Problems i the Applicatio of Jump Detectio Tests to Stock Price Data Michael William Schwert Professor George Tauche, Faculty Advisor Hoors Thesis submitted i partial fulfillmet of the requiremets for Graduatio with Distictio i Ecoomics i Triity College of Duke Uiversity. Duke Uiversity Durham, North Carolia 008
Ackowledgemets I would like to thak George Tauche for leadig the Hoors Fiace Semiar ad for his advice throughout the developmet of this paper. I would also like to thak the members of the Hoors Fiace Semiar for their commets ad feedback o my work. Additioally, I am grateful to Tim Bollerslev, Bjor Eraker, ad G. William Schwert for their helpful suggestios.
Abstract This paper applies several jump detectio tests to itraday stock price data sampled at various frequecies. It fids that the choice of samplig frequecy has a effect o both the amout of jumps detected by these tests, as well as the timig of those jumps. Furthermore, although these tests are desiged to idetify the same pheomeo, they fid differet amouts ad timig of jumps whe performed o the same data. These results suggest that these jump detectio tests are probably idetifyig differet types of jump behavior i stock price data, so they are ot really substitutes for oe aother. 3
1. Itroductio I recet years there has bee a great deal of iterest i studyig jumps i asset price movemets. Reasos why it is importat to kow whe ad how frequetly jumps occur iclude risk maagemet ad the pricig ad hedgig of derivative cotracts. Ivestors would beefit greatly from kowig the properties of jumps, sice large istataeous drops i asset prices result i large istataeous losses. The effect of jumps o derivative pricig is equally sigificat, especially cosiderig the importat role derivatives play i moder fiacial markets. Whe asset price movemets are cotiuous, ivestors ca perfectly hedge derivative cotracts such as optios, but whe jumps occur, they cause a chage i the derivative price that is o-liear to the chage i the price of the uderlyig asset. Thus, jumps itroduce a uhedgeable risk to the holders of derivative cotracts. The ability to idetify realized jumps i the fiacial markets could provide helpful iformatio such as how frequetly jumps occur, how large the jumps are, ad whether they ted to occur i clusters. With this goal i mid, several authors have developed tests to determie whether or ot a asset price movemet is a statistically sigificat jump. These tests take advatage of the high-frequecy itraday price data available today through electroic sources. Bardorff-Nielse ad Shephard (004, 006) use the differece betwee a estimate of variace ad a jump-robust measure of variace to detect jumps over the course of a day. Approachig the problem differetly, Jiag ad Oome (007) exploit high order sample momets of returs to idetify days that iclude jumps. Aїt-Sahalia ad Jacod (008) also exploit high order sample momets of returs to detect jumps by comparig price data sampled at two differet frequecies. Lee ad Myklad (007) test for jumps at idividual price observatios by scalig returs by a local volatility measure. While these tests employ differet strategies for detectig jumps, they are all desiged to idetify the same pheomeo. Due to the distortio of jump test statistics by market microstructure oise 1 i the highestfrequecy itraday price data, it is ecessary to sample prices at a lower frequecy. There is a rage of samplig frequecies i which the effect of market microstructure oise is miimal ad the iformatioal cotet of high-frequecy data is maitaied. Whe performed o price data sampled at differet frequecies withi this rage, oe might expect that these tests would fid similar results, give that all samples are based o the same origial data. However, this paper 1 Market microstructure oise is explaied i Sectio.. 4
fids that this is ot ecessarily the case, as the choice of samplig frequecy affects both the umber of jumps detected by each test ad the timig of those jumps. These fidigs raise questios about the types of jumps that are beig idetified by these differet tests. The remaider of this paper is orgaized as follows. Sectio itroduces the model of equity returs that forms the basis of the jump tests. Icluded i this sectio is a discussio of market microstructure oise. Sectio 3 describes several of the jump tests developed over the past few years. Sectio 4 discusses the high-frequecy stock price data used i this paper. Sectio 5 discusses the results of a relative compariso of the jump tests described i Sectio 3. Sectio 6 iterprets these results ad puts them i cotext with the jump literature. Fially, Sectio 7 summarizes the paper ad highlights importat coclusios.. Model of Equity Returs Before discussig the jump tests ad results, it is importat to uderstad the theoretical model of asset price movemets ad the cocept of market microstructure oise. Sectio.1 itroduces the stochastic price process o which the jump tests i Sectio 3 are based. Sectio. explais the market microstructure oise that creates the ecessity to sample prices at frequecies lower tha the highest available..1 Stochastic Model of Returs ad Volatility This paper uses a model of asset price movemets that icludes jumps i the price evolutio. The followig stochastic differetial equatio, discussed by Merto (1971), is a cotiuous model of logarithmic price movemets: dp( t) = μ ( t) dt + σ ( t) dw ( t), 0 t t = T, (.1.1) where μ(t)dt represets the time-varyig drift compoet ad σ(t)dw(t) represets the timevaryig volatility compoet of the asset price, ad W(t) is a stadard Browia motio. The iclusio of Browia motio i this model cotributes radomess to price movemets. This is appropriate, give Fama s (1965) fidig that stock prices follow a radom walk. Give recet fidigs i the academic fiace literature that suggest there are discotiuities i asset price movemets, it is appropriate to add a jump compoet to the model of logarithmic prices: 5
dp( t) = μ ( t) dt+ σ ( t) dw ( t) + κ ( t) dq( t), 0 t t = T. (.1.) Merto (1976) itroduced this equatio where the discotiuous compoet of the price process is κ(t)dq(t). Here, q(t) is a coutig process ad κ(t) is the magitude of the jump. It is commoly assumed that q(t) is a Poisso process, so the model produces rare large jumps. While it is fairly simple to estimate the drift ad volatility compoets of the price process, it is more difficult to estimate the jump compoet. Several authors have developed tests to determie whe a price movemet is probably a jump, as opposed to a large cotiuous movemet. It is worth otig that oe ca ever be certai that a price movemet is a jump, as oe ca oly observe prices at discrete itervals ad caot form a cotiuous picture of the price path.. Market Microstructure Noise A commo method for calculatig the theoretical price of a equity is to sum the discouted value of all future divided paymets. The costat divided growth model assumes that the firm s divided payout ratio is costat over time ad that the firm s expected earigs per share grow by a costat yearly growth rate g. The expected value of the divided payout i year i is E ) i i ( Di ) = D0 (1 + g) = E0d(1 + g, (..1) where E 0 is the firm s curret level of aual earigs per share ad d is the firm s divided payout ratio. Based o this expectatio, the firm s curret stock price is E Di D + g P = ( ) 0(1 ) 0 =, (..) i = 0 r r g i where r is the required rate of retur as determied by the Capital Asset Pricig Model. For the remaider of this paper, the price of a stock as defied i (..) is referred to as the fudametal price. The model of market microstructure oise describes the deviatio of the observed stock prices from their fudametal values. The observed price of a equity at time t is * p ( t) = p( t)+ ε, (..3) where p(t) is the logarithm of the fudametal price. The evolutio of p(t) is described by (..). I this equatio, ε t represets a short-term deviatio from the fudametal price called t 6
market microstructure oise. Note that this oise term is added after takig the logarithm of the price, so it is proportioal to the stock price. Microstructure oise results from market frictios, such as bid-ask bouce, which cause the price to deviate briefly from its fudametal value. Bardorff-Nielse ad Shephard (005) suggest usig mid-quote data to mitigate the effect of bid-ask bouce, but this is impossible whe full-quote data are uavailable. This paper uses trasactio data, where trades occur at either the bid price or the ask price 3, so bid-ask bouce creates oise i the high-frequecy price series. Market microstructure oise is oticeable whe estimatig variace ad other statistics ad ca have a sigificat impact o tests for jumps i asset prices. Sectio 5 discusses a potetial solutio to this problem. Note that this paper does ot cosider possible medium-term deviatios of stock prices from their fudametal values, such as those discussed by Fama ad Frech (1988). 3. Jump Tests I recet years a umber of papers have bee published itroducig ew statistical tests to detect discotiuities i the price process. Before examiig ad comparig these tests, it is ecessary to provide some backgroud o the tests ad how they detect jumps i asset price movemets. Sectio 3.1 explais the first jump tests itroduced to the literature, the Bardorff-Nielse Shephard (004, 006) tests. Sectio 3. discusses tests created by Jiag ad Oome (007), who were the first authors to attempt to correct the effects of market microstructure oise. Sectio 3.3 describes Lee ad Myklad s (007) test that checks for a jump at every price observatio, rather tha checkig if a jump occurs withi a particular sample period, such as a day. Fially, Sectio 3.4 explais Aїt-Sahalia ad Jacod s (008) test, which compares calculatios of higher momets at two differet samplig frequecies to detect jumps over the course of a day. With full-quote data, oe ca average the bid ad the ask at each quote to create a mid-quote price series. 3 May trades occur betwee the bid ad the ask because of order matchig. However, these trasactios may still deviate from the fudametal price, cotributig to market microstructure oise. 7
3.1 Bardorff-Nielse Shephard Tests Bardorff-Nielse ad Shephard (004, 006) developed a test that uses high-frequecy price data to determie whether there is a jump over the course of a day. Their test compares two measures of variace: Realized Variace, which coverges to the itegrated variace plus a jump compoet as the time betwee observatios approaches zero; ad Bipower Variatio, which coverges to the itegrated variace as the time betwee observatios approaches zero, ad is robust to jumps i the price path, a importat fact for this applicatio. The itegrated variace of a price process is the itegral of the square of the σ(t) term i (..), take over the course of a day. Sice prices caot be observed cotiuously, oe caot calculate itegrated variace exactly, ad must estimate it istead. For the rest of this paper, P(t i ) is the stock price at time t i, ad P( t i ) r i = log ( ) (3.1.1) P ti 1 is the geometric retur from time t i-1 to time t i. The, Realized Variace ad Bipower Variatio are described by the followig equatios: RV = i= r i T 0 σ ( v) dv + κ ( t ) q( t ), (3.1.) i= 1 i i BV T π = ri ri 1 ) 1 i= 3 0 σ ( v dv, (3.1.3) where is the umber of price observatios i a day. The limits i (3.1.) ad (3.1.3) describe the ifill asymptotics of the Realized Variace ad Bipower Variatio, meaig that as the umber of observatios withi a day approaches ifiity, the iterval at which observatios are made approaches zero ad the period over which observatios are made is held costat. Huag ad Tauche (005) also cosider Relative Jump, a measure that approximates the percetage of total variace attributable to jumps: RV BV RJ =. (3.1.4) RV This statistic approximates the ratio of the sum of squared jumps to the total variace ad is useful because it scales out log-term treds i volatility so oe ca compare the relative cotributio of jumps to the variace of two price series with differet volatilities. 8
To develop a statistical test to determie whether there is a sigificat differece betwee RV ad BV, oe eeds a estimate of itegrated quarticity. Aderse, Bollerslev, ad Diebold (004) recommed usig a jump-robust realized Tri-Power Quarticity, TP 3 4 / 3 4 / 3 4 / 3 4 μ 4 / 3 ri ri 1 ri v dv σ ( ), (3.1.5) i= 4 = T 0 where m μ m = E u, ~ N(0,1) u. (3.1.6) Bardorff-Nielse ad Shephard recommed the realized Quad-Power Quarticity, QP 4 4 μ 1 ri ri 1 ri ri 3 v dv σ ( ). (3.1.7) 3 i= 5 = There are umerous potetial test statistics that use these estimates of itegrated variace ad itegrated quarticity, but Huag ad Tauche fid that these two asymptotically stadard ormal test statistics perform best i simulatios: z TP max z QP max = = RJ π 1 TP + π 5 max 1, BV RJ π 1 QP + π 5 max 1, BV T 0, (3.1.8). (3.1.9) The ull hypothesis that there are o jumps i a day is rejected at a.01 level of sigificace if the z-statistic is greater tha.33. 3. Jiag Oome Swap Variace Tests Followig up o the work of Bardorff-Nielse ad Shephard, Jiag ad Oome (007) proposed their ow tests for the presece of jumps i asset prices. Their Swap Variace 4 tests ca ituitively be thought of as measurig the impact of jumps o high order momets of returs by takig the differece betwee the accumulated arithmetic ad geometric returs over the course of a day. Arithmetic ad geometric returs are defied as follows: 4 They call their test the Swap Variace Test because SwV is directly related to the profit/loss fuctio of a variace swap replicatio strategy usig a log cotract. 9
Arithmetic retur: P( ti ) P( ti 1) R i =, (3..1) P( t ) i 1 P( t Geometric retur: i ) r i = log ( ). (3..) P ti 1 The tests rely o a statistic they call Swap Variace, which is equal to the differece betwee accumulated arithmetic ad geometric returs: ( R i r i ) SwV =. (3..3) The tests use the differece betwee Swap Variace ad Realized Variace to detect jumps. A Taylor series expasio of this differece provides further uderstadig of how these tests exploit higher order returs to check for jumps: i= 4 1 3 1 SwV RV = [ r i ] + [ ri ] +... (3..4) 3 1 i= There will be a drastic differece betwee the value of this sum whe jumps are preset ad the value whe there are o jumps over the course of a day, due to the ifluece of large expoets of the retur. Based o this idea, the authors created statistical tests of whether this value is sigificat. The z-statistics that test the ull hypothesis of o jumps i a day are as follows: i= Differece test: JO ( SwV RV ) Diff =, (3..5) Ω Logarithmic test: JO ( log( SwV ) log( RV )) Log SwV V =, (3..6) Ω SwV Ratio test: V RV JORatio = 1, (3..7) Ω SwV SwV where Ω SwV 3 μ6 μ = 9 3 4 3/ i= 5 r i 3/ 3/ ri 1 r i 3/ 3/ ri 3 (3..8) ad V is the itegrated variace, which oe ca estimate well with the Bipower Variatio calculated with prices sampled at a frequecy isesitive to market microstructure oise. Sectio 5 itroduces techiques that idicate what frequecy would be appropriate to use for this calculatio. These scalig factors are ecessary for the JO statistics to be distributed stadard ormal. 10
As a modificatio to their Swap Variace tests, Jiag ad Oome itroduce a ovel approach to the jump detectio literature by creatig tests that accout for the effect of i.i.d. market microstructure oise. The Microstructure Noise Robust z-statistics are quite similar to the oes above, but the Ω SwV term is estimated with a market microstructure oise correctio. Differece test: MNR Diff SwV RV =, (3..9) Ω * SwV * V =, (3..10) Ω Logarithmic test: MNR ( log( SwV ) log( RV )) Log * SwV Ratio test: MNR V * Ratio = 1 * ΩSwV RV SwV, (3..11) where V + * = V ω, (3..1) with V estimated by the Bipower Variatio of prices sampled at a 10-miute iterval ad Ω 6 4 8 5 = 4 ω + 1ω V+ ω Q + X, (3..13) 3 * SwV where ω is a estimate of the variace of the market microstructure oise, V is a estimate of itegrated variace, Q is a estimate of itegrated quarticity, ad X is a estimate of itegrated sexticity. Thus, to calculate the test statistics, estimates of these variables are ecessary. Roll (1984) itroduced the idea of usig the first order egative serial covariace of returs as a estimate of the amout of bid-ask spread over the course of a day. Bid-ask spread is a major cotributor to market microstructure oise, so Jiag ad Oome deem it appropriate to estimate the variace of oise with the first order egative serial correlatio of returs, 1 S( ti ) S( ti 1) ω = log log. (3..14) 1 i= 3 S( ti 1) S( ti ) They itroduce the Noise-corrected Bipower Variatio as a estimate of itegrated variace: * BV V BV =, (3..15) 1 + ( γ ) 1 + γ π γ where c b( γ ) = (1 + γ ) + γ 1 + + γπκ ( λ) ; (3..16) 1 + 3γ (1 + λ) λ + 1 the Noise-corrected Quad-power Quarticity as a estimate of itegrated quarticity: c b 11
* QP Q QP =, (3..17) 1 + ( γ ) c q where c ( γ ) = 5.46648γ + 4γ ; (3..18) q ad the Noise-corrected Sexticity as a estimate of itegrated sexticity: * SP X SP =, (3..19) 1 + ( γ ) c x where 3 c ( γ ) = 13.968γ + 14.455γ + 6λ, (3..0) x κ ( λ) = ω γ =, (3..1) V γ λ =, (3..) 1 + γ ( Φ( x λ ) 1) φ( x dx x Φ( x λ ) ), (3..3) ad V is estimated by the Bipower Variatio of prices sampled at a 10-miute iterval. 3.3 Lee Myklad Test While most of the jump tests developed over the past several years test for jumps durig a set spa of time, Lee ad Myklad (007) developed a test that checks for a jump at each idividual observatio. Their test calculates the ratio of the retur at each price observatio to a measure of istataeous volatility over a period precedig that observatio. If this ratio exceeds a certai threshold, they idetify the observatio as a jump. The statistic L(i) that tests for a jump at time t i is defied as where ri L( i) =, (3.3.1) ˆ( σ t ) i 1 = rj rj 1 K j= i K + i 1 ˆ σ ( t ). (3.3.) i ad r i is the retur as defied i Sectio 3.1. The istataeous volatility measure is similar to Bipower Variatio as defied i Sectio 3.1, with a differet scalig costat, so this test is robust to the presece of jumps i prior 1
periods. Lee ad Myklad suggest that it is appropriate to choose the widow size K betwee 5 ad 5, where is the umber of observatios i a day. For this paper, I use 350, 50, 150, ad 100 as values of K for the 1-miute, 5-miute, 10-miute, ad 15-miute samplig itervals, respectively. These widow sizes are log eough that istataeous volatility maitais the quality of beig a jump-robust Bipower Variatio, ad short eough that it effectively scales the test statistic for treds i volatility. The asymptotic distributio of the test statistic is as follows: L ( i) S C ξ, (3.3.3) x where P( ξ x) = exp( e ). (3.3.4) 1/ ( log ) logπ + log(log ) The costats C =, (3.3.5) 1/ c c( log ) S = 1 1/ c( log ), (3.3.6) ad c = (3.3.7) π scale the L(i) statistic to be expoetially distributed. The threshold for rejectig the ull hypothesis of o jump for a idividual observatio at a.01 sigificace level is 4.6001. For the purpose of comparig this test to other tests, I wat to use this test to detect jumps over the course of a day. However, usig a.01 level of sigificace to reject the ull hypothesis of o jump at each observatio is ot equivalet to usig a.01 level of sigificace to reject the ull hypothesis of o jumps i a day. Thus, to use this statistic for determiig whether there is a jump over the course of a day, I derive rejectio thresholds for performig the Lee Myklad test o price data sampled at a variety of itervals. Here I assume that each observatio is idepedet ad idetically distributed, so the test at each observatio is idepedet of tests at other observatios. Thus, the threshold for rejectig the ull hypothesis of o jumps over the course of a day is x, where x P( ξ x) = exp( e ) (3.3.8) ad is the umber of observatios i a day. These threshold values are 10.554, 8.944, 8.64, ad 7.858 for the 1-miute, 5-miute, 10-miute, ad 15-miute samplig itervals, respectively. 13
3.4 Aїt-Sahalia Jacod Test Aїt-Sahalia ad Jacod (008) recetly itroduced a test that takes advatage of higher retur momets to detect jumps i asset price processes. Their test takes the sum of absolute returs to a power p (p = 4 here) over the course of a day at two differet samplig itervals, kδ ad Δ, ad uses the ratio of these sums to check if there are jumps i the day. Δ is called the base samplig iterval for the remaider of this paper. The test statistics are p Δ S ti Bˆ ( ) ( p, Δ) = log (3.4.1) i= 1 S( t( i 1) Δ ) ad Bˆ( p, kδ) ASJ ( p, k, Δ) =, (3.4.) Bˆ( p, Δ) where is the total umber of observatios available i a day (385 i the miute-by-miute stock price data). Due to the use of higher order returs, whe p >, B ˆ( p, Δ) elimiates the cotiuous compoet of returs as the umber of observatios approaches ifiity ad emphasizes the jump compoet. Thus, the ASJ statistic approaches 1 whe jumps are preset, because regardless of samplig frequecy, B ˆ( p, Δ) will approach the same umber. Whe there are o jumps i the day, B ˆ( p, Δ) ad Bˆ ( p, kδ) will both approach zero, but at differet rates, so the ASJ statistic will coverge to p k 1. These limits are the basis of this jump test. The ull hypothesis of o jumps is rejected whe ASJ < ξ, where ξ = k p 1 z α Vˆ (3.4.3) with ˆ ΔM ( p, k) Aˆ( p, Δ) V = (3.4.4) Aˆ( p, Δ) ad Aˆ( p, Δ) = 1 p Δ μ p i= 1 S( t log S( t iδ ( i 1) Δ ) ) p 1 S ( tiδ ) ϖ log αδ S ( t( i 1) Δ ). (3.4.5) For the choice of p = 4, 16k(k k 1) M (4, k) =, ad the authors suggest usig α = 0. 05 ad 3 ϖ = 0.48as the other costats, as the test performs well i simulatios uder these coditios. The results i Table come from performig the test usig α = 0. 05 ( z =. 33), but for the α 14
purpose of directly comparig jump days with other authors tests, the test is performed usig α = 0.01 ( z = 3. 09) as the level of sigificace for the results i Tables 3 ad 4. α 4. Data The great wealth of high-frequecy fiacial data available to researchers today through olie resources is resposible for the recet iterest i high-frequecy jump detectio techiques. This paper uses price data for the 10 stocks i the S&P 100 Idex with the largest market capitalizatio at the ed of 007, excludig Apple ad Google, for which there are a limited umber of days available. This sample icludes ExxoMobil (NYSE: XOM), Geeral Electric (NYSE: GE), Microsoft (NASDAQ: MSFT), AT&T (NYSE: T), Procter & Gamble (NYSE: PG), Chevro (NYSE: CVX), Johso & Johso (NYSE: JNJ), Bak of America (NYSE: BAC), Cisco Systems (NASDAQ: CSCO), ad Altria Group (NYSE: MO). These price data were dowloaded from a commercial data vedor, price-data.com, ad are collected every miute from 9:35 AM util 4:00 PM every tradig day from 1997 to 007. The umber of tradig days available for each of these stocks rages from 1566 to 686 days. This differece i legth of sample period is due to certai stocks that did ot trade util after 1997, such as ExxoMobil ad Chevro, which first traded followig a merger i 1999 ad a ticker chage i 001, respectively. Additioal Table 1 i the Appedix cotais iformatio o the sample of equities used for the calculatios i this report. 5. Relative Compariso of Jump Tests This sectio cotais the results of performig the jump tests with prices sampled at differet frequecies ad comparisos of these results. First, Sectio 5.1 motivates the idea of samplig prices at differet frequecies for the jump tests. The, Sectio 5. discusses the results of the jump tests performed at differet samplig frequecies, ad the surprisig icoherece amog those results. Fially, Sectio 5.3 discusses the results of a direct compariso of the jump tests ad the icoherece that also exists amog these tests. 15
5.1 Determiig the Proper Samplig Frequecy As discussed i Sectio., the effect of market microstructure oise must be take ito accout whe calculatig statistics for jump tests. Market microstructure oise appears to have a sigificat impact o the Bardorff-Nielse Shephard Z TP-max ad Z QP-max tests performed o miute-by-miute price data. Across the te stocks used i this paper, these tests detect a average of 49.9% ad 53.9% of all days as havig jumps at a.01 level of sigificace. Give the uderlyig model of large, ifrequet jumps, these percetages seem rather high. It is likely that market microstructure oise is ifluecig the test at this high of a samplig frequecy. Cofroted with the problem of determiig the effect of market microstructure oise ad fidig the optimal samplig frequecy at which to calculate Realized Variace, Aderso, Bollerslev, Diebold ad Labys (1999) developed a graphical tool called the volatility sigature plot. The volatility sigature plot graphs the effect of samplig frequecy o volatility, with samplig iterval 5 o the horizotal axis ad volatility o the vertical axis. The key to the volatility sigature plot is that the variace of a price process is idepedet of the frequecy at which prices are observed, so log as prices adhere to the semimartigale process described i (.1.1). However, as market microstructure oise causes the price to deviate from its fudametal value as described i (..), it distorts the calculatio of variace estimates like Realized Variace ad Bipower Variatio. To better uderstad the Bardorff-Nielse Shephard results, I costruct volatility sigature plots for Realized Variace ad Bipower Variatio. Aderso et al. fid that the highest levels of volatility occur at the highest samplig frequecies for a liquid asset, ad that the lowest levels of volatility occur at the highest samplig frequecies for a illiquid asset. Thus, the volatility sigature plot for the liquid asset is a dowward slopig curve ad the volatility sigature plot for the illiquid asset is a upward slopig curve. As the assets i this paper are stocks that trade extremely frequetly, their volatility sigature plots should be similar to that of the highly liquid asset. The plots for Realized Variace (Figure 1) ad Bipower Variatio (Figure ) are fairly difficult to iterpret i the cotext of the Bardorff-Nielse Shephard tests, but as expected, the value of each variace measure decreases as the samplig iterval icreases, particularly i the 1-miute to 5-miute segmet of the graph. For the Microstructure Noise Robust Jiag Oome 5 e.g.: prices sampled at a 5-miute iterval have observatios at 9:35, 9:40, 9:45,, a total of 77 daily observatios 16
tests, it seems appropriate to sample prices at a 10-miute iterval whe estimatig the itegrated variace, give the relatively high levels of Bipower Variatio at the highest samplig frequecies. To further examie the effect of market microstructure oise o the Bardorff-Nielse Shephard results above, I plot the relatio betwee Relative Jump ad samplig frequecy for the te stocks, employig the idea behid the volatility sigature plot. Iterestigly, this Relative Jump sigature plot (Figure 3) is more revealig tha the volatility sigature plots i Figures 1 ad. I this plot, Relative Jump decreases drastically as the samplig frequecy decreases from the highest levels, stabilizes betwee the 5-miute ad 11-miute samplig itervals, ad begis icreasig at lower frequecies, implyig the emergece of samplig error. Give this iformatio, it seems appropriate to sample prices at frequecies i the flat portio of the plot whe performig the Bardorff-Nielse Shephard tests o this data. I also sample prices at these frequecies whe performig the other jump tests for the sake of comparability. While sigature plots for the other jump test statistics might be istructive, they are ot itegral to the results of this paper ad are omitted. 5. Effects of Samplig Frequecy o Jump Tests As discussed above, oe ca sample high-frequecy stock price data at itervals of 5 to 11 miutes to dimiish the effects of market microstructure oise. These sampled data cotai most of the same iformatio as the origial data, oly without market microstructure oise, so oe could reasoably expect that a jump test performed o the same data sampled at two differet frequecies would detect jumps o the same days. However, this is ot the case, as the choice of samplig frequecy affects both the umber of jumps detected as well as the timig of those jumps for each of the tests described i Sectio 3. To examie these effects, I performed each jump test o the stock price data described i Sectio 4 sampled at 1-miute, 5-miute, 10-miute, ad 15-miute itervals. For the Aїt- Sahalia Jacod test, istead of usig the sample itervals above, I use a base samplig iterval of 1 miute ad k values of, 3, ad 4. I make this distictio because they perform their test at samplig itervals of 5 secods to 30 secods i their paper, so samplig itervals loger tha 1 miute seem iappropriate. The compariso of differet values of k for this test will provide similar isight to the compariso of differet samplig frequecies for the other jump tests. 17
Table 1 presets the results of these tests performed at a.01 level of sigificace. The top four rows of each table cotai the average percetage of jump days out of total days. It is worth otig that each test detects that more tha oe percet of the total days have jumps i them, usig data sampled at all four itervals. Thus, the results preseted here are ot solely based o chace. The bottom six rows cotai the average percetage of commo jump days out of total days betwee each pair of samplig itervals. The results i Table 1 are averaged evely across the te stocks. Sice the umber of days of data available differs amog the stocks i the sample, the results i Additioal Table i the Appedix are weighted by the umber of days available for each stock. However, this distictio has a miimal effect o the results. There are varyig patters i the amouts of jump days detected at differet samplig frequecies for each test. As the samplig iterval icreases from 1 to 15 miutes, there is a oticeable decrease i the umber of jump days detected by both of the Bardorff-Nielse Shephard tests ad the Lee Myklad test. While the umber of jump days detected by the three Jiag Oome tests ad the Microstructure Noise Robust Jiag Oome Differece test icreases with samplig iterval, there is a slight decrease i the umber of days detected by the Microstructure Noise Robust Logarithmic ad Ratio tests. For the Aїt-Sahalia Jacod test, the umber of jump days detected decreases as k icreases ad the base samplig iterval is 1 miute. The Bardorff-Nielse Shephard test detects a extremely high umber of jumps with the 1-miute price data, suggestig that it performs particularly poorly i the presece of market microstructure oise. All of the Microstructure Noise Robust Jiag Oome tests detect a similarly high umber of jumps at all samplig frequecies. It is possible that they would perform better at frequecies higher tha the 1-miute level, where there would be more market microstructure oise. Alteratively, there could be a problem with the implemetatio of the market microstructure oise correctio. While it is iterestig that these tests fid differet umbers of jump days at differet samplig frequecies, the most surprisig result of this study is that the timig of these jump days varies betwee samples. The percetage of jump days out of total days that are commo betwee two samplig frequecies is oly a fractio of the percetage of jump days detected at each samplig frequecy for every test ad every pair of samplig frequecies. Furthermore, there does ot appear to be ay tred where high-frequecy or low-frequecy samples perform 18
better amog each other. This result is also surprisig, as the tests are based o statistics that coverge to their distributios as the umber of observatios approaches ifiity, or as the samplig iterval approaches zero. 5.3 Icoherece Amog Jump Tests Sice these jump tests are all desiged to measure the same pheomeo, extreme movemets i asset prices, oe might also reasoably expect that they would detect the same jumps. Tables ad 3 cotai cotigecy matrices for the jump tests performed with prices sampled at 1-miute ad 5-miute itervals, respectively. Each elemet of these matrices represets the average percetage of jump days out of total days that are commo betwee the test o the row ad the test o the colum. The diagoal of each matrix cotais the percetage of jump days out of total days detected by each test idividually. The results are weighted evely across stocks, rather tha by the umber of days of data available for each stock. Results weighted by the umber of days of data available for each stock are preseted i Additioal Tables 3 ad 4 i the Appedix, although this weightig has a miimal effect o the results. There are several oteworthy results of this study. First, the Jiag Oome Logarithmic ad Ratio tests appear to detect the exact same jumps, as evideced by Tables ad 3. Likewise, the Microstructure Noise Robust Logarithmic ad Ratio tests exhibit the same patter. Furthermore, the Jiag Oome Differece test is sigificatly more coheret with the Logarithmic ad Ratio tests tha it is with ay other test. This patter of coherece betwee tests by the same author appears i the results for the Microstructure Noise Robust tests ad the Bardorff-Nielse Shephard tests as well. However, there appears to be little coherece amog tests by differet authors. Sice the jump tests are based o statistics that coverge to their distributios as the umber of observatios approaches ifiity, oe might expect the percetage of commo jump days betwee these tests to icrease as the samplig frequecy icreases. Surprisigly, that patter does ot appear i ay sigificat sese betwee the 5-miute ad 1-miute tables. The umbers are ideed higher i the 1-miute tables, particularly comparig the Jiag Oome tests to the Bardorff-Nielse Shephard ad Lee Myklad tests, but this patter is attributable to the higher percetage of jump days detected by these tests at this frequecy. 19
6. Iterpretatio of Results While the authors of the jump tests described i Sectio 3 spet a great deal of time ruig simulatios to test the accuracy ad cosistecy of their tests, they all spet less time discussig the applicatio of their tests to actual asset price data. Of course, the purpose of theses tests is to apply them to real stock data to lear more about their price paths. The results i Sectio 5 raise doubts about the validity ad cosistecy of these tests whe applied to stock price data. This study demostrates that the jump tests are susceptible to market microstructure oise, as evideced by the Relative Jump volatility sigature plot i Figure 3. This plot idicates that the cotributio of jumps to total variace is 13% for the 1-miute price data, whereas the cotributio of jumps to total variace is oly 7% for the -miute data. Such a large drop i Relative Jump betwee price data sampled at 1-miute ad -miute itervals implies that the Bardorff-Nielse Shephard test statistics will be vastly differet whe performed o those two sets of data. This is a serious problem with the Bardorff-Nielse Shephard tests that the authors might cosider addressig i future revisios of the tests. Furthermore, each jump test is icosistet amog price data sampled at differet frequecies. This is a importat fidig of the study, as the problem caot be easily resolved whe applyig the tests to real stock price data. Samplig prices at frequecies lower tha the highest available appears to mitigate the effect of market microstructure oise, as evideced by the results for the Bardorff-Nielse Shephard tests. However, there is o clear patter i Table 1 that suggests a solutio to the icosistecies i the jump tests due to the frequecy at which price data are sampled. These results cast doubt o the applicability of these jump tests to actual price data, ad have major implicatios for authors who use the tests to detect jumps i the fiacial markets i their research. For istace, Tauche ad Zhou (006) use the Bardorff-Nielse Z TP-max test to detect jumps i S&P 500 Idex prices sampled at 5-miute iterval. They the derive the distributio of the jumps ad use the time-varyig jump volatility to forecast ivestmet grade bod spread idices, fidig that S&P 500 jump volatility is a more effective predictor of bod spreads tha iterest rates, market volatility, ad other risk factors. I a similar paper, Zhag, Zhou ad Zhu (005) use the Bardorff-Nielse Z TP-max test to detect jumps i idividual equity prices sampled at a 5-miute iterval ad use the time-varyig jump volatility to forecast Credit Default Swap spreads. 0
The results i Table 1 suggest that these authors might have foud differet results if they had chose to sample price data at a differet frequecy. While this problem certaily does ot preclude the usefuless of jump detectio tests, it is worth takig ito accout whe applyig the tests to fiacial data. Moreover, these authors coduct their studies uder the assumptio of large ad rare jumps. The results i Tables ad 3 suggest that the differet jump tests are detectig differet types of jump-like movemets, be they large ad rare jumps, small ad frequet jumps, or simply oise. Thus, oe should ot assume that the model of large ad rare jumps fully describes the jump-like pheomea i the fiacial markets. Oe potetial pitfall of my study is that the samplig frequecies chose for the purpose of compariso i this paper may ot all ecessarily be appropriate for every jump test. For istace, Aїt-Sahalia ad Jacod (008) perform their test o stock price data sampled at itervals of 5 secods to 30 secods, fidig that the test statistic is more cosistet with prices sampled at higher frequecies. This suggests that performig the tests o data sampled at lower frequecies will lead to icosistecies i the results. Moreover, Lee ad Myklad (007) perform their test o price data sampled at a 15-miute iterval, so the higher frequecies used i this paper may ot be appropriate. Also, they perform their test at a.05 level of sigificace, so my choice of a.01 level of sigificace may be too far ito the tail of the expoetial distributio of the test statistic. 7. Coclusio This paper applies recetly developed tests for jumps i asset price movemets to a set of highly liquid equities, with prices sampled at a variety of itervals. The surprisig fidigs of this study should motivate authors of jump detectio tests to recosider the applicability of their tests to stock price data ad to address the impact of market microstructure oise ad the choice of samplig frequecy o their tests. Furthermore, authors who apply these tests to stock price data i their research should recosider the validity of their results. To further examie the impact of samplig frequecy o jump detectio tests, it will be ecessary to obtai stock price data collected at samplig frequecies higher tha 385 observatios per day ad perform the tests o these data, to determie whether the results of the tests coverge as the umber of observatios per day approaches ifiity. Beyod this, it will be 1
ecessary to coduct Mote Carlo simulatios to see if this problem is a result of market factors or the properties of the tests.
8. Tables Throughout this sectio, the followig abbreviatios will be used: Z TP : Bardorff-Nielse Shephard test usig Tri-Power Quarticity (3.1.8) Z QP : Bardorff-Nielse Shephard test usig Quad-Power Quarticity (3.1.9) JO D : Jiag Oome Differece test (3..5) JO L : Jiag Oome Logarithmic test (3..6) JO R : Jiag Oome Ratio test (3..7) MNR D : Microstructure Noise Robust Jiag Oome Differece test (3..9) MNR L : Microstructure Noise Robust Jiag Oome Logarithmic test (3..10) MNR R : Microstructure Noise Robust Jiag Oome Ratio test (3..11) LM: Lee Myklad test (3.3.3) ASJ: Aїt-Sahalia Jacod test (3.4.) 3
Table 1* Icosistecy i the Number ad Timig of Jumps Detected by Tests Performed o Stock Prices Sampled at Differet Frequecies Z TP Z QP JO D JO L JO R MNR D MNR L MNR R LM ASJ** 1m jump days.499.539.0406.030.030.3541.516.516.015-5m jump days.0885.1018.0481.0377.0377.498.5057.5057.0505.1530 10m jump days.0694.0761.0593.0457.0457.4481.501.501.0368.18 15m jump days.0638.0698.0739.0563.0563.4559.4994.4994.067.098 1m, 5m days.0519.065.0096.006.006.60.378.378.0346-1m, 10m days.0373.0449.0095.0064.0064.559.3619.3619.09-1m, 15m days.0338.0406.0096.0059.0059.537.3583.3583.0173-5m, 10m days.0119.0151.0150.0101.0101.344.4076.4076.014.0786 5m, 15m days.0085.0108.0148.0097.0097.3316.390.390.0150.0710 10m, 15m days.011.014.001.0137.0137.3575.4038.4038.0159.0676 The rows titled 1m jump days, 5m jump days, 10m jump days, ad 15m jump days cotai the percetage of jump days out of total days for each samplig iterval, averaged over the stocks i the sample. The rows titled 1m, 5m days; 1m, 10m days; 1m, 15m days; 5m, 10m days; 5m, 15m days; ad 10m, 15m days cotai the percetage of commo jump days out of total days amog each pair of samplig itervals, averaged over the stocks i the sample. All of the tests were performed at a.01 level of sigificace. The results preseted i this table are averaged over the te stocks i the sample, with o weightig for the umber of days of data available for each stock. This meas that the percetages are calculated idividually for each stock, the they are summed ad divided by te. A alterative weightig scheme is preseted i Additioal Table i the Appedix. The jump tests all detect differet umbers of jumps whe performed o price data sampled at differet frequecies, as evideced by the top four rows of the table. Furthermore, the jump tests all detect differet timigs of jumps betwee differet frequecies, as evideced by the values i the bottom six rows, which are drastically lower tha the values i the top four rows, meaig that there are few commo days detected by each test amog differet samplig frequecies. *: Explaatios of the abbreviatios used i the colum headers ca be foud at the begiig of this sectio. **: The Aїt-Sahalia Jacod tests were performed at a 1-miute base samplig iterval with k =, 3, 4. This choice is explaied i Sectio 5.. 4
Table * Icoherece amog Tests by Differet Authors i the Number ad Timig of Jumps Detected i Stock Price Data Sampled at a 1-Miute Iterval Z TP Z QP JO D JO L JO R MNR D MNR L MNR R LM ASJ** Z TP.499 Z QP.4917.539 JO D.068.085.0406 JO L.001.015.0319.030 JO R.001.015.0319.030.030 MNR D.1766.1913.0374.094.094.3541 MNR L.530.746.0406.030.030.3541.516 MNR R.530.746.0406.030.030.3541.516.516 LM.1148.148.0180.014.014.0809.0983.0983.015 ASJ**.0895.0956.0046.003.003.064.0940.0940.0178.1416 Each cell of this table cotais the percetage of commo jump days out of total days amog the two jump tests o the row ad colum header, averaged over the stocks i the sample. The diagoal elemets are the percetage of jump days out of total days detected by each test idividually, averaged over the stocks i the sample. The jump tests are performed o price data sampled at a 1-miute iterval. The results preseted i this table are averaged over the te stocks i the sample, with o weightig for the umber of days of data available for each stock. This meas that the percetages are calculated idividually for each stock, the they are summed ad divided by te. A alterative weightig scheme is preseted i Additioal Table 3 i the Appedix. Jump tests by the same author detect similar umbers of jump days, as well as similar timig of these jumps. However, jump tests by differet authors detect differet umbers of jump days with differet timigs, as evideced by the o-diagoal values, which are drastically lower tha the values o the diagoal. *: Explaatios of the abbreviatios used i the row ad colum titles ca be foud at the begiig of this sectio. **: The Aїt-Sahalia Jacod tests were performed at a 1-miute base samplig iterval with k =. 5
Table 3* Icoherece amog Tests by Differet Authors i the Number ad Timig of Jumps Detected i Stock Price Data Sampled at a 5-Miute Iterval Z TP Z QP JO D JO L JO R MNR D MNR L MNR R LM ASJ** Z TP.0885 Z QP.0868.1018 JO D.0147.0158.0481 JO L.0100.0110.0374.0377 JO R.0100.0110.0374.0377.0377 MNR D.0416.0477.0475.0373.0373.498 MNR L.0454.050.0481.0377.0377.498.5057 MNR R.0454.050.0481.0377.0377.498.5057.5057 LM.015.0170.0085.0068.0068.043.050.050.0505 ASJ**.001.006.0018.0015.0015.0191.0.0.0017.079 Each cell of this table cotais the percetage of commo jump days out of total days amog the two jump tests o the row ad colum header, averaged over the stocks i the sample. The diagoal elemets are the percetage of jump days out of total days detected by each test idividually, averaged over the stocks i the sample. The jump tests are performed o price data sampled at a 5-miute iterval. The results preseted i this table are averaged over the te stocks i the sample, with o weightig for the umber of days of data available for each stock. This meas that the percetages are calculated idividually for each stock, the they are summed ad divided by te. A alterative weightig scheme is preseted i Additioal Table 4 i the Appedix. Jump tests by the same author detect similar umbers of jump days, as well as similar timig of these jumps. However, jump tests by differet authors detect differet umbers of jump days with differet timigs, as evideced by the o-diagoal values, which are drastically lower tha the values o the diagoal. *: Explaatios of the abbreviatios used i the row ad colum titles ca be foud at the begiig of this sectio. **: The Aїt-Sahalia Jacod tests were performed at a 5-miute base samplig iterval with k =. 6
9. Figures Figure 1 Volatility Sigature Plot Demostratig the Impact of Market Microstructure Noise o Realized Variace whe Calculated with High-Frequecy Stock Price Data Each poit o this graph represets the average value of Realized Variace (3.1.) over the stocks i the sample, with prices sampled at the iterval o the horizotal axis. The samplig itervals used for this plot rage from 1 miute to 30 miutes. The theory behid this plot is explaied i Sectio 5.1. The dowward slopig curve here is similar to the shape of the volatility sigature plot for a highly liquid asset i Aderso et al. (1999), as expected. The sharp drop i the level of Realized Variace as the samplig iterval icreases from 1 to 5 miutes demostrates the effect of market microstructure oise o the calculatio. 7
Figure Volatility Sigature Plot Demostratig the Impact of Market Microstructure Noise o Bipower Variatio whe Calculated with High-Frequecy Stock Price Data Each poit o this graph represets the average value of Bipower Variatio (3.1.3) over the stocks i the sample, with prices sampled at the iterval o the horizotal axis. The samplig itervals used for this plot rage from 1 miute to 30 miutes. The theory behid this plot is explaied i Sectio 5.1. The dowward slopig curve here is similar to the shape of the volatility sigature plot for a highly liquid asset i Aderso et al. (1999), as expected. The drop i the level of Bipower Variatio as the samplig iterval icreases from 1 to 5 miutes demostrates the effect of market microstructure oise o the calculatio. 8
Figure 3 Relative Jump Sigature Plot Demostratig the Impact of Market Microstructure Noise o Relative Jump whe Calculated with High-Frequecy Stock Price Data Each poit o this graph represets the average value of Relative Jump (3.1.4) over the stocks i the sample, with prices sampled at the iterval o the horizotal axis. The samplig itervals used for this plot rage from 1 miute to 30 miutes. The theory behid this plot is explaied i Sectio 5.1. The sharp drop i the level of Relative Jump as the samplig iterval icreases from 1 to 5 miutes demostrates the effect of market microstructure oise o the calculatio. The relatively flat regio i the plot betwee the 5-miute ad 11-miute samplig itervals suggests that is appropriate to sample prices at these itervals to mitigate the effects of market microstructure oise whe performig the Bardorff-Nielse Shephard jump tests. 9
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11. Appedix Throughout this sectio, the followig abbreviatios will be used: Z TP : Bardorff-Nielse Shephard test usig Tri-Power Quarticity (3.1.8) Z QP : Bardorff-Nielse Shephard test usig Quad-Power Quarticity (3.1.9) JO D : Jiag Oome Differece test (3..5) JO L : Jiag Oome Logarithmic test (3..6) JO R : Jiag Oome Ratio test (3..7) MNR D : Microstructure Noise Robust Jiag Oome Differece test (3..9) MNR L : Microstructure Noise Robust Jiag Oome Logarithmic test (3..10) MNR R : Microstructure Noise Robust Jiag Oome Ratio test (3..11) LM: Lee Myklad test (3.3.3) ASJ: Aїt-Sahalia Jacod test (3.4.) 3