MODELING AUTOREGRESSIVE CONDITIONAL SKEWNESS and KURTOSIS with MULTI-QUANTILE Caviar 1

Working Paper Series No 957 / November 2008

MODELING AUTOREGRESSIVE CONDITIONAL SKEWNESS AND KURTOSIS WITH MULTI-QUANTILE CAViaR

by Halbert White, Tae-Hwan Kim and Simone Manganelli WORKING PAPER SERIES NO 957 / NOVEMBER 2008

MODELING AUTOREGRESSIVE CONDITIONAL SKEWNESS AND KURTOSIS WITH MULTI-QUANTILE CAViaR 1

by Halbert White, 2 Tae-Hwan Kim 3 and Simone Manganelli 4

In 2008 all ECB publications This paper can be downloaded without charge from feature a motif taken from the http://www.ecb.europa.eu or from the Social Science Research Network 10 banknote. electronic library at http://ssrn.com/abstract_id=1291165.

1 The views expressed in this paper are those of the authors and do not necessarily reflect those of the European Central Bank. 2 Department of Economics, 0508 University of California, San Diego 9500 Gilman Drive La Jolla, California 92093-0508, USA; e-mail: [email protected] 3 School of Economics, University of Nottingham, University Park Nottingham NG7 2RD, U.K. and Yonsei University, Seoul 120-749, Korea; e-mail: [email protected] 4 European Central Bank, DG-Research, Kaiserstrasse 29, D-60311 Frankfurt am Main, Germany; e-mail: [email protected] © European Central Bank, 2008

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ISSN 1561-0810 (print) ISSN 1725-2806 (online) CONTENTS

Abstract 4 Non-technical summary 5 1 Introduction 6 2 The MQ-CAViaR process and model 7 3 MQ-CAViaR estimation: consistency and asymptotic normality 9 4 Consistent covariance matrix estimation 13 5 Quantile-based measures of conditional skewness and kurtosis 14 6 Application and simulation 15 6.1 Time-varying skewness and kurtosis for the S&P500 15 6.2 Simulation 18 7 Conclusion 19 References 19 Mathematical appendix 21 Tables and ﬁ gures 32 European Central Bank Working Paper Series 37

ECB Working Paper Series No 957 November 2008 3 Abstract

Engle and Manganelli (2004) propose CAViaR, a class of models suitable for estimating conditional quantiles in dynamic settings. Engle and Manganelli apply their approach to the estimation of Value at Risk, but this is only one of many possible applications. Here we extend CAViaR models to permit joint modeling of multiple quantiles, Multi-Quantile (MQ) CAViaR. We apply our new methods to estimate measures of conditional skewness and kurtosis defined in terms of conditional quantiles, analogous to the unconditional quantile-based measures of skewness and kurtosis studied by Kim and White (2004). We investigate the performance of our methods by simulation, and we apply MQ-CAViaR to study conditional skewness and kurtosis of S&P 500 daily returns.

Keywords: Asset returns; CAViaR; Conditional quantiles; Dynamic quantiles; Kurtosis; Skewness.

JEL Classifications: C13, C32.

ECB Working Paper Series No 957 4 November 2008 Non-technical Summary

Higher moments of distributions of financial variables, such as skewness and kurtosis, can be important to assess the risk of a portfolio, complementing traditional variance measures, as well as for generally improving the performance of various financial models. Responding to this recognition, researchers and practitioners have started to incorporate these higher moments into their models, mostly using the conventional measures, e.g. the sample skewness and/or the sample kurtosis. Models of conditional counterparts of the sample skewness and the sample kurtosis, based on extensions of the GARCH model, have also been developed and used; see, for example, Leon, Rubio, and Serna (2004). Kim and White (2004) point out that because standard measures of skewness and kurtosis are essentially based on averages, they can be sensitive to one or a few outliers - a regular feature of financial returns data - making their reliability doubtful. To deal with this, Kim and White (2004) propose the use of more stable and robust measures of skewness and kurtosis, based on quantiles rather than averages. Nevertheless, Kim and White (2004) only discuss unconditional skewness and kurtosis measures. In this paper, we extend the approach of Kim and White (2004) by proposing conditional quantile-based skewness and kurtosis measures. For this, we extend Engle and Manganelli’s (2004) univariate Conditional Autoregressive Value at Risk (CAViaR) model to a multi-quantile version. This allows for a general vector autoregressive structure in the conditional quantiles, as well as the presence of exogenous variables. We then use this model to specify conditional versions of the more robust skewness and kurtosis measures discussed in Kim and White (2004). We apply our methodology to a sample of S&P500 daily returns. We find that conventional estimates of both skewness and kurtosis tend to be dwarfed by a few outliers, which typically plague financial data. Our more robust measures show more plausible variability, raising doubts about the reliability of unrobust measures. A Monte Carlo simulation is carried out to illustrate the finite sample behavior of our method.

ECB Working Paper Series No 957 November 2008 5 1 Introduction

It is widely recognized that the use of higher moments, such as skewness and kurtosis, can be important for improving the performance of various nancial models. Responding to this recognition, researchers and practitioners have started to incorporate these higher moments into their models, mostly using the conventional measures, e.g. the sample skewness and/or the sample kurtosis. Models of conditional counterparts of the sample skewness and the sample kurtosis, based on extensions of the GARCH model, have also been developed and used; see, for example, Leon, Rubio, and Serna (2004). Nevertheless, Kim and White (2004) point out that because standard measures of skewness and kurtosis are essentially based on averages, they can be sensitive to one or a few outliers — a regular feature of nancial returns data — making their reliability doubtful. To deal with this, Kim and White (2004) propose the use of more stable and robust measures of skewness and kurtosis, based on quantiles rather than averages. Nevertheless, Kim and White (2004) only discuss unconditional skewness and kurtosis measures. In this paper, we extend the approach of Kim and White (2004) by proposing conditional quantile-based skewness and kurtosis measures. For this, we extend Engle and Manganelli’s (2004) univariate CAViaR model to a multi- quantile version, MQ-CAViaR. This allows for both a general vector autoregressive structure in the conditional quantiles and the presence of exogenous variables. We then use the MQ-CAViaR model to specify conditional versions of the more robust skewness and kurtosis measures discussed in Kim and White (2004). The paper is organized as follows. In Section 2, we develop the MQ-CAViaR data generating process (DGP). In Section 3, we propose a quasi-maximum likelihood estimator for the MQ-CAViaR process and prove its consistency and asymptotic normality. In Section 4, we show how to consistently estimate the asymptotic variance-covariance matrix of the MQ-CAViaR estimator. Section 5 speci es conditional quantile-based measures of skewness and kurtosis based on MQ-CAViaR estimates. Section 6 contains an empirical application of our methods to the S&P 500 index. We also report results of a simulation experiment designed to examine the nite sample behavior of our estimator. Section 7 contains a summary and concluding remarks. Mathematical proofs are gathered into the Mathematical Appendix.

ECB Working Paper Series No 957 6 November 2008 2 The MQ-CAViaR Process and Model

We consider data generated as a realization of the following stochastic process.

Assumption 1 The sequence (\w>[0):w =0> 1> 2>===> is a stationary and { w ± ± } ergodic stochastic process on the complete probability space (l> >S0),where\w F is a scalar and [w is a countably dimensioned vector whose rst element is one.

w 1 Let w 1 be the -algebra generated by ] 3 [w> (\w 1>[w 1)> === > i.e. 3 3 3 F w 1 { } w 1 (] 3 ).WeletIw(|) S0[\w ?| w 1] de ne the cumulative distri- F 3 |F 3 bution function (CDF) of \w conditional on w 1. F 3 Let 0 ?1 ?===?s ? 1.Form =1> ===> s> the mth quantile of \w conditional on w 1> denoted tm>wW ,is F 3 tW inf | : Iw(|)=m > (1) m>w { } and if Iw is strictly increasing,

1 tm>wW = Iw3 (m)=

Alternatively, tm>wW can be represented as

tm>wW

gIw(|)=H[1[\w t ] w 1]=m> (2) $ m>wW |F 3 Z3" where gIw(|) is the Lebesgue-Stieltjes probability density function (PDF) of \w conditional on w 1, corresponding to Iw(|)= F 3 Our objective is to jointly estimate the conditional quantile functions tm>wW >m=

1> 2> ===> s. For this we write tW (tW > ===> tW )0 and impose additional appropriate w 1>w s>w structure.

First, we ensure that the conditional distribution of \w is everywhere continuous, with positive density at each conditional quantile of interest, tm>wW .Welet iw denote the conditional probability density function (PDF) corresponding to Iw. In stating our next condition (and where helpful elsewhere), we make explicit the dependence of the conditional CDF Iw on $ by writing Iw($>|) in place of Iw(|)=

Realized values of the conditional quantiles are correspondingly denoted tm>wW ($)=

Similarly, we write iw($> |) in place of iw(|)= After ensuring this continuity, we impose speci c structure on the quantiles of interest.

Assumption 2 (i) \w is continuously distributed such that for each w and each

$ l>Iw($> ) and iw($> ) are continuous on R; (ii) For given 0 ?1 ? === ? 5 · ·

ECB Working Paper Series No 957 November 2008 7 s ? 1 and tW as de ned above, suppose: (a) For each w and m =1>===>s> { m>w} iw($>tm>wW ($)) A 0; (b) For given nite integers n and p> there exist a stationary

ergodic sequence of random n 1 vectors [w > with [w measurable w 1> and × { } F 3 real vectors W (W > ===> W )0 and W (W > ===> W )0 such that for all w and m m>1 m>n m m 1 m s m =1> ===> s> p

tm>wW = [w0 mW + twW0 m W = (3) 3 =1 X The structure of eq. (3) is a multi-quantile version of the CAViaR process

introduced by Engle and Manganelli (2004). When W =0for l = m> we have the m l 6 standard CAViaR process. Thus, we call processes satisfying our structure "Multi- Quantile CAViaR" (MQ-CAViaR) processes. For MQ-CAViaR, the number of relevant lags can dier across the conditional quantiles; this is re ected in the

possibility that for given m,elementsofm W may be zero for values of greater than some given integer. For notational simplicity, we do not represent p as

depending on m= Nevertheless, by convention, for no p do we have W equal m to the zero vector for all m.

The nitely dimensioned random vectors [w may contain lagged values of \w,

as well as measurable functions of [w and lagged [w and \w= In particular, [w may contain Stinchcombe and White’s (1998) GCR transformations, as discussed in White (2006).

For a particular quantile, say m,thecoe!cients to be estimated are mW and

W (W0 > ===> W0 )0= Let W0 (W0>W0),andwriteW =(W0> ===> W0)0> an c 1 m m1 mp m m m 1 s × vector, where c s(n +ps)= We will call W the "MQ-CAViaR coe!cient vector." We estimate W using a correctly speci ed model of the MQ-CAViaR process. First, we specify our model.

c Assumption 3 Let A be a compact subset of R = (i) The sequence of functions tw : s { l A R is such that for each w and each A>tw( >) is measurable w 1; × $ } 5 · F 3 for each w and each $ l>tw($> ) is continuous on A; and for each w and m =1>===>s> 5 · p

tm>w( >)=[w0 m + tw ( >)0m = 3 · =1 · X Next, we impose correct speci cationandanidenti cation condition. As- sumption 4(i.a) delivers correct speci cation by ensuring that the MQ-CAViaR

coe!cient vector W belongs to the parameter space, A. This ensures that W optimizes the estimation objective function asymptotically. Assumption 4(i.b) de-

livers identi cation by ensuring that W is the only such optimizer. In stating the

ECB Working Paper Series No 957 8 November 2008 identi cation condition, we de ne m>w(> W) tm>w( >) tm>w( >W) and use the · · norm maxl=1>===>c l = || || | |

Assumption 4 (i)(a) There exists W A such that for all w 5

tw( >W)=tW; (4) · w (b) There exists a non-empty set M 1> ===> s such that for each A0 there { } exists A 0 such that for all A with W A, 5 || ||

S [ m M m>w(> W) A ] A 0= ^ M {| | }

Among other things, this identi cation condition ensures that there is su!cient variation in the shape of the conditional distribution to support estimation of asu!cient number (#M) of variation-free conditional quantiles. In particular, distributions that depend on a given nite number of parameters, say n, will generally be able to support n variation-free quantiles. For example, the quantiles of the Q(> 1) distribution all depend on alone, so there is only one "degree of freedom" for the quantile variation. Similarly the quantiles of scaled and shifted w distributions depend on three parameters (location, scale, and kurtosis), so there are only three "degrees of freedom" for the quantile variation.

3 MQ-CAViaR Estimation: Consistency and Asymptotic Normality

We estimate W by the method of quasi-maximum likelihood. Speci cally, we construct a quasi-maximum likelihood estimator (QMLE) ˆW as the solution to the following optimization problem:

W s ¯ 1 min VW () W 3 (\w tm>w( >)) > (5) m M w=1 { m=1 · } X X where (h)=h#(h) is the standard "check function," de ned using the usual quantile step function, #(h)= 1[h 0]= We thus view $ s

Vw() m (\w tm>w( >)) { m=1 · } X as the quasi log-likelihood for observation w= In particular, Vw() is the log-likelihood of a vector of s independent asymmetric double exponential random variables

ECB Working Paper Series No 957 November 2008 9 (see White, 1994, ch. 5.3; Kim and White, 2003; Komunjer, 2005). Because

\w tm>w( >W)>m =1> ===> s need not actually have this distribution, the method is · quasi maximum likelihood.

We can establish the consistency of ˆW by applying results of White (1994). For this we impose the following moment and domination conditions. In stating this next condition and where convenient elsewhere, we exploit stationarity to omit explicit reference to all values of w=

Assumption 5 (i) H \w ? ; (ii) let G0>w maxm=1>===>s sup tm>w( >) > | | 4 M | · | w =1> 2> === = Then H(G0>w) ? = 4

We now have conditions su!cient to establish the consistency of ˆW =

Theorem 1 Suppose that Assumptions 1> 2(l> ll)> 3(l)> 4(l)> and 5(l> ll) hold. Then d=v= ˆW W. $

1@2 Next, we establish the asymptotic normality of W (ˆW W).Weuseamethod originally proposed by Huber (1967) and later extended by Weiss (1991). We rst sketch the method before providing formal conditions and results.

Huber’s method applies to our estimator ˆW > provided that ˆW satis es the asymptotic rst order conditions

W s 1 1@2 W 3 tm>w( > ˆW ) #m (\w tm>w( > ˆW )) = rs(W )> (6) w=1 { m=1 u · · } X X

where tm>w( >) is the c 1 gradient vector with elements (C@Cl)tm>w( >)>l = u · × · 1> ===> c> and # (\w tm>w( > ˆW )) is a generalized residual. Our rsttaskisthusto m · ensure that eq. (6) holds. Next, we de ne

() H[ tm>w( >)#m (\w tm>w( >))]= m=1 u · · X With continuously dierentiable at W interior to A, we can apply the mean value theorem to obtain

()=(W)+T0( W)> (7)

where T0 is an c c matrix with (1 c) rows T0>l = 0(¯(l)),where¯(l) is a mean × × u value (dierent for each l) lying on the segment connecting and W>l =1>===>c.

ECB Working Paper Series No 957 10 November 2008 It is straightforward to show that correct speci cation ensures that (W) is zero. We will also show that

T0 = TW + R( W )> (8) || || s where TW H[im>w(0) tm>w( >W) 0tw( >W)] with im>w(0) the value at zero of m=1 u · u · the density im>w of %m>w \w tm>w( >W)> conditional on w 1= Combining eqs. (7) P · F 3 and (8) and putting (W)=0,weobtain

2 ()= TW( W)+R( W )= (9) || || Thenextstepistoshowthat

1@2 W (ˆW )+KW = rs(1)> (10)

1@2 W s where KW W 3 W> with W tm>w( >W)# (%m>w). Eqs. (9) and (10) w=1 w w m=1 u · m then yield the following asymptotic representation of our estimator ˆ : P P W W 1@2 1 1@2 W (ˆW W)=TW3 W 3 wW + rs(1)= (11) w=1 X As we impose conditions su!cient to ensure that W> w is a martingale dif- { w F } ference sequence (MDS), a suitable central limit theorem (e.g., theorem 5.24 in

White, 2001) applies to (11) to yield the desired asymptotic normality of ˆW :

1@2 g 1 1 W (ˆW W) Q(0>TW3 Y WTW3 )> (12) $ where Y W H(WW0). w w We now strengthen the conditions above to ensure that each step of the above argument is valid.

Assumption 2 (iii) (a) There exists a nite positive constant i0 such that for each w> each $ l> and each | R, iw($> |) i0 ? ; (b) There exists a 5 5 4 nite positive constant O0 such that for each w> each $ l> and each |1>|2 R, 5 5 iw($> |1) iw($>|2) O0 |1 |2 . | | | |

Next we impose su!cient dierentiability of tw with respect to .

Assumption 3 (ii) For each w and each $ l>tw($> ) is continuously dier- 5 · entiable on A; (iii) For each w and each $ l>tw($> ) is twice continuously 5 · dierentiable on A;

ECB Working Paper Series No 957 November 2008 11 To exploit the mean value theorem, we require that W belongs to the interior of A>lqw(A).

Assumption 4 (ii) W lqw(A)= 5

Next, we place domination conditions on the derivatives of tw=

Assumption 5 (iii) Let G1>w maxm=1>===>s maxl=1>===>c sup (C@Cl)tm>w( >) >w= M 2 | · | 1> 2> === .Then(a)H(G1>w) ? ; (b) H(G1>w) ? ; (iv) Let G2>w maxm=1>===>s 42 4 maxl=1>===>c maxk=1>===>c sup (C @ClCk)tm>w( >) >w=1> 2> === .Then(a)H(G2>w) ? M | · | ; (b) H(G2 ) ? = 4 2>w 4 s Assumption 6 (i) TW H[im>w(0) tm>w( >W) 0tm>w( >W)] is positive de nite; m=1 u · u · (ii) Y W H(WW0) is positive de nite. w w P Assumptions 3(ii) and 5(iii.a) are additional assumptions helping to ensure that eq. (6) holds. Further imposing Assumptions 2(iii), 3(iii.a), 4(ii), and 5(iv.a) su!ces to ensure that eq. (9) holds. The additional regularity provided by Assumptions 5(iii.b), 5(iv.b), and 6(i) ensures that eq. (10) holds. Assumptions 5(iii.b) and 6(ii) help ensure the availability of the MDS central limit theorem. We now have conditions su!cient to ensure asymptotic normality of our MQ- CAViaR estimator. Formally, we have

Theorem 2 Suppose that Assumptions 1-6 hold. Then

1@2 1@2 g Y W3 TWW (ˆW W) Q(0>L)= $

Theorem 2 shows that our QML estimator ˆW is asymptotically normal with 1 1 asymptotic covariance matrix TW3 Y WTW3 . There is, however, no guarantee that

ˆW is asymptotically e!cient. There is now a considerable literature investigating e!cient estimation in quantile models; see, for example, Newey and Powell (1990), Otsu (2003), Komunjer and Vuong (2006, 2007a, 2007b). So far, this literature has only considered single quantile models. It is not obvious how the results for single quantile models extend to multi-quantile models such as ours. Nevertheless, Komunjer and Vuong (2007a) show that the class of QML estimators is not large enough to include an e!cient estimator, and that the class of M-estimators, which strictly includes the QMLE class, yields an estimator that attains the e!ciency bound. Speci cally, they show that replacing the usual quantile check function

m appearing in eq.(5) with

W (\w tm>w( >)) = ( 1[\w tm>w( >) 0])(Iw(\w) Iw(tm>w( >))) m · 3 · $ ·

ECB Working Paper Series No 957 12 November 2008 will deliver an asymptotically e!cient quantile estimator under the single quantile restriction. We conjecture that replacing m with Wm in eq.(5) will improve estimator e!ciency. We leave the study of the asymptotically e!cient multi-quantile estimator for future work.

4 Consistent Covariance Matrix Estimation

To test restrictions on W or to obtain con dence intervals, we require a consistent 1 1 estimator of the asymptotic covariance matrix FW TW3 Y WTW3 .First,wepro- vide a consistent estimator YˆW for Y W;thenwegiveaconsistentestimatorTˆW for ˆ ˆ 1 ˆ ˆ 1 TW= It follows that FW TW3 YW TW3 is a consistent estimator for FW= s Recall that Y W H(WW0),withW tm>w( >W)# (%m>w). A straightfor- w w w m=1 u · m ward plug-in estimator of Y is W P W ˆ 1 YW W 3 ˆwˆw0 > with w=1 s X

ˆw tm>w( > ˆW )#m (ˆ%m>w) m=1 u · X ˆ%m>w \w tm>w( > ˆW )= ·

We already have conditions su!cient to deliver the consistency of YˆW for Y W= Formally, we have

s Theorem 3 Suppose that Assumptions 1-6 hold. Then YˆW Y W= $

Next, we provide a consistent estimator of

TW H[im>w(0) tm>w( >W) 0tm>w( >W)]= m=1 u · u · X

We follow Powell’s (1984) suggestion of estimating im>w(0) with 1[ fˆW ˆ%m>w fˆW ]@2ˆfW for 3 $ $ a suitably chosen sequence fˆW = ThisisalsotheapproachtakeninKimandWhite { } (2003) and Engle and Manganelli (2004). Accordingly, our proposed estimator is

W s ˆ 1 TW =(2ˆfW W )3 1[ fˆW ˆ%m>w fˆW ] tm>w( > ˆW ) 0tm>w( > ˆW )= 3 $ $ w=1 m=1 u · u · X X

To establish consistency, we strengthen the domination condition on tm>w and u impose conditions on fˆW . { }

ECB Working Paper Series No 957 November 2008 13 Assumption 5 (iii.c) H(G3 ) ? = 1>w 4

Assumption 7 fˆW is a stochastic sequence and fW is a non-stochastic sequence { }s { 1 } 1@2 such that (i) fˆW @fW 1; (ii) fW = r(1); and (iii) f3 = r(W ). $ W

s Theorem 4 Suppose that Assumptions 1-7 hold. Then TˆW TW= $

5 Quantile-Based Measures of Conditional Skew- ness and Kurtosis

Moments of asset returns of order higher than two are important because these permit a recognition of the multi-dimensional nature of the concept of risk. Such higher order moments have thus proved useful for asset pricing, portfolio construction, and risk assessment. See, for example, Hwang and Satchell (1999) and Harvey and Siddique (2000). Higher order moments that have received particular attention are skewness and kurtosis, which involve moments of order three and four, respectively. Indeed, it is widely held as a "stylized fact" that the distribution of stock returns exhibits both left skewness and excess kurtosis (fat tails); there is a large amount of empirical evidence to this eect. Recently, Kim and White (2004) have challenged this stylized fact and the conventional way of measuring skewness and kurtosis. As moments, skewness and kurtosis are computed using averages, speci cally, averages of third and fourth powers of standardized random variables. Kim and White (2004) point out that averages are sensitive to outliers, and that taking third or fourth powers greatly enhances the in uence of any outliers that may be present. Moreover, asset returns are particularly prone to containing outliers, as the result of crashes or rallies. According to Kim and White’s simulation study, even a single outlier of a size comparable to the sharp drop in stock returns caused by the 1987 stock market crash can generate dramatic irregularities in the behavior of the traditional moment-based measures of skewness and kurtosis. Kim and White (2004) propose using more robust measures instead, based on sample quantiles. For example, Bowley’s (1920) coe!cient of skewness is given by

t3W + t1W 2t2W VN2 = > t t 3W 1W 1 1 1 where tW = I 3 (0=25)>tW = I 3 (0=5)> and tW = I 3 (0=75),whereI(|) S0[\w ? 1 2 3 |] is the unconditional CDF of \w. Similarly, Crow & Siddiqui’s (1967) coe!cient

ECB Working Paper Series No 957 14 November 2008 of kurtosis is given by t4W t0W NU4 = 2=91> t t 3W 1W 1 1 where t0W = I 3 (0=025) and t4W = I 3 (0=975).(ThenotationsVN2 and NU4 correspond to those of Kim and White (2004).) A limitation of these measures is that they are based on unconditional sample quantiles. Thus, in measuring skewness or kurtosis, these can neither incorporate useful information contained in relevant exogenous variables nor exploit the dynamic evolution of quantiles over time. To avoid these limitations, we propose constructing measures of conditional skewness and kurtosis using conditional quantiles tm>wW in place of the unconditional quantiles tmW. In particular, the conditional Bowley coe!cient of skewness and the conditional Crow & Siddiqui coe!cient of kurtosis are given by

t3W>w + t1W>w 2t2W>w FVN2 = > tW tW 3>w 1>w t4W>w t0W>w FNU4 = 2=91= t t 3W>w 1W>w Another quantile-based kurtosis measure discussed in Kim and White (2004) is Moors’s (1988) coe!cient of kurtosis, which involves computing six quantiles. Because our approach requires joint estimation of all relevant quantiles, and, in our model, each quantile depends not only on its own lags, but also possibly on the lags of other quantiles, the number of parameters to be estimated can be quite large.

Moreover, if the m’s are too close to each other, then the corresponding quantiles may be highly correlated, which can result in an analog of multicollinearity. For these reasons, in what follows we focus only on VN2 and NU4> as these require jointly estimating at most ve quantiles.

6 Application and Simulation

6.1 Time-varying skewness and kurtosis for the S&P500

In this section we obtain estimates of time-varying skewness and kurtosis for the S&P 500 index daily returns. Figure 1 plots the S&P 500 daily returns series used for estimation. The sample ranges from January 1, 1999 to September 28, 2007, for a total of 2,280 observations. First, we estimate time-varying skewness and kurtosis using the GARCH-type model of Leon, Rubio and Serna (2004), the LRS model for short. Letting uw

ECB Working Paper Series No 957 November 2008 15 denote the return for day w, we estimate the following speci cation of their model:

1@2 uw = kw w 2 kw = 1 + 2uw 1 + 3kw 1 3 3 3 vw = 4 + 5w 1 + 6vw 1 3 3 4 nw = 7 + 8w 1 + 9nw 1> 3 3 2 3 4 where we assume that Hw 1(w)=0, Hw 1(w )=1, Hw 1(w )=vw,andHw 1(w )= 3 3 3 3 nw> where Hw 1 denotes the conditional expectation given uw 1>uw 2> === The likeli- 3 3 3 hood is constructed using a Gram-Charlier series expansion of the normal density

function for w, truncated at the fourth moment. We refer the interested reader to Leon, Rubio, and Serna (2004) for technical details. The model is estimated via (quasi-) maximum likelihood. As starting values

for the optimization, we use estimates of 1>2> and 3 from the standard GARCH

model. We set initial values of 4 and 7 equal to the unconditional skewness and kurtosis values of the GARCH residuals. The remaining coe!cients are initialized at zero. The point estimates for the model parameters are given in Table 1.

Figures 3 and 5 display the time-series plots for vw and nw respectively.

Next, we estimate the MQ-CAViaR model. Given the expressions for FVN2

and FNU4,werequire ve quantiles, i.e. those for m =0=025> 0=25> 0=5> 0=75> and 0=975.WethusestimateanMQ-CAViaRmodelforthefollowingDGP:

t0W=025>w = 11W + 12W uw 1 + twW0 11W | 3 | 3 t0W=25>w = 21W + 22W uw 1 + twW0 12W | 3 | 3 . .

t0W=975>w = 51W + 52W uw 1 + twW0 15W> | 3 | 3

where twW 1 (t0W=025>w 1>t0W=25>w 1>t0W=5>w 1>t0W=75>w 1>t0W=975>w 1)0 and mW (mW1>mW2>mW3> 3 3 3 3 3 3 mW4>mW5)0>m =1> ===> 5= Hence, the coe!cient vector W consists of all the coe!cients

mnW and mnW > as above. Estimating the full model is not trivial. We discuss this brie y before present- ing the estimation results. We perform the computations in a step-wise fashion as follows. In the rst step, we estimate the MQ-CAViaR model containing just the 2=5% and 25% quantiles. The starting values for optimization are the individual CAViaR estimates, and we initialize the remaining parameters at zero. We repeat this estimation procedure for the MQ-CAViaR model containing the 75% and 97=5% quantiles. In the second step, we use the estimated parameters of the rst step as starting values for the optimization of the MQ-CAViaR model containing

ECB Working Paper Series No 957 16 November 2008 the 2=5%, 25%, 75%> and 97=5% quantiles, initializing the remaining parameters at zero. Third and nally, we use the estimates from the second step as starting values for the full MQ-CAViaR model optimization containing all ve quantiles of interest, again setting to zero the remaining parameters. The likelihood function appears quite at around the optimum, making the optimization procedure sensitive to the choice of initial conditions. In particular, choosing a dierent combination of quantile couples in the rst step of our estimation procedure tends to produce dierent parameter estimates for the full MQ-CAViaR model. Nevertheless, the likelihood values are similar, and there are no substantial dierences in the dynamic behavior of the individual quantiles associated with these dierent estimates. Table 2 presents our MQ-CAViaR estimation results. In calculating the standard errors, we have set the bandwidth to 1. Results are slightly sensitive to the choice of the bandwidth, with standard errors increasing for lower values of the bandwidth. We observe that there is interaction across quantile processes. This is particularly evident for the 75% quantile: the autoregressive coe!cient associated with the lagged 75% quantile is only 0=04, while that associated with the lagged 97=5% quantile is 0=29. This implies that the autoregressive process of the 75% quantile is mostly driven by the lagged 97=5% quantile, although this is not sta- tistically signi cant at the usual signi cance level. Figure 2 displays plots of the ve individual quantiles for the time period under consideration.

Next, we use the estimates of the individual quantiles t0W=025>w> ===> t0W=975>w to calcu- late the robust skewness and kurtosis measures FVN2 and FNU4. The resulting time-series plots are shown in Figures 4 and 6, respectively. We observe that the LRS model estimates of both skewness and kurtosis do not vary much and are dwarfed by those for the end of February 2007. The market was doing well until February 27, when the S&P 500 index dropped by 3=5%,asthe market worried about global economic growth. (The sub-prime mortgage asco was still not yet public knowledge.) Interestingly, this is not a particularly large negative return (there are larger negative returns in our sample between 2000 and 2001), but this one occurred in a period of relatively low volatility. Our more robust MQ-CAViaR measures show more plausible variability and con rm that the February 2007 market correction was indeed a case of large negative conditional skewness and high conditional kurtosis. This episode appears to be substantially aecting the LRS model estimates for the entire sample, raising doubts about the reliability of LRS estimates in general, consistent with the ndings of Sakata and White (1998).

ECB Working Paper Series No 957 November 2008 17 6.2 Simulation

In this section we provide some Monte Carlo evidence illustrating the nite sample behavior of our methods. We consider the same MQ-CAViaR process estimated in the previous subsection,

t0W=025>w = 11W + 12W uw 1 + twW0 11W | 3 | 3 t0W=25>w = 21W + 22W uw 1 + twW0 12W | 3 | 3 . .

t0W=975>w = 51W + 52W uw 1 + twW0 15W= (13) | 3 | 3 For the simulation exercise, we set the true coe!cients equal to the estimates reported in Table 2. Using these values, we generate the above MQ-CAViaR process 100 times, and each time we estimate all the coe!cients, using the procedure described in the previous subsection. t >m =1> ===> 5 Data were generated as follows. We initialize the quantiles Wm >w at w =1using the empirical quantiles of the rst 100 observations of our S&P 500

data. Given quantiles for time w, we generate a random variable uw compatible

with these using the following procedure. First, we draw a random variable Xw>

uniform over the interval [0> 1].Next,we nd m such that m 1 ?Xw ?m.This 3 determines the quantile range within which the random variable to be generated

should fall. Finally, we generated the desired random variable uw by drawing it [t >t ] from a uniform distribution within the interval Wm 1>w Wm >w . The procedure can 3 be represented as follows:

s+1

uw = L(m 1 ?Xw ?m)[tWm 1>w +(tWm >w tWm 1>w)Yw]> 3 3 3 m=1 X where Xw and Yw are i.i.d. U(0,1), 0 =0, s+1 =1, tW = tW 0=05 and 0>w 1>w tWs+1>w = tWs>w +0=05. It is easy to check that the random variable uw has the desired quantiles by construction. Further, it doesn’t matter that the distribution within the quantiles is uniform, as that distribution has essentially no impact on the

resulting parameter estimates. Using these values of uw and twW> we apply eq.(13) to generate conditional quantiles for the next period. The process iterates until w = W= Once we have a full sample, we perform the estimation procedure described in the previous subsection. Tables 3 and 4 provide the sample means and standard deviations over 100 replications of each coe!cient estimate for two dierent sample sizes, W =1> 000 and W =2> 280 (the sample size of the S&P 500 data), respectively. The mean

ECB Working Paper Series No 957 18 November 2008 estimates are fairly close to the values of Table 2, showing that the available sample sizes are su!cient to recover the true DGP parameters. (To obtain standard error estimates for the means, divide the reported standard deviations by 10.) A potentially interesting experiment that one might consider is to generate data from the LRS process and see how the MQ-CAViaR model performs in revealing underlying patterns of conditional skewness and kurtosis. Nevertheless, we leave this aside here, as the LRS model depends on four distributional shape parameters, but we require ve variation-free quantiles for the present exercise. As noted in Section 2, the MQ-CAViaR model will generally not satisfy the identi cation condition in such circumstances.

7Conclusion

In this paper, we generalize Engle and Manganelli’s (2004) single-quantile CAViaR process to its multi-quantile version. This allows for (i) joint modeling of multiple quantiles; (ii) dynamic interactions between quantiles; and (iii) the use of exogenous variables. We apply our MQ-CAViaR process to de ne conditional versions of existing unconditional quantile-based measures of skewness and kurtosis. Because of their use of quantiles, these measures may be much less sensitive than standard moment-based methods to the adverse impact of outliers that regularly appear in nancial market data. An empirical analysis of the S&P 500 index demonstrates the use and utility of our new methods.

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Mathematical Appendix

Proof of Theorem 1: We verify the conditions of corollary 5.11 of White (1994), which delivers ˆW W,where $ W 1 ˆW arg max W 3 *w(\w>tw( >))> M w=1 · X

ECB Working Paper Series No 957 November 2008 21 s and * (\w>tw( >)) (\w tm>w( >)). Assumption 1 ensures White’s w · m=1 m · Assumption 2.1. Assumption 3(i) ensures White’s Assumption 5.1. Our choice of P

m satis es White’s Assumption 5.4. To verify White’s Assumption 3.1, it su!ces

that *w(\w>tw( >)) is dominated on A by an integrable function (ensuring White’s · Assumption 3.1(a,b)) and that for each in A, *w(\w>tw( >)) is stationary { · } and ergodic (ensuring White’s Assumption 3.1(c), the strong uniform law of large numbers (ULLN)). Stationarity and ergodicity is ensured by Assumptions 1 and 3(i). To show domination, we write

*w(\w>tw( >)) m (\w tm>w( >)) | · | m=1 | · | Xs

= (\w tm>w( >))(m 1[\w tm>w( >) 0]) 3 · $ m=1 | · | Xs

sup *w(\w>tw( >)) 2s( \w + G0>w )= | · | | | | | M Thus, 2s( \w + G0>w ) dominates * (\w>tw( >)) and has nite expectation by | | | | | w · | Assumption 5(i,ii). It remains to verify White’s Assumption 3.2; here this is the condition that

W istheuniquemaximizerofH(* (\w>tw( >))= Given Assumptions 2(ii.b) and w · 4(i), it follows by argument directly parallel to that in the proof of White (1994, corollary 5.11) that for all A> 5

H(* (\w>tw( >)) H(* (\w>tw( >W))= w · w ·

Thus, it su!ces to show that the above inequality is strict for = W= Letting s 6 {() H({m>w()) with {m>w() (\w tm>w( >)) (\w tm>w( >W)),it m=1 m · m · su!ces to show that for each A0> {() A 0 for all A such that W A= P 5 || || Pick A0 and A such that W A=With m>w(> W) tw(m>) 5 || || tw(m>W)> by Assumption 4(i.b), there exist M 1>===>s and A 0 such that { } S [ m M m>w(> W) A ] A 0.Forthis and all m, some algebra and Assumption ^ M {| | }

ECB Working Paper Series No 957 22 November 2008 2(ii.a) ensure that

m>w(>W) H({m>w()) = H[ (m>w(> W) v) im>w(v)gv] Z0 1 2 1 2 H[ 1[ m>w(> ) A] + m>w(> W) 1[ m>w(> ) ])] 2 | W | 2 | W |$ 1 2 H[1[ m>w(> ) A]]= 2 | W | The rst inequality above comes from the fact that Assumption 2(ii.a) implies that for any A 0 su!ciently small, we have im>w(v) A for v ?.Thus, | | s s 1 2 {() H({m>w()) H[1[ m>w(> ) A]] 2 | W | m=1 m=1 X s X 1 2 1 2 = S[ m>w(> W) A] S[ m>w(> W) A] 2 | | 2 | | m=1 m M X XM 1 2 S [ m M m>w(> W) A ] 2 ^ M {| | } A 0> where the nal inequality follows from Assumption 4(i.b). As A0 and are arbitrary, the result follows.

ProofofTheorem2: As outlined in the text, we rst prove

W s 1@2 W 3 tm>w( > ˆW ) #m (\w tm>w( > ˆW )) = rs(1)= (14) w=1 m=1 u · · X X

The existence of tm>w is ensured by Assumption 3(ii). Let hl be the c 1 unit u × vector with lwk element equal to one and the rest zero, and let

W s 1@2 Jl(f) W 3 m (\w tm>w( > ˆW + fhl))> w=1 m=1 · X X for any real number f. Then by the de nition of ˆW , Jl(f) is minimized at f =0.

Let Kl(f) bethederivativeofJl(f) with respect to f from the right. Then

W s 1@2 Kl(f)= W 3 ltm>w( > ˆW + fhl) #m (\w tm>w( > ˆW + fhl))> w=1 m=1 u · · X X wk where ltm>w( > ˆW +fhl) is the l element of tm>w( > ˆW +fhl). Using the facts that u · u · (i) Kl(f) is non-decreasing in f and (ii) for any A0, Kl( ) 0 and Kl() 0,

ECB Working Paper Series No 957 November 2008 23 we have

Kl(0) Kl() Kl( ) | | W s 1@2 W 3 ltm>w( > ˆW ) 1[\w tm>w( >ˆW )=0] 3 · w=1 m=1 |u · | X X W s 1@2 W 3 max G1>w 1[\w tm>w( >ˆW )=0]> 1 w W 3 · $ $ w=1 m=1 X X where the last inequality follows by the domination condition imposed in Assump- 1@2 tion 5(iii.a). Because G1>w is stationary, W 3 max1 w W G1>w = rs(1).Thesecond $ $ W s term is bounded in probability: w=1 m=1 1[\w tm>w( >ˆW )=0] = Rs(1) given Assump- 3 · tion 2(i,ii.a) (see Koenker and Bassett, 1978, for details). Since K (0) is the lwk P P l 1@2 W s element of W 3 tm>w( > ˆW ) # (\w tm>w( > ˆW )),theclaimin(14)is w=1 m=1 u · m · proved. P P Next, for each A, Assumptions 3(ii) and 5(iii.a) ensure the existence and 5 niteness of the c 1 vector × s

() H[ tm>w( >) #m (\w tm>w( >))] m=1 u · · X s 0 = H[ tm>w( >) im>w(v)gv]> u · m=1 m>w(>W) X Z where m>w(> W) tm>w( >) tm>w( >W) and im>w(v)=(g@gv)Iw(v + tm>w( >W)) rep- · · · resents the conditional density of %m>w \w tm>w( >W) with respect to Lebesgue · measure= The dierentiability and domination conditions provided by Assump- tions 3(iii) and 5(iv.a) ensure (e.g., by Bartle, corollary 5.9??) the continuous dierentiability of on A,with s 0 ()= H[ 0tm>w( >) im>w(v)gv ]= u u{u · } m=1 m>w(>W) X Z Since W is interior to A by Assumption 4(ii), the mean value theorem applies to each element of to yield

()=(W)+T0( W)> (15)

for in a convex compact neighborhood of W,where T0 is an c c matrix with × (1 c)rowsTl(¯(l))= 0l(¯(l)),where¯(l) is a mean value (dierent for each × u l) lying on the segment connecting and W>l =1> ===> c. The chain rule and an 0 application of the Leibniz rule to im>w(v)gv then give m>w(>W)

Tl()=R Dl() El()>

ECB Working Paper Series No 957 24 November 2008 where

s 0 Dl() H[ l 0tm>w( >) im>w(v)gv] u u · m=1 Zm>w(>W) Xs

El() H[im>w(m>w(> W)) ltm>w( >) 0tm>w( >)]= m=1 u · u · X Assumption 2(iii) and the other domination conditions (those of Assumption 5) then ensure that

Dl(¯(l))=R( W ) || || El(¯(l))=TW + R( W )> l || || s s where TW H[im>w(0) ltm>w( >W) 0tm>w( >W)]= Letting TW H[im>w(0) l m=1 u · u · m=1 tm>w( >W) 0tm>w( >W)],weobtain × u · Pu · P

T0 = TW + R( W )= (16) || ||

Next, we have that (W)=0= To show this, we write

(W)= H[ tm>w( >W) #m (\w tm>w( >W))] m=1 u · · Xs

= H(H[ tm>w( >W) # (\w tm>w( >W)) w 1]) m 3 m=1 u · · |F Xs

= H( tm>w( >W) H[# (\w tm>w( >W)) w 1]) m 3 m=1 u · · |F Xs

= H( tm>w( >W) H[# (%m>w) w 1]) m 3 m=1 u · |F X =0>

as H[# (%m>w) w 1]=m H[1[\w t ] w 1]=0> by de nition of tm>wW >m =1> ===> s m |F 3 $ m>wW |F 3 (see eq. (2)). Combining (W)=0with eqs. (15) and (16), we obtain

2 ()= TW( W)+R( W )= (17) || || Thenextstepistoshowthat

1@2 W (ˆ)+KW = rs(1) (18)

1@2 W s where KW W 3 W> with W (W)> () tm>w( >) # (\w w=1 w w w w m=1 u · m tm>w( >)).Letxw(> g) sup : g w( ) w() . By the results of Huber · P { || 3 ||$ } || || P

ECB Working Paper Series No 957 November 2008 25 (1967) and Weiss (1991), to prove (18) it su!ces to show the following: (i) there

exist dA0 and g0 A 0 such that () d W for W g0;(ii)there || || || || || || exist eA0>g0 A 0> and g 0 such that H[xw(> g)] eg for W + g g0; || 2|| and (iii) there exist fA0>g0 A 0> and g 0 such that H[xw(> g) ] fg for W + g g0. || || The condition that TW is positive-de nite in Assumption 6(i) is su!cient for (i). For (ii), we have that for given (small) gA0

xw(> g) sup tm>w( > )#m (\w tm>w( > )) tm>w( >)#m (\w tm>w( >)) : g ||u · · u · · || { || 3 ||$ } m=1 s X

sup #m (\w tm>w( > )) sup tm>w( > ) tm>w( >) : g || · || × : g ||u · u · || m=1 { || 3 ||$ } { || 3 ||$ } Xs

+ sup #m (\w tm>w( >)) #m (\w tm>w( > )) sup tm>w( >) : g || · · || × : g ||u · || m=1 { || 3 ||$ } { || 3 ||$ } X s

sG2>wg + G1>w 1[ \w tm>w( >) ?G2>wg] (19) | 3 · | m=1 X

using the following; (i) # (\w tm>w( > )) 1> (ii) # (\w tm>w( >)) # (\w || m · || || m · m tm>w( > )) 1[ \w tm>w( >) ? tm>w( > ) tm>w( >) ], and (iii) the mean value theorem applied · || | 3 · | | · 3 · | to tm>w( > ) and tm>w( >). Hence, we have u · ·

H[xw(> g)] sF0g +2sF1i0g

for some constants F0 and F1 given Assumptions 2(iii.a), 5(iii.a), and 5(iv.a).

Hence, (ii) holds for e = sF0 +2sF1i0 and g0 =2g= The last condition (iii) can

be similarly veri ed by applying the fu inequality to eq. (19) with g?1 (so that g2 ?g) and using Assumptions 2(iii.a), 5(iii.b), and 5(iv.b).Thus,eq.(18)is veri ed . Combining eqs. (17) and (18) thus yields

W 1@2 1@2 TWW (ˆW W)=W 3 wW + rs(1) w=1 X

But wW> w is a stationary ergodic martingale dierence sequence (MDS). In par- { F } s ticular, wW is measurable w,andH(wW w 1)=H( m=1 tm>w( >W)# (%m>w) w 1)= 3 m 3 s F |F u · |F m=1 tm>w( >W)H(# (%m>w) w 1)=0,asH[# (%m>w) w 1]=0for all m =1>===>s= u · m |F 3 m P|F 3 Assumption 5(iii.b) ensures that Y W H(WW0) is nite. The MDS central limit P w w theorem (e.g., theorem 5.24 of White, 2001) applies, provided Y W is positive de - 1 W nite (as ensured by Assumption 6(ii)) and that W 3 w=1 wWwW0 = Y W +rs(1),which P

ECB Working Paper Series No 957 26 November 2008 is ensured by the ergodic theorem. The standard argument now gives

1@2 1@2 g Y W3 TWW (ˆW W) Q(0>L)> $ which completes the proof.

ProofofTheorem3:Wehave

W W W ˆ 1 1 1 YW Y W =(W 3 ˆwˆw0 W 3 wWwW0)+(W 3 wWwW0 H[wWwW0])> w=1 w=1 w=1 X X X s ˆ s ˆ where ˆ tˆm>w# and W tW #W > with tˆm>w tm>w( > ˆW )> # w m=1 u m>w w m=1 u m>w m>w u u · m>w # (\w tm>w( > ˆW ))> tW tm>w( >W),and#W # (\w tm>w( >W))= Assump- m P· u m>w u · P m>w m · tions 1 and 2(i,ii) ensure that WW0 is a stationary ergodic sequence. Assump- { w w } tions 3(i,ii), 4(i.a), and 5(iii) ensure that H[wWwW0] ? = It follows by the er- 1 W 4 godic theorem that W 3 w=1 wWwW0 H[wWwW0]=rs(1)= Thus, it su!ces to prove 1 W 1 W W ˆ ˆ0 W = r (1)= 3 w=1 w w 3 w=1P wW wW0 s 1 W 1 W The (k> l) element of W 3 ˆ ˆ0 W 3 WW0 is P P w=1 w w w=1 w w W s s P P 1 ˆ ˆ W 3 # # ktˆm>w ltˆn>w #W #W ktW ltW = { m>w n>wu u m>w n>wu m>wu n>w} w=1 m=1 n=1 X X X Thus,itwillsu!ce to show that for each (k> l) and (m> n) we have

W 1 ˆ ˆ W 3 #m>w#n>w ktˆm>w ltˆn>w #m>wW #n>wW ktm>wW ltn>wW = rs(1)= w=1 { u u u u } X By the triangle inequality,

W 1 ˆ ˆ W 3 #m>w#n>w ktˆm>w ltˆn>w #m>wW #n>wW ktm>wW ltn>wW DW + EW > | w=1 { u u u u }| X where

W 1 ˆ ˆ DW = W 3 #m>w#n>w ktˆm>w ltˆn>w #m>wW #n>wW ktˆm>w ltˆn>w | w=1 u u u u | XW 1 EW = W 3 #m>wW #n>wW ktm>wW ltn>wW #m>wW #n>wW ktˆm>w ltˆn>w = | w=1 u u u u | X We now show that DW = rs(1) and EW = rs(1), delivering the desired result.

For DW > the triangle inequality gives

DW D1W + D2W + D3W >

ECB Working Paper Series No 957 November 2008 27 where

W 1 D1W = W 3 m 1[%m>w 0] 1[ˆ%m>w 0] ktˆm>w ltˆn>w $ $ w=1 | ||u u | XW 1 D2W = W 3 n 1[% 0] 1[ˆ% 0] ktˆm>w ltˆn>w n>w$ n>w$ w=1 | ||u u | XW 1 D3W = W 3 1[%m>w 0]1[% 0] 1[ˆ%m>w 0]1[ˆ% 0] ktˆm>w ltˆn>w = $ n>w$ $ n>w$ w=1 | ||u u | X 1@2 Theorem 2, ensured by Assumptions 1 6, implies that W ˆW W = Rs(1)= || || This, together with Assumptions 2(iii,iv) and 5(iii.b), enables us to apply the

same techniques used in Kim and White (2003) to show D1W = rs(1), D2W =

rs(1)>D3W = rs(1), implying DW = rs(1)=

It remains to show EW = rs(1). By the triangle inequality,

EW E1W + E2W > where

W 1 E1W = W 3 #m>wW #n>wW ktm>wW ltn>wW H[#m>wW #n>wW ktm>wW ltn>wW ] | w=1 u u u u | XW 1 E2W = W 3 #m>wW #n>wW ktˆm>w ltˆn>w H[#m>wW #n>wW ktm>wW ltn>wW ] = | w=1 u u u u | X Assumptions 1, 2(i,ii), 3(i,ii), 4(i.a), and 5(iii) ensure that the ergodic theorem

applies to #W #W ktW ltW > so E1W = rs(1)= Next, Assumptions 1, 3(i,ii), and { m>w n>wu m>wu n>w} 5(iii) ensure that the stationary ergodic ULLN applies to #W #W ktm>w( >) ltn>w( >) = { m>w n>wu · u · } This and the result of Theorem 1 (ˆW W = rs(1)) ensure that E2W = rs(1) by e.g., White (1994, corollary 3.8), and the proof is complete.

ProofofTheorem4: We begin by sketching the proof. We rst de ne

W s 1 TW (2fW W)3 1[ fW %m>w fW ] tm>wW 0tm>wW > 3 $ $ w=1 m=1 u u X X and then we will show the following:

s TW H(TW ) 0> (20) $s H(TW ) TW 0> (21) s $ TW TˆW 0= (22) $

ECB Working Paper Series No 957 28 November 2008 s Combining the results above will deliver the desired outcome: TˆW TW 0. $ For (20), one can show by applying the mean value theorem to Im>w(fW ) Im>w( fW ),whereIm>w(f) 1 v f im>w(v)gv,that { $ } W s R s 1 H(TW )=W 3 H[im>w(m>W ) tm>wW 0tm>wW ]= H[im>w(m>W ) tm>wW 0tm>wW ]> w=1 m=1 u u m=1 u u X X X where is a mean value lying between fW and fW > and the second equality m>W follows by stationarity. Therefore, the (k> l) element of H(TW ) TW satis es | | s

O0H m>W ktm>wW ltm>wW m=1 | ||u u | X © 2 ª sO0fW H[G ]> 1>w which converges to zero as fW 0. The second inequality follows by Assumption $ 2(iii.b), and the last inequality follows by Assumption 5(iii.b). Therefore, we have the result in eq.(20). To show (21), it su!ces simply to apply a LLN for double arrays, e.g. theorem 2 in Andrews (1988).

Finally,for(22),weconsiderthe(k> l) element of TˆW TW ,whichisgivenby | | s s 1 W 1 W 1[ fˆW ˆ%m>w fˆW ] ktˆm>w ltˆm>w 1[ fW %m>w fW ] ktm>wW ltm>wW 2ˆf W 3 $ $ 2f W 3 $ $ | W w=1 m=1 u u W w=1 m=1 u u | X X X X W s fW 1 = (1[ fˆW ˆ%m>w fˆW ] 1[ fW %m>w fW ]) ktˆm>w ltˆm>w fˆ 2f W 3 $ $ 3 $ $ W | W w=1 m=1 u u XsX 1 W + 1[ fW %m>w fW ]( ktˆm>w ktm>wW ) ltˆm>w 2f W 3 $ $ W w=1 m=1 u u u X Xs 1 W + 1[ fW %m>w fW ] ktm>wW ( ltˆm>w ltm>wW ) 2f W 3 $ $ W w=1 m=1 u u u X X W s 1 fˆW + (1 ) 1[ fW %m>w fW ] ktm>wW ltm>wW 2f W f 3 $ $ W W w=1 m=1 u u | X X fW fˆW [D1W + D2W + D3W +(1 )D4W ]> fˆW fW

ECB Working Paper Series No 957 November 2008 29 where

s 1 W D1W 1[ fˆW ˆ%m>w fˆW ] 1[ fW %m>w fW ] ktˆm>w ltˆm>w 2f W 3 $ $ 3 $ $ W w=1 m=1 | |×|u u | X Xs 1 W D2W 1[ fW %m>w fW ] ktˆm>w ktm>wW ltˆm>w 2f W 3 $ $ W w=1 m=1 |u u |×|u | X Xs 1 W D3W 1[ fW %m>w fW ] ktm>wW ltˆm>w ltm>wW 2f W 3 $ $ W w=1 m=1 |u |×|u u | X Xs 1 W D4W 1[ fW %m>w fW ] ktm>wW ltm>wW = 2f W 3 $ $ W w=1 m=1 |u u | X X

It will su!ce to show that D1W = rs(1)>D2W = rs(1)>D3W = rs(1)> and D4W = s s Rs(1)= Then, because fˆW @fW 1, we obtain the desired result: TˆW TW 0. $ $ We rst show D1W = rs(1). It will su!ce to show that for each m,

1 W 1[ fˆW ˆ%m>w fˆW ] 1[ fW %m>w fW ] ktˆm>w ltˆm>w = rs(1)= 2f W 3 $ $ 3 $ $ W w=1 | |×|u u | X Let W lie between ˆW and W> and put gm>w>W tm>w( >W ) ˆW W + fˆW fW = ||u · ||×|| || | | Then

W 1 (2fW W)3 1[ fˆW ˆ%m>w fˆW ] 1[ fW %m>w fW ] ktˆm>w ltˆm>w XW + YW > 3 $ $ 3 $ $ w=1 | |×|u u | X where

W 1 XW (2fW W )3 1[ %m>w fW ?gm>w>W ] ktˆm>w ltˆm>w | 3 | w=1 |u u | XW 1 YW (2fW W )3 1[ %m>w+fW ?gm>w>W ] ktˆm>w ltˆm>w = | | w=1 |u u | X s s It will su!ce to show that XW 0 and YW 0= Let A0 and let } be an arbitrary $ $ positive number. Then, using reasoning similar to that of Kim and White (2003, lemma 5), one can show that for any A0>

W 1 S(XW A) S ((2fW W)3 1[ %m>w fW ?( tw(m >0) +1)}fW ]) ktˆm>w ltˆm>w A) | 3 | ||Q || w=1 |u u | X }i W 0 H ( t ( > ) +1) tˆ tˆ W w m W k m>w l m>w w=1 { ||u || |u u |} X 3 2 }i0 H G + H G @> { | 1>w| | 1>w|}

ECB Working Paper Series No 957 30 November 2008 where the second inequality is due to the Markov inequality and Assumption 2(iii.a), and the third is due to Assumption 5(iii.c). As } can be chosen arbitrarily s small and the remaining terms are nite by assumption, we have XW 0.The s $ same argument is used to show YW 0= Hence, D1W = rs(1) is proved. $ 1 W Next, we show D2W = rs(1).Forthis,itsu!ces to show D2W>m 2fW W w=1 1[ fW %m>w fW ] ktˆm>w ktm>wW ltˆm>w = rs(1) for each m.Notethat 3 $ $ |u u |×|u | P 1 W D tˆ t tˆ 2W>m 2f W k m>w k m>wW l m>w W w=1 |u u |×|u | X 1 W 2 t ( > ˜) ˆ tˆ 2f W k m>w W W l m>w W w=1 ||u · || × || || × |u | X 1 1 W ˆ G G 2f W W W 2>w 1>w W || || w=1 X 1 1 W = W 1@2 ˆ G G > 2f W 1@2 W W W 2>w 1>w W || || w=1 X 2 where ˜ is between ˆW and W,and tm>w( > ˜) is the rst derivative of ktˆm>w uk · u with respect to > which is evaluated at ˜. The last expression above is rs(1) 1@2 1 W because (i) W ˆW W = Rs(1) by Theorem 2, (ii) W 3 G2>wG1>w = Rs(1) || || w=1 by the ergodic theorem and (iii) 1@(f W 1@2)=r (1) by Assumption 7(iii). Hence, W s P D2W = rs(1). The other claims D3W = rs(1) and D4W = Rs(1) can be analogously and more easily proven. Hence, they are omitted. Therefore, we nally have s s TW TˆW 0> which, together with (20) and (21), implies that TˆW TW 0. $ $ The proof is complete.

ECB Working Paper Series No 957 November 2008 31 Table 1. S&P500 index: Estimation results for the LRS model

E1 E 2 E3 E 4 E 5 E 6 E 7 E8 E 9 0.01 0.05 0.94 -0.04 0.01 0.01 3.25 0.00 0.00 (0.18) (0.19) (0.04) (0.15) (0.01) (0.02) (0.04) (0.00) (0.00) Note: standard errors are in parentheses.

Table 2. S&P500 index : Estimation results for the MQ-CAViaR model

T j E j1 E j2 J j1 J j2 J j3 J j4 J j5 0.025 -0.04 -0.11 0.93 0.02 0 0 0 (0.05) (0.07) (0.12) (0.10) (0.29) (0.93) (0.30) 0.25 0.001 -0.01 0 0.99 0 0 0 (0.02) (0.05) (0.06) (0.03) (0.04) (0.63) (0.20) 0.50 0.10 0.04 0.03 0 -0.32 0 -0.02 (0.02) (0.04) (0.04) (0.02) (0.02) (0.52) (0.17) 0.75 0.03 -0.01 0 0 0 0.04 0.29 (0.31) (0.05) (0.70) (0.80) (2.33) (0.84) (0.34) 0.975 0.03 0.24 0 0 0 0.03 0.89 (0.06) (0.07) (0.16) (0.16) (0.33) (0.99) (0.29) Note: standard errors are in parentheses.

Table 3. Means of point estimates through 100 replications (T = 1,000) True parameters

T j E j1 E j2 J j1 J j2 J j3 J j4 J j5 0.025 -0.05 -0.10 0.93 0.04 0.00 0.00 0.00 (0.08) (0.02) (0.14) (0.34) (0.00) (0.01) (0.00) 0.25 -0.05 -0.01 0.04 0.81 0.00 0.00 0.00 (0.40) (0.05) (0.17) (0.47) (0.01) (0.00) (0.00) 0.50 -0.08 0.02 0.00 0.00 -0.06 0.00 0.00 (0.15) (0.06) (0.01) (0.00) (0.81) (0.00) (0.01) 0.75 0.20 0.05 0.00 0.00 0.00 0.38 0.13 (0.42) (0.11) (0.02) (0.02) (0.00) (0.63) (0.19) 0.975 0.06 0.22 0.00 0.00 0.00 0.10 0.87 (0.16) (0.03) (0.00) (0.01) (0.00) (0.56) (0.16) Note: standard deviations are in parentheses.

ECB Working Paper Series No 957 32 November 2008 Table 4. Means of point estimates through 100 replications (T = 2,280) True parameters

T j E j1 E j2 J j1 J j2 J j3 J j4 J j5 0.025 -0.04 -0.10 0.93 0.03 0.00 0.00 0.00 (0.03) (0.01) (0.07) (0.21) (0.00) (0.00) (0.00) 0.25 -0.04 -0.01 0.03 0.88 0.00 0.00 0.00 (0.18) (0.02) (0.12) (0.38) (0.00) (0.00) (0.00) 0.50 -0.01 0.02 0.00 0.00 -0.03 0.00 0.00 (0.11) (0.04) (0.00) (0.01) (0.75) (0.00) (0.02) 0.75 0.09 0.01 0.00 0.00 0.00 0.33 0.19 (0.21) (0.07) (0.00 (0.01) (0.00) (0.58) (0.18) 0.975 0.05 0.24 0.00 0.00 0.00 0.18 0.83 (0.13) (0.02) (0.00) (0.03) (0.00) (0.69) (0.22) Note: standard deviations are in parentheses.

-2

-4

-6

-8 1999 2000 2001 2002 2003 2004 2005 2006 2007

Figure 1: S&P500 daily returns: January 1, 1999 — September 30, 2007.

ECB Working Paper Series No 957 November 2008 33 6

-2

-4

-6 1999 2000 2001 2002 2003 2004 2005 2006 2007

Figure 2: S&P500 conditional quantiles: January 1, 1999 - September 30, 2007.

0.5

-0.5

-1

-1.5

-2 1999 2000 2001 2002 2003 2004 2005 2006 2007

Figure 3: S&P 500: Estimated conditional skewness, LRS model.

ECB Working Paper Series No 957 34 November 2008 0.5

-0.5

-1

-1.5

-2 1999 2000 2001 2002 2003 2004 2005 2006 2007

Figure 4: S&P 500: Estimated conditional skewness, MQ-CAViaR model.

6.5

5.5

4.5

3.5

2.5 1999 2000 2001 2002 2003 2004 2005 2006 2007

Figure 5: S&P 500: Estimated conditional kurtosis, LRS model.

ECB Working Paper Series No 957 November 2008 35 6.5

5.5

4.5

3.5

2.5 1999 2000 2001 2002 2003 2004 2005 2006 2007

Figure 6: S&P 500: Estimated conditional kurtosis, MQ-CAViaR model.

ECB Working Paper Series No 957 36 November 2008 European Central Bank Working Paper Series

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ECB Working Paper Series No 957 38 November 2008 Working Paper Series No 957 / November 2008

MODELING AUTOREGRESSIVE CONDITIONAL SKEWNESS AND KURTOSIS WITH MULTI-QUANTILE CAViaR

by Halbert White, Tae-Hwan Kim and Simone Manganelli