Attitudes Toward and Perceptions of the Ambiguity of House and Stock Prices

Transcription

1 Attitudes Toward ad Perceptios of the Ambiguity of House ad Stock Prices By YU ZHANG AND DONALD R. HAURIN* This study estimates idividuals' attitudes toward ad perceptios of ambiguity of house prices ad stock prices, usig experimet data from the Rad America Life Pael (ALP) survey. We estimate two importat parameters i multiple prior models ad α -MaxMi ambiguity prefereces: the degree of ambiguity aversio ad the degree of cofidece i the referece prior distributio of future prices, this beig a measuremet of the perceived level of ambiguity. Regardig attitudes, we fid that idividuals are slightly ambiguity seekig with regard to house prices while they are slightly ambiguity averse with regard to stock prices. Their degree of cofidece i the referece distributio for stocks is lower tha for house prices. We also fid that icreased state-level house price volatility durig the past year ad growth of house price i the past three years icrease perceived ambiguity. Moreover, ambiguity matters i that ambiguityaverse reters are less likely to buy a house. Correspodigly, ambiguity-averse stock ivestors ted to have less stock holdigs. (JEL: D8, R11, G11) * Zhag: Uiversity of Iteratioal Busiess ad Ecoomics, School of Isurace ad Ecoomics, Rm. 1320, Bldg. Boxue, 10 Huixi Dogjie, Chaoyag District, Beijig, , Chia, ( zhagy2119@uibe.edu.c). Hauri: Ohio State Uiversity, Departmet of Ecoomics, 1945 N High Street, Columbus, Ohio 43210, ( hauri.2@osu.edu). We thak semiar participats at the Real Estate ad Housig PhD coferece at Ohio State Uiversity, AEDE PhD semiar, CES coferece, MEA coferece ad AREUEA-ASSA coferece, All errors are our ow.

2 This paper develops a method to estimate idividual s perceptios of ad attitudes toward ambiguity with regard to future house ad stock prices. We idetify idividuals socio-ecoomic characteristics ad price treds that correlate with house ad stock price ambiguity ad the test how ambiguity iflueces reters home purchase behavior ad the stock holdigs of ivestors. Iovatios iclude our method that simultaeously idetifies ambiguity attitudes ad perceptios from two experimets ad the applicatio to the housig market. The housig market i the period differed from prior ad subsequet periods i multiple ways. The most otable differece was the large icrease i house prices followed by a substatial dowtur. At the atioal level, the aual house price icrease averaged from the begiig of 2000 through 2006, but fell by aually from 2007 through Apart from the price level, the volatility of house prices also peaked durig this period. The stadard deviatio was i the 1980s ad i the 1990s. It rose to i the first decade of the ew milleium ad the fell back to from 2010 to We argue that this history of house price chages icreased idividuals perceptios of risk ad the level of ambiguity of house prices. The empirical questio addressed is whether this potetial icrease i ambiguity affected housig market behaviors such as the home purchase decisio. Throughout the aalysis, we compare our results regardig the ambiguity of the housig market to the same idividuals perceptios of ambiguity i the stock market. Idividuals attitudes toward ad perceptios of ambiguity have bee defied ad measured i the fiace ad psychology literatures (e.g. i fiace see Ellsberg 1961; Curly ad Yates 1985; Machia ad Siiscalchi 2013; Dimmock et al. 2015, 1 The data used are the Freddie Mac House Price Idex, which icludes all coformig trasactios (purchases ad refiacig appraisals) from Freddie Mac ad Faie Mae. The method used to compute the price idex is based o the repeat sales method. Data at the MSA, state, ad atioal level are available from

3 2016; i psychology see Budescu et al. 1988; Budescu ad Wallste 1995; Du ad Budescu 2005). The differece betwee risk ad ambiguity depeds o the amout of kowledge about the probability distributio(s) of outcomes of a ucertai evet. Risk correspods to situatios where the possible outcomes of a future evet are subject to chace ad the odds of each outcome are uiquely determied. Ambiguity describes the situatios where the odds of each outcome are ot uiquely specified. If a idividual has a sigle prior distributio of outcomes with uiquely specified probabilities, the level of ambiguity is zero. If a idividual caot uiquely specify the odds of each outcome of the evet, the ambiguity is ozero. The source of perceived ambiguity i house prices could derive from two sources. A idividual may perceive multiple distributios of outcomes whe the set of house price outcomes is cotiget o other outcomes. Give that house prices are the outcome of the iteractio of supply ad demad ad both supply ad demad are affected by may ucertai ecoomic factors (icome, household formatio, buildig regulatios, materials costs), a idividual s aticipated house price chage may require may distributios to describe. Aother factor that could result i the perceptio of house price ambiguity is the presece of a diversity of opiios about future house prices by housig market experts or jouralistic sources resultig i idividuals receivig mixed iformatio about the likely future path of house prices. For example, i the 2017 Zillow Home Price Expectatios Survey (composed of about 100 experts) the respodets' cumulative expected price chage after five years raged from -23 to 37 percet. 2 Such diversity of opiios could ifluece a idividual ad result i the perso havig multiple distributios of expected future house price outcomes. 2 The resultig stadard deviatio is 9 percet. The Zillow Home Price Expected Survey is available at

4 Idividuals have prefereces toward ambiguity, just as they have prefereces toward risk. They may be ambiguity averse, eutral, or seekig. Ambiguity-averse idividuals prefer a future evet where the outcomes ca be described by oe or a few probability distributios of outcomes. Ambiguity-seekig idividuals prefer a evet for which there are relatively may distributios of outcomes. Our simple assumptio is that a idividual has a particular preferece for ambiguity, applicable to all ucertai evets, this beig typical of the assumptio made regardig risk preferece. Prefereces (attitudes) toward risk ca be measured through a idividual participatig i a experimet. Similarly, a experimet ca be used to measure prefereces toward ambiguity. Usig the data of RAND America Life Pael (ALP) ( Dimmock et al. (2015) coducted a Ellsberg-like experimet to elicit ambiguity prefereces. They asked respodets to choose betwee a ambiguous eviromet ad a uambiguous, yet risky, eviromet. We follow their method to elicit respodets' geeral ambiguity preferece by measurig the matchig probability of ambiguity aversio (MP AA ), defied as the probability at which the respodet is idifferet betwee a ambiguous eviromet ad the zero ambiguity but risky case. It is reasoable to assume that idividuals are heterogeeous i terms of their attitudes toward risk ad ambiguity. We correlate idividuals socio-ecoomic characteristics with their prefereces toward ambiguity ad fid that females, seiors, college graduates ad Hispaics ted to be relatively risk averse, while males, youth, college graduates, ad idividuals with relatively more kowledge about fiace ted to be more ambiguity averse. We also study the similarity of idividual level prefereces toward risk ad ambiguity. The correlatio is relatively low, 0.144, but is statistically sigificat at the 1 percet level. Next, we differece the measures of aversio toward ambiguity ad risk ad regress this differece o a set of socio-ecoomic characteristics of idividuals. We fid that youger

5 idividuals, males, Hispaics ad fiacially kowledgeable idividuals are relatively more averse to ambiguity tha risk. The measuremet of a idividual s perceptio of the level of ambiguity of house or stock prices is more complex. Aderso, Ghysels ad Juerges (2009) employ the dispersio of professioal predictios of the stock prices from the Survey of Professioal Forecasters to measure the ambiguity of stock prices i time series data. However, this method is uable to measure the perceived ambiguity at a idividual level. Dimmock et al. (2015) overcome this drawback by usig data from a experimet to estimate idividuals degree of cofidece i the referece probability assigmets of two possible outcomes of a gamble, this the is used to measure the perceptio of ambiguity. However, their estimatio of perceived ambiguity of a gamble is ot applicable to a specific ucertai evet such as future house or stock prices. This paper is ovel i that we estimate ambiguity perceptios ad prefereces toward house ad stock prices at the idividual level. We exted Dimmock et al.'s (2015) method, combiig the multiple prior models ad the assumptio of α MaxMi ambiguity prefereces, to estimate idividuals degree of cofidece i oe s subjective referece prior distributio for future house prices as the measuremet of perceptio of ambiguity ad the degree of ambiguity aversio, α. After measurig perceived ambiguity, we search for correlates that arguably are causal, fidig that past house price treds at the state level affect idividuals perceived ambiguity of house prices. Moreover, we fid that ambiguity-averse reters are less likely to buy a house ad ambiguity-averse stock ivestors ted to hold less stocks. This is the first study to augmet the explaatio of home purchase behavior by icludig the ambiguity ad ambiguity aversio of house prices i the model. The rest of this paper is orgaized as follows. Sectio I describes the data ad explais the methodology used to estimate a idividual's attitude toward ad

6 perceptios of ambiguity with regard to house ad stock prices. Sectio II reports the results of the estimatio that yield the measures of ambiguity, first for a represetative idividual ad the allowig for heterogeeity. We the relate ambiguity ad risk prefereces ad perceptios with idividuals characteristics. We measure itertemporal variatios i the perceptio of house price ambiguity ad relate these chages to variatios i state level house prices. I sectio III, we estimate the effect of attitudes toward ad perceptios of ambiguity o reters' purchase behavior ad ivestors stock holdigs betwee Jauary 2009 ad Jauary 2016 usig pael data ad a fixed effects model. Sectio IV cocludes the whole paper. I. Methodology A. Data Source The dataset we employ i the study iclude various modules of RAND America Life Pael (ALP). The ALP is a atioally represetative iteret survey of more tha 6,000 respodets aged 18 or above. I order to be atioally represetative, ALP provides sample weights, these used i our descriptive statistics. 3 The data measurig attitudes toward ambiguity are from the Netspar Ucertaity (NU) module, which is a cross-sectio coducted from March 20, 2012 to April 16, I this module, a ambiguity experimet was coducted, cosistig of 2,367 respodets. 4 The data we use for measurig the referece prior distributio is from the Effects of Fiacial Crisis (EFC) module. This module cosists of 61 waves from November 2008 to Jauary We use six waves of the module whe a experimet, called bis-ad-balls, was coducted betwee April 2011 ad April 3 Detailed iformatio about ALP weightig is available at 4 Respodets are icluded if they took at least two miutes to complete the experimet ad provided iformatio about their characteristics.

7 2013. We joitly employ the bis-ad-balls ad the ambiguity attitudes experimet i NU module to measure the level of perceived ambiguity. B. Measurig Geeral Attitudes Toward Ambiguity We follow Dimmock et al. (2016) ad costruct a cotiuous measuremet of a idividual s attitude toward ambiguity. I the NU module, all respodets are asked to choose betwee a uambiguous Box K with kow distributio of purple balls ad orage balls, ad a ambiguous Box U with ukow distributio of the balls for each color. There are a total of 100 balls i each box. After the respodet selects the box, oe ball is radomly draw from the box chose by the respodet who wis $15 if a purple ball is draw. Up to four rouds of the experimet is coducted. I the first roud, the probability of wiig $15 i Box K is exactly 50% (see Figure 1), while the probability of wiig i Box U is ot give. FIGURE 1: FIRST ROUND QUESTION ABOUT ATTITUDES TOWARD AMBIGUITY Note: The figure is borrowed from Dimmock et. al (2015) The experimet cotiues as follows. If the respodet selects Box K i the first roud, its probability of wiig falls to 25%, while if the respodet selects Box U i the first roud, the probability wiig i Box K icreases to 75%. I roud

8 three, the kow probabilities are 12.5% ad 87.5%. The experimet cocludes whe the respodet is idifferet betwee Box K ad U, or the fial fourth roud. The value whe idifferece is achieved is desigated as the matchig probability for ambiguity aversio (MP AA ) (Wakker 2010; Dimmock et al. 2016). 5 We use MP AA as the cotiuous measure of attitude toward ambiguity. The respodets with MP AA > 50% are ambiguity-seekig, those with MP AA = 50% are ambiguity-eutral ad those with MP AA < 50% are ambiguity-averse. 6 I Dimmock et al. s (2015) weighted sample, 52.4% of respodets were ambiguity averse, 9.9% eutral, ad 37.7% ambiguity seekig. I order to separate attitudes toward ambiguity from attitudes toward risk, a similar four-roud experimet was coducted i the NU module, focusig o risk. I this experimet, respodets were asked to choose betwee Box A which has a 100% chace of wiig the icetive ad Box B with a risky, but well-defied probability of wiig. The expected returs i the two boxes were equivalet. Based o this experimet, we costruct a cotiuous measure of the attitude toward risk: the matchig probability for risk aversio (MP RA ). 7 The respodets with MP RA > 50% are defied as risk-seekig, those with MP RA = 50% are riskeutral, ad those with MP RA < 50% are risk-averse. Figure 2 shows the distributios of MP AA ad MP RA i our sample. We fid that ambiguity aversio is approximately ormally distributed i the sample, while highly risk averse ad risk 5 If the respodet selects box K i the first roud ad the reports idifferece i the secod, third, or fourth rouds, the assiged MP AA values are 0.25, 0.12, ad 0.06, respectively. If the respodet selects box K i the fial roud, the value assiged is Each sequece of choices of boxes geerates a correspodig MP AA value betwee 0 ad 1. The precise values are displayed i Table A.1, i a o-lie appedix to Dimmock et al. (2016), which is available at 6 Details about ad validatio of the ALP experimet is cotaied i Dimmock et al. (2015). They also describe additioal experimets that address the issue of a idividual s attitude toward ambiguity appearig to differ depedig o whether the evet is likely or ot, ad o whether the payoff is a gai or loss. 7 We defie the matchig probability for risk aversio as the probability at which the respodet is idifferet betwee a risk-free box ad risky box.

9 seekig idividuals are overrepreseted compared to a ormal distributio of risk prefereces. 8 FIGURE 2: DISTRIBUTIONS OF MATCHING PROBABILITY FOR AMBIGUITY AVERSION AND RISK AVERSION C. Measurig the Referece Prior Distributio We use ALP s Bis ad Balls questio to measure a idividual s referece prior distributio about future house prices. The we use a extesio of Dimmock et al. s (2015) theoretical model to measure perceptios about the ambiguity of house prices. The bis ad balls questio was desiged to capture a respodet s expectatios about future housig prices. Each respodet was assiged 20 balls ad 6 bis were preseted. Each bi represets a rage of percetage price chage i the future ad the respodets were asked to allocate the 20 balls ito the 6 bis. The umber of balls allocated to each bi represets the likelihood s/he believes that the price chage will be i the rage correspodig to the bi. These questios were asked regardig the expectatios of housig prices oe ad five years after the survey date. 8 The extreme values resulted from a idividual either selectig the safe bet i all four rouds of the experimet o matter how high the expected value was of the gamble, or always selectig the gamble o matter how poor the expected payout. Experimet coducted by Taaka, Camerer ad Nguye (2010) also shows that extreme risk-averse ad extreme risk-seekig values are overrepreseted (see Figure 1, Taaka, Camerer ad Nguye (2010)).

10 Figure 3 shows a example of the bis ad balls questio. Here, the respodet allocated balls ito three bis, with the 0 to 10 percet price icrease bi receivig half of the balls, the 10 percet to 20 percet price icrease bi receivig six balls ad the 0 to 10 percet price decrease bi receivig four balls. We use the umber of balls i the bis to represet the referece probability assigmet for each idividual. FIGURE 3: EXAMPLE OF ALP BINS-AND-BALLS SURVEY QUESTION D. Theoretical Model We exted Dimmock et al. s (2015) theoretical model i order to measure both a idividual s attitude toward ambiguity ad perceived level of ambiguity. They specify a tractable versio of the α-maxmi utility optimizatio model, which is a weighted average of the MaxMi ad MaxMax models. As described below, this model has two key parameters, oe beig α, which represets the attitude toward ambiguity ad the other beig δ, which represets the perceived level of ambiguity of a evet. Suppose we have a state space S cosistig of a fiite umber of possible states of house price growth rates ext year: S = (s 1, s 2,, s ), where s i represets oe state of the growth rate of house prices. I the bis-ad-balls questios, = 6 because the umber of bis preseted to the respodets is six. A probability

11 measure P is a assigmet fuctio: P: s i R, for all i that assigs a probability value to each possible state. The probability assigmet has the followig properties: P(s i ) [0,1] for all i, P( ) = 0, P(S) = 1, P(s i s i i ) = P(s i) + P(s i i ), P(s i s i i ) = 0. The last two properties reflect the assumptio that all possible states of the future house price are mutually exclusive. These properties are cosistet with the requiremets of the bis-ad-balls questios. Ambiguity i projectig house prices occurs whe a idividual caot form a uique assigmet for P(s i ) for all future states. Derived from the MaxMi model, Gilboa ad Schmeidler (1989) propose the multiple prior model, which assumes the aget has a covex set C for all possible probability assigmets P C. Suppose u( ) is a vo Neuma-Morgester utility fuctio. A ambiguity-averse aget s actio-cotiget value is characterized as V(φ) = mi ( P C s i S u(φ(s i)) dp(s i )), where φ represets a aget s decisio as a real-valued fuctio defied o the state space S. Ambiguityaverse agets select the decisio φ that maximizes the value V( ) = max mi φ ( P C s i S u(φ(s i)) dp(s i )). Ituitively, this meas that ambiguity-averse agets select the act whose most ufavorable prior is the best. A aget is ambiguity-seekig if we replace the mi operator with max, where this aget selects the act based o the prior distributio givig the highest expected utility, which is called the MaxMax model. A more geeral model, called α-maxmi model, weights the MaxMi ad MaxMax models, α [0,1], yieldig a actiocotiget value fuctio of: (1) V(φ) = α mi ( P C s i S u(φ(s i)) dp(s i )) + (1 α) max ( u(φ(s i)) dp(s P C s i S i ))

12 As oted by Dimmock et al. (2015), the maximum ambiguity aversio occurs at the value α = 1 (MaxMi), ad maximum ambiguity seekig at α = 0 (MaxMax). I order to estimate perceived ambiguity, we must idetify the set of probability assigmets C. Epstei ad Wag (1994) assume that a idividual has a sigle subjective referece probability assigmet π(s i ) for each possible state, this assumptio beig part of their ε-cotamiatio model. 9 Chateaueuf et al. (2007) assume the decisio-maker has a degree of cofidece (1 δ) [0,1] i their referece prior distributio. Usig this framework, Dimmock et al. (2015) derive the set of possible subjective probability assigmets: (2) C δ = {P: 0 (1 δ)π P (1 δ)π + δ 1} The probability assigmet P amog the various sets varies i a iterval of legth δ aroud the referece probability π. Importatly, the degree of cofidece i the referece probability distributio, 1 δ, is used to measure the level of perceived ambiguity.(see Dimmock for the proof) Dimmock et al. (2015) used data from the ALP Ellsberg ur experimet to estimate α ad δ i a gamble (just 2 states). However, their experimet provides a pre-determied objective referece wiig probability to the respodets for the gamble ad thus their estimates are ot specific to ay real-world situatio where each idividual has a subjective referece prior probability but does ot kow the true probability. We exted the Dimmock model to measure perceived ambiguity usig the ALP bis ad balls questio, applied to future house price chages. We geeralize the set of possible subjective probability assigmets to a state-specific set of possible probability assigmet o each state s i, which is characterized as: (we have 6 states) 9 ε measures the degree that the referece probability is cotamiated by other probability beliefs.

13 (3) C δ = {P i : 0 (1 δ)π(s i ) P i (1 δ)π(s i ) + δ 1, s i S} This meas that the domai of the set of probability assigmet for each possible state represeted by each bi is depedet o the referece probability assigmet ad the degree of cofidece i the referece probability assigmet. Without loss of geerality we ca ormalize the utility obtaied from state 1 to 0, u(s 1 ) = 0, ad assume u(s i ) > 0 for i = {2,3, }. Give these assumptios, the most ufavorable prior distributio is P i,mi = (1 δ)π(s i ) for i = {2,3, }, ad P 1,mi = 1 i=2 P i,mi = 1 (1 δ) i=2 π(s i ). 10 Oe ca show that P i,mi is i the feasible domai (3) for all i. 11 However, we are ot able to aalytically derive the most favorable prior distributio by simply assumig P i,max P 1,max = (1 δ)π(s i ) + δ for i = {2,3, } ad = 1 i=2 P i,max, which assigs greater values to P i 1 but lower values to P 1, give that u(s 1 ) = 0 ad u(s i 1 ) > 0. That is because P 1,max i this case is out of domai i (3) if Therefore, for ay evet that has two or two above possible states, the probability assigmets P i,max the domai i (3). Eve though P i,max for i = {1,2,3, } are out of caot be reached, it is the upper limit of the max situatio ad the true most favorable prior distributio P i,max will satisfy i=1 P i,max u(s i ) < i=1 P i,max u(s i ). Thus, the most favorable distributio (which ca be solved umerically but ot aalytically) has a greater value tha the aalytical solutio we use below. We accout for this with a iequality. The α-maxmi model evaluates a actio-cotiget value fuctio as: 10 Coditioal o u(s1 ) = 0 ad u(s i ) > 0 for i = {2,3, }, the most ufavorable prior distributio that gives the lowest expected utility, i=1 P i u(s i ), is to assig greater P 1 but lower P i Pi,mi = (1 δ)π(s i ) for i = {2,3, } reaches the lower boud of P i i (3). Ad (1 δ)π(s 1 ) P 1,mi = 1 (1 δ) i=2 π(s i ) (1 δ)π(s 1 ) + δ holds for all δ [0,1]. 12 P 1,max = 1 i=2 P i,max = 1 (1 δ) π(s i ) i=2 ( 1)δ = 1 (1 δ) π(s i ) (1 δ)π(s 1 ) (1 δ)π(s 1 ) = 1 (1 δ) ( 1)δ = (2 )δ, which is smaller tha 0 if 2. i=2 ( 1)δ +

14 (4) α mi i P i u(s i ) + (1 α) max i P i u(s i ) P i C δ P i C δ = α P i,mi u(s i ) i < α P i,mi u(s i ) i + (1 α) P i,max u(s i ) i + (1 α) P i,max u(s i ) = α (1 δ)π(s i )u(s i ) + α [1 (1 δ)π(s i )] u(s 1 ) i=2 i i=2 + (1 α) [(1 δ)π(s i ) + δ]u(s i ) i=2 + (1 α) [1 (1 δ)π(s i ) ( 1)δ] u(s 1 ) i=2 = i=2[(1 δ)π(s i ) + (1 α)δ] u(s i ). (because u(s 1 ) = 0) = [(1 δ)π(s i ) + (1 α)δ] u(s i ). i=1 Let m be a positive value such that m i u(s i ) α)δ] u(s i ). Thus, we ca derive: (5) m = [(1 δ)π(s i )+(1 α)δ] i u(s i ) i u(s i ) = [(1 δ)π(s i )] i u(s i ) i u(s i ) = i [(1 δ)π(s i ) + (1 + (1 α)δ Let w i = u(s i ) i u(s i ) deote the weight of the utility obtaied from the state s i relative to the sum of the utility obtaied from every state, thus i w i = 1. Moreover, i the referece prior distributio, we have i π(s i ) i=2 = 1. Substitute w i ito (5) ad replace π(s 1 ) by 1 π(s i ), which yields the m as a fuctio of

15 the referece probability assigmet for state s i, give the parameter set (w i for i, δ, α): (6) m = i [(1 δ)w i π(s i )] = (1 δ)w 1 [1 π(s i ) i=2 + (1 α)δ ] + [(1 δ)w i π(s i )] + (1 α)δ i=2 = [(1 δ)(w i w 1 )π(s i )] + (1 δ)w 1 + (1 α)δ. i=2 Suppose m i u(s i ) = α mi i P i u(s i ) + (1 α) max i P i u(s i ), which P i C δ P i C δ meas that m is the matchig probability such that a idividual is idifferet betwee a uambiguous evet (left-had side of the equatio) ad a ambiguous evet (right-had side). 13 Therefore, m ca be used to measure a idividual s degree of ambiguity aversio for house prices if it is observable. Ufortuately, m is ot observable ad we eed idetify α to measure the degree of ambiguity aversio for house prices. The idetificatio procedure will rely o the aalytical relatioship betwee m ad π(s i ) derived i equatio (6), while the aalytical relatioship betwee m ad π(s i ) does ot exist as we discussed above. We should ote that m < m based o the iequality i (4). E. Ecoometric Model ad Idetificatio I the bis ad balls ALP questio, s 1 represets the state that house price ext year will decrease by 20% or greater, s 2 represets a decrease by 10%-20%, s 3 a 13 The defiitio of m is aalogous to the defiitio of matchig probability for ambiguity aversio. However, the true matchig probability should satisfy the coditio that i m (s i )u(s i ) = α mi i P i u(s i ) + (1 α) max i P i u(s i ), P i C δ P i C δ which meas that there exists a matchig probability for each possible state s i ad m (s i ) is the certai umerical probability i state s i such that the idividual is idifferet betwee the uambiguous evet ad the ambiguous evet. Nevertheless, we are ot able to observe m (s i ) for all states, but use a sigular value m such that m i u(s i ) = α mi i P i u(s i ) + P i C δ (1 α) max i P i u(s i ) to represet the matchig probability. P i C δ

16 decrease by 0%-10%, s 4 a icrease by 0%-10%, s 5 a icrease by 10%-20%, ad s 6 represets a icrease by 20% or more. Therefore, each respodet provides a uique subjective probability assigmet for each state s i. We assume that the subjective probability assigmet represets their referece probability for each state ad thus we ca observe π j (s i ), s i S, for every idividual j i the data. Give m j for every idividual j, the we ca estimate the followig regressio: (7) m j = i=2 β i π j (s i ) + β 0 + ε j From this model, we ca obtai the estimates of β i, i = {2,3,, } ad β 0, otig that β 1 is omitted because each respodet is required to allocate exactly 20 balls. Comparig (6) with (7), we fid: (8) β 0 = (1 δ)w 1 + (1 α)δ ad (9) β i = (1 δ)(w i w 1 ) Because u(s 1 ) = 0, we kow w 1 = 0. Applyig this to (8) ad (9) yields: (8 ) β 0 = (1 α)δ ad (9 ) β i = (1 δ)w i Next, the slope coefficiets over i are summed, ad ote that the sum of the weights equals oe.

17 (9 ) i=2 β i = (1 δ) From (9 ), the degree of cofidece i the referece distributio (perceived ambiguity) is foud, ad from (8 ), the attitude toward ambiguity is determied. Thus, the parameters of the α-maxmi model are idetified from the regressio as δ = 1 i=2 β i ad α = 1 β 0 /δ. Oce m j is observed δ ad α ca be idetified. Eve though we caot observe m j i the dataset for house price, we ca observe m j, the matchig probability i a gamble for each respodet j based o the ambiguity attitude experimet i the Netspar Ucertaity (NU) module. It is arguable whether the matchig probability i a gamble (m j) is equivalet to the matchig probability i the case of house prices (m j), which meas whether a idividual s degree of ambiguity aversio is costat across differet evets. If oe believes so, which meas that she believes m j = m j for all idividual j, we ca simply use m j to measure the degree of ambiguity aversio of the respodet for all evets. However, some literature argue that oe s degree of ambiguity aversio is varyig across evets (Heath ad Tversky, 1991). If so, usig ambiguity aversio i a gamble to measure the respodet s ambiguity aversio i house prices will geerate measuremet error problem. Therefore, we eed idetify α for house prices. As we metioed above, the idetificatio procedure will rely o the aalytical relatioship betwee m j ad π j (s i ) derived i (6). Suppose m j m j = μ j for idividual j, ad we assume E(μ j ) = k. Therefore, the relatioship betwee m j ad π j (s i ) is characterized as: (10) m j = m j μ j = β i π j (s i ) + β 0 + ε j μ j i=2

18 = β i π j (s i ) + β 0 k + ε j i=2 where E(ε j )=0. We the estimate the followig regressio: (11) m j = i=2 β iπ j (s i ) + β 0 + ε j Note that μ j is uobservable i the regressio. Comparig equatios (10) ad (11), we fid β i = β i ad β 0 = β 0 k. Thus, give that μ j is ucorrelated with π j (s i ), β i is a ubiased estimator of β i ad β 0 is a ubiased estimator of β 0 if k = 0, but the estimate of β 0 is a biased estimator of β 0 if k Because the idetificatio of δ oly depeds o β i, we ca get the ubiased estimatio of δ based o: (12) δ = 1 i=2 β i We also fid a estimate of: (13) α = 1 β 0 δ = 1 β 0 δ Notice that α α = (1 β 0 δ ) (1 β 0 δ ) = k δ. This meas that α will overestimate the true α by k if k > 0 ad uderestimate the true α if k < 0.15 δ 14 If k = 0, the E(m j) = E(m j ) > E(m j). Because greater values i matchig probability represets lower degree of ambiguity aversio, the ituitio of k = 0 is that populatio s ambiguity aversio toward the gamble is smaller tha the populatio s ambiguity aversio toward house prices o average. It is possible because the gai or loss i housig market is much greater tha the gamble i the experimet, which may make people more ambiguity-averse with regard to house prices. 15 If oe believes that ambiguity aversio i the gamble is equivalet to the ambiguity aversio i house prices (m j = m j), the α estimated by usig m j will overestimate the true α. Because we kow m j = m j < m j, we have μ j = m j m j > 0 for all j ad, thus, k = E(μ j ) > 0. However, if oe believes m j = m j, she does t eed estimate α, but ca simply use m j to measure respodet j s ambiguity aversio for all evets.

19 The theoretical framework is summarized i Table 1. This framework geerates a ubiased estimatio of δ ad a ubiased estimatio α if μ j is urelated with π j (s i ) ad E(μ j ) = 0. However, the estimate of α will be biased if E(μ j ) 0 ad the estimate of δ will be biased if μ j is correlated with π j (s i ). TABLE 1 THE IDENTIFICATION STRATEGY FOR δ AND α Matchig probability m j (Uobserved) m j (Observed) The relatioship with π j (s i ) m j = β i π j (s i ) + β 0 + ε j m j = β iπ j (s i ) + β 0 + ε j Idetificatio of δ i=2 δ = 1 β i i=2 i=2 δ = 1 β i Idetificatio of α α = 1 β 0 /δ α = (1 β 0 δ ) Coclusio 1 i=2 The true δ ad true α ca be idetified if m j is observable. The relatioship betwee m j ad m j m j m j = μ j Assumptio μ j is urelated with π j (s i ) Coclusio 2 δ = δ α = α E(μ j) δ δ is the ubiased estimator of true δ α is the ubiased estimator of true α if E(μ j ) = 0; α overestimates the true α if E(μ j ) > 0; α uderestimates the true α if E(μ j ) < 0; F. Estimatig Heterogeeous Attitudes toward ad Perceptios of Ambiguity We ext estimate idividuals perceived ambiguity, δ j, ad attitude toward ambiguity, α j, with regard to house prices by allowig the two parameters to vary across idividual characteristics. Because the NU ambiguity attitude experimet is a cross-sectioal experimet coducted i April 2012, we ca oly sample the 697

20 respodets who completed both the NU ambiguity attitude experimet ad the bis-ad-balls experimet at the same time period i this sectio. Suppose that the relatioship betwee m j ad π j (s i ) ca be expressed as: Q x=1 Q x=1 (14) m j = i=2 π j (s i )( β ix X xj + c i ) + β 0 + ρ x X xj + ε j X xj deotes a idividual j s characteristics x, ad Q is the total umber of idividual characteristics. The idepedet variables are the iteractio terms betwee the referece probability i each state s i ad the idividual characteristics. The umber of estimated β ix thus will be Q. The, based o equatio (6), we fid that: Q (15) x=1 β ix X xj + c i = (1 δ j )(w ij w 1j ) ad Q (16) β 0 + x=1 ρ x X xj = (1 δ j )w 1j + (1 α j )δ j From equatio (15) ad (16), we ca derive: Q x=1 +c i ) 1 w 1j (17) δ j = 1 i=2 ( β ixx xj ad (18) α j = 1 β 0+ x=1 ρ x X xj Q δ j (1 δ j )w 1j Equatios (17) ad (18) idicate that perceived ambiguity ad attitudes toward ambiguity about house prices are affected by the idividual characteristics. Similar

21 to the model for represetative aget, we ormalize w 1j = 0 for every respodet j. The ituitio of the ormalizatio is that every respodet treats the state 1, s 1, as the worst state. 16 The, (17 ) δ j = 1 i=2 ( x=1 β ix X xj + c i ) ad Q (18 ) α j = 1 β Q 0+ x=1 ρ x X xj δ j I the ambiguity attitude experimet, we caot observe m j, but ca observe m j. Based o m j = m j μ j, we ca derive Q x=1 Q x=1 (19) m j = i=2 π j (s i )( β ix X xj + c i ) + β 0 + ρ x X xj + ε j μ j Q Q = π j (s i )( β ix X xj + c i ) + β 0 + ρ x X xj k + ε j i=2 x=1 x=1 Therefore, we ca estimate the followig regressio: Q x=1 Q x=1 (20) m j = i=2 π j (s i )( β ix X xj + c i) + β 0 + ρ xx xj + ε j Ad we ca get: (21) β ix = β ix ad c i = c i 16 Oe may argue that reters may ot treat the state 1 as the worst state because they ca afford a house i the future if the house price ext year ca decrease by 20% or above. However, the housig bust may adversely affect the reters welfare by affectig their icome. Therefore, assumig u j (s 1 ) = 0 may ot be a bad assumptio.

22 ad Q Q (22) β 0 + x=1 ρ xx xj = β 0 + x=1 ρ x X xj k Equatios (21) ad (22) show that δ j ad α j that are estimated based o m j are ubiased estimators of true δ j ad α j if, first, μ j is ucorrelated with referece probability assigmets, π j (s i ), ad idividual characteristics, X xj ; ad secod, E(μ j ) = 0. G. Measurig Perceived Risk Because we ca observe the etire referece prior distributio of each respodet i bis-ad-balls, we follow Delavade ad Rohwedder (2008) to calculate the expectatio ad variace of the prior by assumig that the probability assigmet withi each bi is uiformly distributed. The, we use the expectatio of the prior to measure a idividual s expected growth rate of house price, ad use the variace of the prior to measure a idividual s perceived risk. Suppose [D i, D i+1 ] represets the iterval of the growth rates of house price i bi i = {1,2,, } ; s S represets the possible growth rate. The, the expectatio of the growth rate derived from bis-ad-balls questios is characterized as: D i+1 (23) E(S) = sf(s) ds i D i Assumig that the probability assigmet withi each bi is uiformly distributed, the we fid the probability desity fuctio withi each bi f(s) =

23 π i D i+1 D i, for s [D i, D i+1 ]. π i represets the probability assigmet withi i s bi, which equals to the umber of balls i i s bi times 5%. Therefore, we have: D sπ (24) E(S) = i+1 i i ds = 1 [ s2 π i 2 i D i D i+1 D i D ] i+1 D i+1 D Di i 2 Di 2 ) = 1 π i(d i+1 2 i = 1 π 2 i i(d i+1 + D i ) D i+1 D i The, the variace of the distributio ca be calculated as (see Appedix 1 for more details): (25) Var(S) = π i [ D 3 3 i+1 Di i D i+1 D i 3 2 (D i+1 D 2 i )E(S) + E(S) 2 (D i+1 D i )] E(S), Var(S) ad δ are used to evaluate idividuals price expectatios, the level of perceived risk, ad the level of perceived ambiguity of i the housig market. II. Results A. α-maxmi Model Estimatio Results for Represetative We apply our theoretical model to estimate the level of perceived ambiguity for oe-year house prices ad oe-year stock prices. We first estimate a represetative aget model without cotrollig ay idividual characteristics. Table 2 shows the results of the regressio usig ALP data from 2011 to 2013: For each respodet j, the matchig probability is computed from the Ellsberg ur experimet ad the πj (s i ) are determied by the umber of balls this respodet places i each bi.

24 TABLE 2 REGRESSION RESULTS FOR REPRESENTATIVE AGENT MODEL Dep variable: MPAA 1-year house prices 1-year stock prices (1) (2) β * 0.056* (0.038) (0.032) β ** (0.031) (0.026) β ** 0.049* (0.030) (0.025) β *** 0.081*** (0.033) (0.028) β *** 0.080** (0.038) (0.033) Costat β *** 0.417*** (0.030) (0.024) Wave * (0.011) (0.011) Wave (0.010) (0.010) Wave (0.011) (0.011) Wave (0.010) (0.010) Wave (0.010) (0.010) Obs. 4,148 4,131 R Note: Stadard errors are i the paretheses. ***: sigificace at 1%; **: sigificace at 5%; *: sigificace at 10% Colum 1 ad 2 show the results for house prices ad stock prices, respectively. For oe-year house prices, the sum of the slope coefficiets equals 0.395, meaig that the represetative aget has a degree of cofidece i the referece prior

25 distributio is δ is 1 less this value, equalig The attitude toward ambiguity measure, α, equals 0.411, suggestig that a represetative idividual i this sample was ambiguity seekig with regard to house prices. For oe-year stock prices, the δ ad degree of cofidece are ad (icludig the poit estimate for β 3, which is ot statistically sigificat), showig that the idividual perceives greater ambiguity about stock prices tha house prices. The attitude toward ambiguity with regard to stock prices is 0.480, slightly ambiguity-seekig but closer to ambiguity-eutral tha the attitude toward ambiguity with regard to house prices. If we set β 3 = 0, the =0.734, ad α =0.510, idicatig that the represetative aget is slightly ambiguity-averse about stock prices. B. α-maxmi Model Estimatio Results for Heterogeeous Agets Because some estimated δ j ad α j are out of the domai of [0,1], we use the followig method to ormalize the estimated δ j ad α j ito the domai of [0,1], ad we use the ormalized δ j ad α j for the followig aalysis: δ j mi (δ j ) Normalized δ j = max (δ j ) mi (δ j ) Table 3 shows the summary statistics of attitude toward ambiguity, level of perceived ambiguity ad level of perceived risk with regard to house prices ad stock prices, respectively. Similar to the results for represetative aget model, we fid that people perceive larger ambiguity about stock prices tha house prices. Moreover, people are ambiguity-seekig with regard to house prices, but ambiguity-averse with regard to stock prices. The mea of idividual α j ad of δ j

26 for house prices are ad 0.541, which are close to the ad for represetative aget model, ad The mea of idividual δ j for stock prices is 0.705, which is close to the i represetative aget model, (excludig isigificat β 3 ). Nevertheless, the mea idividual α j for stock prices is 0.610, which is more ambiguity-averse tha the i the represetative aget model, We also fid that people perceive larger risk about stock prices tha house prices, too. The idex to measure perceived risk, VAR(S), is for stock prices, but for house prices. TABLE 3 SUMMARY STATISTICS OF AMBIGUITY ATTITUDES, AMBIGUITY AND RISK Obs. 18 Mea Std.Dev. Mi Media Max 1-year house prices Attitude toward ambiguity ( ) Level of perceived ambiguity ( ) Level of perceived risk (VAR(S)) year stock prices Attitude toward ambiguity ( ) Level of perceived ambiguity ( ) Level of perceived risk (VAR(S)) C. Socio-ecoomic Correlates with Attitudes toward Ambiguity, Ambiguity ad Risk I this sectio, we show the relatioship of attitude toward ambiguity, perceived ambiguity ad perceived risk with idividual characteristics for house prices ad stock prices. Table 4 shows the results of house prices ad Table 5 shows the results of stock prices. 18 The bottom 1% ad top 1% estimated αj are excluded because the scale of the outliers is too large.

27 We estimate a OLS model to idetify which idividual characteristics are sigificatly correlated with attitudes toward ambiguity. Appedix 2 reports variables defiitios, meas, ad stadard deviatios. We fid the followig attributes are sigificat ad icrease the tedecy for a idividual to be ambiguity-averse toward house prices: smaller age ad worse health. We also estimate a Tobit model trucated a 0 ad 1 for the estimated but o-ormalized α j as a robustess check (see appedix 3 for more details). The followig attributes are statistically sigificat ad icrease the tedecy for a idividual to be ambiguity-averse toward house prices i the Tobit model: smaller age, married, Hispaic, employed, fiacial literacy ad worse health. No effect is foud for geder, icome, educatio, White, Black, havig retiremet accout or ot, the umber of household members ad wealth. 19 We also fid that followig idividual characteristics are positively correlated with the level of perceived ambiguity about house prices: greater age, male, icome, advaced educatioal attaimet, Black, havig retiremet accout, greater household size, greater wealth ad better health. The idividual characteristics that are egatively correlated with the level of perceived ambiguity are married, White, Hispaic, employed ad greater fiacial literacy. 20 Moreover, we fid that males ad idividuals with higher educatioal attaimet perceive less risk about house prices. TABLE 4 THE RELATIONSHIP OF INDIVIDUAL CHARACTERISTICS WITH AMBIGUITY ATTITUDES, PERCEIVED AMBIGUITY AND PERCEIVED RISK (HOUSE PRICES) Depedet variable Ambiguity attitude Ambiguity Risk (α) (δ) Var(S) (1) (2) (3) 19 Our results differ from Dimmock et al. (2015) i that he foud a more limited umber of sigificat variables. They icluded age ad Hispaic, with their sigs agreeig with ours. For these cases the coefficiets are similar i size. 20 Dimmock et al. (2015) ad our results agree o household size ad educatioal attaimet, but disagree o White ad fiacial literacy.

28 Age ** *** (0.0003) (0.0001) (0.0000) Male *** *** (0.0061) (0.0019) (0.0004) Married *** (0.0071) (0.0022) (0.0005) Log (icome) *** (0.0050) (0.0015) (0.0004) High school *** *** (0.0184) (0.0057) (0.0014) College *** *** (0.0188) (0.0058) (0.0014) Graduate *** *** (0.0197) (0.0060) (0.0014) White *** (0.0154) (0.0047) (0.0011) Black *** (0.0189) (0.0058) (0.0014) Hispaic *** (0.0144) (0.0044) (0.0010) Retiremet accout *** (0.0075) (0.0023) (0.0005) Household members *** (0.0024) (0.0007) (0.0002) Employed *** (0.0069) (0.0021) (0.0005) Fiacial kowledge *** (0.0040) (0.0012) (0.0003) Wealth (1 millio) *** (0.0081) (0.0022) (0.0001) Good health *** *** (0.0192) (0.0058) (0.0014) Fair health *** *** (0.0190) (0.0057) (0.0013) Costat *** *** *** (0.0545) (0.0166) (0.0040) Obs R

29 Note: Stadard errors are i the paretheses. ***: sigificace at 1%; **: sigificace at 5%; *: sigificace at 10%. We use similar method to estimate relatioship for stock prices, which is showed i Table 5. We fid that greater age ad greater household size are sigificat ad icrease the tedecy for a idividual to be ambiguity-averse toward stock prices accordig to the result of OLS model. Based o the Tobit model, we fid that greater age, male, higher icome are sigificat ad icrease the tedecy to be ambiguity-averse with regard to stock prices, ad higher wealth ad better health are sigificat ad decrease the tedecy to be ambiguity-averse with regard to stock prices. We also fid that followig idividual characteristics are positively correlated with the level of perceived ambiguity about stock prices: married, advaced educatioal attaimet, White, Black, employed, greater wealth ad better health. The idividual characteristics that are egatively correlated with the level of perceived ambiguity are age, male, icome, Hispaic, havig retiremet accout, greater household size ad greater fiacial literacy. Moreover, we fid that idividuals with higher educatioal attaimet perceive less risk about stock prices. TABLE 5 THE RELATIONSHIP OF INDIVIDUAL CHARACTERISTICS WITH AMBIGUITY ATTITUDES, PERCEIVED AMBIGUITY AND PERCEIVED RISK (STOCK PRICES) Depedet variable Ambiguity attitude Ambiguity Risk (α) (δ) Var(S) (1) (2) (3) Age *** *** (0.0002) (0.0001) (0.0000) Male *** (0.0044) (0.0014) (0.0005) Married ***

30 (0.0053) (0.0017) (0.0007) Log (icome) *** (0.0037) (0.0012) (0.0004) High school *** *** (0.0136) (0.0044) (0.0015) College *** *** (0.0141) (0.0045) (0.0015) Graduate *** ** (0.0146) (0.0047) (0.0016) White *** (0.0101) (0.0036) (0.0012) Black *** (0.0138) (0.0045) (0.0015) Hispaic *** (0.0108) (0.0034) (0.0012) Retiremet accout *** (0.0054) (0.0017) (0.0006) Household members * *** (0.0017) (0.0006) (0.0002) Employed *** (0.0050) (0.0016) (0.0005) Fiacial kowledge *** (0.0030) (0.0010) (0.0003) Wealth (1 millio) *** (0.0057) (0.0018) (0.0006) Good health *** (0.0139) (0.0044) (0.0015) Fair health (0.0137) (0.0044) (0.0015) Costat *** *** ** (0.0405) (0.0130) (0.0044) Obs R Note: Stadard errors are i the paretheses. ***: sigificace at 1%; **: sigificace at 5%; *: sigificace at 10%

31 Whe comparig the results betwee Table 4 ad 5, we fid a more importat fidig that the relatioship of the variables of ambiguity ad risk with idividual characteristics are heterogeeous across housig market ad stock market. For example, we fid that age has sigificat opposite relatioship to the attitude toward ambiguity about stock prices ad house prices. The effect of age, geder, marriage status, icome, White, retiremet accout, household members ad employmet status o the level of perceived ambiguity is varyig across markets, too. Geder has sigificat effect o the perceived risk about house prices but o sigificat effect o the perceived risk about stock prices. People perceive heterogeeous ambiguity toward differet markets may be because of their various prefereces o differet markets so that people with similar characteristics are more likely to collect the iformatio about a particular market. Accordig to Dimmock et al. (2016), White, o-hispaic, married, people livig with less umber of childre, healthier people ad wealthier people are more likely to participate i stock market, which implies that they may be more used to collect iformatio about stock market. However, at the era of iformatio explosio, receivig too much iformatio about oe market may trigger larger ambiguity perceptio about the market because of the diverse professioal forecasts o future s developmet of the market (Viscusi ad Chesso, 1999). Therefore, people with such characteristics perceive large ambiguity about stock market accordig to Table 5 i our results. 21 Moreover, we also fid that people s attitude toward ambiguity is heterogeeous across markets. Oe possible explaatio to the pheomeo is competece hypothesis proposed by Heath ad Tversky (1991), which claims that people are more ambiguity-seekig toward the evet they cosider themselves kowledgeable. 21 Dimmock et al. (2016) also fid that high-icome ad fiacial kowledgeable people are more likely to participate stock market, but high-icome ad fiacial kowledgeable people perceive less ambiguity about stock prices i our results.

32 For example, elder people perceive larger ambiguity about house prices tha youger people, but less ambiguity about stock prices tha youger people, which may be because they collect more iformatio ad cosider themselves kowledgeable about housig market. Therefore, elder people may be more familiar with housig market, while youg people may be more familiar with stock market. Based o the competece hypothesis, elder people are hypothesized to be more ambiguity-seekig toward housig market, but more ambiguity-averse toward stock market, which is validated by our results. Competece hypothesis ca explai why certai people perceivig larger (smaller) ambiguity about a market are more ambiguity-seekig (ambiguity-averse) toward the market showed i our results. Moreover, comparative igorace hypothesis proposed by Fox ad Tversky (1995) ca explai the pheomeo partially as well. 22 If elder people cosiders themselves more kowledgeable tha youger people about housig prices, they should be more ambiguity-seekig tha youger people about housig market. D. Itertemporal Variatio of Perceived Ambiguity The bis ad balls questios were coducted six times betwee April 2011 ad April 2013 ad thus we are able to observe the itertemporal variatio i perceptios about the house prices. I Table 6 we show the itertemporal variatio of average expected house price growth rates, perceptios of house price risk ad ambiguity toward ext year s house prices. I Pael A, we show the results derived from bis ad balls. I Pael B, we show the results from Wall Street Joural Ecoomic Forecastig Survey ad i Pael C, the results are from the Zillow/Pulseomics survey of housig experts. The average predicted growth rates of the three sources are relatively close. Pael D cotais oe-year growth rates 22 Comparative igorace hypothesis: ambiguity aversio is triggered whe the decisio makers compare the prospect with more familiar evets or whe they feel igorat to the evet compared with other kowledgeable idividuals.