Intrinsic Bubbles and Fat Tails in Stock Prices

Transcription

1 Inrinsic Bubbles and Fa Tails in Sock Prices Prasad V. Bidarkoa Deparmen of Economics, Universiy Park DM 320A, Florida Inernaional Universiy, Miami, FL 3399, USA; Tel: ; Fax: address: Absrac: We sudy he consan discoun rae presen value model for sock pricing in a sochasic seing where he exogenous dividend sream is modeled as a random walk wih innovaions drawn from he family of sable disribuions. We derive an exac analyical soluion for he fundamenal sock price. We evaluae he abiliy of he model fundamenals and he dividends-driven inrinsic bubbles o explain he observed variaion in annual US sock prices. We compare resuls obained in his seing wih hose from he radiional model where all sochasic processes are driven by Gaussian shocks. Key phrases: Sock prices; presen-value model; inrinsic bubbles; fa ails; normal disribuions; sable disribuions. JEL classificaion: G2 Sepember 8, 2003

2 . INTRODUCTION Financial models of asse pricing radiionally have no done a very good job of explaining observed variaion in sock prices. The failure seems o sem from an inabiliy of pricing models o generae sufficien variaion in implied price-dividend raios. For insance, he consan discoun facor presen value model wih a random walk process for dividends implies a consan price-dividend raio whereas he observed series shows grea flucuaions over ime (Leroy and Porer, 98; Shiller, 98). One mehod o generae non-consan price-dividend raios in his model is o enerain soluions o asse prices ha do no saisfy he ransversaliy condiion. Such soluions ha are fully raional and depend only on he fundamenals of he model and no on any exraneous facors (such as calendar ime) are ermed as raional inrinsic bubbles by Froo and Obsfeld (99). In he linear presen value model wih exogenous dividends as he only fundamenals, inrinsic bubbles urn ou o be non-linear deerminisic funcions of dividends. Their non-linear naure allows bubble soluions o generae non-consan price-dividend raios, and allows hem o capure excess sensiiviy of sock prices o movemens in dividends. However, he non-linear naure of bubbles also implies ha hey are explosive in naure. Wih high values of dividends he bubble componen of sock prices will be very large. This remains an unsaisfacory feaure of bubble soluions o he presen value model. 2

3 Recen lieraure suggess ha he degree of non-lineariy required o generae observed variaion in a daa series is reduced when one accouns for any fa ails ha may exis in he empirical disribuions of he daa (Bidarkoa, 2000). There is a long and celebraed lieraure documening fa ails in sock prices, going back o early work by Mandelbro (963). McCulloch (996a) provides a summary of evidence on fa ails in sock prices. In a more recen paper, Lux and Sornee (999) demonsrae heoreically ha sock prices driven by processes wih raional bubble componens exhibi fa ails. Mandelbro (963) advocaed he use of sable disribuions for modeling hese fa ails. McCulloch (996a) provides a comprehensive survey on he financial applicaions of sable disribuions. These disribuions are he naural exensions of Gaussian disribuions, which are widely used on accoun of heir convenience and ease of analyical racabiliy. Gaussian errors are also ofen moivaed by heir Cenral Limi aribues. If financial markes evolve as an oucome of several individually unimporan decisions of a large number of invesors, hen one may appeal o he Cenral Limi Theorem and assume ha sock prices can be modeled as Gaussian processes. However, exacly he same argumen holds in he sable case as well since he Generalized Cenral Limi Theorem dicaes ha he limiing disribuion of such a process mus belong o he more general class of sable disribuions of which he Gaussian is jus one member (Zoloarev, 986). In his paper, we sudy raional inrinsic bubbles in he consan discoun facor presen value model where he only exogenous fundamenals (dividends) evolve as a 3

4 random walk sochasic process wih innovaions ha have sable disribuions. We derive an exac formula for he presen value sock prices in such a seing. We apply his model o analyze annual US sock price daa over he las cenury. We sudy o wha exen he presen value sock prices, derived in a sochasic seing ha admis fa ails in dividend realizaions, maches observed movemens in sock prices. We hen explore he role of inrinsic bubbles in such a seing. Because our assumed sochasic processes are able o model fa ails in dividends and price daa, we expec he conribuion of he nonlinear bubble erm in raionalizing observed sock prices o be diminished. We compare our resuls wih hose obained wihin a Gaussian seing ha does no accoun for fa ails. Driffill and Sola (998) also sudy inrinsic bubbles in he presen value model, assuming ha dividends follow a Markov swiching process proposed by Hamilon (989). They find ha he incremenal explanaory power of he bubble componen over he presen value fundamenal componen is significanly reduced when discree regime changes are allowed in he mean of he dividends process. The paper is organized as follows. We se ou he presen value model for sock prices in secion 2 and describe wha we mean by he fundamenal soluion and a bubble soluion. In secion 3 we derive he fundamenal sock price and he bubble under he assumpion ha dividends evolve as a random walk process wih sably disribued innovaions. We also compare his soluion wih ha obained under a Gaussian random walk for dividends. In secion 4, we underake an empirical sudy o deermine o wha 4

5 exen he presen value model, wih and wihou he bubble componens, explains he observed variaion in sock prices in sable and Gaussian seings. We summarize our main findings in he concluding secion. 2. THE PRESENT VALUE MODEL The presen value model wih a consan discoun rae is given by: r e + P = E [D + P ]. () Here, P D r E is he real price of a share a he beginning of period are he real dividends per share paid ou over period is he non-sochasic and consan discoun rae, equal o he real rae of ineres is he mahemaical expecaion, condiioned on informaion available a he sar of period. On forward ieraion, he presen value equaion yields: P r(s + ) e E s + s= s rs ( D ) lim e E ( P s = ). (2) One soluion o sock prices in he above equaion, denoed he ransversaliy condiion: lim s e rs E ( P ) = 0 s Imposing he ransversaliy condiion on Equaion (2) gives: pv P, is obained by imposing. (3) 5

6 pv s= r(s + ) ( D ) P = e E. (4) s Thus, his equaion provides he fundamenal value of he sock price. One specifies an exogenous sochasic process for dividends and evaluaes pv P. There exis oher soluions o he presen value model given in Equaion () ha do no saisfy he ransversaliy condiion in Equaion (3). For insance, le { } be any sequence of random variables ha saisfy: r B e E { B+ B =0 = }. (5) pv One can easily show ha ( P + B ) saisfies Equaion () bu violaes Equaion (3) for all B 0. If B is consruced as a funcion of he fundamenals alone, i.e. as a funcion of he dividends D alone in he presen value model of Equaion (), i is ermed an inrinsic raional bubble by Froo and Obsfeld (99). Inrinsic bubbles urn ou o be a non-linear funcion of dividends. Their exac funcional form depends on he assumed sochasic process for he dividends. 3. SOLUTION TO THE MODEL price In his secion, we obain an exac analyical soluion for he presen value sock pv P when he dividend growh rae follows a random walk wih drif wih 6

7 innovaions drawn from he family of sable disribuions. The Gaussian random walk emerges as a special case. We also derive condiions under which a posied funcional form for B saisfies all he condiions for a raional inrinsic bubble. 3a. Specificaion of he Dividends process moion: We assume ha log-dividends sochasically evolve according o he law of ln( D ) µ + ln(d ) = + ξ, ξ ~ iids(, β,c,0). (6) Here, S (,β, c,0) represens a sable disribuion wih characerisic exponen, skewness parameer β, scale parameer c, and locaion parameer se o zero. Appendix A defines hese disribuions and liss some of heir properies. For s, Equaion (6) implies ha: [( s ) µ + ξ + ξ + + ξ ] Ds D exp =. (7) + (s ) Subsiuing his ino he soluion for he fundamenal sock price given in Equaion (4) yields: [ exp( ξ + ξ + + ξ )] pv r(s + ) + (s ) µ = D e E s= P. (8) + (s ) In deriving Equaion (8), we assume ha D is conained in he informaion se available a he sar of period on which he expecaions E are based. 7

8 3b. Finieness of Condiional Expecaions Given he iid naure of he innovaions { ξ } o he dividends process, he expecaions erm on he righ hand side of Equaion (8) reduces o: E [ exp( ξ + ξ ξ )] = E exp( ξ ) [ ] E exp( ξ ) [ ]...E [ exp( )] (s ) ξ+ (s ). (9) ξ } When { is iid normal, each of he condiional expecaions on he righ hand side of he above equaion are finie and are given by he momen generaing funcion. However, when ξ } is iid non-normal sable, i.e. when he exponen characerizing { hese innovaions is less han 2, each of he condiional expecaions is infinie, unless he skewness parameer β = (see Appendix A). Equaion (A8) gives he formulae for hese condiional expecaions. 3c. Solving for he Presen Value Sock Price Thus, under he assumpion ha dividends evolve according o he sochasic process given in Equaion (6) wih β =, one can now derive he presen value sock price by evaluaing he righ hand side of Equaion (8). The expression for pv P differs in he case when he characerisic exponen = from ha when. In he res of his X This arises because of wo reasons. One reason is ha he expressions for Ee differ in he wo cases (see Equaion (A8) in Appendix A). A second reason is ha when we aggregae iid random variables wih sable disribuions, he expressions for he locaion 8

9 paper we focus our aenion on he more general case. All he resuls ha follow for are also applicable for = wih appropriae modificaions. The required derivaions for = do no pose any addiional difficulies, and can be easily adaped from hose given for in his paper. Appendix B shows ha he presen value sock price is given by: pv P = κd (0) where: κ = [ / { exp() r exp( µ c sec( π / 2))}]. () For convergence of he infinie summaion in Equaion (8), we need r > µ c sec ( π / 2). 3d. Inrinsic Raional Bubbles Le us posulae ha inrinsic raional bubbles ake he form given in Froo and Obsfeld (99): B( D ) = a 0 D λ. (2) Here, λ > 0 for he bubble o grow wih an increase in dividends and a 0 > 0 o ensure non-negaiviy of sock prices. parameer δ for he aggregae random variable also differ in he wo cases (see Equaion (A7) in Appendix A). 9

10 Appendix C shows ha he funcional form for he inrinsic bubble in Equaion (2) saisfies Equaion (5) defining a bubble, provided ha λ is chosen o saisfy: r = λµ ( λc) sec( π / 2 ). (3) The inequaliy r > µ c sec( π / 2 ) can be used o show ha λ > whenever he characerisic exponen >. 3e. Soluion under Gaussian Random Walk If he process for dividend growh raes is a Gaussian random walk plus drif, i.e. if he innovaions in Equaion (6) were Gaussian, hen he soluion for he presen value sock price is easily obained by seing = 2 in Equaions (0) and () above. One can readily show ha he expression obained for he sock prices in his case is idenical o he one given in Froo and Obsfeld (99). The condiions needed for convergence of he price-dividend raio as well as he ( condiions for B D ) o be a raional inrinsic bubble are also idenical o hose in Froo and Obsfeld (99). 4. EMPIRICAL ASSESSMENT OF THE MODEL 4a. Characerisics of he Daa All daa series used are aken from Shiller s (986) daa appendix. The nominal sock prices are annual series of January values of he Sandard and Poor Composie Sock Price Index (series in Shiller s daase). The nominal dividend series are 0

11 dividends per share (series 2 in Shiller s daase). The producer price index is used as he deflaor o obain real values (series 5 in Shiller s daase). This choice gives us he longes sample lengh spanning he period Alhough all hree series are available going back o 87, we sar he series in 900 because Froo and Obsfeld (99) use daa saring a his ime poin. They provide reasons for omiing daa from he earlier hree decades. Figure plos real sock prices, real dividends, real dividend growh raes, and he price-dividend raios. Table presens summary saisics on he dividend growh raes and on he price-dividend raios. A feaure ha emerges srongly from hese saisics is he lepokuric naure of boh series. Kurosis is saisically significanly greaer han hree indicaing fa ails in he empirical hisograms. Normaliy is srongly rejeced for boh series. This provides he basis for our empirical specificaion ha follows in he nex subsecion. 4b. Economeric Specificaion The empirical evaluaion of he presen value model requires specificaion of an exogenous sochasic process for dividends. From Equaion (6) and wih he assumed β=, we ge: ln( D ) µ + ln(d ) = + ξ, ξ ~ iids( ξ,,c,0). (4) ξ From he discussion immediaely following Equaion (5), a complee soluion o he presen value model can be wrien as:

12 pv P = P + B. (5) This saisfies he presen value model given by Equaion () bu violaes he ransversaliy condiion given in Equaion (3) for all B 0. Using Equaions (0), (2) and (3), one can wrie: λ = κd + a 0D P. (6) prices: Moivaed by his, one can hen wrie down an economeric model for sock P = b 0 D + b D λ + ε. (7) This can be rewrien afer dividing hrough by D as: P D = b 0 + b D λ + η, η ~ iids(,0,c,0). (8) η η where 0,b, λ > b 0. The error erm η is assumed o be independen of he innovaions ξ, and of he dividends D, a all leads and lags. The empirical assessmen of he presen value model proceeds wih esimaion of Equaions (4) and (8), subjec o: ( λc ) sec( π / 2) r = λµ ξ. (9) ξ ξ The null hypohesis of no bubbles implies ha alernaive hypohesis of a bubble implies ha b = κ and b = 0, whereas he 0 b = κ and b >

13 4c. Random Walk Model Esimaes for Real Dividends Table 2 presens empirical resuls on maximum likelihood esimaes of Equaion (4). 2 The firs panel repors resuls on fiing a random walk wih sable innovaions o real dividends and he second panel repors resuls on fiing a Gaussian random walk. The characerisic exponen is esimaed o be.86, well below he bound of 2 ha characerizes Gaussian disribuions. Following Froo and Obsfeld (99), he consan discoun facor is chosen o be r = Using he esimaes from maximum likelihood esimaion of he random walk model, we verify ha he convergence condiion required o obain he presen value sock price in Equaions (0 and () is saisfied. The model-implied price-dividend raio pv P / D κ is repored in Table 3 o be This agrees closely wih he mean price-dividend raio of repored in he second row of Table. Solving he nonlinear Equaion (9) yields λ = Compuing he probabiliy densiies for sable disribuions poses a challenge. One way o evaluae hese is by using Zoloarev s (986, p.74, 78) proper inegral represenaions or by aking he inverse Fourier ransform of heir characerisic funcion given in Equaions (A2) and (A3) in Appendix A. Here, we use he compuaional algorihm developed by J.P. Nolan (2000), archived a hp:// 3

14 From Table 3, wih Gaussian innovaions driving he random walk for dividends, he model-implied price-dividend raio κ is only 4.998, considerably below he empirically observed raio. The exponen defining he bubble componen λ is higher a For comparison we noe ha Froo and Obsfeld (99), wih a shorer and somewha differen daa series, obain an esimae of κ = 4 and λ = Thus, he sable model for dividends implies a consan heoreical price-dividend raio ha is close o he empirically observed mean. The Gaussian model also implies a consan heoreical price-dividend raio bu is value is low when compared o he empirically observed mean. Also, he sable model gives a bubble componen ha is considerably less nonlinear (as measured by he value of he exponen λ ) han ha under he Gaussian model. This is in accord wih he fac ha accouning for fa ails reduces he degree of nonlineariy required o explain observed variaion in price-dividend raios (see, for insance, Bidarkoa, 2000). 4d. Price-Dividend Raio Regression Resuls As noed a he end of subsecion (4b), he empirical evaluaion of he presen value model could proceed by esimaing Equaions (4) and (8), subjec o he resricion given in Equaion (9). One could esimae all he parameers of he model joinly, by simulaneous esimaion of he wo equaions. Or, alernaively, one could 4

15 esimae Equaion (4) firs, se b = κ and λ equal o he value obained by solving Equaion (9), and hen esimae Equaion (8). 0 In wha follows, we always esimae Equaions (4) and (8) individually raher han simulaneously. The reason is echnical. As noed in foonoe 2, compuing he probabiliy densiies for sable disribuions poses a challenge. While he innovaions o he log-dividends in Equaion (4) have a skewness coefficien of, he error erm in he price-dividend regression Equaion (8) has a skewness coefficien of 0. We use McCulloch s (996b) GAUSS code for esimaing he probabiliy densiies of he sable shocks in Equaion (8), bu his only works for errors ha are symmeric. To esimae he random walk wih maximally skewed sable errors in Equaion (4), we use Nolan s (2000) compuer program available in digial Forran (see foonoe 2 for furher deails). In our esimaion of various versions of he price-dividend raio regression ha we repor on below, we always se he exponen on he bubble erm λ a is value obained by solving Equaion (9). Froo and Obsfeld (99) do esimae he price-dividend raio regression his way and also alernaively by esimaing λ along wih he oher parameers of Equaions (4) and (8) simulaneously. However, heir inferences on he saisical significance of he bubble componen in he wo insances are qualiaively similar. Finally, we esimae Equaion (8) boh by esimaing b 0 as a free parameer and alernaively resricing b 0 = κ. We repor on boh resuls below. 5

16 Table 4 presens empirical resuls on maximum likelihood esimaion of he nonlinear price-dividend regression given in (8). The firs panel presens regression resuls wih sable errors and a sable random walk process for dividends. Resuls are presened boh for an unresriced model in which he coefficien on he bubble componen b is esimaed and a resriced model in which we se b = 0. Furher, wihin he unresriced and resriced models, resuls are presened boh for a version in which he inercep erm b is esimaed as a free parameer and a resriced model in 0 which we se b = κ. 0 For he fully unresriced model, we find from he firs row ha he characerisic exponen η is esimaed o be.76, suggesing subsanial fa ails compared o he Gaussian disribuion. The inercep erm b 0 is esimaed o be This is considerably lower han he heoreical price-dividend raio κ of The likelihood raio (LR) es indicaes ha he esimaed b is saisically significanly differen from κ. 0 The coefficien on he bubble componen b is esimaed o be The LR es for b = 0 is srongly rejeced. On accoun of he explosive naure of he bubble erm in Equaion (8), Froo and Obsfeld (99) show ha he -saisic for he hypohesis b = 0 will have he normal disribuion only if he regression residuals η are normally disribued. In Equaion (8), we have modeled hese as being drawn from he sable disribuion, however. In order o see wheher our saisical inference on he exisence of bubbles is affeced by his assumpion, we also esimaed he price-dividend regression 6

17 equaion wih η assumed normal, bu he dividends process is sill a random walk wih sable innovaions. Resuls are presened in he hird panel of Table 4. As we can see, none of our inferences change qualiaively from hose wih sable regression errors. Finally, panel 2 presens regression resuls wih a Gaussian random walk process for dividends and Gaussian errors η in he price-dividend regression. The esimaed of 2.50 is now much closer in value o he heoreical price-dividend raio κ of b 0 under a Gaussian random walk for dividends. Noneheless, he hypohesis ha b = κ is 0 sill srongly rejeced. The coefficien on he bubble componen b is lower a only Once again, one canno rejec he exisence of bubbles. Thus, all our saisical inferences are qualiaively unchanged across all hree panels of Table 4. Figures 2-4 plo he observed price-dividend raios and prices, along wih he fied values from he fully unresriced models in panels -3 of Table 4, respecively. The conribuion of he fundamenal presen value componen and ha of he bubble in accouning for observed variaion in P / D raios and sock prices is clearly eviden in he figures. There does no appear o be much of an improvemen in overall fi of he model when one goes from Gaussian o sable disribuions. 4e. Discussion of Resuls The P / D regression resuls repored in he previous subsecion indicae ha he hypohesis ha b = κ is rejeced across all hree panels. This conrass sharply wih he 0 7

18 resuls in Froo and Obsfeld (99). The difference is likely due o our longer daa series conaining observaions from he bull marke of he 990s. However, he fac ha we can rejec he absence of bubbles across all hree panels is in line wih he inference in Froo and Obsfeld (99). Thus, accouning for fa ails does no affec he qualiaive oucome of esing his hypohesis. The mos significan difference beween our resuls wih and wihou fa ails is he esimae of he exponen on he bubble componen λ. As repored in subsecion (4c), he Gaussian random walk for log-dividends yields an esimae for λ of whereas he esimae implied by he sable random walk is only.836. The sensiiviy of prices wih respec o dividends, measured by dp / dd, works ou o be wih he fully unresriced sable P / D regression resuls and wih he corresponding Gaussian regression. Thus, he sable model implies a lower sensiiviy of prices o dividends. 5. CONCLUSIONS We sudied he presen value model wih a consan discoun facor. The exogenous dividends are assumed o evolve as a random walk wih innovaions drawn from he family of sable innovaions. We derived an analyical formula for he presen value sock price in such a seing. Furher exending he analysis in Froo and Obsfeld (99) ha developed a Gaussian framework, we derived a funcional form for inrinsic bubble ha violaes he ransversaliy condiion. 8

19 We esimaed he model wih annual US sock price and dividends daa over he las cenury. Our saisical rejecion of he absence of a bubble componen in annual US sock price daa is unchanged when we accoun for fa ails in dividends and sock price daa. However, accouning for fa ails leads o an inrinsic bubble componen ha is less non-linear, and consequenly less explosive, han in he Gaussian seup. This seup also yields lower sensiiviy of prices o changes in dividends han is implied by he Gaussian framework. 9

20 APPENDIX A Sable Disribuions and Their Properies This secion draws heavily from McCulloch (996a). Sable disribuions S(x;, β,c, δ) are deermined by four parameers. The locaion parameer δ (, ) shifs he disribuion o he lef or righ, while he scale parameer c (0, ) expands or conracs i abou δ, so ha S(x;, β,c, δ) = S((x δ) / c;, β,,0). (A) The sandard sable disribuion funcion has c = and δ = 0. If a random variable X has a sable disribuion, i is represened as X ~ S(, β, c, δ ). The characerisic exponen ( 0, 2 ] governs he ail behavior, and herefore he degree of lepokurosis. When = 2, he normal disribuion resuls, wih variance 2c 2. For < 2, he variance is infinie. When >, E (X) = δ ; bu if, he mean is undefined. The skewness parameer β [ ], is defined such ha β > 0 indicaes posiive skewness. If β = 0, he disribuion is symmeric sable. As 2, β loses is effec and becomes unidenified. funcions: Sable disribuions are defined mos concisely in erms of heir log-characerisic ln Eexp( ix) = iδ+ ψ β ( c) (A2), where ψ, β ( iβsign() an( π / 2)) () = ( + iβ(2 / π)sign() ln ) for for = (A3) is he log-characerisic funcion for S (, β,,0 ). 20

21 When < 2, sable disribuions have ails ha behave asympoically like and give he sable disribuions infinie absolue populaion momens of order greaer han or equal o. x Le X ~ S( β,, c, δ) and a be any real consan. Then (A2) imples: ax ~ S(, sign( a) β, a c, aδ ). (A4) Le X ~ (, β,c, δ) and X2 ~ (, β2,c2, δ2 ) be independen drawings from sable disribuions wih a common. Then Y = X + X2 ~ S(, β,c, δ), where c = c + c (A5) 2 β = β c + β ( 2 c 2 )/c (A6) δ + δ2 for δ = δ + δ2 + 2( βcln( c) βcln( c) β2c2 ln( c2)) / π for =. (A7) When β = β, β equals heir common value, so ha Y has he same shaped disribuion 2 X 2 as and. This is he sabiliy propery of sable disribuions ha leads direcly o X heir role in he cenral limi heorem, and makes hem paricularly useful in financial porfolio heory. When β β 2, β lies beween β and β 2. For < 2 and β >, he long upper Pareian ail of X ~ S(, β, c, δ ) makes Ee X infinie. However, when β =, ln Ee X = δ c sec( π/ 2), δ+ ( 2c/ π)ln c, = (A8) This formula grealy faciliaes asse pricing under log-sable uncerainy. See also Zoloarev (986, p.2) and McCulloch (996a). 2

22 APPENDIX B Derivaion of he Presen Value Sock Price In his appendix we derive he soluion for he presen value sock price given by Equaions (0) and (). As noed in he firs paragraph of subsecion (3c), we only derive he formula for he sock price in he case. From Equaion (8), P [ exp( ξ + ξ + + ξ )] pv r(s + ) + (s ) µ = D e E s= Subsiuing Equaion (9) ino he above equaion yields: s= [ + (s ) + (s ). (B) pv r(s + ) + (s ) µ P = D e E [ exp( ξ )] E [ exp( ξ )]...E exp( ξ )]. (B2) From Equaion (6), ξ ~ iids(, β,c,0). Wih β = assumed in he derivaion of Equaion (0) and using Equaion (A8) in Appendix A, we ge: [ exp( ξ )] = E exp( ξ ) ( ) [ ] =... = E [ exp( ξ )] = exp c sec( π / 2) E (s ) Subsiuing Equaion (B3) ino Equaion (B2) yields: [ exp( c sec( π / ))] pv r(s + ) + (s ) µ = D e 2 s= s. (B3) P. (B4) This can be rewrien as: { exp( s ) ( r + µ c sec( π / ))} pv r P = De + 2. (B5) s= + 22

23 The infinie summaion in he above equaion converges only if r > µ c sec π / 2. In his case, from he sum of an infinie geomeric progression, we find: P pv = [ / { exp() r exp( µ c sec( π / 2) )} ] D ( ). (B6) or P pv = κd (B7) where: κ = [ / { exp() r exp( µ c sec( π / 2) )} ]. (B8) APPENDIX C Inrinsic Bubbles under Sable Random Walk plus Drif In his appendix we demonsrae ha B given by Equaion (2) is an inrinsic raional bubble when log-dividends evolves as a sable random walk plus drif as given in Equaion (6). Now, From Equaion (2), B( D ) ( ) λ B D a D, a 0 > 0. (C) = 0 is a raional inrinsic bubble if i saisfies Equaion (5), which is given as: r B e E { B+ } =. (C2) Equaion (6) implies ha: D + D exp[ µ + ξ+ ] =. (C3) 23

24 Therefore, λ D + [ + λ = D exp λµ + λξ ]. (C4) From Equaion (6), ξ ~ iids(, β,c,0). Wih β = and λ > 0, Equaion (A4) from Appendix A yields ξ ~ iids(,, λc,0). Using Equaion (A8) in Appendix A, we ge: [ ] { exp( λξ )} = exp ( λc) sec( π / 2) E +. (C5) Now, using Equaion (C) one can wrie he righ hand side of Equaion (C2) as: r r λ { B } e E { a D } e E. (C6) + = 0 + Subsiuing Equaion (C4) ino (Equaion (C6) yields: r λ { B } a e D E { exp[ λµ + λξ ]} r e E = (C7) Now, subsiuing Equaion (C5) ino Equaion (C7) gives: e r E [ ] λ { B } = a D exp r + λµ ( λc) sec( π / 2) + Thus, Equaion (C2) is saisfied, provided ha: r = λµ 0 ( λc) sec( π / 2 ). (C8). (C9) 24

25 Table : Summary Saisics of he Daa Mean Variance Skewness Kurosis Normaliy es Dividend growh raes.500e-2.590e (.26e-2) (2.24e-3) (0.99) (2.75e-0) (2.77e-0) Price-Dividend raios (0.92) (.92) (4.77e-26) (2.0e-94) (.32e-6) Noes o Table :. Numbers in parenheses in he firs wo columns are he sandard errors for he mean and variance. 2. Numbers in parenheses in he hird and fourh columns are he p-values for he null hypohesis of no skewness and no excess kurosis, respecively. 3. The normaliy es gives he Jarque-Bera es saisic and he p-value in parenheses. 25

26 Table 2: Maximum Likelihood Model Esimaes ln( D ) µ + ln(d ) = + ξ, ξ ~ iids( ξ,,c,0). (4) ξ Panel : Sable Random Walk ξ.859 (0.94) c µ ξ (0.02) (0.026) Panel 2: Gaussian Random Walk 2 σ ξ µ 2 (resriced) 0.06 (2.24e 3) 0.05 (.26e-2) Noes o Table 2: 2 2. When = 2, errors are Gaussian wih variance σ = 2c. 2. Numbers in parenheses for panel are he 95 percen confidence inerval esimaes. 3. Numbers in parenheses for panel 2 are he sandard errors. 26

27 Table 3: Implied Parameer Values Discoun facor r λ κ Sable random walk for Dividends Gaussian random walk for Dividends

28 Table 4: Price-Dividend Raio Regression Esimaes P D Panel : Sable Random Walk plus Sable Errors = b 0 + b D λ + η, η ~ iids(,0,c,0). (8) η η b 0 b η c η log L 2 log L 2 log L for for b 0 = κ b = 0 Unresriced model (.359) (0.306) (0.097) (0.260) (4.9e-5) (4.8e-9) (resriced o κ ) (0.4) (0.095) (0.34) (2.0e-6) Resriced model (0.694) (0.32) (0.385) (0.04) (resriced o κ ) (0.82) (0.44) Noes o Table 3:. Unresriced model is one in which b is esimaed. Resriced model ses b = Numbers in parenheses for he parameer esimaes are he Hessian-based sandard errors log L gives he likelihood raio (LR) es saisics. P-values from he χ disribuion are in parenheses. 28

29 Panel 2: Gaussian Random Walk plus Gaussian Errors P D = b 0 + b D λ + η 2 η, η ~ iid N(0, σ ). b 0 b 2 σ η log L 2 log L 2 log L for for b 0 = κ b = 0 Unresriced model (0.977) (0.053) (4.43) (0.0) (8.7e-25) (resriced o κ ) (0.03) (4.44) (2.6e-37) Resriced model (0.98) (.920) (.6e-5) (resriced o κ ) (22.506) Noes o Table 3:. Unresriced model is one in which b is esimaed. Resriced model ses b = Numbers in parenheses for he parameer esimaes are he Hessian-based sandard errors log L gives he likelihood raio (LR) es saisics. P-values from he χ disribuion are in parenheses. 29

30 Panel 3: Sable Random Walk plus Gaussian Errors P D = b 0 + b D λ + η 2 η, η ~ iid N(0, σ ). b 0 b 2 σ η log L 2 log L 2 log L for for b 0 = κ b = 0 Unresriced model (.703) (0.364) (4.977) (.8e-5) (9.0e-2) (resriced o κ ) (0.74) (9.365) (7.4e-9) Resriced model (0.98) (.920) (2.3e-3) (resriced o κ ) (3.08) Noes o Table 3:. Unresriced model is one in which b is esimaed. Resriced model ses b = Numbers in parenheses for he parameer esimaes are he Hessian-based sandard errors log L gives he likelihood raio (LR) es saisics. P-values from he χ disribuion are in parenheses. 30

31 Figure. Plos of Raw Daa 3

32 Figure 2. Resuls wih Sable Random Walk and Sable Price-Dividend Raio Regression 32

33 Figure 3. Resuls wih Gaussian Random Walk and Gaussian Price-Dividend Raio Regression 33

34 Figure 4. Resuls wih Sable Random Walk and Gaussian Price-Dividend Raio Regression 34

35 REFERENCES Bidarkoa, P.V., Asymmeries in he condiional mean dynamics of real GNP: robus evidence, The Review of Economics and Saisics, Vol.82, Issue, Driffill, J. and M. Sola, 998. Inrinsic bubbles and regime-swiching. Journal of Moneary Economics 42, Froo, K.A., Obsfeld, M., 99. Inrinsic bubbles: The case of sock prices, The American Economic Review 8, No.5, Hamilon, J.D A new approach o he economic analysis of nonsaionary ime series and he business cycle. Economerica 57, No.2, LeRoy, S.F. and R.D. Porer, 98. The presen value relaion: Tess based on implied variance bounds, Economerica 49, Lux, T. and D. Sornee, 999. On raional bubbles and fa ails. Unpublished manuscrip, Deparmen of Economics, Universiy of Bonn, Germany. Mandelbro, B., 963. The variaion of cerain speculaive prices, Journal of Business 36,

36 McCulloch, J.H., 996a. Financial applicaions of sable disribuions, in: Maddala, G.S., Rao, C.R., eds., Handbook of Saisics, Vol.4 (Elsevier, Amserdam) , 996b. Numerical approximaion of he symmeric sable disribuion and densiy, in: Adler, R., Feldman, R., and Taqqu, M.S. eds., A Pracical Guide o Heavy Tails: Saisical Techniques for Analyzing Heavy Tailed Disribuions, (Birkhauser, Boson). Nolan, J.P., Program for maximum likelihood esimaion of parameers for general (non-symmeric) sable disribuions, mle.ps a hp:// Shiller, R.J., 98. Do sock prices move oo much o be jusified by subsequen changes in dividends? American Economic Review 7, , 989. Marke Volailiy, Chaper 26 (Daa Appendix), MIT Press, Cambridge MA. Zoloarev, V.M., 986. One-dimensional sable disribuions (American Mahemaical Sociey, Providence, Rhode Island). (Translaion of Odnomernye Usoichivye Raspredeleniia, NAUKA, Moscow, 983.) 36