The Dynamic Power Law Model

Transcription

1 The Dynamic Power Law Model Bryan Kelly Extremely Preliminary and Incomplete Abstract I propose a new measre of common, time varying tail risk in large cross sections. It is motivated by asset pricing theory and is directly estimable from the cross section of retrns. The model, which is derived from extreme vale theory and estimated via maximm likelihood, allows the power law exponent to transition smoothly throgh time as a fnction of recent data. Empirically, my measre has strong predictive power for tail risk of aggregate market retrns implied from options prices, and has predicts the freqency and magnitde of extremal retrns for individal stocks.

2 Introdction The mere potential for infreqent events of extreme magnitde can have important effects on asset prices. Tail risk, by natre, is an elsive qantity, which presents economists with the danting task of explaining market behavior with rarely observed phenomena. This crx has led to notions sch as peso problems (Krasker 980) and the rare disaster hypothesis (Rietz 988; Barro 2006), as well as skepticism abot these theories de to the difficlty in testing them. The goal of this paper is to investigate the effects of time-varying extreme event risk in asset markets. The chief obstacle to this investigation is a viable measre of tail risk over time. To overcome this, I devise a panel approach to estimating economy-wide conditional tail risk. I assme that tail risks of all firms are driven by a common nderlying process. Becase individal retrns contain information abot the likelihood of market-wide extremes, the cross section of firms can be sed to accrately measre prevailing tail risk in the economy. I elicit a conditional tail estimate by trning to the cross section of extreme events at each point in time, rather than waiting to accmlate a sfficient nmber of extreme observations in nivariate time series. This bypasses data limitations faced by alternative estimators, for example those relying on options prices or intra-daily data. My framework, which fses asset pricing theory with extreme vale econometrics, distills to a central postlate for the tail distribtion of retrns. Define the tail as the set of retrn events exceeding some high threshold. I assme that the tail of asset retrn i behaves according to P (R i,t+ > r Ri,t+ > and F t ) = ( ) ai ζ r t. () Eqation states that extreme retrn events obey a power law. Since at least Mandelbrot (963) and Fama (963), economists have arged that nconditional tail distribtions of financial retrns are aptly described by a power law. The key parameter of the model, a i ζ t, determines the shape of the tail and is referred to as the tail exponent. High vales of a i ζ t correspond to fat tails and high probabilities of extreme retrns. In contrast to past power law research, Eqation is a statement abot the conditional retrn tail. The exponent varies over time becase ζ t is a fnction of the conditioning information set F t. While different assets have different levels of tail risk (determined by the constant a i ), dynamics are the same for all assets becase they are driven by a common conditional

3 process. Ths, ζ t may be thoght of as economy-wide extreme event risk in retrns. I refer to the tail strctre in () as the dynamic power law model. I bild an econometric estimator for the dynamic power law strctre. The intition from strctral models is that tail risks of individal assets are closely related to aggregate tail risk. In a sfficiently large cross section, enogh stocks will experience tail events each period to provide accrate information abot the prevailing level of tail risk. I se this cross-sectional extreme retrn information to estimate economy-wide tail risk at each point in time. This avoids having to accmlate years of tail observations from the aggregate market time series, and therefore avoids sing stale observations that carry little information abot crrent tail risk. My procedre applies Hill s (975) tail risk estimator to the cross section of extreme events each day. The model then optimally averages recent cross-sectional Hill estimates to provide conditional tail risk forecasts. A major obstacle in estimation is the model s potentially enormos nmber of nisance parameters. I overcome this with a strategy based on qasimaximm likelihood theory. The idea is to find a simpler version of the infeasible model that has the same maximm likelihood first order conditions. Estimation may then be based on this mis-specified, yet feasible, model. I redce the complexity of the problem to three parameters by treating observations as thogh they are i.i.d. I then prove that maximizing the reslting qasi-likelihood provides consistent and asymptotically normal estimates of the data s tre dynamics, and show how to calclate standard errors for inference. 2 Empirical Methodology 2. The Dynamic Power Law Model In this section I propose a procedre for estimating the dynamic power law model. My approach exploits the comparatively rich information abot tail risk in the cross section of retrns, as opposed to relying, for example, on short samples of high freqency nivariate data or options prices. Estimating flly-specified versions of the tail models from Section 2 is extremely difficlt, and essentially infeasible withot mlti-step estimation. It reqires specifying a dependence strctre among retrn tails and estimating stock-specific a i parameters. Incorporating both 2

4 considerations adds an enormos nmber of parameters: Estimating the a i constants adds n parameters while imposing dependence strctres like those implied by the models in Section 2 adds another nk parameters, where K is the nmber of factors in a given model. Frthermore, these parameters are nisances since the goal is to measre the common element of tail risk, not nivariate distribtions. The stochastic natre of the tail risk process frther complicates estimation. Contemporaneos shocks to the tail exponent can be thoght of as the extreme vale eqivalent of stochastic volatility. It rles ot simple likelihood methods, instead reqiring comptation-intensive procedres like simlation-based estimation. I propose several simplifications of the dynamic power law model that isolate the common component of tail risk with a tractable, accrate procedre. My simplification reqires estimating only three parameters instead of several thosand, and redces estimation time to nder one minte despite working with a daily cross section of several thosand stocks. To begin, I more flly specify the statistical model, inclding an evoltion eqation for the tail exponent (which was left nspecified in Eqation ). Assmption (Dynamic Power Law Model). Let R t = (R,t,..., R n,t ) denote the cross section of retrns in period t. 2 Let K t denote the nmber of R t elements exceeding threshold in period t. 3 The tail of individal retrns for stock i (i =,..., n), conditional pon exceeding and given information F t, obeys the probability distribtion 4 with corresponding density F,i,t (r) = P (R i,t+ > r R i,t+ >, F t ) = f,i,t (r) = a iζ t+ ( ) (+ai ζ r t+ ). ( ) ai ζ r t+ In Section 2, retrns obey a tail factor strctre de to the factor strctre in heavy-tailed cash flow shocks. The common heavy-tailed shock enters via each firm s exposre to consmption growth while the heavy-tailed idiosyncrasy comes from firm-specific dividend shocks. In that case, K =. 2 R denotes arithmetic retrn, which directly maps the tail distribtion here with theoretical reslts in Section 2. This is withot loss of generality as the model is eqally applicable to log retrns. In estimation, I work with daily retrns. Becase of the small scale of daily retrns, the approximation ln( + x) x is accrate to a high order and the distinction between arithmetic and log retrns is negligible. 3 I assme for notational simplicity that these are the first K t elements of R t. This is immaterial since, in the treatment here, elements of R t are exchangeable. 4 This formlation applies similarly to the lower tail of retrns. When estimating the lower tail, I reverse the sign of log retrns and estimate the pper tail of the model. This is cstomary in extreme vale statistics, as it streamlines exposition of models as well as compter code sed in estimation. 3

5 The common element of exponent processes, ζ t+, evolves according to 5 and the observable pdate of ζ t+ is = π 0 + π + π ζ t+ ζ pd 2 (2) ζ t t ζ pd t = K t ln R k,t K t. k= The evoltion of ζ t is designed to captre atoregressive time series behavior with a parsimonios parameterization. The pdate term /ζ pd t is a smmary statistic calclated from the cross section of tail observations on date t, which I discss in more detail below. Recrsively sbstitting for ζ t shows that ζ t+ = π 0 π 2 + π Ths, /ζ t+ is simply an exponentially-weighted moving average of daily pdates based on observed extreme retrns. When the π coefficients are estimated with maximm likelihood, j=0 π j 2 ζ pd t j this moving average is an optimal forecast of ftre tail risk. The role of the pdate is to smmarize information abot prevailing tail risk from recent extreme retrn observations. This conditioning information enters the evoltion of /ζ t+ via the smmary statistic to refresh the conditional tail measre. I calclate a smmary of tail risk from the cross section each period sing Hill s (975) estimator. The Hill estimator is a maximm likelihood estimator of the cross-sectional tail distribtion. It takes the form ζ Hill t = K t K t k= ln R k,t. To see why this makes sense as an pdate, note that when -exceedences (i.e., R i,t /) obey a power law with exponent a i ζ t, the log exceedence is exponentially distribted with scale parameter a i ζ t. By the properties of an exponential random variable, E t [ln(r i,t /)] = /(a i ζ t ). As a conseqence, the expected vale of pdate /ζ pd t is the cross-sectional har- 5 The time convention sing in expressions for the distribtion and density fnctions are consistent with the t-measrability of ζ t+, as seen in expression 2.. 4

6 monic average tail exponent, E t [ K t K t k= ln R k,t ] = āζ t, where ā n n i= a i. (3) The left hand side is an average over the entire cross section de to the fact that the identities of the K t+ exceedences are nknown at time t 6 This important property will be sed to establish consistency and asymptotic normality of the dynamic power law estimation procedre that follows. Before proceeding to the estimation approach, note that Eqation 2 is a stochastic process becase ζ pd t+ is a fnction of time t retrns. However, ζ t+ is deterministic conditional pon time-t information. This is different than the specification in the strctral models of Section 2, which imply that ζ t+ is sbject to a t + shock. I arge that this discrepancy is largely innocos. In the limit of small time intervals, tail risk processes in the strctral models and the exponent process in the econometric model can be specified to line p exactly. A conditionally deterministic tail exponent process, then, can be thoght of as a discrete time approximation to a continos time stochastic process. The advantage of the approximation is that straight-forward likelihood maximization procedres can be sed for estimation. This property is the tail analoge to the relation between GARCH models (in which volatility is conditionally deterministic) and stochastic volatility models. Nelson (990) shows that a discrete GARCH(,) retrn process converges to a stochastic volatility process as the time interval shrinks to zero. An important reslt of Drost and Werker (996) proves that estimates of a GARCH model at any discrete freqency completely characterize the parameters of its continos time stochastic volatility eqivalent. The same notion lies behind treating the process in (2) as a discrete time approximation to a continos time stochastic process for the tail exponent. 7 6 While the identities of the exceedences are nknown, the nmber of exceedences is known since the tail is defined by a fixed fraction of the cross section size. 7 I condct a Monte Carlo experiment to examine the tail analoge of the Drost and Werker reslt. I find that the dynamic power law estimator based on the model in Assmption contines to provide accrate estimates of the tail exponent process when the exponent follows a Gassian atoregression. I discss this more at the end of the section, and provide a detailed description of the experiment and its reslts in Appendix A. 5

7 2.2 Estimating the Dynamic Power Law Model My estimation strategy ses a qasi-likelihood techniqe, and is an example of a widely sed econometric method with early examples dating at least back to Neyman and Scott (948), Berk (966), and the in-depth development of White (982). The general idea is to se a partial or even mis-specified likelihood to consistently estimate an otherwise intractable model. The proofs that I present can also be thoght of as a special case of Hansen s (982) GMM theory. To avoid the nisance parameter problem, I treat assets as thogh they are independent with identical tail distribtions each period. The independence assmption avoids the need to estimate factor loadings for each stock, and the identical assmption avoids having to estimate each a i coefficient. These simplifications, however, alter the likelihood from the tre likelihood associated with Assmption, to a qasi -likelihood, written below. I show that maximizing the qasi-likelihood prodces consistent and asymptotically normal estimates for the parameters that govern tail dynamics, π and π 2. Ultimately, the estimated ζ t series is shown to be the fitted cross-sectional harmonic average tail exponent. Since the average exponent series differs from ζ t only by the mltiplicative factor ā, the two are perfectly correlated. Before stating the main proposition I discss two important objects, the log qasi-likelihood and the score fnction (the derivative of the log qasi-likelihood with respect to model parameters). I refer to the tail model in Assmption as the tre model. Sppose, conterfactally, that all retrns in the cross section share the same exponent, which is eqal to the cross-sectional harmonic average exponent. Then the tail distribtion of all assets becomes with corresponding density ( ) āζt+ Ri,t+ F,i,t (R i,t+ ) = f,i,t (R i,t+ ) = āζ t+ ( Ri,t+ ) (+āζt+ ). Tildes signify that this distribtion is different than the tre marginal, F,i,t. Under crosssectional independence, the corresponding (scaled) log qasi-likelihood is L({R t } T t=; π) = T T ln f(r t+ ; π, F t ) = T t=0 T t=0 K t+ k= ( ln āζ t+ ( + āζ t+) ln R ) k,t+, (4) 6

8 where -exceedences are inclded in the likelihood and non-exceedences are discarded. Define the gradient of ln f t (R t+ ; π) with respect to π (the time-t element of the score fnction) as s t (R t+ ; π) π ln f t (R t+ ; π) = ln f t (R t+ ; π) π ζ t+ ζ t+ = ( Kt+ ζ t+ K t+ ā k= ln R k,t+ With these expressions in place, I present my central econometric reslt. ) π ζ t+. (5) Let the tre data generating process of {R t } T t= satisfy the dynamic power law model of Assmption with parameter vector π. Define the qasi-likelihood estimator ˆπ QL as ˆπ QL = arg max π Π L({R t} T t=; π). If the following conditions are satisfied i. π is interior to the parameter space Π over which maximization occrs; ii. for π π, E[s t (R t+ ; π)] 0, and iii. E[sp π Π s t (R t+ ; π) ] <, Then ˆπ QL p π. Frthermore, if iv. E[sp π Π π s t (R t+ ; π) ] <, v. T T t=0 s t(r t+ ; π ) d N(0, G) and vi. E[ π s t (R t+ ; π )] is fll colmn rank, Then T (ˆπ QL π ) d N(0, Ψ), where Ψ = S GS, S = E[ π s t (R t+ ; π )], and G = E[s t (R t+ ; π )s t (R t+ ; π ) ]. Proof. The proof follows Newey and McFadden (994). Before proceeding, I establish a key lemma pon which the remainder of the proposition relies. It shows that s t (R t+ ; π) (which is based on the mis-specified model F,t ) has expectation eqal to zero given that the tre 7

9 data generating process satisfies Assmption. Lemma. Under Assmption, E[s t (R t+ ; π)] = 0. By the law of iterated expectations, E[s t (R t+ ; π)] = E [ E t [s t (R t+ ; π)] ] [ K t+ ] ] K t+ = E [E t ā ln R k,t+ π ζ t+ ζ t+ k= [( Kt+ = E K ) ] t+ π ζ t+ ζ t+ ζ t+ = 0. The second eqality follows from expression 5 and the t-measrability of ζ t+. The third eqality follows from Eqation 3, proving the lemma. Observe that the first order condition for maximization of (4) is T T t=0 s t(r t+ ; π) = 0. That is, maximization of the log qasi-likelihood prodces a valid moment condition pon which estimation may be based. With this insight in hand, the approach of Newey and McFadden may be employed to establish asymptotic properties of the dynamic power law qasi-likelihood estimator. By condition (i) and the fact that the tre generating process satisfies Assmption, Lemma shows that the moment condition arising from maximization of L({R t } T t=; π) is satisfied. Adding condition (ii), π is niqely identified. By the dominance condition (iii), the niform law of large nmbers may be invoked to establish convergence in probability. To establish convergence to normality, I se a mean vale expansion of the moment condition sample analoge arond π (a vale between ˆπ and π ), which gives (ˆπ π) = [ T T t π s t (R t+ ; π) ] T s t (R t+ ; π ). This expansion is performed noting that π is interior to the parameter space and, by the fnctional form of the qasi-likelihood, s t (R t+ ; π) is continosly differentiable over Π. Becase π is between ˆπ and π, π is also consistent for π by the convergence in probability reslt jst shown. Using this fact together with condition (iv) delivers T t πs t (R t+ ; π) p S. By condition (v), T T t=0 s t(r t+ ; π ) d N(0, G). This, together with Sltzky s theorem and assmption (vi), proves the reslt. 8

10 2.3 Volatility and Heterogeneos Exceedence Probabilities Implicit in the formlation above is that each element of the vector R t has an eqal probability of exceeding threshold. However, heterogeneity in individal stock volatilities affects the likelihood that a particlar stock will experience an exceedence. power law variable sch that P (X > ) = b ζ. Let X be a The -exceedence distribtion of X is P (X > x X > ) = ( x ) ζ. Now consider a volatility rescaled version of this variable, Y = σx. The exceedence probability of Y eqals b ( σ) ζ, different than that of X. When σ >, Y has a higher exceedence probability than X. However, the shape of Y s -exceedence distribtion is identical to that of X. A reformlation of the estimator to allow for heterogeneos volatilities is easily established. Let each stock have niqe -exceedence probability p i, and consider the effect of this heterogeneity on the expectation of the tail exponent pdate. In this case, the expectation is no longer the harmonic average tail exponent, bt is instead the exceedence probability-weighted average exponent, E t [ K t+ K t+ k= ln R k,t+ ] = ζ t+ where ω i = p i / j p j. The entire estimation approach and consistency argment otlined above proceeds identically after establishing this point. The ltimate reslt is that the fitted ζ t series is no longer an estimate of the eqal-weighted average exponent, bt takes on a volatility-weighted character de to the effect that volatility has on the probability of tail occrrences. Another potential concern is contamination of tail estimates de to time-variation in volatility. I address this by allowing the threshold to vary over time. My procedre selects as a fixed q% qantile, û t (q) = inf { R (i),t R t : n i= ω i a i, q 00 (i) } n where (i) denotes the i th order statistic of (n ) vector R t. In this case, expands and contracts with volatility so that a fixed fraction of the most extreme observations are sed for estimation each period, nllifying the effect of volatility dynamics on tail estimates. My estimates are based on q = 5 (or 95 for the pper tail). 8 8 Threshold choice can have important effects on reslts. An inappropriately low threshold will contaminate tail exponent estimates by sing data from the center of the distribtion, whose behavior can vary markedly from tail data. A very high threshold can reslt in noisy estimates reslting from too few data 9

11 2.4 Monte Carlo Evidence Appendix A describes a series of Monte Carlo experiments designed to assess finite sample properties of the dynamic power law estimator. Table shows reslts confirming that the asymptotic properties derived above serve as accrate approximations in finite samples. They also demonstrate the estimator s robstness to dependence among tail observations and volatility heterogeneity across stocks, both of which are sggested by the strctral models. Table 2 explores the estimator s performance when the tre tail exponent is conditionally stochastic. Even thogh the estimator presented here relies on a conditionally deterministic exponent process, its estimates achieve over 80% correlation with the tre tail series on average. 3 Empirical Reslts 3. Tail Risk Estimates Estimates for the dynamic power law model se daily CRSP data from Agst 962 to December 2008 for NYSE/AMEX/NASDAQ stocks with share codes 0 and. Accracy of extreme vale estimators typically reqires very large data sets becase only a small fraction of data is informative abot the tail distribtion. Since the dynamic power law estimator relies on the cross section of retrns, I reqire a large panel of stocks in order to gather sfficient information abot the tail at each point in time. Figre plots the nmber of stocks in CRSP each month. The sample begins with jst nder 500 stocks in 926, and has fewer than,000 stocks for the next 25 years. In Jly 962, the sample size roghly dobles to almost 2,000 stocks with the addition of AMEX. In December 972, NASDAQ stocks enter the sample and the stock cont leaps above 5,000, flctating arond this size throgh The dramatic cross-sectional expansion of CRSP beginning in Agst 962 points. While sophisticated methods for threshold selection have been developed (Dpis 999; Matthys and Beirlant 2000; among others), these often reqire estimation of additional parameters. In light of this, Gabaix et al. (2006) advocate a simple rle fixing the -exceedence probability at 5% for nconditional power law estimation. I follow these athors by applying a similar simple rle in the dynamic setting. Unreported simlations sggest that q = to 5 (or 95 to 99 for the pper tail) is an effective qantile choice in my dynamic setting. 9 The dynamic power law estimator in Section 2 accommodates changes in size of the cross section over time, highlighting another attractive featre of the estimator. 0

12 leads to my focs on the 963 to 2008 sample. Other data sed in my analysis are daily Fama-French retrn factors, monthly risk free rates and size/vale-sorted portfolio retrns from Ken French s Data Library 0, market retrn predictor variables from Ivo Welch s website, variance risk premim estimates from Hao Zho s website 2, and macroeconomic data from the Federal Reserve. I focs my empirical analysis on the tails of raw retrns. In each test I consider tail risk estimated sing data from the lower tail only, from the pper tail only, and from combining lower and pper tail data. I refer to the latter case as both tails, which in the tests that follow shold not be nderstood as simltaneosly inclding separate estimates of the lower and pper tail in regressions. For robstness, I also explore how reslts change when residals from the Fama-French three-factor model are sed to estimate tail risk. Factor model residals offer a means of mitigating the effects of dependence on the estimator s efficiency. 3 Threshold t is chosen to be the 5% cross-sectional qantile each period. Standard errors are estimated based on the form of the asymptotic covariance matrix Ψ derived in Proposition 2.2. In particlar, Ŝ is the log qasi-likelihood Hessian evalated at ˆπ parameters and Ĝ is the sample average log qasi-likelihood gradient oter prodct evalated at ˆπ. Frther details on standard error estimation via likelihood Hessians and gradient oter prodcts may be fond in Hayashi (2000). Table reports estimates for the dynamic power law evoltion (2). The lower tail exponent of raw retrns ζ t varies arond a mean of 2.20, with ˆπ = and ˆπ 2 = The p-vales for all statistics reported in Table 2 are below 0.00, rejecting the nll hypothesis of constant tail risk and spporting Testable Implication??. The fact that ˆπ + ˆπ 2 > 0.99 implies that the tail exponent is highly persistent. The pper tail is slightly fatter (consistent with the reslts of Jansen and de Vries 99) and less persistent, thogh with more time series variability (given the higher vale of ˆπ ). When stock retrns are converted to factor model residals, estimation reslts are qalitatively nchanged. 0 URL: library.html. URL: 2 URL: 3 As my asymptotic theory reslts and Monte Carlo evidence show, abstracting from dependence does not affect the estimator s consistency. It may, however, affect the variance of estimates. The asymptotic covariance derived in Proposition 2.2 acconts for this decreased efficiency, hence test statistics maintain appropriate size.

13 I plot the fitted tail series ( ζ t ) based on raw retrns in Figre 2. The estimates from both tails together are shown in Figre 2a, and separate estimates for the lower and pper tails are shown in Figre 2b. These are plotted alongside the log price-dividend ratio for the aggregate market. The estimated tail risk series appears moderately contercyclical, sharing a monthly correlation of -4%, -5% and -4% with the log price-dividend ratio based on estimates from both tails, the lower tail and the pper tail, respectively. The beginning of the sample sees high tail risk, immediately following a drop in the vale-weighted index of 28% in the first half of 962 (the first major market decline dring the post-war period). Tail risk declines steadily ntil December of 968, when it reaches its lowest levels in the sample. This corresponds to a late 960 s bll market peak, the level of which is not reached again ntil the mid-970 s. Tail risk rises throghot the 970 s, accelerating its ascent dring the oil crisis. It remains arond its mean vale of -2 to -2.5 for most of the remaining sample. Tail risk recedes in the for bll market years leading p to the 987 crash, rising qickly in the following months. Dring the technology boom tail risk retreats sharply bt briefly, spiking to its highest post-2000 level amid the early 2003 market trogh. At this time the vale-weighted index was down 49% from its 2000 high and NASDAQ was 78% of its peak. Finally, dring the late 2008 financial crisis, tail risk sees a modest climb after falling in the first half of Figre 3 plots the tail series estimated from Fama-French three-factor model residals. The broad dynamic patterns are the same as those in Figre 2. Figre 4 shows the threshold series t for raw retrns alongside monthly realized volatility of the CRSP vale-weighted index. The threshold based on both tails has a 60% correlation with volatility. As the figre and correlation show, the threshold appears to sccessflly absorb volatility changes. In Table 2, I report monthly correlations between tail risk estimates and macroeconomic variables. Lower (pper) tail estimates have correlation with nemployment of 53% (39%) and -0% (-7%) with the Chicago Fed National Activity Index (CFNAI), once again sggesting some cyclicality in tail risk. As a brief exploration into the determinants of tail risk, I estimate a regression of the tail process on its own lag and lags of a collection of macroeconomic variables, inclding realized eqity volatility, nemployment, inflation, growth in indstrial prodction, CFNAI and the aggregate stock market retrn. I show the estimated impacts of these variables on ftre tail risk in Table 3. Coefficients have been scaled to be interpreted as the response of tail risk ( ζ t, in nmber of standard deviations) to a one standard deviation increase in the dependent 2

14 variable. The most important variable for determining tail risk is the past market retrn. A retrn one standard deviation above its mean predicts that the lower tail becomes thinner by 0.75 standard deviations (Newey-West t=7.), while a high past retrn increases the weight of the pper tail by standard deviations (t=4.6). Other potentially important determinants of tail risk are past nemployment and eqity volatility. 3

15 Proof of Proposition?? Proof. The expected retrn on asset i is E t [r i,t+ ] = κ i,0 + φ i µ + µ i + A i,0 (κ i, ) + A i,σ (κ i, ρ σ )σ 2 t + [A i,λ (κ i, ρ Λ ) δ(φ i + q i )]Λ t. Sbstitting the wealth-consmption ratio from Proposition?? into the stochastic discont factor (via the Campbell-Shiller identity) and evalating the expectation gives r f,t = r f,0 + b f,σ σ 2 t + b f,λ Λ t where b f,λ = A Λ ( θ)(κ ρ Λ ) c( γ). Assembling the preceding expressions prodces the reslt. Proof of Proposition?? Proof. The proof proceeds as in Proposition??. Let S = φ i Λ t V c,t+ and let Y = exp(s). The density of S is g S (s) = ( ) s exp, s 0. φ i Λ t φ i Λ t The derivative of S with respect to Y is /Y, which implies that the density of Y is g Y (y) = g S (s) ds dy = φ i Λ t y /(φ iλ t) with corresponding cmlative distribtion fnction G Y (y) = y /(φ iλ t). Acconting for the interaction between S and the Bernolli variable ι c,t+ amonts to deriving the distribtion of Y ι c,t+, which is G (Y ι )(y) = δ y= + ( δ)y /(φ iλ t) ( δ)y /(φ iλ t). Repeating this argment for exp( q i ι i,t+ Λ t V i,t+ ) shows that its lower tail is also power lawdistribted with exponent /(q i Λ t ). Applying the asymptotic tail aggregation properties referenced in Proposition?? and noting that the remaining shocks to r i,t+ are Gassian (ths they have no asymptotic contribtion to the tail distribtion) delivers the reslt. 4

16 A Monte Carlo Evidence A. Correct Specification of Tail Parameter Evoltion The first Monte Carlo experiment I rn is designed to assess the finite sample properties of the dynamic power law qasi-maximm likelihood estimator nder different dependence and heterogeneity conditions. In all cases, the evoltion of the parameter governing tail risk follows Eqation 2, and therefore the statistical model s specification of this process is correct. I allow for mis-specification in terms of dependence and in the level of the tail exponent across stock. In particlar, data is generated by the following process: R i,t = b i R m,t + e i,t where R m,t and e i,t, i =,..., n, are independent Stdent t variates with a i ζ t degrees of freedom. A well-known property of the Stdent t is that its tail distribtion is asymptotically eqivalent to a power law with tail exponent eqal to (mins) the degrees of freedom. In generating data I therefore set the degrees of freedom eqal to ζ t, whose transition is described by Eqation 2. The b i coefficients control cross section dependence and heterogeneity in volatility. The a i coefficients control the tail risk heterogeneity across observations. I consider for cases:. Independent and identically distribted observations: b i = 0 and a i = for all i, 2. Dependent and identically distribted observations: b i N(,.5 2 ) and a i = for all i, 4 3. Independent and heterogeneosly distribted observations: b i = 0 and a i N(,.2 2 ) for all i, 4. Dependent and heterogeneosly distribted observations: b i N(,.5 2 ) and a i N(,.2 2 ) for all i. The cross section size is n=000 or 2500, and the time series length is T =000 or Parameters sed to generate data are fixed at π = 0.05 and π 2 = 0.93, with an intercept that ensres the mean vale of ζ t is three. 4 Observations in this case are identical only in terms of their tail exponent. Differences in b i across stocks introdces volatility and dependence heterogeneity. 5

17 In each simlation, the qasi-maximm likelihood procedre described in Section 2 is sed to estimate the model and its asymptotic standard errors. Smmary statistics for parameter and asymptotic standard error estimates are reported in Table. Also reported is the time series correlation and mean absolte deviation between the fitted tail series and the tre ζ t series, averaged across simlations. The general conclsion of the experiment is that the asymptotic theory of Section 2 is a good approximation for the finite sample behavior of the dynamic power law estimator. This is tre not only when data are i.i.d., bt also when observations are dependent and heterogenos. In all cases, the fitted tail series achieves a correlation of at least 97% with the tre tail series. A.2 Correct Specification of Tail Parameter Evoltion The next experiment proceeds as in the i.i.d. case above, bt the tre tail diverges from that assmed in the statistical model. In particlar, the tre degrees of freedom parameter ζ t is conditionally stochastic and follows a first order Gassian atoregression, ζ t+ = ζ( ρ) + ρζ t + ση t+, η t+ N(0, ). (6) I fix ρ = 0.99, n=000 and T =000, and let σ =0.005 or The parameter σ governs the variability of the process, and ths the range of tail risk vales that the data can experience. In each simlation, the qasi-maximm likelihood procedre described in Section 2 is sed to estimate the model and its asymptotic standard errors. Smmary statistics for the tre process and the fitted process are reported in Table 2, as well as smmary statistics for π parameter estimates and their asymptotic standard errors. The deterministic tail process provides accrate estimates even when the tre tail parameter is stochastic. The mean absolte error between the fitted and tre series ranges from 0.73 to 0.307, and their correlation ranges from 8.8% to 87.5%. 6

18 Table. Dynamic Power Law Estimates for the NYSE/AMEX/NASDAQ Panel. The table reports estimates for the dynamic power law model sing the panel of NYSE/AMEX/NASDAQ stocks from Agst 962 to December Estimation follows the qasi-maximm likelihood procedre described in Section 3. The model is estimated separately sing data from both tails together, the lower tail alone and the pper tail alone. Reslts in Panel A are based on raw retrns and reslts in Panel B are based on residals from the Fama-French three-factor model. Standard errors of coefficients are reported in parentheses, and are calclated from estimates of the qasi-maximm likelihood asymptotic covariance strctre derived in Section 3. The parameter ζ is the implied nconditional mean tail exponent based on estimates of π 0, π and π 2 (the reported standard error estimate for ζ has been appropriately transformed based on standard errors of π 0, π and π 2 ). Both Tails Lower Tail Upper Tail Panel A: Raw Retrns ζ (0.02) (0.044) (0.08) π (0.04) (0.00) (0.058) π (0.05) (0.0) (0.092) Panel B: Factor Model Residals ζ (0.024) (0.052) (0.07) π (0.03) (0.007) (0.04) π (0.04) (0.007) (0.07) Table 2. Tail Risk Correlation with Macroeconomic Variables. The table reports correlations between tail risk estimates and macroeconomic variables. Tail risk (-ζ t ) is estimated for the pper and lower tail separately and for both tails together by the dynamic power law estimator on raw retrn data for NYSE/AMEX/NASDAQ stocks. Macroeconomic variables inclded are the log dividend-price ratio, nemployment rate, inflation rate, growth rate of indstrial prodction, Chicago Fed National Activity Index and the variance risk premim (VIX 2 mins realized S&P 500 variance). Since tail risk is measred daily, correlations are calclated based on month-end vales. The sample horizon is 963 to 2008 ( for the variance risk premim). Lower Upper Both Ind. Prod. -ζ -ζ -ζ ln D/P Unemp. Infl. Growth CFNAI VRP Lower -ζ.00 Upper -ζ Both -ζ ln D/P Unemp Inflation Ind. Prod. Gr CFNAI VRP

19 Table. Dynamic Power Law Monte Carlo Reslts. The table reports simlation reslts based on the Monte Carlo experiment described in Appendix B. In all cases, data is generated by the process R i,t =b i R m,t +e i,t where R m,t and e i,t, i=,...,n are independent Stdent t variates with a i ζ t degrees of freedom. I consider for cases: i) Independent and identically distribted observations: b i =0 and a i = for all i, ii) dependent and identically distribted observations: b i ~ N(,.5 2 ) and a i = for all i, iii) independent and heterogeneosly distribted observations: b i =0 and a i ~ N(,.2 2 ) for all i, and iv) dependent and heterogeneosly distribted observations: b i ~ N(,.5 2 ) and a i ~ N(,.2 2 ) for all i. The cross section size is n=,000 or 2,500 and the time series length is T=,000 or 5,000. Parameters sed to generate the data are shown in the Tre Vale row. I report the mean, median and standard deviation of parameter estimates across simlations, as well as the mean asymptotic standard error estimate. In the last row of each set of reslts, I report the mean absolte error and the correlation between the fitted and tre ζ t series. The colmn heading d.o.f. denotes estimates of the intercept parameter (transformed to be interpreted as the time series mean of ζ t ). Reslts are based on,000 replications. Independent, Identical Dependent, Identical Independent, Heterogeneos Dependent, Heterogeneos π π 2 d.o.f. π π 2 d.o.f. π π 2 d.o.f. π π 2 d.o.f. T=,000 Tre Vale n =,000 Mean Median Mean ASE Std. Dev MAE, Corr n =2,500 Mean Median Mean ASE Std. Dev MAE, Corr T=5,000 n =,000 Mean Median Mean ASE Std. Dev MAE, Corr n =2,500 Mean Median Mean ASE Std. Dev MAE, Corr

20 Table 2. Stochastic Tail Exponent Monte Carlo Reslts. The table reports simlation reslts based on the Monte Carlo experiment described in Appendix B. In all cases, data is generated as a vector of n i.i.d. Stdent t variates with ζ t degrees of freedom over T periods, where ζ t+ =ζ(-ρ)+ρζ t +ση t+, η t+ is standard normal, n=,000, T=,000, and ρ= The standard deviation of the tail risk process is σ=0.005 or 0.00 (Panels A and B, respectively). I report smmary statistics for the tre and fitted tail processes, as well as their mean absolte error and correlation, averaged over all simlations. I also report smmary statistics of parameter estimates and the mean asymptotic standard error estimate. The colmn heading d.o.f. denotes estimates of the intercept parameter (transformed to be interpreted as the time series mean of ζ t ). Reslts are based on,000 replications. Panel A: σ =0.005 Mean Std. Dev. Max Min Tre ζ t Fitted ζ t Tre/Fitted MAE Tre/Fitted Correlation 0.88 Parameter Estimates π π 2 d.o.f. Mean Median Mean ASE Std. Dev Panel B: σ =0.00 Mean Std. Dev. Max Min Tre ζ t Fitted ζ t Tre/Fitted MAE Tre/Fitted Correlation Parameter Estimates π π 2 d.o.f. Mean Median Mean ASE Std. Dev