THE DATA-BASED CHOICE OF BANDWIDTH FOR KERNEL QUANTILE ESTIMATOR OF VAR

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1 Iteratioal Joural of Iovative Maagemet, Iformatio & Productio ISME Iteratioal c2013 ISSN Volume 4, Number 1, Jue 2013 PP THE DATA-BASED CHOICE OF BANDWIDTH FOR KERNEL QUANTILE ESTIMATOR OF VAR XIN YANG 1, SHANCHAO YANG 2 College of Iteratioal Educatio 1 GuagXi Uiversity of Fiace ad Ecoomics Naig, , P. R. Chia College of Mathematical Sciece 2 Guagxi Normal Uiversity Guili, , P. R. Chia ABSTRACT. Value-at-Risk is a importat risk measure, has bee wildly applied i market practice ad fiacial risk measuremet. We use the smooth kerel estimator for the quatile proposed by Parze (1979) as a VaR estimator ad propose a data-based choice method of optimal badwidth via ormal referece distributio, which is easy to compute. The simulatios show that the choice method of badwidth is effective, ad the smooth kerel estimator has better performace tha the sample quatile estimator. Keywords: VaR; Kerel Quatile Estimator; Choice of Badwidth 1. Itroductio. Suppose that { Xi, i 1} is a sequece of idepedet ad idetically distributed radom variables. Deote the distributio fuctio of X 1 by F( x ) ad the 1 p-quatile of the distributio fuctio by Q( p) = F ( p) = i { fx: F( x) p} for 0< p < 1. Let the empirical distributio fuctio of a sample X1, X2,, X be 1 F( x) = i= 1I( Xi x) ad the empirical quatile fuctio be 1 Q( p) = F ( p) = i { fx: F( x) p}. Assume that X(1), X(2),, X( ) are the order statistics of the sample. The Q ( p) = X ([ p] + 1), where [ x ] is the iteger part of x. It is well-kow that Q ( p ) is a sample estimator of the p -quatile Q( p ). The sample quatile is a very importat statistics. Its applicatios are beyod the area of statistics. For example, the sample quatile is used to estimate Value-at-Risk (VaR), which is a risk measure ad has bee used widely i fiacial istitutios for their risk maagemet. Let X be a radom variable describig the retur of a asset or a portfolio. Give a positive value p (0,1), the 1 p level VaR is VaRp = Q( p), which measures the maximum potetial loss of a give asset or portfolio over a prescribed holdig period at a give cofidece level. Early estimators of VaR are based o parametric models for the retur distributio F, for istace, Gaussia or t distributios, oe ca refer to Dowd (2000), Giot ad Lauret (2003), Lauret ad Peters (2002), ad the refereces therei. The advatage of parametric methods is that it is easy to calculate, but actual distributio of retur serials is t always

2 XIN YANG AND SHANCHAO YANG 18 kow. No-parametric methods, without assumig statistical distributio, ca deal with asymmetry ad fat-tail problems of retur serials effectively. This has caused wide attetio amog the world of fiacial maagemet. Dowd (2001) used the sample quatile X ([ p]) to estimate VaR. Of course, X ([ p]) ca be also used as a VaR estimator because it has the same asymptotical properties as X ([ p] + 1). However, Berkowitz ad O Brie (2002) ad Iui et al.(2005) reported that X ([ p]) teds to be coservative. I fact, the estimator X ([ p] + 1) teds to be less tha the real value of VaR while X ([ p]) teds to be more tha the value of VaR. For other oparametric methods, Gourieroux et al.(2000) first itroduced VaR o-parametric kerel estimator. Che ad Tag (2005) proposed VaR o-parametric estimatio based o kerel quatile estimator. Wei et al.(2010) used the kerel quatile estimator proposed by Parze(1979) as a VaR estimator ad studied its Bahadur represetatio, asymptotical mea square error uder strogly mixig assumptio. I this paper, we will cosider the estimator. Let us to recall Parze s smooth kerel estimator for the quatile Q( p) as follows where h is a smoothig badwidth ad K() is a kerel fuctio. The estimator is a liear combiatio of order statistics with kerel weights. Falk (1984) ivestigated the asymptotical mea square error of the estimator. Yag(1985) established a Bahadur represetatio i seses of probability ad a mea squared covergecy rate. Later, Sheather ad Marro (1990) improved Falk s asymptotical mea square error, gave a asymptotical optimal badwidth, ad studied the data-based choice method of the optimal badwidth by usig the estimators of Q ' ( p ) ad Q '' ( p ) as follows ad where K () * is a kerel fuctio of order m ad symmetric about 0, ad a, b are also two badwidths. At the same time, Sheather ad Marro (1990) compared the performace of the kerel quatile estimator with the estimators of Harrel ad Davis (1982) ad Kaigh ad Lachebruch (1982) by usig the samples of size 50 ad 100 from the double-expoetial, expoetial, logormal ad ormal distributios. Uder radom cesorship or radomly trucated data, Xiag(1995) ad Zhou (2006) established a Bahadur represetatio of T ( p ), the latter also gave the strog cosistece, asymptotic ormality of the estimator. For α -mixig radom variables, Wei et al.(2010) showed Bahadur represetatio i seses of almost surely covergece, strog cosistece ad mea square error. By simulatios, they also reported that the mea absolute biases of

3 THE DATA-BASED CHOICE OF BANDWIDTH 19 the estimator are much less tha those of sample quatile for samples of size 100, 200, 500 ad 1000 from ormal distributio ad studet t distributio, wheever samples are idepedet or depedet, ad from light tail distributio or fat tail distributio. Recetly, Ajami et al.(2011) established the Bahadur-type represetatio of the kerel smooth estimator uder strog mixig ad cesored data, ad showed that this estimator is strogly cosistet from the Bahadur represetatio. These studies metioed above idicate that the kerel estimator possesses may ice properties ad its efficiecy is better tha that of the sample quatile. Therefore, it is a reasoable estimator for the quatile. It is well kow that badwidth selectio for kerel estimator is a hard work. For the estimator T ( p ), the data-based choice method of the asymptotical optimal badwidth by usig the estimators (2) ad (3) was studied i Sheather ad Marro (1990). However, ' because the estimators Qˆ ( p ) '' ad Qˆ ( p ) are also kerel estimatio, oe further faces ew work to choose the badwidths. To simplify the procedure, i this paper, we propose a data-based choice method of badwidth via ormal referece distributio. Our simulatio results report that the choice method of badwidth has good performace. 2. Mea Square Error ad Optimal Badwidth. Here let us recall the results about asymptotical mea square error ad asymptotical optimal badwidth i Sheather ad Marro (1990). Theorem 1: (Sheather ad Marro, 1990, Theorem 1) Suppose that Q" is cotiuous i a eighborhood of p ad that K is a desity with compactly supported set [ cc, ], x symmetric about 0. Let K( x) = K() t dt. The for all fixed p (0,1), apart from p = 0.5 whe F is symmetric, ad for p = 0.5 whe F is symmetric, Theorem 2: (Sheather ad Marro, 1990, Corollary 1) Supposed that the coditios give i Theorem 1 hold. The for all fixed p (0,1), apart from p = 0.5 whe F is symmetric, the asymptotically optimal badwidth is give by Ad whe h = hopt,

4 XIN YANG AND SHANCHAO YANG 20 Supposed that the distributio fuctio F( x ) has a desity fuctio f( x ). The Hece Because F is symmetric, so f( x ) has a maximal value at x = 0, which implies that f ' ( Q (0.5)) = 0. From (9), Q ' ( p) Q " ( p ) will be ifiity at p = 0.5. Therefore, these theorems require that p is apart from Data-based Choice Method via Normal Referece Distributio. From Theorem 2, we see that the asymptotical optimal badwidth h opt depeds o the first ad secod derivatives of the quatile fuctio, ad also o the kerel fuctio. Here, we will use ormal distributio as the populatio distributio to estimate the the quatile fuctio Q( p ), while quadratic fuctio as kerel fuctio Quadratic Kerel Fuctio ad Weight Fuctio. Let the quadratic kerel fuctio x x ad 2 3 The K( x) = K( u) du = 3 (1 u ) du = 3 ( x 1 x + 2 ) i Deote the weight ω, ( p i ) = K (( p t ) h ) dt. Let a 1, i = max{( i 1), p h} ad a2, i = mi{ i, p+ h}. The ( i 1) It is clear that ω i, ( p) = 0 whe a1, i a2, i. Ad whe a1, i < a2, i we have

5 THE DATA-BASED CHOICE OF BANDWIDTH 21 where gx ( ) = hx+ ( p x) Optimal Badwidth Based o Normal Distributio. We use ormal distributio as a referece distributio of the populatio distributio. That is to assume that 2 F( x) N ( µσ, ). I the case, ' ( ) µ x f x = ( ) 2 f x. From (9), σ Thus, the optimal badwidth Therefore, we have the estimator of the optimal badwidth where X ad S are sample mea ad sample stadard deviatio, respectively, ad Q ( p ) is the quatile of 2 N( X, S ). 4. Simulatios. I this sectio, some umerical simulatios are carried out to evaluate the performace of the kerel quatile estimator (1) by usig the data-based badwidth via ormal referece distributio (16), ad to compare it with the sample quatile estimator. We kow that ormal distributio is light tail distributio, while studet t distributio is fat tail distributio. So we cosider the samples that are from ormal distributio with stadard deviatios 1 ad 3, studet t distributio with degrees of freedom 3 ad 5. The sample sizes are 200, 500, 800 ad 1000, ad repeated 1000 times for each oe. The probability level p is from 0.01 to 0.4 by step 0.01, i.e. pi = 0.01 i for i = 1, 2,, m, where m = 40.

6 XIN YANG AND SHANCHAO YANG 22 FIGURE 1. The estimatio of Quatile for stadard ormal populatio. For stadard ormal populatio, Figure 1 reports the kerel quatile estimator T ( p ) (dashed lie), the sample quatile estimator Q ( p ) (logdash lie) ad the quatile Q( p ) (solid lie) for four cases of the sample sizes 200, 500, 800 ad The figures illustrate that the estimator values are ear to the real values, ad they are gettig better as sample size grows. Moreover, the kerel quatile estimator is more smooth ad closer to the real value tha the sample quatile estimator. For other populatios, there are similar figures. Due to the space, we do t show the figures. Now we observe the relatively absolute biases of the estimators. Let for i = 1, 2,, m, ad RQ m 1 m RT i= 1 i =, which is the mea of the relatively absolute biases over p for the kerel quatile estimator. Aalogously let for i = 1, 2,, m, ad 1 m RQ m RQ i= 1 i =. It is the mea of the relatively absolute biases over p for the sample quatile estimator. The the meas of the relatively absolute biases for each populatio ad each sample size are listed i Table 1. These show that the meas of the relatively absolute biases are less tha 5% for most cases, whether light or fat tail distributio. Ad the meas of the

7 THE DATA-BASED CHOICE OF BANDWIDTH 23 relatively absolute biases of the kerel quatile estimator are less tha those of the sample quatile estimator. These results show that the data-based badwidth via ormal referece distributio (16) is effective. TABLE 1: Mea of the relatively absolute biases (%) 5. Coclusio. Value-at-Risk, a very importat risk measure, has bee wildly applied i market practice ad fiacial risk measuremet. Ad estimatig VaR has received a cosiderable amout of attetio i literature. I this paper, we cosider the smooth kerel estimator for the quatile proposed by Parze (1979) as a VaR estimator. I view of the complexity to choose badwidth for the kerel estimator, we propose a data-based choice method of optimal badwidth via ormal referece distributio. The choice method is easy to compute ad the simulatios show that it is effective. Ackowledgmet. This project is supported by the Natioal Sciece Foudatio of Chia( ), the Social Sciece Foudatio of Chia(10CTJ004) ad the Natioal Sciece Foudatio of Guaxi(2011GXNSFA018133). REFERENCES [1] B. M. Falk (1984), Relative deficiecy of kerel type estimators of quatile, The Aals of Statistics, vol.12, pp [2] C. Gourieroux, O. Scaillet ad J. P. Lauret (2000), Sesitivity aalysis of values at risk, Joural of Empirical Fiace, vol.7, pp [3] E. Parze (1979), Noparametric statistical data modelig, America Statistical Associatio, vol.74, pp [4] F. E. Harrell ad C. E. Davis (1982), A ew distributio-free quatile estimator, Biometrika, vol.69, pp

8 XIN YANG AND SHANCHAO YANG 24 [5] J. Berkowitz ad J. O Brie (2002), How accurate are value-at-risk models at commercial baks, J. Fiace, vol.57, pp [6] K. Dowd (2000), Assessig VaR accuracy, Derivatives Quarterly, vol.6, o.3, pp [7] K. Dowd (2001), Estimatig VaR with order statistics, Joural of Derivatives, vol.8, o.3, pp [8] K. Iui, M. Kijima ad A. Kitao (2005), VaR is subject to a sigificat positive bias, Statistics ad Probability Letters, vol.72, pp [9] M. Ajami, V. Fakoor ad S. Jomhoori (2011), The Bahadur represetatio for kerel-type estimator of the quatile fuctio uder strog mixig ad cesored data, Statistics ad Probability Letters, vol.81, pp [10] P. Giot ad S. Lauret (2003), Value-at-risk for log ad short tradig positios, Joural of Applied Ecoometrics, vol.18, pp [11] S. Lauret ad J. P. Peters (2002), GARCH 2.2: A Ox package for estimatig ad forecastig various ARCH models, Joural of Ecoomic Surveys, vol.16, pp [12] S. Sheather ad J. Marro (1990), Kerel quatile estimators, Joural of the America Statistical Associatio, vol.85, pp [13] S. S. Yag (1985), A smooth oparametric estimator of a quatile fuctio, Joural of America Statistical Associatio, vol.80, pp [14] S. X. Che ad C. Y. Tag (2005), Noparametric iferece of value-at risk for depedet fiacial returs, Joural of Fiacial Ecoometrics, vol.3, pp [15] W. D. Kaigh ad P. A. Lachebruch (1982), A geeralized quatile estimator, Commuicatios i Statistics, Part A-Theory ad Methods, vol.11, pp [16] X. J. Xiag (1995), Bahadur represetatio of the kerel quatlie estimator uder radom cesorship, Joural of Multivariate Aalysis, vol.54, pp [17] X. Wei, S. Yag, K. Yu, X. Yag ad G. Xig (2010), Bahadur represetatio of liear kerel quatile estimator of VaR uder α-mixig assumptios, Joural of Statistical Plaig ad Iferece, vol.140, pp [18] Y. Zhou (2006), A kerel-type estimator of quatile fuctio uder radomly trucated data, Acta Mathematica Scietia, vol.26, pp