Local Characterization of a Time Series Model Using

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1 It. Statistical Ist.: Proc. 58th Worl Statistical Cogress, 2011, Dubli (Sessio CPS020) p.6204 Local Characterizatio of a Time Series Moel Usig Geeralize Tukey Depth Kosiorowski, Daiel Cracow Uiversity of Ecoomics, Departmet of Statistics Ul. Rakowicka 27 Cracow , Pola aiel.kosiorowski@uek.krakow.pl I. Itrouctio From a practical view, the primary aims of ay ecoomic time series aalysis woul be to provie a isight ito the short term probabilistic features of the possible uerlyig moel o base of costatly upate sample path of a moerate legth. O base of such a geerally imprecise kowlege are mae various ecoomic ecisios or preictios. We mea here for example evaluatio of a portfolio of securities, optios for sale of purchase shares maagemet. I practice mai tools for time series aalysis a moel ietificatio are mea, autocovariace, partial autocovariace or cross-correlatio fuctios. They are extremely sesible to various kis of outliers that may occur i time series. Their estimates critically epes o statioarity, ergoicity of the uerlyig moel. Several authors stress that observe time series almost always cosist of atypical observatios (see (Pea (1990) or Maroa et all (2006)). These atypical poits ca be prouce by osystematic chages i the variables that are rivig the series or affectig them. Sice the forecast from ay time series moel are base o the extrapolatio of the historical patters, if the parameters of the moel are very epeet o a few atypical observatios resultig from isolate or orepeatable evets, the the quality of the forecasts ca be expecte to be poor. Moreover, whe these parameters have or ecoomic iterpretatios, the presece of uetecte ifluetial observatios ca lea the ecoomist to wrog ecisios. I this paper we stuy certai properties of the geeralize Tukey epth (locatio, locatio-scale, regressio epths) iuce proceures a look ito the probabilistic iformatio of the uerlyig time series moel carrie by the proceures. We focus our attetio o short term multivariate quatile base escriptio of the possible time series moel. We give several examples of easy a user friely epth iuce statistical proceures for robust short term ecoomic ecisio makig. II. Data Depth Proceures Statistical epth fuctios origiate with the otio of halfspace epth which has became much stuie as a tool i oparametric multivariate locatio iferece. Tukey a Dooho a Gasko (see Zuo a Serflig (2000)) efie the halfspace epth of a poit x with respect to a empirical istributio P o base o ata { y1,..., y } as the smallest proportio of ata poits i ay close halfspace with x o 1 the bouary. I etail let u be a vector o uit sphere S of the the Tukey epth of a poit x ca be writte as 1

2 It. Statistical Ist.: Proc. 58th Worl Statistical Cogress, 2011, Dubli (Sessio CPS020) p.6205 (1) where T T ( x, P ) mi # i : u y u x mi # i : ( y x ) H, 1 i 1 i u u S u S P is the empirical istributio base o ata { y1, y2,..., y }, #{} eotes the umber of ata T poits i {}, a H u { x : u x 0} is the close halfspace cotaiig 0 o its bouary with u poitig isie the halfspace a orthogoal to the bouary. The Tukey epth is iepeet of the cooriate system, that is it is affie ivariat. The poit(s) with the maximum Tukey epth provies a measure of cetrality kow as Tukey meia. For p (0,1), the p-th Tukey epth cotour Dp ( ) is the collectio of x such that () x p ; it meas D( p) x : ( x ) p. Cotours (some authors use term cetral regios) form a sequece of este covex sets (for etails see Rousseeuw a Struyf (1999) or Zuo a Serflig (2000)). Oe useful applicatio of the cotours is to provie a oparametric escriptio of the ispersio of istributio usig the volumes of the eclose regios. A example cocerig a relatio betwee iflatio a uemploymet rate i Pola is presete o Figure 1 a Figure 2. Struyf a Rousseeuw (1999) prove that the Tukey epth completely etermies empirical istributios by actually recostructig the ata poits from the epth cotours. Also Kog a Zuo (2010) stuie properties of the Tukey epth cotours a looke ito the probabilistic iterpretatio carrie by the cotours a show that ay istributio with smooth epth cotours is completely escribe by its Tukey epth. Iovative extesio of the Tukey epth to uivariate multiple regressio was propose by Rousseeuw a Hubert (1999). Mizera (2002) ecompases otio of halfspace epth a regressio epth withi a geeral framework taget epth efie with respect to graiet probability fiels a equippe with ifferetial calculus. His efiitio of the epth i geeral moels is motivate by theoretical cosieratios with a ecisio theoretic flavor. Geeral halfspace epth ca be efie as a measure of ata aalytic amissibility the simplest versio of this priciple efies epth of a parameter ata poits whose omissio causes as the proportio of the to become a ofit, a fit that ca be uiformly omiate by aother oe. Mizera a Muller (2004) apply the taget epth to the classical uivariate locatio scale problem through a locatio scale epth efie o a bivariate parameter space. Mizera a Muller itrouce ot oe but a family of epths epeig o the choice of the uerlyig esity. I a cotext of robust shortterm aalysis of relatios betwee the mea a the ispersio of the ecoomic time series their Stuet epth seems to be especially iterestig. The Stuet epth of (, ) [0, ) with respect to a probability measure P o is efie (2) 2 2 (,, P ) if P y : u1 ( y ) u2 (( y ) ) 0, T ( u1, u2) 0 the Stuet epth with respect to the ata y1,..., y is obtaie by applyig the efiitio to the empirical probability measure P supporte by the ata poits. The locatio of the Stuet meia lies relatively close to the sample meia i particular for ata exhibitig symmetry. For asymmetric uimoal istributios, we may observe that the Stuet meia locatio scale shriks from the sample meia towar the moe. We observe also that the stuet meia is usually shruk ow from the MAD. Results presete i Mizera (2002) imply for the Stuet 2

It. Statistical Ist.: Proc. 58th Worl Statistical Cogress, 2011, Dubli (Sessio CPS020) p.6206 epth that breakow poit of the Stuet meia is ot less tha 3. This meas cosierable robustess.

1: Tukey epth cotour plot iflatio vs. uemploymet i Pola. Fig. 2: Liear regressio fits iflatio vs. uemploymet i Pola Source: Ow calculatios, ata GUS Source: Ow calculatios, ata GUS III.

3 It. Statistical Ist.: Proc. 58th Worl Statistical Cogress, 2011, Dubli (Sessio CPS020) p.6206 epth that breakow poit of the Stuet meia is ot less tha 3. This meas cosierable robustess. We ca say that the Stuet epth plots iicate asymmetry icluig that preset i the core of the ata rather tha just i the tails, but they are capable of etectig heavy taile behavior too. Fig. 1: Tukey epth cotour plot iflatio vs. uemploymet i Pola. Fig. 2: Liear regressio fits iflatio vs. uemploymet i Pola Source: Ow calculatios, ata GUS Source: Ow calculatios, ata GUS III. Propositios Depth base statistical methos are proviig short term multivariate quatile base escriptio of the possible time series moel. Although such the escriptio is rather imprecise but very ofte gives us a base for a ecisio makig. Data epth cocept offers a variety of easy a user friely aalytic tools for a prelimiary aalysis of time series a ecoomic ecisio makig. We mea here i particular: A. We ca use movig multiimesioal meia as a alterative to oe-imesioal movig mea or meia filter. I a cotrary to the mea a the meia, the multiimesioal meia takes ito accout multiimesioal geometry of the ata a hece the atural epeece of poits i time series aalysis. We avocate here usig a movig projectio meia which has very goo properties i the cotext of a balace betwee robustess a efficiecy (for etails see Zuo (2003)). I case of liear autoregressio estimatio we strogly recomme usig maximum regressio epth estimator (Maxepth) istea of least squares, maximum likelihoo base methos. We uerlie here relatively high breakow (BP) poit of Maxepth estimator but also relatively small sesitivity of the maximum epth estimator for a ata subset for a majority of the ata (see Visek (2002)). Usig the autoregressio estimator with relatively high BP we protect our aalysis agaist a effect of propagatio of a outlier. 3

We calculate sample epths of poits assumig say first 25% a last 25% poits of the cosiere time series. Next we compare calculate epths for each poit usig scatter plot.

4 It. Statistical Ist.: Proc. 58th Worl Statistical Cogress, 2011, Dubli (Sessio CPS020) p.6207 B. We ca use simple (see Liu et all (1999)) for a prelimiary aalysis of statioarity of multiimesioal time series. We calculate sample epths of poits assumig say first 25% a last 25% poits of the cosiere time series. Next we compare calculate epths for each poit usig scatter plot. Departures from iagoal lie of the scatter plot shoul iform us about iffereces of the probability istributio geeratig time series. Figure 3 presets four epth vs. epth plots prepare o base of two-imesioal time series of the mothly log returs of IBM stock a the S&P 500 iex from Jauary 1926 to December 1999 with 888 observatios (see Tsay (2010)). We ivie series ito four approximately equal size parts followig each aother. We ca otice a sigificat iffereces betwee first a seco a first a fourth perio. Fig. 3: Tukey epth cotour plot iflatio vs. uemploymet i Pola. Fig. 4: Tukey epth cotour plot iflatio vs. uemploymet i Pola. Mothly log returs of IBM stock p2 p3 p2 time perio1 perio2 Mothly log returs of S&P500 iex p4 p4 time perio 3 perio1 Source: Ow calculatios, ata Tsay (2010) Source: Ow calculatios, ata Tsay (2010) C. I orer to iicate a moel geeratig time series it is useful to aalyze a behavior of a movig Stuet meia or the Stuet meia calculate for short followig each aother perios. Scatter iagrams of the locatio a the scale cooriates of the Stuet meias coul be very helpful tools for a ivestigatio of relatios betwee the mea a the ispersio of the uerlyig process geeratig series. We recomme this tool for a prelimiary iscrimiatio betwee simple GARCH, SV a ARMA moels i cases of the samples of a short or moerate legth cosistig outliers. Figure 5 presets 10-miute FX log returs of Mark-Dollars exchage rate. Figure 6 presets the locatios a the scales for Stuet meias calculate o base of 6-hours perios followig oe aother subtracte from the origial series (30 observatios for each meia calculatio). Figure 7 presets scatter iagram of the Stuet meia scale i the perio t versus the Stuet meia scale i the preceig perio (t-1) with maximal regressio fit represete by re lie. Figure 8 presets scatter iagram of the Stuet meia locatio i the perio t versus Stuet meia scale i the preceig perio (t-1) with maximal regressio fit represete by re lie. Figure 7 a Figure 8 together focus our further attetio o MGARCH (GARCH i mea) class of moels geeratig the cosiere time series. 4

5 It. Statistical Ist.: Proc. 58th Worl Statistical Cogress, 2011, Dubli (Sessio CPS020) p.6208 D. I a cotext of the autocorrelatio coefficiet estimatio we recomme usig slope calculate by meas of maximal regressio epth metho ajuste by meas of meia of absolute eviatio from the meia. For autocorrelatio of a moerate orer we avoi a effect of propagatio of a outlier. Fig. 5: DM/USD 10-mi log returs. Fig. 6: 6 hours movig Stuet meia. 10-mi log retur 6-hours Stuet locatio 6-hours Stuet scale Source: Ow calculatios, ata Tsay (2010) Source: Ow calculatios, ata Tsay (2010) Fig. 7: Stuet meia scale i the perio t vs. Stuet meia scale i the perio (t-1). Fig. 8: Stuet meia locatio i the perio t vs. Stuet meia scale i the perio (t-1). Source: Ow calculatios, ata Tsay (2010) Source: Ow calculatios, ata Tsay (2010) IV. Coclusios I the short term forecastig a behavior of a ecoomic system the goal is to preict future values of a time series base o the ata collecte to the preset. Very ofte time series cotai ifluetial outliers 5

6 It. Statistical Ist.: Proc. 58th Worl Statistical Cogress, 2011, Dubli (Sessio CPS020) p.6209 misleaig the ecoomist about the properties of the cosiere process. I these situatios ata epth base exploratory techiques coul provie us sufficiet basis for the ecisio makig. The Locatio scale epth propose by Mizera & Muller (2003) seems to be especially worth further stuies i the cotext of a robust time series aalysis. REFERENCES Kog L., Zuo Y. (2010), Smooth Depth Cotours Characterize the Uerlyig Distributio, Joural of Multivariate Aalysis 101, Kosiorowski D. (2010). Depth Base Proceures for Estimatio ARMA a GARCH Moels, Y. Lechevallier, G. Saporta (Re.) Proceeigs of COMPSTAT'2010, , Physica - Verlag. Liu, R. Y., Parelius, J. M., Sigh, K. (1999) Multivariate Aalysis by Data Depth: Descriptive statistics, Graphics a Iferece (with iscussio). The Aals of Statistics 27, Maroa, R. A., Marti, R. D., Yohai, V. J. (2006). Robust Statistics - Theory a Methos. Chichester: Joh Wiley & Sos Lt. Mizera, I. (2002). O Depth a Depth Pois: a Calculus. The Aals of Statistics (30), Mizera, I., Muller, C. H. (2004). Locatio Scale Depth (with Discussio a Rejoier). Joural of the America Statistical Associatio 99(4), Pea, D. (1990). Ifluetial Observatios i Time Series, Joural of Busiess & Ecoomic Statistics 8(2), Rousseeuw, J. P., Hubert, M. (1998). Regressio Depth. Joural of The America Statistical Associatio (94), Tsay R. S. (2010), Aalysis of Fiacial Time Series, Wiley Itersciece, Hoboke, New Yersey Tukey, J. (1975). Mathematics a Picturig Data. R. James (Re.), Proceeigs of the 1974 Iteratioal Cogress of Mathematicias. 2, stroy Caaia Math. Cogress. Struyf a Rousseeuw (1999), Halfspace Depth a Regressio Depth Characterize the Empirical Distributio, Joural of Multivariate Aalysis 69, Visek, J. A. (2002) Sesitivity Aalysis of M estimates of Noliear Regressio Moel: Ifluece of Data Subsets. The Aals of the Istitute of Statistical Mathematics 54(2), Zuo, Y. (2003). Projectio-base Depth Fuctios a Associate Meias, Aals of Statistics 31, Zuo, Y., Serflig, R. (2000). Geeral Notios of Statistical Depth Fuctio. The Aals of Statistics (28),

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