Cambridge-INET Institute

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1 Faculty of Economics Cambridge-INET Institute Cambridge-INET Working Paer Series No: 4/3 Cambridge Working Paer in Economics: 45 THE CROSS-QUANTILOGRAM: MEASURING QUANTILE DEPENDENCE AND TESTING DIRECTIONAL PREDICTABILITY BETWEEN TIME SERIES Heejoon Han Oliver Linton Tatsushi Oka Yoon-Jae Whang (Kyung Hee University) (University of Cambridge) (University of Singaore) (Seoul National University) ABSTRACT This aer rooses the cross-quantilogram to measure the quantile deendence between two time series. We aly it to test the hyothesis that one time series has no directional redictability to another time series. We establish the asymtotic distribution of the cross quantilogram and the corresonding test statistic. The limiting distributions deend on nuisance arameters. To construct consistent confidence intervals we emloy the stationary bootstra rocedure; we show the consistency of this bootstra. Also, we consider the self-normalized aroach, which is shown to be asymtotically ivotal under the null hyothesis of no redictability. We rovide simulation studies and two emirical alications. First, we use the cross-quantilogram to detect redictability from stock variance to excess stock return. Comared to existing tools used in the literature of stock return redictability, our method rovides a more comlete relationshi between a redictor and stock return. Second, we investigate the systemic risk of individual financial institutions, such as JP Morgan Chase, Goldman Sachs and AIG. This article has sulementary materials online.

2 The Cross-: Measuring Quantile Deendence and Testing Directional Predictability between Time Series Heejoon Han Oliver Linton Tatsushi Oka Yoon-Jae Whang January 4 Abstract This aer rooses the cross-quantilogram to measure the quantile deendence between two time series. We aly it to test the hyothesis that one time series has no directional redictability to another time series. We establish the asymtotic distribution of the cross quantilogram and the corresonding test statistic. The limiting distributions deend on nuisance arameters. To construct consistent con dence intervals we emloy the stationary bootstra rocedure; we show the consistency of this bootstra. Also, we consider the self-normalized aroach, which is shown to be asymtotically ivotal under Heejoon Han is Associate Professor, Deartment of Economics, Kyung Hee University, Seoul, Reublic of Korea ( heejoon@khu.ac.kr). Oliver Linton is Fellow of Trinity College and Professor of Political Economy, Faculty of Economics, University of Cambridge, Cambridge, UK ( obl@cam.ac.uk). Tatsushi Oka is Assistant Professor, Deartment of Economics, National University of Singaore, Singaore, Reublic of Singaore ( oka@nus.edu.sg). Yoon-Jae Whang is Professor, Deartment of Economics, Seoul National University, Seoul, Reublic of Korea ( whang@snu.ac.kr). Han s work was suorted by the National Research Foundation of Korea (NRF-35). Linton s work was suorted by Cambridge INET and the ERC. Whang s work was suorted by the National Research Foundation of Korea (NRF--34-B4).

3 the null hyothesis of no redictability. We rovide simulation studies and two emirical alications. First, we use the cross-quantilogram to detect redictability from stock variance to excess stock return. Comared to existing tools used in the literature of stock return redictability, our method rovides a more comlete relationshi between a redictor and stock return. Second, we investigate the systemic risk of individual nancial institutions, such as JP Morgan Chase, Goldman Sachs and AIG. This article has sulementary materials online. Keywords: Quantile, Correlogram, Deendence, Predictability, Systemic risk.

4 Introduction Linton and Whang (7) introduced the quantilogram to measure redictability in di erent arts of the distribution of a stationary time series based on the correlogram of "quantile hits". They alied it to test the hyothesis that a given time series has no directional redictability. More seci cally, their null hyothesis was that the ast information set of the stationary time series fx t g does not imrove the rediction about whether x t will be above or below the unconditional quantile. The test is based on comaring the quantilogram to a ointwise con dence band. This contribution ts into a long literature of testing redictability using signs or rank statistics, including the aers of Cowles and Jones (937), Dufour et al. (998), and Christoffersen and Diebold (). The quantilogram has several advantages comared to other test statistics for directional redictability. It is concetually aealing and simle to interret. Since the method is based on quantile hits it does not require moment conditions like the ordinary correlogram and statistics like the variance ratio that are derived from it, Mikosch and Starica (), and so it works well for heavy tailed series. Many nancial time series have heavy tails, see, e.g., Mandelbrot (963), Fama (965), Rachev and Mittnik (), Embrechts et al. (997), Ibragimov et al. (9), and Ibragimov (9), and so this is an imortant consideration in ractice. Additionally, this tye of method allows researchers to consider very long lags in comarison with regression tye methods, such as Engle and Manganelli (4). There have been a number of recent works either extending or alying this methodology. Davis and Mikosch (9) have introduced the extremogram, which is essentially the quantilogram for extreme quantiles. Hagemann () has introduced a Fourier domain version of the quantilogram, see also Dette et al. (3) for an alternative aroach. The quantilogram has recently been alied to stock returns

5 and exchange rates, Laurini et al. (8) and Chang and Shie (). Our aer addresses two outstanding issues with regard to the quantilogram. First, the construction of con dence intervals that are valid under general deendence structures. Linton and Whang (7) derived the limiting distribution of the samle quantilogram under the null hyothesis that the quantilogram itself is zero, in fact under a secial case of that where the rocess has a tye of conditional heteroskedasticity structure. Even in that very secial case, the limiting distribution deends on model seci c quantities. They derived a bound on the asymtotic variance that allows one to test the null hyothesis of the absence of redictability (or rather the secial case of this that they work with). Even when this model structure is aroriate, the bounds can be quite large esecially when one looks into the tail of the distribution. The quantilogram is also useful in cases where the null hyothesis of no redictability is not thought to be true - one can be interested in measuring the degree of redictability of a series across di erent quantiles. We rovide a more comlete solution to the issue of inference for the quantilogram. Seci cally, we derive the asymtotic distribution of the quantilogram under general weak deendence conditions, seci cally strong mixing. The limiting distribution is quite comlicated and deends on the long run variance of the quantile hits. To conduct inference we roose the stationary bootstra method of Politis and Romano (994) and rove that it rovides asymtotically valid con dence intervals. We investigate the nite samle erformance of this rocedure and show that it works well. We also rovide R code that carries out the comutations e ciently. We also de ne a self-normalized version of the statistic for testing the null hyothesis that the quantilogram is zero, following Lobato (). This statistic has an asymtotically ivotal distribution whose critical values have been tabulated so that there is no need for long run variance estimation or even bootstra.

6 Second, we develo our methodology inside a multivariate setting and exlicitly consider the cross-quantilogram. Linton and Whang (7) brie y mentioned such a multivariate version of the quantilogram but they rovided neither theoretical results or emirical results. In fact, the cross correlogram is a vitally imortant measure of deendence between time series: Cambell et al. (997), for examle, use the cross autocorrelation function to describe lead lag relations between large stocks and small stocks. We aly the cross-quantilogram to the study of stock return redictability; our method rovides a more comlete icture of the redictability structure. We also aly the cross quantilogram to the question of systemic risk. Our theoretical results described in the revious aragrah are all derived for the multivariate case. The Cross- Let fx t : t Zg be a two dimensional strictly stationary time series with x t (x t ; x t ) and let F i () denote the distribution function of the series x it with density function f i () for i = ;. The quantile function of the time series x it is de ned as q i ( i ) = inffv : F i (v) i g for i (; ); for i = ;. Let q = (q ( ); q ( )) for ( ; ). We consider a measure of serial deendence between two events fx t q ( )g and fx t k q ( )g for arbitrary quantiles. In the literature, f[x it q i ()]g is called the quantile-hit or quantile-exceedance rocess for i = ;, where [] denotes the indicator function taking the value one when its argument is true, and zero otherwise. The cross-quantilogram is de ned as the cross-correlation of the quantile-hit rocesses (k) = E (x t q ( )) (x t k q ( )) q E (x t q ( )) q E (x t q ( )) () for k = ; ; ; : : : ; where a (u) [u < ] a. The cross-quantilogram catures 3

7 serial deendency between the two series at di erent quantile levels. In the secial case of a single time series, the cross-quantilogram becomes the quantilogram roosed by Linton and Whang (7). Note that it is well-de ned even for rocesses with in nite moments. Like the quantilogram, the cross-quantilogram is invariant to any strictly monotonic transformation alied to both series, such as the logarithmic transformation. To construct the samle analogue of the cross-quantilogram based on observations fx ; : : : ; x T g, we rst estimate the unconditional quantile functions by solving the following minimization roblems, searately: ^q ( ) = arg min v R TX (x t t= v ) and ^q ( ) = arg min v R TX (x t v ); t= where a (u) u(a [u < ]). Then, the samle cross-quantilogram is de ned as ^ (k) = q PT P T t=k+ (x t ^q ( )) (x t k ^q ( )) q PT ; () t=k+ (x t ^q ( )) t=k+ (x t k ^q ( )) for k = ; ; ; : : : : Given a set of quantiles, the cross-quantilogram considers deendency in terms of the direction of deviation from quantiles and thus measures the directional redictability from one series to another. This can be a useful descritive device. By construction, ^ (k) [ ; ] with ^ (k) = corresonding to the case of no directional redictability. The form of the statistic generalizes to the d dimensional multivariate case and the (i; j)th entry of the corresonding cross-correlation matrices (k) is given by alying () for a air of variables (x it ; x jt k ) and a air of quantiles (^q i ( i ); ^q j ( j ))) for = ( ; : : : ; d ). The cross-correlation matrices ossess the usual symmetry roerty (k) = ( k) when = = d : We may be interested in testing for the absence of directional redictability over a set of quantiles. Let A A A, where A i (; ) is a quantile range for 4

8 each time series fx i ; : : : ; x it g (i = ; ). We are interested in testing the hyothesis H : () = = = ; 8 A; against the alternative hyothesis that (k) 6= for some k f; : : : ; g and some A with xed. This is a test for the directional redictability of events u to lags fx t k q ( ) : k = ; : : : ; g for x t : To discriminate between these hyotheses we will use the test statistic su A ^Q = su T jj^ jj = su A A T X ^ (k); k= where ^Q is the quantile seci c Box-Pierce tye statistic and ^ [^ (); : : : ; ^ ] : To test the directional redictability in a seci c quantile, or to rovide con dence intervals for the oulation quantities, we use ^ (k) or ^Q, which are secial cases of the su-tye test statistic. In ractice, we have found that the Box-Ljung version ^Q T (T + ) P k= ^ (k)=(t k) yields some small samle imrovements. 3 Asymtotic Proerties Here we resent the asymtotic roerties of the samle cross-quantilogram and related test statistics. Since these quantities contain non-smooth functions, we emloy techniques widely used in the literature on quantile regression, see Koenker and Bassett (978) and Pollard (99) among others. We imose the following assumtions. Assumtion A. fx t : t Zg is a strictly stationary and strong mixing with coe cients f(n)g that satisfy (n) = O(n a ) for a >. A. The distribution functions F i () for i = ; have continuous densities f i () uniformly bounded away from and at q i ( i ) uniformly over i A i. A3. For any > there exists Hong (996) established the roerties of the Box-Pierce statistic in the case that = n! : after a location and scale adjustment the statistic is asymtotically normal. No doubt our results can be extended to accommodate this case, although in ractice the desirability of such a test is questionable, and our limit theory may rovide better critical values for even quite long lags. 5

9 a () such that su Ai su s:jsj() jf i (q i ()) f i (q i () + s)j < for i = ;. A4. The joint distribution G k of (x t ; x t k ) has a bounded, continuous rst derivative for each argument uniformly in the neighborhood of quantiles of interest for every k f; : : : ; g. Assumtion A imoses a mixing rate used in Rio (, Chater 7). For a strong mixing rocess, (k)! as k! for all (; ): Assumtion A ensures that the quantile functions are uniquely de ned. Assumtion A3 imlies that the densities are smooth in some neighborhood of the quantiles of interest. Assumtion A4 ensures that the joint distribution of (x t ; x t k ) is continuously di erentiable. To describe the asymtotic behavior of the cross-quantilogram, we de ne a set of 3-dimensional mean-zero Gaussian rocess fb k () : [; ] g k= with covariancematrix function given by X kk (; ) E[B k ()B k ( )] = cov l (; k); ( ; k ) ; for k; k f; : : : ; g and for ; A, where l= t (; k) = [ t ( ) ; t k ( ) ] G k (q ); ( t ( )); ( t k ( )) ; with it () = x it q i () for i = ; and for t = ; : : : ; T. De ne the 3-dimensional zeromean Gaussian rocess B () = [B () ; : : : ; B () ] with the covariance-matrix function denoted by (; ). The next theorem establishes the asymtotic roerties of the cross-quantilogram. Theorem Suose that Assumtions A-A4 hold for some nite integer > : Then, in the sense of weak convergence of the stochastic rocess we have: T ^ ) B (); (3) 6

10 where = diag( ; : : : ; ) and k = ( ) ( ) diag ; with the gradient vector rg k () of G k (). f (q ( )) ; 6 4 f (q ( )) rg k (q ) ; (4) Under the null hyothesis that () = = = for every A, it follows that su A ^Q by the continuous maing theorem. =) su k B ()k (5) A 3. Inference Methods 3.. The Stationary Bootstra The asymtotic null distribution resented in Theorem deends on nuisance arameters. To estimate the critical values from the limiting distribution we could use nonarametric estimation, but that may su er from a slow convergence rate. We address this issue by using the stationary bootstra (SB) of Politis and Romano (994). The SB is a block bootstra method with blocks of random lengths. The SB resamle is strictly stationary conditional on the original samle. Let fl i g in denote a sequence of block lengths, which are iid random variables having the geometric distribution with a scalar arameter T (; ): P (L i = l) = ( ) l ; 8i N; for each ositive integer l, where P denotes the conditional exectation given the original samle. We assume that the arameter satis es the following growth condition 7

11 Assumtion A5. + ( T )! as T!. Let fk i g in be iid random variables, which are indeendent of both the original data and fl i g in, and have the discrete uniform distribution on fk + ; : : : ; T g. We set B Ki ;L i = f(x t ; x t k )g L i t=k i reresenting the block of length L i starting with the K i -th air of observations. The SB rocedure generate samles f(x t; x t by taking the rst (T k )gt t=k+ k) observations from a sequence of the resamled blocks fb Ki ;L i g in. In this notation, when t > T, (x t ; x t k ) is set to be (x j ; x j k ), where j = k + (t mod (T k)) and (x k ; x ) = (x T ; x T +k ), where mod denotes the modulo oerator. Using the SB resamle, we obtain quantile estimates ^q = (^q ( ); ^q ( )) for q = (q ( ); q ( )) by solving the minimization roblems: ^q ( ) = arg min TX v R t=k+ (x t v ) and ^q ( ) = arg min TX v R t=k+ (x t k v ): We construct ^q by using (T k) SB observations, while ^q is based on T observations. The di erence of samle sizes is asymtotically negligible given the nite lag order k. The cross-quantilogram based on the SB resamle is de ned as follows: ^ (k) = q PT P T t=k+ (x t ^q ( )) (x t k ^q ( )) q PT : t=k+ (x t ^q ( )) t=k+ (x t k ^q ( )) We consider the SB bootstra to construct a con dence intervals for each statistic of cross-quantilograms f^ (); : : : ; ^ g for a nite ositive integer and subsequently construct a con dence interval for the omnibus test based on the statistics. To maintain ossible deendence structures, we use (T ) airs of observations f(x t ; : : : ; x t )g T t=+ to resamle the blocks of random lengths. Given a vector cross-quantilogram ^, we de ne the omnibus test based on the For any ositive integers a and b, the modulo oeration a mod b is equal to the remainder, on division of a by b. 8

12 SB resamle as ^Q = T (^ ^ ) (^ validity of the SB rocedure for the cross-quantilogram. ^ ). The following lemma shows the Theorem Suose that Assumtion A-A5 hold. Then, in the sense of weak convergence conditional on the samle we have: (a) T ^ ^ =) B () in robability; (b) Under the null hyothesis that () = = = for every A, su zr su P A ^Q z P su A ^Q z! : In ractice, reeating the SB rocedure B times, we obtain B sets of crossquantilograms and f^ ;b = [^ ;b(); : : : ; ^ ;b] g B b= f ^Q ;b gb b= with ^Q ;b = T (^ ;b ^ ) (^ ;b and B sets of omnibus tests ^ ). For testing jointly the null of no directional redictability, a critical value, c Q;, corresonding to a signi cance level is give by the ( )% ercentile of B test statistics fsu ^Q A ;b gb b=, that is, c Q; = inf c : P su A ^Q ;b c : For the individual cross-quantilogram, we ick u ercentiles (c k; ; c k; ) of the bootstra distribution of f T (^ ;b(k) ^ (k))g B b= such that P (c k T (^ ;b(k) ^ (k)) c k ) =, in order to obtain a ( )% con dence interval for (k) given by [^ (k) T = c k; ; ^ (k) + T = c k; ]: In the following theorem, we rovide a ower analysis of the omnibus test statistic su A ^Q when we use a critical value c Q;. That is, we examine the ower of the omnibus test by using P (su A ^Q > c Q; ). We consider xed and local alternatives. The xed alternative hyothesis against the null of no directional redictability is H : (k) is nonzero constant for some A and for some k f; : : : ; g; (6) 9

13 and the local alternative H T : (k) = = T ; for some A and for some k f; : : : ; g; (7) where is a nite non-zero scalar. a -dimensional vector non-zero element for some A. such that Thus, under the local alternative, there exist = T = with having at least one While we consider the asymtotic ower of test for the directional redictability over a range of quantiles with multile lags in the following theorem, the results can be alied to test for a seci c quantile or a seci c lag order. The following theorem shows that the cross-quantilogram rocess has non-trivial local ower against T -local alternatives. Theorem 3 Suose that Assumtions A-A5 hold. Then: (a) Under the xed alternative in (6), lim P su T! A (b) Under the local alternative in (7) lim P su T! A ^Q ^Q > c Q;! : > c Q; = P su k B () + k c Q; ; A where c Q; = inffc : P (su A k B ()k c)) g. 3.. The Self-Normalized Cross- The self-normalized aroach was roosed in Lobato () for testing the absence of autocorrelation of a time series that is not necessarily indeendent. The idea was recently extended by Shao () to a class of asymtotically linear test statistics. Kuan and Lee (6) aly the aroach to a class of seci cation tests, the so-called M tests, which are based on the moment conditions involving unknown arameters.

14 Chen and Qu () roose a rocedure for imroving the ower of the M test, by dividing the original samle into subsamles before alying the self-normalization rocedure. The self-normalized aroach has a tight link with the xed-b asymtotic framework, which was roosed by Kiefer et al. () and has been studied by Bunzel et al. (), Kiefer and Vogelsang (, 5), Sun et al. (8), Kim and Sun () and Sun and Kim () among others. As discussed in section. of Shao (), the self-normalized and the xed-b aroach have better size roerties, comared with the standard aroach involving a consistent asymtotic variance estimator, while it may be asymtotically less owerful under local alternatives (see Lobato () and Sun et al. (8) for instance). The relation between size and ower roerties is consistent with simulation results reorted in the cited aers above. We use recursive estimates to construct a self-normalized cross-quantilogram. Given a subsamle fx t g s t=, we can estimate samle quantile functions by solving minimization roblems ^q s ( ) = arg min v R sx (x t t= v ) and ^q s ( ) = arg min v R sx (x t v ): t= We consider the minimum subsamle size s larger than [T!], where! (; ) is an arbitrary small ositive constant. The trimming arameter,!, is necessary to guarantee that the quantiles estimators based on subsamles have standard asymtotic roerties and lays a di erent role to that of smoothing arameters in long-run variance estimators. Our simulation study suggests that the erformance of the test is not sensitive to the trimming arameters. A key ingredient of the self-normalized statistic is an estimate of cross-correlation

15 based on subsamles: ^ ;s (k) = q Ps P s t=k+ (x t ^q s ( )) (x t k ^q s ( )) q Ps ; t=k+ (x t ^q s ( )) t=k+ (x t k ^q s ( )) for [T!] s T. For a nite integer >, let ^ ;s = [^ ;s (); : : : ; ^ ;s ]. We construct an outer roduct of the cross-quantilogram using the subsamle ^A = T TX s=[t!] s ^ ;s ^ ^ ;s ^ : We can obtain the asymtotically ivotal distribution using ^A as the asymtotically random normalization. For testing the null of no directional redictability, we de ne the self-normalized omnibus test statistic ^S = T ^ ^A ^ : The following theorem shows that ^S is asymtotically ivotal. To distinguish the rocess used in the following theorem from the one used in the revious section, let f B ()g denote a 3-dimensional, standard Brownian motion on D 3 ([; ]). Theorem 4 Suose that Assumtions A-A4 hold. Then, for each A, ^S =) B () A B (); where A = R B (r) rb ()! B (r) rb () dr. The joint test based on nite multile quantiles can be constructed in a similar manner, while the test based on a range of quantiles has a limiting distribution deending on the Kiefer rocess: this may be di cult to imlement in ractice. The asymtotic null distribution in the above theorem can be simulated and a critical value, c S;, corresonding to a signi cance level is tabulated by using the

16 ( )% ercentile of the simulated distribution. 3 In the theorem below, we consider a ower function of the self-normalized omnibus test statistic, P ( ^S > c S; ). For a xed A, we consider a xed alternative H : (k) is nonzero constant for some k f; : : : ; g; (8) and a local alternative H T : (k) = = T ; and for some k f; : : : ; g; (9) where is a nite non-zero scalar. This imlies that there exist a -dimensional vector such that = T = with having at least one non-zero element. Theorem 5 (a) Suose that the xed alternative in (8) and Assumtions A-A4 hold. Then, lim T! P ^S > c S;! : (b) Suose that the local alternative in (9) is true and Assumtions A-A4 hold. Then, nb o lim P ^S > c S; = P () + ( ) T! n o A B () + ( ) c S; ; where is a 3 3 matrix with ( ) (; ). 4 The Partial Cross- We de ne the artial cross-quantilogram, which measures the relationshi between two events fx t q ( )g and fx t k q ( )g, while controlling for intermediate events between t and t k as well as whether some state variables exceed a given 3 We rovide the simulated critical values in our R ackage. 3

17 quantile. Let z t (z t ; : : : ; z lt ) be an l-dimensional vector for l, which may include some of the lagged redicted variables fx t ; : : : ; x t k g, the intermediate redictors fx t ; : : : ; x t k g and some state variables that may re ect some historical events u to t. We use q zi ( i ) to denote the i th quantile of z i given i (; ) for i = ; : : : ; l and de ne q z; = (q z ( ); : : : ; q zl ( l )), with = ( ; : : : ; l ). To ease the notational burden in the rest of this section, we suress the deendency of q zi ( i ) on i for i = ; : : : ; l and use q z q z; and q zi q zi ( i ). We resent results for the single quantile and a single lag k, although the results can be extended to the case of a range of quantiles and multile lags. We introduce the correlation matrix of the hit rocesses and its inverse matrix R k = E h t;k h t;k and Pk = R k ; where h t;k = [ ( t ( )); ( t k ( )); (z t q z ) ] with (u) = [ (u ); : : : ; l (u l )] for u = (u ; : : : ; u l ) R l. For i; j f; : : : ; lg, let r k;ij and k;ij be the (i; j) element of R k and P k, resectively. Notice that the cross-quantilogram is (k) = r k; = r k; r k; ; and the artial cross-quantilogram is de ned as jz (k) = k; k; k; : To obtain the samle analogue of the artial cross-quantilogram, we rst construct a vector of hit rocesses, ^h t;k, by relacing the oulation quantiles in h t;k by the samle analogues (^q ( ); ^q ( ); ^q z ; : : : ; ^q zl ). Then, we obtain the estimator for the correlation matrix and its inverse as ^R k = T TX t=k+ ^h t;k^h t;k and ^P k = ^R k ; 4

18 which leads to the samle analogue of the artial cross-quantilogram ^ jz (k) = ^ k; ^k;^ k; ; () where ^ k;ij denotes the (i; j) element of ^P k for i; j f; : : : ; lg. In Theorem 6 below, we show that ^ jz (k) asymtotically follows the normal distribution, while the asymtotic variance deends on nuisance arameters as in the revious section. To address the issue of the nuisance arameters, we emloy the stationary bootstra or the self-normalization technique. For the bootstra, we can use airs of variables f(x t ; x t k ; z t )g T t=k+ to generate the SB resamle f(x t; x t k ; z t )g T t=k+ and we then obtain the SB version of the artial cross-quantilogram, denoted by ^ jz(k), using the formula in (). When we use the self-normalized test statistics, we estimate the artial cross-quantilogram ;sjz (k) based on the subsamle u to s, recursively and then we use TX ^A jz = T s ^ ;sjz ^ ;T jz ; s=[t!] to normalize the cross-quantilogram, thereby obtaining the asymtotically ivotal statistics. To obtain the asymtotic results, we imose the following conditions on the distribution function F zi () and the density function F zi () of each controlling variable z i for i = ; : : : ; l. Assumtion A6. For every i f; : : : ; lg: (a) fz it g tz is strictly stationary and strong mixing as assumed in Assumtion A; (b) The conditions in Assumtion A and A3 hold for the F zi () and f zi () at the relevant quantile; (c) G zi (v ; v ) P (x t v ; z it v 3 ) for (v ; v ) R is continuously di erentiable. Assumtion A6(a) requires the control variables z t to satisfy the same weak deendent roerty as x t and x t. Assumtion A6(b)-(c) ensure the smoothness of the 5

19 distribution, density function and the joint distribution of x t and z it. De ne the covariance matrix k;z() = X l= cov ~l (; k); ~ (; k) ; where ~ t (; k) = [ ~ t (; k) ; ~ t (; k) ] ~ t (; k) = 6 4 ( t( )) ( t k ( )) ( t k ( )) (z t q z ) with and ~ t (; k) = 6 4 ( t ( )) ( t k ( )) (z t q z ) 3 7 : 5 Also, let k;z = ( k; k; ) = ( () k;z () k;z ) ; where () k;z = 6 4 z z and () k;z = f 3 6 B4 rg k(q ) 7 l 6 4 z z (v; q z )=@vj v=q ( ) z (q ( ); w)=@w j w=qz 3 7C 5A ; with f = diagff (q ( )); f (q ( )); f z (q z ); : : : ; f zl (q zl )g, l = (; : : : ; ) R l, z = E[ ( t ( )) (z t q z )], z = E[ ( t k ( )) (z t q z )], z = E[ (z t q z ) (z t q z ) ] and G z (v; w) = [G z (v; w ); : : : ; G zl (v; w l )] for (v; w) RR l with w = (w ; : : : ; w l ). We now state the asymtotic roerties of the artial cross-quantilogram and the related inference methods. Theorem 6 (a) Suose that Assumtion A-A4 and A6 hold. Then, T (^jz (k) jz (k))! d N(; jz); for each A and for a nite ositive integer k, where jz = k;z k;z() k;z. 6

20 (b) Suose that Assumtion A-A6 hold. Then, su P ^ jz(k) s P ^ jz (k) s! ; sr for each A and for a nite ositive integer k. (c) Suose that Assumtion A-A4 and A6 hold. Then, under the null hyothesis that jz (k) =, we have T ^jz (k) ^A = jz! d B() n R o = ; fb()! rb(r)g dr for each A and for a nite ositive integer k. We can show that the artial cross-quantilogram has non-trivial local ower against T -local alternatives, alying the similar arguments used in Theorem 3 and Theorem 5, and thus we omit the details. 5 Monte Carlo Simulation We investigate the nite samle erformance of our test statistics in the following two data generating rocesses: DGP: (x t ; x t ) iid N (; I ) where I is a identity matrix. DGP: x t x t C A = t C B " t where (" t ; " t ) iid N (; I ) and t = : + :x t + : t + x t : " t C A Under DGP, there is no redictability from x t k to x t for all quantiles: 4 Under DGP, (x t ) is de ned as the GARCH-X rocess, where its conditional variance is the 4 Even when we allow for moderate deendence in the DGP such as AR rocesses, we obtain similar results and the results are available uon request. 7

21 GARCH(,) rocess with an exogenous covariate. The GARCH-X rocess is commonly used for modeling volatility of economic or nancial time series in the literature (see Han (3) and references therein). Under DGP, there is no redictability for the median because x t is symmetrically distributed but, for other quantiles, there exists redictability from x t k to x t for k through t: 5. Results Based on the Bootstra Procedure We rst examine the nite-samle erformance of the Box-Ljung test statistics based on the stationary bootstra rocedure. To save sace, only the results for the case where = are reorted here because the results for cases where 6= are similar. The Box-Ljung test statistics ^Q are based on ^ (k) for = :5; :; :; :3; :5; :7; :8; :9,:95 and k = ; ; : : : ; 5. Tables and reort the emirical rejection frequency of the Box-Ljung test statistics based on bootstra critical values at the 5% level. We chose samle sizes T = 5; ; and ; : We comuted ns = ; relications. The bootstra critical values are based on ; bootstraed relicates. The tuning arameter = is chosen by adating the rule suggested by Politis and White (4) (and later corrected in Patton et al. (9)). Since it is for univariate data, we aly it searately to each time series and de ne as the average value. In general, our simulation results in Tables and show that the test has relatively good size and ower roerties. Table reorts the simulation results for the DGP, which show the size erformance. The rejection frequencies are mostly close to :5: Excet for = :5 and :95 cases, the rejection frequency is uniformly within simulation error of the nominal value. For all quantiles including = :5 and :95 cases, the rejection frequency becomes closer to :5 as samle size increases. 8

22 Table. (size) Emirical rejection frequency of the Box-Ljung test statistic (DGP and the nominal level: 5%) Quantiles ( = ) T Notes: The rst and second columns reort the samle size T and the number of lags for the Box-Ljung test statistics ^Q. The rest of columns show emirical rejection frequencies based on bootstra critical values at the 5% signi cance level. The tuning arameter is chosen as exlained above. ^Q Table reorts the simulation results for the DGP. As exlained above, the null hyothesis of no redictability is satis ed at the median under the DGP and the rejection frequency is close to :5 at the median and moreover ower is lower when quantile is closer to median in Table. However, for other quantiles, the rejection frequency aroaches one in the largest samle size. 5. Results for the Self-Normalized Statistics We also examine the erformance of the self-normalized version of ^Q under the same setu as above. We let the trimming value! be.. We also considered.3 and.5 for! and the results are similar to those for! = :: We rovide the results 9

23 Table. (ower) Emirical rejection frequency of the Box-Ljung test statistic (DGP: GARCH-X rocess) Quantiles ( = ) T Notes: Same as Table. ^Q for! = :5 in the sulementary material. The number of reetitions is 3;. The emirical sizes of the test are reorted in Table 3, where the underlying rocess is DGP and the nominal size is 5%. The rejection frequencies for quantiles between. and.9 are close to 5% nominal level or a little conservative regardless of samle sizes. At.5 and.95 quantiles, the result shows size distortions for T = 5, but this becomes moderate for T = and emirical sizes become close to the nominal size for T =. Using the GARCH-X rocess as in DGP, we obtain emirical owers and resent the result in Table 4. With a one-eriod lag ( = ), the self-normalized quantilogram at ; f:; :; :8; :9g rejects the null by about %, % and % for samle sizes 5,, and,, resectively. In general, the rejection frequencies increase as the samle size increases, decline as the maximum number of lags increases, and is not sensitive to the choice of the trimming value.

24 Our results show that the self-normalized statistics have lower ower in nite samles comared with the stationary bootstra. 6 Emirical Studies 6. Stock Return Predictability We aly the cross-quantilogram to detect directional redictability from an economic state variable to stock returns. The issue of stock return redictability has been very imortant and extensively investigated in the literature, see Lettau and Ludvigson () for an extensive review. A large literature has considered redictability of the mean of stock return. The tyical mean return forecast examines whether the mean Table 3. (size) Emirical Rejection Frequencies of the Self-Normalized Statistics (DGP and the nominal level: 5%) Quantiles ( = ) T Notes: The rst and second columns reort the samle size T and the number of lags for the test statistics ^Q. The rest of columns show emirical rejection frequencies given simulated critical values at 5% signi cance level. The trimming value! is set to be..

25 Table 4. (ower) Emirical Rejection Frequencies of the Self-Normalized Statistics (DGP: GARCH-X rocess) Quantiles ( = ) T Notes: Same as Table 3. of an economic state variable is helful in redicting the mean of stock return (meanto-mean relationshi). Recently, Cenesizoglu and Timmermann (8) considered whether the mean of an economic state variable is helful in redicting di erent quantiles of stock returns reresenting left tail, right tail or shoulders of the return distribution. They show that while the OLS estimates of regression coe cients for the mean return forecast are mostly insigni cant even at the % level (see Table ), the mean of some redictors is helful in redicting certain quantiles of stock returns (see Table 3). Seci cally, they considered the following linear quantile regression q (r t+ jf t ) = ; + ; x t + ; q (r t jf t ) + 3; jr t j ; where r t and x t are stock returns and an economic state variable, resectively, and q (r t+ jf t ) is the conditional quantile of stock return given the information F t at

26 time t. They include q (r t jf t ) and jr t j in the model following Engle and Manganelli (4). Maynard et al. () investigate inferences in such linear quantile regressions when a redictive regressor has a near-unit root. Our cross-quantilogram adds one more dimension to analyze redictability comared with the linear quantile regression, and so it shows a more comlete relationshi between a redictor and stock returns. Moreover, as ointed out by Linton and Whang (7), we can consider very large lags in the framework of the quantilogram. For examle, in our alication, We consider lags k = 6, i.e. 5 years. We use the data set of monthly stock returns and the redictor variables reviously analyzed in Goyal and Welch (8). Stock returns are measured by the S&P 5 index and include dividends. A Treasury-bill rate is subtracted from stock returns to give excess returns. There are a total of 6 variables considered as redictors in Goyal and Welch (8). However, most time series used as redictors are highly ersistent. For examle, both the dividend-rice ratio and the earnings-rice ratio have autoregressive coe cients being estimated to be :99; and unit root tests show that there exists a near-unit root in each time series. This motivated the work by Maynard et al. (). Since our aer establishes asymtotic and bootstra theories for the cross-quantilogram for strictly stationary time series, we do not consider these variables. Extension of the quantilogram for such ersistent time series will be interesting and useful. We leave it as future work. In our alication, we consider stock variance as the redictor because the autoregressive coe cient for stock variance is estimated to be :59 and the unit root hyothesis is clearly rejected. The stock variance is a volatility estimate based on daily squared returns and is treated as an estimate of equity risk in the literature. The risk-return relationshi is a very imortant issue in the literature. See Lettau and Ludvigson () for an extensive review. The cross-quantilogram can rovide 3

27 quantile-to-quantile relationshi from risk to return, which cannot be examined using existing methods. The samle eriod is from Feb. 885 to Dec. 5 with samle size,45. The samle mean and median of stock return are :8 and :3; resectively: In Figures (a)-3(b), we rovide the cross-quantilogram ^ (k) and the ortmanteau tests ^Q (we use the Box-Ljung versions throughout) to detect directional redictability from stock variance, reresenting risk, to stock return. For the quantiles of stock return q ( ), we consider a wide range for = :5; :; :; :3; :5; :7; :8; :9 and :95. For the quantiles of stock variance q ( ), we consider = :, :5 and :9: The tuning arameter is chosen as exlained in Section 5.. In each grah, we show the 95% bootstra con dence intervals for no redictability based on ; bootstraed relicates. Figures (a) and (b) are for the case when the stock variance is in the lower quantile, i.e. q ( ) for = :. The cross-quantilogram ^ (k) for = :5 is negative and signi cant for some lags. It means that when risk is very low, it is less likely to have a very large negative loss. The cross-quantilograms ^ (k) for = :8, :9 and :95 are ositive and signi cant for some lags. For examle, for = :95; the cross-quantilogram is signi cantly ositive for about a half year. This means that when risk is very low, it is less likely to have a very large ositive gain for the next half year. Figure (b) shows that the Box-Ljung test statistics are signi cant for quantiles = :5; :8, :9 and :95: Figures (a) and (b) are for the case when the stock variance is in the median, i.e. q ( ) for = :5. If the distributions of stock returns and the redictor are symmetric, the median return forecast will be equal to the mean return forecast. In the case of = :5 and = :5, the cross-quantilograms are insigni cant for almost all lags. This result corresonds to the mean return forecast result given in Table 4

28 Portmantau Portmantau Portmantau. α =.5. α =.. α = α = α = α = α = α = α = Figure (a). The samle cross quantilogram ^ (k) for = : to detect directional redictability from stock variance to stock return. Bar grahs describe samle cross-quantilograms and lines are the 95% bootstra con dence intervals. α =.5 α =. α =. 4 6 α = α = α = α = α = α = ^Q Figure (b). Box-Ljung test statistic for each lag and quantile using ^ (k) with = :. The dashed lines are the 95% bootstra con dence intervals. 5

29 Portmantau Portmantau Portmantau. α =.5. α =.. α = α = α = α = α = α = α = Figure (a). The samle cross quantilogram ^ (k) for = :5 to detect directional redictability from stock variance to stock return. Same as Figure (a).. α =.5 4 α =. 4 α = α = α = α = α = α = α = Figure (b). Box-Ljung test statistic = :5. Same as Figure (b). ^Q for each lag and quantile using ^ (k) with 6

30 Portmantau Portmantau Portmantau. α =.5. α =.. α = α = α = α = α = α = α = Figure 3(a). The samle cross quantilogram ^ (k) for = :9 to detect directional redictability from stock variance to stock return. Same as Figure (a). α =.5 4 α =. 4 α = α = α = α = α = α = α = Figure 3(b). Box-Ljung test statistic = :9. Same as Figure (b). ^Q for each lag and quantile using ^ (k) with 7

31 by Cenesizoglu and Timmermann (8), where the OLS estimate for the mean return forecast is also insigni cant. However, if we consider other quantiles of stock returns, the stock variance aears to have some redictability for stock returns. It should be noted that the results of the linear quantile regression given in Table 3 by Cenesizoglu and Timmermann (8) corresond to our results for the rst lag in Figure (a): For = :5, : and :; the cross-quantilograms are negative and signi cant for some lags. This imlies that when risk is lower than the median, it is less likely to have large negative losses. For = :8, :9 and :95; the cross-quantilograms are ositive and signi cant for some lags. For examle, for = :95; the crossquantilogram is signi cantly ositive for about one and a half years. This means that when risk is lower than median, it is less likely to have a very large ositive gain for the next one and a half years. Figure (b) shows that the Box-Ljung test statistics are signi cant for both tails and both shoulders of the stock return distribution. Figures 3(a) and 3(b) are for the case when stock variance is in the higher quantile, i.e. q ( ) for = :9. Comared to the revious case of = :5; the cross quantilograms have similar trends but larger absolute values. For = :5; the cross-quantilograms are negative and signi cant for about two years. This imlies that when risk is higher than the :9 quantile, there is an increased likelihood of having very large negative losses u to two years. For = :95; the cross quantilograms are ositive and signi cant for about three years. This imlies that when risk is very high (higher than the :9 quantile), there is an increased likelihood of having a very large ositive gain for the next three years. Figure 3(b) shows that the Box-Ljung test statistics are signi cant for some lags in most quantiles of stock returns excet for = :5. The results in Figures (a)-3(b) show that stock variance is helful in redicting stock return and detailed features deend on each quantile of stock variance and stock 8

32 return. Stock variance is helful in redicting articularly both tails and both shoulders of stock return distribution. When stock variance is in higher quantile, in general the absolute value of the cross-quantilogram is higher and the cross-quantilogram is signi cantly di erent from zero for larger lags. Our results exhibit a more comlete quantile-to-quantile relationshi between risk and return and additionally show how the relationshi changes for di erent lags. 6. Systemic Risk The Great Recession of 7-9 has motivated researchers to better understand systemic risk the risk that the intermediation caacity of the entire nancial system can be imaired, with otentially adverse consequences for the suly of credit to the real economy. Bisias et al. () summarized that the current aroaches to measure systemic risk can be categorized as ) tail measures, ) contingent claims analysis, 3) network models and 4) dynamic stochastic macroeconomic models. Among these, the rst aroach is related to the cross-quantilogram and it measures co-deendence in the tails of equity returns of an individual nancial institution and the nancial system. Prominent examles include the work of Adrian and Brunnermeier (), Brownlees and Engle () and White et al. (). Since the cross-quantilogram measures quantile deendence between time series, we aly it to measure systemic risk. We use the daily CRSP market value weighted index return as the market index return as in Brownlees and Engle (). We consider returns on JP Morgan Chase (JPM), Goldman Sachs (GS) and AIG as individual nancial institutions. JPM belongs to the Deositories grou and Brownlees and Engle () ut GS with Broker-Dealers grou. AIG belongs to the Insurance grou. The samle eriod is 9

33 from Jan. to 3 Dec. with samle size ; 767. We investigate the cross-quantilogram ^ (k) between an individual institution s stock return and the market index return for k = 6 and = = :5. In each grah, we show the 95% bootstra con dence intervals for no quantile deendence based on ; bootstraed relicates. Figure 4. (a)-(a3) shows the cross-quantilogram from an individual institution to the market. The cross-quantilograms are ositive and signi cant for large lags. The cross-quantilogram from JPM to the market reaches its eak (:39) at k = and declines steadily afterwards. This means that it takes about two weeks for the systemic risk from JPM to reach its eak once JPM is in distress. From GS to the market, it shows a similar attern by reaching its eak (:47) at k = : However, from AIG to the market, it exhibits a di erent trend by reaching its eak (:36) at k = 5: When AIG is in distress, the systemic risk from AIG takes a longer time (more than two months) to reach its eak. Figure 4. (b)-(b3) shows the cross-quantilograms from the market to an individual institution. The cross-quantilogram for this case is a measure of an individual institution s exosure to system wide distress, and is similar to the stress tests erformed by individual institutions. From the market to JPM, it reaches its eak (:53) at k = : From the market to GS, it reaches its eak (:6) at k = and declines afterwards. Meanwhile, from the market to AIG, it reaches its eak (:33) at k = 7: When the market is in distress, each institution is in uenced by its imact in a di erent way. As shown in Figure 4, the cross-quantilogram is a measure for either an institution s systemic risk or an institution s exosure to system wide distress. Comared to existing methods, one imortant feature of the cross-quantilogram is that it rovides in a simle manner how such a measure changes as the lag k increases. For 3

34 Partial Partial Partial Partial Partial Partial (a) JPM to Market (b) Market to JPM (a) GS to Market (b) Market to GS (a3) AIG to Market (b3) Market to AIG Figure 4. The samle cross quantilogram ^ (k). Bar grahs describe samle cross-quantilograms and dotted lines are the 95% bootstra con dence intervals. examle, White et al. () adots an additional imulse resonse function within the multivariate and multi-quantile framework to consider tail deendence for a large k. Moreover, another feature of the cross-quantilogram is that it does not require any modeling. For examle, the aroach by Brownlees and Engle () is based on 3

35 Partial Partial Partial Partial Partial Partial (a) JPM to Market (b) Market to JPM (a) GS to Market (b) Market to GS (a3) AIG to Market (b3) Market to AIG Figure 5. The samle artial cross-quantilogram ^ jz (k). Bar grahs describe samle artial quantilograms and dotted lines are the 95% bootstra con dence intervals. the standard multivariate GARCH model and it requires the modeling of the entire multivariate distribution. Next, we aly the artial cross-quantilogram to examine the systemic risk after controlling for an economic state variable. Following Adrian and Brunnermeier () 3

36 and Bedljkovic (), we adot the VIX index as the economic state variable. Since the VIX index itself is highly ersistent and can be modeled as an integrated rocess, we instead use the VIX index change, the rst di erence of the VIX index level, as the state variable. For the quantile of the state variable, i.e. in (), we let = :95 because a low quantile of a stock return is generally associated with a raid increase of the VIX index. Figure 5 shows that the artial cross-quantilograms are still signi cant in some cases even if their values are generally smaller than the values of the cross-quantilograms in Figure 4. This indicates that there still remains systemic risk from an individual institution after controlling for an economic state variable. These signi cant artial cross-quantilograms will be of interest for the management of the systemic risk of an individual nancial institution. Sulementary Materials Section A: includes all roofs of theorems. Section B: contains additional Figures. Section C: rovides comuter codes and datasets. References Adrian, T. and M.K. Brunnermeier () CoVaR, Tech. re., Federal Reserve Bank of New York, Sta Reorts Bisias, D., M. Flood, A.W. Lo and S. Valavanis () A survey of systemic risk analytics, working aer, O ce of Financial Research. Brownlees, C. and R.F. Engle () Volatility, correlation and tails for systemic risk management, Tech. re., SSRN. Bedljkovic, M. () Nonarametric test of conditional quantile indeendence with an alicationi to banks systemic risk, working aer. 33