t +2 + (1 − δ ) p t +2 )]= E t [ M t +1 × ( d t +1 )] + (1 − δ ) E t [ M t +1 ,t +2 ( d t +2 )] + (1 − δ ) E t [ M t +1 ,t +2 p t +2 ]= E t (cid:34) ∞ (cid:88) j =1 M t +1 ,t + j × ( d t + j ) (1 − δ ) j − (cid:35) = E t (cid:34) ∞ (cid:88) j =1 M t +1 ,t + j × ( s t + j + CSF t + j ) (1 − δ ) j − (cid:35) = E t (cid:34) ∞ (cid:88) j =1 M t +1 ,t + j × (cid:0) s t + j (1 − δ ) j − (cid:1)(cid:35) + E t (cid:34) ∞ (cid:88) j =1 M t +1 ,t + j × (cid:0) CSF t + j (1 − δ ) j − (cid:1)(cid:35) Given a stochastic process x ≡ { x t + j } j = ∞ j =1 we can define the PDV-operator P DV t ( x t + j ) = E t [ M t +1 ,t + j x t + j ] P DV t ( x ) = E t (cid:34) ∞ (cid:88) j =1 M t +1 ,t + j × ( x t + j ) (cid:35) = ∞ (cid:88) j =1 P DV t ( x t + j )= P DV t ( x t +1 ) + P DV t (cid:16) { x t + j } j = ∞ j =2 (cid:17) p t = E t (cid:34) ∞ (cid:88) j =1 M t +1 ,t + j × (cid:0) s t + j (1 − δ ) j − (cid:1)(cid:35) + E t (cid:34) ∞ (cid:88) j =1 M t +1 ,t + j × (cid:0) CSF t + j (1 − δ ) j − (cid:1)(cid:35) p t (cid:124)(cid:123)(cid:122)(cid:125) price = P DV t (cid:16) { s t + j } j = ∞ j =1 (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) pdv housing service flow + P DV t (cid:16) { CSF t + j } j = ∞ j =1 (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) pdv collateral service flow s ≡ { s t + j (1 − δ ) j − } j = ∞ j =1 will be written s ≡ { s t + j } j = ∞ j =1 to save space. F.2 Price before the law
We can re-write the price before the law assuming no expectations of the law: p NLt = P DV t (cid:0) s NLt +1 (cid:1) + P DV t (cid:16)(cid:8) s NLt + j (cid:9) j = ∞ j =2 (cid:17) p NLt +1 = P DV t +1 (cid:16)(cid:8) s NLt + j (cid:9) j = ∞ j =2 (cid:17) F.3 Price change before the law p NLt +1 − p NLt = P DV t +1 (cid:16)(cid:8) s NLt + j (cid:9) j = ∞ j =2 (cid:17) − (cid:16) P DV t (cid:0) s NLt +1 (cid:1) + P DV t (cid:16)(cid:8) s NLt + j (cid:9) j = ∞ j =2 (cid:17)(cid:17) = ( P DV t +1 − P DV t ) (cid:16)(cid:8) s NLt + j (cid:9) j = ∞ j =2 (cid:17) − P DV t (cid:0) s NLt +1 (cid:1) = news about future cash flow − current period cash flow F.4 Price Change at the time of the law
The law is a surprise and occurs at t + 1. Homeowners can borrow at t + 1, but the pricewill only reflect CSF starting at t + 2. p Lt +1 − p NLt = P DV t +1 (cid:16) { CSF t + j } j = ∞ j =2 (cid:17) + P DV t +1 (cid:16)(cid:8) s Lt + j (cid:9) j = ∞ j =2 (cid:17) − (cid:16) P DV t (cid:0) s NLt +1 (cid:1) + P DV t (cid:16)(cid:8) s NLt + j (cid:9) j = ∞ j =2 (cid:17)(cid:17) = PDV(CSF) + news about future cash flow − current period cash flow60 .5 Impact of law changes on house prices Compare the price change if the law change occurred at t + 1 versus if it didn’t:∆ Law p t +1 ≡ p Lt +1 − p NLt = P DV t +1 (cid:16) { CSF t + j } j = ∞ j =2 (cid:17) + P DV t +1 (cid:16)(cid:8) s Lt + j (cid:9) j = ∞ j =2 (cid:17) − P DV t (cid:16)(cid:8) s NLt + j (cid:9) j = ∞ j =1 (cid:17) ∆ NoLaw p t +1 ≡ p NLt +1 − p NLt = (cid:16) P DV t +1 (cid:16)(cid:8) s NLt + j (cid:9) j = ∞ j =2 (cid:17) − P DV t (cid:16)(cid:8) s NLt + j (cid:9) j = ∞ j =1 (cid:17)(cid:17) ∆ Law p t +1 − ∆ NoLaw p t +1 = p Lt +1 − p NLt +1 = P DV t +1 (cid:16) { CSF t + j } j = ∞ j =2 (cid:17) + P DV t +1 (cid:16)(cid:8) s Lt + j − s NLt + j (cid:9) j = ∞ j =2 (cid:17) If we assume the law had no impact on rents s Lt + j = s NLt + j , then the second term cancels out∆ Law p t +1 − ∆ NoLaw p t +1 = P DV t +1 (cid:16) { CSF t + j } j = ∞ j =2 (cid:17) ∆ Law p t +1 p t − ∆ NoLaw p t +1 p t = P DV t +1 (cid:16) { CSF t + j } j = ∞ j =2 (cid:17) p t In the dynamic difference-in-differences regression the coefficient is η = ∆ Law p t +1 p t − ∆ NoLaw p t +1 p t = P DV t +1 ( CSF ) p t = E t +1 (cid:104)(cid:80) ∞ j =1 (1 − δ ) j − M t +1 ,j × CSF t +1+ j (cid:105) p t Hence, if we assume that the law change had no impact on rents then η is not only theimpact of the law change on house prices, but also the percent of a house price due to thecollateral option value.If the law change increases the supply of housing and h t + j rises more than it would with-out the law, then rents should be lower s Lt + j ≤ s NLt + j implying the coefficient underestimates61he collateral option value η = ∆ Law p t +1 p t − ∆ NoLaw p t +1 p t ≤ P DV t +1 ( CSF ) p t = E t +1 (cid:104)(cid:80) ∞ j =1 (1 − δ ) j − M t +1 ,j × CSF t +1+ j (cid:105) p t The more elastic housing supply is in a given location, the bigger the rise in h t + j , the morewe would be underestimating the collateral option value. However, regardless of the impacton rent (service flow), as long as the parallel trends assumption holds, we are still able toidentify the total impact of the law on house prices.In general η k = E ( y T exas,k − y Control,k ) − E ( y T exas, − y Control, ) η k = ∆ treatment p t + k p t − ∆ control p t + k p t For Online Publication: Disentangling Purchase VSHEL Collateral Service Flow
The objective of this note is to disentangle purchase debt capacity from HEL debt capacity.Consider a household who lives for three periods. The household: buys a home with apurchase mortgage ( b P M ) at t = 0, borrows via a HEL ( b HEL ) at t = 1, sells the home andrepays the loans at t = 2.The household maximizes: U = u ( c , h ) + βu ( c , h ) + β B ( c )subject to: c + p h ≤ y + b P M b P M ≤ κ P M p h c ≤ y + b HEL b HEL ≤ κ HEL p h − b P M ⇔ b PM + b HEL p h ≤ κ HEL
Note: κ HEL is the combined LTV limit for after purchase loans based on the appraised p h . c + b P M + b HEL ≤ y + p h We impose: b P Mt , b
HELt ≥
0: households don’t want to save b P Mt +1 = Rb P Mt and b HELt +1 = Rb HELt : households pay principal and interest at the end h = h = h ≡ h : housing is chosen once ( t = 0) and doesn’t depreciateThe household’s Lagrangian is: L = U + λ (cid:0) y + b P M − c − p h (cid:1) + µ (cid:0) κ P M p h − b P M (cid:1) + λ (cid:0) y + b HEL − c (cid:1) + µ (cid:0) κ HEL p h − Rb P M − b HEL (cid:1) + λ (cid:0) y + p h − c − R b P M − Rb HEL (cid:1) FOCs, Primal Feasibility, Dual Feasibility, and Complementary Slackness: L c : U c − λ = 0 L c : U c − λ = 0 L c : U c − λ = 0 L λ : (cid:0) y + b P M − c − p h (cid:1) = 0 L λ : (cid:0) y + b HEL − c (cid:1) = 0 L λ : (cid:0) y + p h − c − R b P M − Rb HEL (cid:1) = 0 L h : U h − λ p + µ κ P M p + µ κ HEL p + λ p = 0 L b HEL : λ − µ − λ R = 0 L b PM : λ − µ − µ R − λ R = 0 L µ : (cid:0) κ P M p h − b P M (cid:1) ≥ µ ≥ µ × (cid:0) κ P M p h − b P M (cid:1) = 0 L µ : (cid:0) κ HEL p h − Rb P M − b HEL (cid:1) ≥ µ ≥ µ × (cid:0) κ HEL p h − Rb P M − b HEL (cid:1) = 063he housing FOC can be rearranged: p (cid:124)(cid:123)(cid:122)(cid:125) price = 1 λ × U h (cid:124)(cid:123)(cid:122)(cid:125) housingservice flow (rent) + µ κ P M p (cid:124) (cid:123)(cid:122) (cid:125) purchase collateralservice flow + µ κ HEL p (cid:124) (cid:123)(cid:122) (cid:125) HEL collateralservice flow + λ p (cid:124)(cid:123)(cid:122)(cid:125) resale price Observe that both the purchase collateral service flow and the HEL collateral service flowenter the equation. 64
For Online Publication: Household’s Problem WithOwning and Renting
This appendix solves the household’s problem presented in Section 3 except the house-hold can own or rent a unit of housing at prices p own t , p rent t . Both owner-occupied housingand rental housing provide shelter, however-owner occupied housing also provides collateralservice flows.max E ∞ (cid:88) t =0 β t u ( c t , h t = h own t + h rent t )s.t. c t + p own t h own t +1 + p rent t h rent t +1 + a t +1 ≤ y t + p own t h own t (1 − δ t ) + (1 + r t ) a t (DBC λ t ) − a t +1 ≤ κ t p own t h own t (CC µ t )Savings: a t +1 at time t . If a t +1 >
0, save, if a t +1 <
0, borrow.The multiplier on the dynamic budget constraint (DBC): λ ( s t ) β t π ( s t ) > µ ( s t ) β t π ( s t ) ≥ µ ( s t ) β t π ( s t ) [ κ ( s t ) p own ( s t ) h own t ( s t − ) + a t +1 ( s t )] = 0State variables in s t : y ( s t ), h own t ( s t − ), h rent t ( s t − ), a t ( s t − ), r t ( s t − ), δ ( s t ), κ ( s t ).Choice variables in s t : c ( s t ), h own t +1 ( s t ), h rent t +1 ( s t ), a t +1 ( s t ). L = ∞ (cid:88) t =0 (cid:88) s t ∈ S t β t π ( s t ) u ( c ( s t ) , h ( s t ))+ λ ( s t ) β t π ( s t ) (cid:2) y ( s t ) + p own ( s t ) h own t ( s t − )(1 − δ ( s t )) + (cid:0) r t ( s t − ) (cid:1) a t ( s t − ) (cid:3) − λ ( s t ) β t π ( s t ) (cid:2) c ( s t ) + p own ( s t ) h own t +1 ( s t ) + p rent ( s t ) h rent t +1 ( s t ) + a t +1 ( s t ) (cid:3) + µ ( s t ) β t π ( s t ) (cid:2) κ ( s t ) p own ( s t ) h own t ( s t − ) + a t +1 ( s t ) (cid:3) L c ( s t ) = β t π ( s t ) u ( s t ) + λ ( s t ) β t π ( s t ) [ − ⇔ λ ( s t ) = u ( s t ) 65 h own t +1 ( s t ) = (cid:88) s t +1 | s t β t +1 π ( s t +1 ) u ( s t +1 )+ λ ( s t ) β t π ( s t ) (cid:2) − p own ( s t ) (cid:3) + (cid:88) s t +1 | s t λ ( s t +1 ) β t +1 π ( s t +1 ) (cid:2) p own ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + (cid:88) s t +1 | s t µ ( s t +1 ) β t +1 π ( s t +1 ) (cid:2) κ ( s t +1 ) p own ( s t +1 ) (cid:3) = 0 L h rent t +1 ( s t ) = (cid:88) s t +1 | s t β t +1 π ( s t +1 ) u ( s t +1 )+ λ ( s t ) β t π ( s t ) (cid:2) − p rent ( s t ) (cid:3) = 0 L a t +1 ( s t ) = λ ( s t ) β t π ( s t ) [ − µ ( s t ) β t π ( s t ) [1]+ (cid:88) s t +1 | s t λ ( s t +1 ) β t +1 π ( s t +1 ) (cid:0) r t +1 ( s t ) (cid:1) = 0 ⇔ λ ( s t ) β t π ( s t ) = µ ( s t ) β t π ( s t ) + (cid:88) s t +1 | s t λ ( s t +1 ) β t +1 π ( s t +1 ) (cid:0) r t +1 ( s t ) (cid:1) ⇔ λ ( s t ) = µ ( s t ) + β (cid:0) r t +1 ( s t ) (cid:1) E t [ λ ( s t +1 )]recall λ ( s t ) = u ( s t ) (consumption foc) ⇔ u ( s t ) = µ ( s t ) + β (cid:0) r t +1 ( s t ) (cid:1) E t [ u ( s t +1 )]66sing λ ( s t ) = u ( s t ), we combine the consumption and owner-occupied housing FOCs. u ( s t ) β t π ( s t ) (cid:2) p own ( s t ) (cid:3) = (cid:88) s t +1 | s t β t +1 π ( s t +1 ) u ( s t +1 )+ (cid:88) s t +1 | s t u ( s t +1 ) β t +1 π ( s t +1 ) (cid:2) p own ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + (cid:88) s t +1 | s t µ ( s t +1 ) β t +1 π ( s t +1 ) (cid:2) p own ( s t +1 ) κ ( s t +1 ) (cid:3) u ( s t ) (cid:2) p own ( s t ) (cid:3) = E t βu ( s t +1 )+ E t u ( s t +1 ) β (cid:2) p own ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + E t µ ( s t +1 ) β (cid:2) p own ( s t +1 ) κ ( s t +1 ) (cid:3) p own ( s t ) = E t β u ( s t +1 ) u ( s t )+ E t β u ( s t +1 ) u ( s t ) (cid:2) p own ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + E t β µ ( s t +1 ) u ( s t ) (cid:2) p own ( s t +1 ) κ ( s t +1 ) (cid:3) p own ( s t ) = E t β u ( s t +1 ) u ( s t ) u ( s t +1 ) u ( s t +1 )+ E t β u ( s t +1 ) u ( s t ) (cid:2) p own ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + E t β u ( s t +1 ) u ( s t ) µ ( s t +1 ) u ( s t +1 ) (cid:2) p own ( s t +1 ) κ ( s t +1 ) (cid:3) p own ( s t ) = E t (cid:20) β u ( s t +1 ) u ( s t ) (cid:18) u ( s t +1 ) u ( s t +1 ) + (cid:2) p own ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + µ ( s t +1 ) u ( s t +1 ) (cid:2) p own ( s t +1 ) κ ( s t +1 ) (cid:3)(cid:19)(cid:21) own t = E t β u ( t + 1) u ( t ) (cid:124) (cid:123)(cid:122) (cid:125) sdf × u ( t + 1) u ( t + 1) (cid:124) (cid:123)(cid:122) (cid:125) housing service flow + µ ( t + 1) u ( t + 1) κ t +1 p own t +1 (cid:124) (cid:123)(cid:122) (cid:125) collateral service flow + (1 − δ t +1 ) p own t +1 (cid:124) (cid:123)(cid:122) (cid:125) resale price Using λ ( s t ) = u ( s t ), we combine the consumption and rental housing FOCs. p rent ( s t ) = (cid:88) s t +1 | s t β t +1 π ( s t +1 ) β t π ( s t ) u ( s t +1 ) λ ( s t )= (cid:88) s t +1 | s t β t +1 π ( s t +1 ) β t π ( s t ) u ( s t +1 ) u ( s t ) p rent t = E t β u ( t + 1) u ( t ) (cid:124) (cid:123)(cid:122) (cid:125) sdf × u ( t + 1) u ( t + 1) (cid:124) (cid:123)(cid:122) (cid:125) housing service flow p own t = p rent t + E t β u ( t + 1) u ( t ) (cid:124) (cid:123)(cid:122) (cid:125) sdf × µ ( t + 1) u ( t + 1) κ t +1 p own t +1 (cid:124) (cid:123)(cid:122) (cid:125) collateral service flow + (1 − δ t +1 ) p own t +1 (cid:124) (cid:123)(cid:122) (cid:125) resale price For Online Publication: Household’s Problem WithOwning and Renting 2
This appendix solves the household’s problem presented in Section 3 except the householdcan own or rent a unit of housing at prices p own t , p rent t . Both owner-occupied housing andrental housing provide shelter, h units of rental housing provides h units of housing serviceswhile h units of owner-occupied housing provides ωh units of housing services, where ω ≥ E ∞ (cid:88) t =0 β t u ( c t , h t = ωh own t + h rent t )s.t. c t + p own t h own t +1 + p rent t h rent t +1 + a t +1 ≤ y t + p own t h own t (1 − δ t ) + (1 + r t ) a t (DBC λ t ) − a t +1 ≤ κ t p own t h own t (CC µ t )Savings: a t +1 at time t . If a t +1 >
0, save, if a t +1 <
0, borrow.The multiplier on the dynamic budget constraint (DBC): λ ( s t ) β t π ( s t ) > µ ( s t ) β t π ( s t ) ≥ µ ( s t ) β t π ( s t ) [ κ ( s t ) p own ( s t ) h own t ( s t − ) + a t +1 ( s t )] = 0State variables in s t : y ( s t ), h own t ( s t − ), h rent t ( s t − ), a t ( s t − ), r t ( s t − ), δ ( s t ), κ ( s t ).Choice variables in s t : c ( s t ), h own t +1 ( s t ), h rent t +1 ( s t ), a t +1 ( s t ). L = ∞ (cid:88) t =0 (cid:88) s t ∈ S t β t π ( s t ) u ( c ( s t ) , h ( s t ))+ λ ( s t ) β t π ( s t ) (cid:2) y ( s t ) + p own ( s t ) h own t ( s t − )(1 − δ ( s t )) + (cid:0) r t ( s t − ) (cid:1) a t ( s t − ) (cid:3) − λ ( s t ) β t π ( s t ) (cid:2) c ( s t ) + p own ( s t ) h own t +1 ( s t ) + p rent ( s t ) h rent t +1 ( s t ) + a t +1 ( s t ) (cid:3) + µ ( s t ) β t π ( s t ) (cid:2) κ ( s t ) p own ( s t ) h own t ( s t − ) + a t +1 ( s t ) (cid:3) c ( s t ) = β t π ( s t ) u ( s t ) + λ ( s t ) β t π ( s t ) [ − ⇔ λ ( s t ) = u ( s t ) L h own t +1 ( s t ) = (cid:88) s t +1 | s t β t +1 π ( s t +1 ) ωu ( s t +1 )+ λ ( s t ) β t π ( s t ) (cid:2) − p own ( s t ) (cid:3) + (cid:88) s t +1 | s t λ ( s t +1 ) β t +1 π ( s t +1 ) (cid:2) p own ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + (cid:88) s t +1 | s t µ ( s t +1 ) β t +1 π ( s t +1 ) (cid:2) κ ( s t +1 ) p own ( s t +1 ) (cid:3) = 0 L h rent t +1 ( s t ) = (cid:88) s t +1 | s t β t +1 π ( s t +1 ) u ( s t +1 )+ λ ( s t ) β t π ( s t ) (cid:2) − p rent ( s t ) (cid:3) = 0 70 a t +1 ( s t ) = λ ( s t ) β t π ( s t ) [ − µ ( s t ) β t π ( s t ) [1]+ (cid:88) s t +1 | s t λ ( s t +1 ) β t +1 π ( s t +1 ) (cid:0) r t +1 ( s t ) (cid:1) = 0 ⇔ λ ( s t ) β t π ( s t ) = µ ( s t ) β t π ( s t ) + (cid:88) s t +1 | s t λ ( s t +1 ) β t +1 π ( s t +1 ) (cid:0) r t +1 ( s t ) (cid:1) ⇔ λ ( s t ) = µ ( s t ) + β (cid:0) r t +1 ( s t ) (cid:1) E t [ λ ( s t +1 )]recall λ ( s t ) = u ( s t ) (consumption foc) ⇔ u ( s t ) = µ ( s t ) + β (cid:0) r t +1 ( s t ) (cid:1) E t [ u ( s t +1 )]Using λ ( s t ) = u ( s t ), we combine the consumption and owner-occupied housing FOCs. u ( s t ) β t π ( s t ) (cid:2) p own ( s t ) (cid:3) = (cid:88) s t +1 | s t β t +1 π ( s t +1 ) ωu ( s t +1 )+ (cid:88) s t +1 | s t u ( s t +1 ) β t +1 π ( s t +1 ) (cid:2) p own ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + (cid:88) s t +1 | s t µ ( s t +1 ) β t +1 π ( s t +1 ) (cid:2) p own ( s t +1 ) κ ( s t +1 ) (cid:3) u ( s t ) (cid:2) p own ( s t ) (cid:3) = E t βωu ( s t +1 )+ E t u ( s t +1 ) β (cid:2) p own ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + E t µ ( s t +1 ) β (cid:2) p own ( s t +1 ) κ ( s t +1 ) (cid:3) own ( s t ) = E t βω u ( s t +1 ) u ( s t )+ E t β u ( s t +1 ) u ( s t ) (cid:2) p own ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + E t β µ ( s t +1 ) u ( s t ) (cid:2) p own ( s t +1 ) κ ( s t +1 ) (cid:3) p own ( s t ) = E t βω u ( s t +1 ) u ( s t ) u ( s t +1 ) u ( s t +1 )+ E t β u ( s t +1 ) u ( s t ) (cid:2) p own ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + E t β u ( s t +1 ) u ( s t ) µ ( s t +1 ) u ( s t +1 ) (cid:2) p own ( s t +1 ) κ ( s t +1 ) (cid:3) p own ( s t ) = E t (cid:20) β u ( s t +1 ) u ( s t ) (cid:18) ω u ( s t +1 ) u ( s t +1 ) + (cid:2) p own ( s t +1 )(1 − δ ( s t +1 )) (cid:3) + µ ( s t +1 ) u ( s t +1 ) (cid:2) p own ( s t +1 ) κ ( s t +1 ) (cid:3)(cid:19)(cid:21) p own t = E t β u ( t + 1) u ( t ) (cid:124) (cid:123)(cid:122) (cid:125) sdf × ω u ( t + 1) u ( t + 1) (cid:124) (cid:123)(cid:122) (cid:125) housing service flow + µ ( t + 1) u ( t + 1) κ t +1 p own t +1 (cid:124) (cid:123)(cid:122) (cid:125) collateral service flow + (1 − δ t +1 ) p own t +1 (cid:124) (cid:123)(cid:122) (cid:125) resale price Using λ ( s t ) = u ( s t ), we combine the consumption and rental housing FOCs. p rent ( s t ) = (cid:88) s t +1 | s t β t +1 π ( s t +1 ) β t π ( s t ) u ( s t +1 ) λ ( s t )= (cid:88) s t +1 | s t β t +1 π ( s t +1 ) β t π ( s t ) u ( s t +1 ) u ( s t ) p rent t = E t β u ( t + 1) u ( t ) (cid:124) (cid:123)(cid:122) (cid:125) sdf × u ( t + 1) u ( t + 1) (cid:124) (cid:123)(cid:122) (cid:125) housing service flow own t = ωp rent t + E t β u ( t + 1) u ( t ) (cid:124) (cid:123)(cid:122) (cid:125) sdf × µ ( t + 1) u ( t + 1) κ t +1 p own t +1 (cid:124) (cid:123)(cid:122) (cid:125) collateral service flow + (1 − δ t +1 ) p own t +1 (cid:124) (cid:123)(cid:122) (cid:125) resale price If ω > ω >
1, rent remains avalid proxy as long as the additional utility from shelter was not affected by the law change.73
For Online Publication: Comparison of CollateralValue Models in the Literature
The model in Section 3 assumes the collateral constraint is − a t +1 ≤ κ t p t h t . This sectionshows the results are preserved under different specifications of the collateral constraintwhich appear in the literature.The collateral constraint can be written in the following ways: − a t +1 ≤ κ t p t h t − a t +1 ≤ κ t E t [ p t +1 ] h t − a t +1 ≤ κ t p t h t +1 − a t +1 ≤ κ t E t [ p t +1 ] h t +1 Depending on whether lenders let you borrow against the (1) present value of the asset youcurrently own, (2) expected future value of the asset you currently own, (3) present valueof the asset you bought for tomorrow, (4) expected future value of the asset you bought fortomorrow.(1) Households borrow against the present value of housing they have today. c t + p t h t +1 + a t +1 ≤ y t + p t h t (1 − δ t ) + (1 + r t ) a t (DBC λ t ) − a t +1 ≤ κ t p t h t (CC µ t ) p t (cid:124)(cid:123)(cid:122)(cid:125) price = E t β u ( t + 1) u ( t ) (cid:124) (cid:123)(cid:122) (cid:125) sdf × u ( t + 1) u ( t + 1) (cid:124) (cid:123)(cid:122) (cid:125) service flow + µ ( t + 1) u ( t + 1) κ t +1 p t +1 (cid:124) (cid:123)(cid:122) (cid:125) collateral value + (1 − δ t +1 ) p t +1 (cid:124) (cid:123)(cid:122) (cid:125) resale price (2) Households borrow against the expected future value of new housing. c t + p t h t +1 + a t +1 ≤ y t + p t h t (1 − δ t ) + (1 + r t ) a t (DBC λ t ) − a t +1 ≤ κ t E t [ p t +1 ] h t +1 (CC µ t )74 t (cid:124)(cid:123)(cid:122)(cid:125) price = E t β u ( t + 1) u ( t ) (cid:124) (cid:123)(cid:122) (cid:125) sdf × u ( t + 1) u ( t + 1) (cid:124) (cid:123)(cid:122) (cid:125) service flow + µ t κ t βu ( t + 1) p t +1 (cid:124) (cid:123)(cid:122) (cid:125) collateral value + (1 − δ t +1 ) p t +1 (cid:124) (cid:123)(cid:122) (cid:125) resale price (3) Households borrow against the present value of land they buy at t for t + 1 (Bianchiet al., 2012). q t k t +1 + c t + b t +1 R t ≤ q t k t + b t + ε t Y ( k t ) (DBC λ t ) − b t +1 R t ≤ κ t q t k t +1 (CC µ t ) q t (cid:124)(cid:123)(cid:122)(cid:125) price = q t µ t κ t u (cid:48) ( t ) (cid:124) (cid:123)(cid:122) (cid:125) collateral value + E st β u (cid:48) ( t + 1) u (cid:48) ( t ) (cid:124) (cid:123)(cid:122) (cid:125) sdf × ε t +1 Y k ( k t +1 ) (cid:124) (cid:123)(cid:122) (cid:125) service flow + q t +1 (cid:124)(cid:123)(cid:122)(cid:125) resale price This equation is not in their paper, this is a rearrangement of their FOC to illustratecollateral value.(4) Households borrow against the expected future value of new housing collateral purchasedat time t (Iacoviello, 2005). c t + q t h t + R t − b t − π t + w (cid:48) t L t ≤ Y t X t + q t h t − + b t (DBC) b t ≤ m E t (cid:20) q t +1 h t π t +1 R t (cid:21) (CC λ t ) q t (cid:124)(cid:123)(cid:122)(cid:125) price = E t γ u (cid:48) ( c t +1 ) u (cid:48) ( c t ) (cid:124) (cid:123)(cid:122) (cid:125) sdf × ν Y t +1 X t +1 h t (cid:124) (cid:123)(cid:122) (cid:125) service + q t +1 (cid:124)(cid:123)(cid:122)(cid:125) resale price + λ t mπ t +1 q t +1 (cid:124) (cid:123)(cid:122) (cid:125) collateral value Farmers borrow against the land they buy at t at next period’s price (Kiyotaki and Moore,75997). x t + q t k t + Rb t − ≤ ( a + c ) k t − + q t k t − + b t (DBC λ t ) Rb t ≤ q t +1 k t (CC µ t )The collateral value of asset j in state s to agent i is the marginal benefit from being ableto take out loans backed by asset j (Fostel and Geanakoplos, 2008). CV is,j = (cid:20)
11 + r s −
11 + ω is
11 + r s (cid:21) φ is,j = 11 + r s ω is ω is φ is,j Where ω is is the liquidity wedge and φ is,j is the collateral capacity.He et al. (2015) have a model where housing loans can only be used as collateral fornon-housing consumption in a separate “KM” market. ψ t (cid:124)(cid:123)(cid:122)(cid:125) price = β (cid:124)(cid:123)(cid:122)(cid:125) sdf × U ( x t +1 , h t +1 ) (cid:124) (cid:123)(cid:122) (cid:125) service flow + αD ψ t +1 λ ( y t +1 ) (cid:124) (cid:123)(cid:122) (cid:125) collateral value + ψ t +1 (cid:124)(cid:123)(cid:122)(cid:125) resale price Their work is closely related to the current paper. Housing is used as collateral for futurenon-housing consumption, not to buy housing. When the liquidity value is positive, theyfind that house prices can display fascinating dynamics. Figure 2 in their paper shows thatsteady state house prices are hump shaped in the LTV ratio. In Section 6 they allow housesto depreciate and construction. ψ t (cid:124)(cid:123)(cid:122)(cid:125) price = β (cid:124)(cid:123)(cid:122)(cid:125) sdf × Ω ( h t +1 ) (cid:124) (cid:123)(cid:122) (cid:125) service flow + αD (1 − δ ) ψ t +1 Λ [(1 − δ ) ψ t +1 h t +1 ] (cid:124) (cid:123)(cid:122) (cid:125) collateral value + (1 − δ ) ψ t +1 (cid:124) (cid:123)(cid:122) (cid:125) resale price or Online Publication: Appendix Referencesor Online Publication: Appendix References