fat tails, tall tales, puppy dog tails

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Fat Tails, Tall Tales, Fat Tails, Tall Tales, Puppy Dog Tails Puppy Dog Tails Dan diBartolomeo Dan diBartolomeo Annual Summer Seminar Annual Summer Seminar Newport, RI Newport, RI June 8, 2007 June 8, 2007

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Fat tailed distributions in the stock market

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Page 1: Fat Tails, Tall Tales, Puppy Dog Tails

Fat Tails, Tall Tales, Fat Tails, Tall Tales, Puppy Dog TailsPuppy Dog Tails

Dan diBartolomeoDan diBartolomeoAnnual Summer Seminar Annual Summer Seminar –– Newport, RINewport, RI

June 8, 2007June 8, 2007

Page 2: Fat Tails, Tall Tales, Puppy Dog Tails

Goals for this TalkGoals for this Talk

•• Survey and navigate the enormous literature in Survey and navigate the enormous literature in this areathis area

•• Review the debate on assumed distributions for Review the debate on assumed distributions for stock returnsstock returns

•• Consider the implications of the various possible Consider the implications of the various possible conclusions on asset pricing, portfolio conclusions on asset pricing, portfolio construction and risk managementconstruction and risk management

Page 3: Fat Tails, Tall Tales, Puppy Dog Tails

Return DistributionsReturn Distributions•• While traditional portfolio theory assumes that returns While traditional portfolio theory assumes that returns

for equity securities and market are normally distributed, for equity securities and market are normally distributed, there is a vast amount of empirical evidence that the there is a vast amount of empirical evidence that the frequency of large magnitude events frequency of large magnitude events seems much seems much greater than is predicted by the normal distribution with greater than is predicted by the normal distribution with observed sample variance parametersobserved sample variance parameters

•• Three broad schools of thought:Three broad schools of thought:–– Equity returns have stable distributions of infinite variance.Equity returns have stable distributions of infinite variance.–– Equity returns have specific, identifiable distributions that haEquity returns have specific, identifiable distributions that have ve

significant kurtosis (fat tails) relative to the normal distribusignificant kurtosis (fat tails) relative to the normal distribution tion (e.g. a gamma distribution)(e.g. a gamma distribution)

–– Distributions of equity returns are normal at each instant of Distributions of equity returns are normal at each instant of time, but look fat tailed due to time series fluctuations in thetime, but look fat tailed due to time series fluctuations in thevariancevariance

Page 4: Fat Tails, Tall Tales, Puppy Dog Tails

Stable Pareto DistributionsStable Pareto Distributions

•• Mandelbrot (1963) argues that extreme events are far Mandelbrot (1963) argues that extreme events are far too frequent in financial data series for the normal too frequent in financial data series for the normal distribution to hold. He argues for a stable distribution to hold. He argues for a stable ParetianParetianmodel, which has the uncomfortable property of infinite model, which has the uncomfortable property of infinite variancevariance

•• Mandelbrot (1969) provides a compromise, allowing for Mandelbrot (1969) provides a compromise, allowing for “locally Gaussian processes”“locally Gaussian processes”

•• FamaFama (1965) provides empirical tests of Mandelbrot’s (1965) provides empirical tests of Mandelbrot’s idea on daily US stock returns. Finds fat tails, but also idea on daily US stock returns. Finds fat tails, but also volatility clusteringvolatility clustering

•• Lau, Lau and Lau, Lau and WingenderWingender (1990) reject the stable (1990) reject the stable distribution hypothesisdistribution hypothesis

•• RachevRachev (2000, 2003) deeply explores the mathematics (2000, 2003) deeply explores the mathematics of stable distributionsof stable distributions

Page 5: Fat Tails, Tall Tales, Puppy Dog Tails

A Bit on Stable DistributionsA Bit on Stable Distributions

•• General stable distributions have four parametersGeneral stable distributions have four parameters–– Location (replaces mean)Location (replaces mean)–– Scale (replaces standard deviation)Scale (replaces standard deviation)–– SkewSkew–– Tail Fatness Tail Fatness

•• Some moments are infiniteSome moments are infinite•• Except for some special cases (e.g. normal) there are no Except for some special cases (e.g. normal) there are no

analytical expressions for the likelihood functionsanalytical expressions for the likelihood functions•• Estimation of the parameters is very fragile. Many, Estimation of the parameters is very fragile. Many,

many different combinations of the four parameters can many different combinations of the four parameters can fit data equally wellfit data equally well

•• These distributions do have time scaling (you should be These distributions do have time scaling (you should be able to scale from daily observations to monthly able to scale from daily observations to monthly observations, etc.)observations, etc.)

Page 6: Fat Tails, Tall Tales, Puppy Dog Tails

Specific Fat Tailed DistributionSpecific Fat Tailed Distribution

•• Gulko (1999) argues that an efficient market Gulko (1999) argues that an efficient market corresponds to a state where the informational entropy corresponds to a state where the informational entropy of the system is maximizedof the system is maximized

•• Finds the riskFinds the risk--neutral probabilities that maximize entropyneutral probabilities that maximize entropy•• The entropy maximizing risk neutral probabilities are The entropy maximizing risk neutral probabilities are

equivalent to returns having the Gamma distributionequivalent to returns having the Gamma distribution•• Gamma has fat tails but only two parameters and finite Gamma has fat tails but only two parameters and finite

momentsmoments•• Has finite lower bound which fits nicely with the lower Has finite lower bound which fits nicely with the lower

bound on returns (i.e. bound on returns (i.e. --100%)100%)•• Derives an option pricing model of which BlackDerives an option pricing model of which Black--ScholesScholes

is a special caseis a special case

Page 7: Fat Tails, Tall Tales, Puppy Dog Tails

Time Varying VolatilityTime Varying Volatility•• The alternative to stable fatThe alternative to stable fat--tailed distributions is that tailed distributions is that

returns are normally distributed at each moment in time, returns are normally distributed at each moment in time, but with time varying volatility, giving the illusion of fat but with time varying volatility, giving the illusion of fat tails when a long period is examinedtails when a long period is examined

•• Rosenberg (1974?)Rosenberg (1974?)–– Most kurtosis in financial time series can be explained by Most kurtosis in financial time series can be explained by

predictable predictable time series variation in the volatility of a normal time series variation in the volatility of a normal distributiondistribution

•• Engle and Engle and BollerslevBollerslev: ARCH/GARCH models: ARCH/GARCH models–– Models that presume that volatility events occur in clustersModels that presume that volatility events occur in clusters–– Huge literature. I stopped counting when I hit 250 papers in Huge literature. I stopped counting when I hit 250 papers in

referred journals as of 2003referred journals as of 2003•• LeBaronLeBaron (2006)(2006)

–– Extensive empirical analysis of stock returnsExtensive empirical analysis of stock returns–– Finds strong support for time varying volatility, but very weak Finds strong support for time varying volatility, but very weak

evidence of actual kurtosisevidence of actual kurtosis

Page 8: Fat Tails, Tall Tales, Puppy Dog Tails

The Remarkable Rosenberg PaperThe Remarkable Rosenberg Paper

•• Unpublished paper by Barr Rosenberg (1974?), under US Unpublished paper by Barr Rosenberg (1974?), under US National Science Foundation Grant 3306National Science Foundation Grant 3306

•• Builds detailed model of timeBuilds detailed model of time--varying volatility in which varying volatility in which long run kurtosis arises from two sourceslong run kurtosis arises from two sources–– The kurtosis of a population is an accumulation of the kurtosis The kurtosis of a population is an accumulation of the kurtosis

across each sample subacross each sample sub--periodperiod–– Time varying volatility and serial correlation can induce the Time varying volatility and serial correlation can induce the

appearance of kurtosis when the distribution at any one moment appearance of kurtosis when the distribution at any one moment in time is normalin time is normal

–– Predicts more kurtosis for high frequency dataPredicts more kurtosis for high frequency data•• An empirical test on 100 years of monthly US stock index An empirical test on 100 years of monthly US stock index

returns shows an Rreturns shows an R--squared of .86squared of .86•• Very reminiscent of subsequent ARCH/GARCH modelsVery reminiscent of subsequent ARCH/GARCH models

Page 9: Fat Tails, Tall Tales, Puppy Dog Tails

ARCH/GARCHARCH/GARCH

•• Engle (1982) for ARCH, Engle (1982) for ARCH, BollerslevBollerslev (1986) for GARCH(1986) for GARCH•• Conditional heteroscedasticity models are standard Conditional heteroscedasticity models are standard

operating procedure in most financial market operating procedure in most financial market applications with high frequency dataapplications with high frequency data

•• They assume that volatility occurs in clusters, hence They assume that volatility occurs in clusters, hence changes in volatility are predictablechanges in volatility are predictable

•• Andersen, Andersen, BollerslevBollerslev, , DieboldDiebold and and LabysLabys (2000)(2000)–– Exchange rate returns are GaussianExchange rate returns are Gaussian

•• Andersen, Andersen, BollerslevBollerslev, , DieboldDiebold and and EbensEbens (2001)(2001)–– The distribution of stock return variance is right skewed for The distribution of stock return variance is right skewed for

arithmetic returns, normal for log returnarithmetic returns, normal for log return–– Stock returns must be Gaussian because the distribution of Stock returns must be Gaussian because the distribution of

returns/volatility is unit normalreturns/volatility is unit normal

Page 10: Fat Tails, Tall Tales, Puppy Dog Tails

Recent Empirical ResearchRecent Empirical Research

•• LebaronLebaron, , SamantaSamanta and and CecchettiCecchetti (2006)(2006)•• Exhaustive MonteExhaustive Monte--Carlo bootstrap tests of various fat Carlo bootstrap tests of various fat

tailed distributions to daily Dow Jones Index data using tailed distributions to daily Dow Jones Index data using robust estimatorsrobust estimators

•• Propose a novel adjustment for time scaling volatilities to Propose a novel adjustment for time scaling volatilities to account for kurtosis, in order to use daily data to account for kurtosis, in order to use daily data to forecast monthly volatilityforecast monthly volatility

•• Conclusion: “No compelling evidence that 4Conclusion: “No compelling evidence that 4thth moments moments exist”exist”–– If variance is unstable, then its difficult to estimateIf variance is unstable, then its difficult to estimate–– High frequency data is less usefulHigh frequency data is less useful–– Use robust estimators of volatilityUse robust estimators of volatility–– Estimation error of expected returns dominates variance in Estimation error of expected returns dominates variance in

forming optimal portfoliosforming optimal portfolios

Page 11: Fat Tails, Tall Tales, Puppy Dog Tails

More Work on Fat TailsMore Work on Fat Tails

•• Japan Stock ReturnsJapan Stock Returns–– AggarwalAggarwal, , RaoRao and and HirakiHiraki (1989) (1989) –– Watanabe (2000)Watanabe (2000)

•• France Stock ReturnsFrance Stock Returns–– NavatteNavatte, , ChristopheChristophe Villa (2000)Villa (2000)

•• Option implied kurtosisOption implied kurtosis–– CorradoCorrado and Su (1996, 1997a, 1997b)and Su (1996, 1997a, 1997b)–– Brown and Robinson (2002)Brown and Robinson (2002)

•• Sides of the debateSides of the debate–– Lee and Wu (1985)Lee and Wu (1985)–– Tucker (1992)Tucker (1992)–– GhoseGhose and and KronerKroner (1995)(1995)–– MittnikMittnik, , PaolellaPaolella and and RachevRachev (2000)(2000)–– RockingerRockinger and and JondeauJondeau (2002)(2002)

Page 12: Fat Tails, Tall Tales, Puppy Dog Tails

The Time Scale IssueThe Time Scale Issue

•• Almost all empirical work shows that fat tails are more Almost all empirical work shows that fat tails are more prevalent with high frequency (i.e. daily rather than prevalent with high frequency (i.e. daily rather than monthly) return observationsmonthly) return observations

•• Lack of fat tails in low frequency data is problem for Lack of fat tails in low frequency data is problem for proponents of stable distributions, proponents of stable distributions, –– the tail properties should time scalethe tail properties should time scale–– maybe we just don’t have enough observations when we use maybe we just don’t have enough observations when we use

lower frequency data for apparent kurtosis to be statistically lower frequency data for apparent kurtosis to be statistically significantsignificant

•• Or the observed differences in higher moments could be Or the observed differences in higher moments could be a mathematical artifact of the way returns are being a mathematical artifact of the way returns are being calculatedcalculated–– Lau and Lau and WingenderWingender (1989) call this the “(1989) call this the “intervalingintervaling effect”effect”

Page 13: Fat Tails, Tall Tales, Puppy Dog Tails

The Curious Compromise of The Curious Compromise of FinanalyticaFinanalytica•• The basic concepts of stable fat tailed distributions and The basic concepts of stable fat tailed distributions and

timetime--varying volatility models are clearly mutually varying volatility models are clearly mutually exclusive as explanations for the observed empirical dataexclusive as explanations for the observed empirical data

•• From the From the FinanalyticaFinanalytica website:website:–– “uses proprietary generalized multivariate stable (“uses proprietary generalized multivariate stable (GMstableGMstable) )

distributions as the central foundation of its risk management distributions as the central foundation of its risk management and portfolio optimization solutions”and portfolio optimization solutions”

–– “Clustering of volatility effects are well known to anyone who “Clustering of volatility effects are well known to anyone who has traded securities during periods of changing market has traded securities during periods of changing market volatility. volatility. FinanalyticaFinanalytica uses advanced volatility clustering models uses advanced volatility clustering models such as stable GARCH…”such as stable GARCH…”

•• SvetlozarSvetlozar RachevRachev and Doug Martin are really smart guys and Doug Martin are really smart guys so I’m putting this down to pragmatism rather than so I’m putting this down to pragmatism rather than schizophreniaschizophrenia

Page 14: Fat Tails, Tall Tales, Puppy Dog Tails

Kurtosis versus SkewKurtosis versus Skew

•• So far we’ve talked largely about 4So far we’ve talked largely about 4thth momentsmoments•• We haven’t done much in terms of economic arguments We haven’t done much in terms of economic arguments

about why fat tails exist, and at least appear to be more about why fat tails exist, and at least appear to be more prevalent with higher frequency dataprevalent with higher frequency data

•• Many of the same arguments apply to skew (one fat Many of the same arguments apply to skew (one fat tail), tail), –– consistent prevalence of negative skew in financial data seriesconsistent prevalence of negative skew in financial data series

•• Harvey and Harvey and SiddiqueSiddique (1999) find that skew can be (1999) find that skew can be predicted using an autoregressive scheme similar to predicted using an autoregressive scheme similar to GARCHGARCH

Page 15: Fat Tails, Tall Tales, Puppy Dog Tails

CrossCross--Sectional DispersionSectional Dispersion

•• When we think about “fat tails” we are usually thinking When we think about “fat tails” we are usually thinking about time series observations of returnsabout time series observations of returns

•• For active managers, the crossFor active managers, the cross--section of returns may be section of returns may be even more important, as it defines the opportunity set even more important, as it defines the opportunity set

•• DeSilvaDeSilva, , SapraSapra and and ThorleyThorley (2001) (2001) –– if asset specific risk varies across stocks, the crossif asset specific risk varies across stocks, the cross--section section

should be expected to have a should be expected to have a unimodalunimodal, fat, fat--tailed distributiontailed distribution

•• AlmgrenAlmgren and and ChrissChriss (2004)(2004)–– provides a substitute for “alpha scaling” that sorts stocks by provides a substitute for “alpha scaling” that sorts stocks by

attractiveness criteria, then maps the sorted values into a fatattractiveness criteria, then maps the sorted values into a fat--tailed multivariate distribution using copula methodstailed multivariate distribution using copula methods

Page 16: Fat Tails, Tall Tales, Puppy Dog Tails

What’s the Problem What’s the Problem with Daily Returns Anyway?with Daily Returns Anyway?

•• Financial markets are driven by the arrival of information Financial markets are driven by the arrival of information in the form of “news” (truly unanticipated) and the form in the form of “news” (truly unanticipated) and the form of “announcements” that are anticipated with respect to of “announcements” that are anticipated with respect to time but not with respect to content. time but not with respect to content.

•• The time intervals it takes markets to absorb and adjust The time intervals it takes markets to absorb and adjust to new information ranges from minutes to days. to new information ranges from minutes to days. Generally much smaller than a month, but up to and Generally much smaller than a month, but up to and often larger than a day. That’s why US markets were often larger than a day. That’s why US markets were closed for a week at September 11closed for a week at September 11thth..

Page 17: Fat Tails, Tall Tales, Puppy Dog Tails

Investor Response to InformationInvestor Response to Information

•• Several papers have examined the relative market Several papers have examined the relative market response to response to ““newsnews”” and and ““announcementsannouncements””–– EderingtonEderington and Lee (1996)and Lee (1996)–– KwagKwag ShrievesShrieves and Wansley(2000)and Wansley(2000)–– Abraham and Taylor (1993) Abraham and Taylor (1993)

•• Jones, Lamont and Jones, Lamont and LumsdaineLumsdaine (1998) show a (1998) show a remarkable result for the US bond marketremarkable result for the US bond market–– Total returns for long bonds and Treasury bills are not differenTotal returns for long bonds and Treasury bills are not different t

if announcement days are removed from the data setif announcement days are removed from the data set•• Brown, Harlow and Brown, Harlow and TinicTinic (1988) provide a framework for (1988) provide a framework for

asymmetrical response to “good” and “bad” news asymmetrical response to “good” and “bad” news –– Good news increases projected cash flows, bad news decreasesGood news increases projected cash flows, bad news decreases–– All new information is a “surprise”, decreasing investor All new information is a “surprise”, decreasing investor

confidence and increasing discount ratesconfidence and increasing discount rates–– Upward price movements are muted, while downward Upward price movements are muted, while downward

movements are accentuatedmovements are accentuated

Page 18: Fat Tails, Tall Tales, Puppy Dog Tails

Implications for Asset PricingImplications for Asset Pricing•• If investors price skew and/or kurtosis, there are If investors price skew and/or kurtosis, there are

implications for asset pricingimplications for asset pricing•• Harvey (1989) finds relationship between asset prices Harvey (1989) finds relationship between asset prices

and time varying and time varying covariancescovariances•• Kraus and Kraus and LitzenbergerLitzenberger (1976) and Harvey and (1976) and Harvey and SiddiqueSiddique

(2000) find that investors are averse to negative skew(2000) find that investors are averse to negative skew–– diBartolomeo (2003) argues that the value/growth relationship diBartolomeo (2003) argues that the value/growth relationship

in equity returns can be modeled as option payoffs, implying in equity returns can be modeled as option payoffs, implying skew in distributionskew in distribution

–– If the value/growth relationship has skew and investors price If the value/growth relationship has skew and investors price skew, then an efficient market will show a value premiumskew, then an efficient market will show a value premium

•• DittmarDittmar (2002) find that non(2002) find that non--linear asset pricing models linear asset pricing models for stocks work if a kurtosis preference is includedfor stocks work if a kurtosis preference is included

•• BarroBarro (2005) finds that the large equity risk premium (2005) finds that the large equity risk premium observed in most markets is justified under a “rare observed in most markets is justified under a “rare disaster” scenariodisaster” scenario

Page 19: Fat Tails, Tall Tales, Puppy Dog Tails

Portfolio Construction and Risk Portfolio Construction and Risk ManagementManagement

•• Kritzman and Rich (1998) define risk management Kritzman and Rich (1998) define risk management function when nonfunction when non--survival is possiblesurvival is possible

•• SatchellSatchell (2004)(2004)–– Describes the Describes the thethe diversification of skew and kurtosisdiversification of skew and kurtosis–– Illustrates that plausible utility functions will favor Illustrates that plausible utility functions will favor

positive skew and dislike kurtosispositive skew and dislike kurtosis

•• Wilcox (2000) shows that the importance of higher Wilcox (2000) shows that the importance of higher moments is an increasing function of investor gearingmoments is an increasing function of investor gearing

Page 20: Fat Tails, Tall Tales, Puppy Dog Tails

Optimization with Higher MomentsOptimization with Higher Moments•• ChamberlinChamberlin, Cheung and Kwan(1990) derive portfolio , Cheung and Kwan(1990) derive portfolio

optimality for multioptimality for multi--factor models under stable factor models under stable paretianparetianassumptionsassumptions

•• Lai (1991) derives portfolio selection based on Lai (1991) derives portfolio selection based on skewnessskewness•• Davis (1995) derives optimal portfolios under the Davis (1995) derives optimal portfolios under the

Gamma distribution assumptionGamma distribution assumption•• HlawitsckaHlawitscka and Stern (1995) show the simulated and Stern (1995) show the simulated

performance of mean variance portfolios is nearly performance of mean variance portfolios is nearly indistinguishable from the utility maximizing portfolioindistinguishable from the utility maximizing portfolio

•• CremersCremers, Kritzman and Paige (2003), Kritzman and Paige (2003)–– Use extensive simulations to measure the loss of utility Use extensive simulations to measure the loss of utility

associated with ignoring higher moments in portfolio associated with ignoring higher moments in portfolio constructionconstruction

–– They find that the loss of utility is negligible except for the They find that the loss of utility is negligible except for the special cases of concentrated portfolios or “kinked” utility special cases of concentrated portfolios or “kinked” utility functions (i.e. when there is risk of nonfunctions (i.e. when there is risk of non--survival). survival).

Page 21: Fat Tails, Tall Tales, Puppy Dog Tails

ConclusionsConclusions•• The fat tailed nature of high frequency returns is well The fat tailed nature of high frequency returns is well

establishedestablished•• The nature of the process is usually described as being a The nature of the process is usually described as being a

fat tailed stable distribution or a normal distribution with fat tailed stable distribution or a normal distribution with time varying volatilitytime varying volatility

•• The process that creates fat tailed distributions probably The process that creates fat tailed distributions probably has to do with rate at which markets can absorb new has to do with rate at which markets can absorb new informationinformation

•• The existence of fat tails and skew has important The existence of fat tails and skew has important implications for asset pricingimplications for asset pricing

•• Fat tails probably have relatively lesser importance for Fat tails probably have relatively lesser importance for portfolio formation, unless there are special conditions portfolio formation, unless there are special conditions such as gearing that imply nonsuch as gearing that imply non--standard utility functionsstandard utility functions

Page 22: Fat Tails, Tall Tales, Puppy Dog Tails

ReferencesReferences

•• Mandelbrot, Benoit. "LongMandelbrot, Benoit. "Long--Run Linearity, Locally Gaussian Process, Run Linearity, Locally Gaussian Process, HH--Spectra And Infinite Variances," International Economic Review, Spectra And Infinite Variances," International Economic Review, 1969, v10(1), 821969, v10(1), 82--111.111.

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•• Lau, Amy Lau, Amy HingHing--Ling, HonLing, Hon--ShiangShiang Lau And John R. Lau And John R. WingenderWingender. "The . "The Distribution Of Stock Returns: New Evidence Against The Stable Distribution Of Stock Returns: New Evidence Against The Stable Model," Journal of Business and Economic Statistics, 1990, v8(2)Model," Journal of Business and Economic Statistics, 1990, v8(2), , 217217--224.224.

•• RachevRachev, S.T. and S. , S.T. and S. MittnikMittnik (2000). Stable (2000). Stable ParetianParetian Models in Models in Finance, Wiley.Finance, Wiley.

Page 23: Fat Tails, Tall Tales, Puppy Dog Tails

ReferencesReferences•• RachevRachev, S.T. (editor) Handbook of Heavy Tailed Distributions in , S.T. (editor) Handbook of Heavy Tailed Distributions in

Finance. Elsevier.Finance. Elsevier.•• Gulko, L. "The Entropy Theory Of Stock Option Pricing," Gulko, L. "The Entropy Theory Of Stock Option Pricing,"

International Journal of Theoretical and Applied Finance, 1999, International Journal of Theoretical and Applied Finance, 1999, v2(3,Jul), 331v2(3,Jul), 331--356.356.

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•• LebaronLebaron, Blake. , Blake. RitirupaRitirupa SamantaSamanta, and Stephen , and Stephen CecchettiCecchetti. “Fat Tails . “Fat Tails and 4and 4thth Moments: Practical Problems of Variance Estimation”, Moments: Practical Problems of Variance Estimation”, Brandeis University Working Paper, 2006. Brandeis University Working Paper, 2006.

•• Engle, Robert F. "Autoregressive Conditional Heteroscedasticity Engle, Robert F. "Autoregressive Conditional Heteroscedasticity With With Estimates Of The Variance Of United Kingdom Inflations," Estimates Of The Variance Of United Kingdom Inflations," EconometricaEconometrica, 1982, v50(4), 987, 1982, v50(4), 987--1008.1008.

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Page 24: Fat Tails, Tall Tales, Puppy Dog Tails

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