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# SAS MTHSC 981

Clemson

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This 23 page Class Notes was uploaded by Jazmyn Braun on Saturday September 26, 2015. The Class Notes belongs to MTHSC 981 at Clemson University taught by Staff in Fall. Since its upload, it has received 71 views. For similar materials see /class/214288/mthsc-981-clemson-university in Mathematics (M) at Clemson University.

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Date Created: 09/26/15

A Large Sample Independence Test for Finite Mean Processes Colin Gallagher Mafhemafical Sciences Clemson University Email cgallagcesclemsonedu We use the sample covariation to develop asymptotic tests for independence for data in the normal domain of attraction of a stable law The tests can be used for nite or in nite variance processes In a simulation study we compare the nite sample performance of the proposed tests to the Portmanteau test commonly used in time series modeling The null convergence ofthetest 39 39 39 I 39 d39 39h quot than that of the Portmanteau statistic especially when the data is fattailed In the nite variance case I I I 39 I I I 39 Portmanteau test and have similar small sample empirical power Simulations indicate the covariation test has a higher power for small sample sizes when the process is symmetric stable with tail decay parameter a gt 16 Key Words time series independence heavytails covariation 1 INTRODUCTION In this paper we use the sample covariation described below to develop Portman teau type tests for linear dependence in symmetric time series data We compare D R A F T October 5 2001 319pm D R A F T 2 c GALLAGHER the proposed tests to the traditional Portm anteau test for both in nite and nite variance data We will see that the test statistics described below seem to converge to their null asymptotic distribution faster than the Portm anteau statistic especially when the data is fattailed In the nite variance case the proposed test shares sim ilar power properties with the usual test In the in nite variance case the proposed test can have signi cantly higher empirical power for small and moderate sample sizes in many cases Let X be a strictly stationary ergodic stochastic process with zero mean satisfying Enigma win r k le 1 Condition 1 is satis ed by any symmetric astable process as well as any rst order autoregression XI ltgtXz i Z 2 where 2 is an independent identically distributed iid zerom ean sequence For a zeromeaniid sequence X l is satis ed with Mk 2 O for k 1 2 E h We consider testing the null hypothesis that X is an iid sequence against H1 Mk 55 O forsome k 1E2h Under Hg we will assume X1 has a symmetric density in the normal domain of attraction see Feller 1971 of an astable law for some a E 1 2 ie WW 2 X gt Sm 3 1 D R A F T October 5 2001 319pm D R A F T FINTTE MEAN INDEPENDENCE TEST 3 Where gt denotes convergence in distribution and S has a stable distribution with characteristic function Eews e d When a 2 EX lt 00 is a suf cient condition for 3 If X1 has a nite variance the above hypothesis test is usually performed using the sample autocorrelation function j For example one may use the Portman teau test statistic h 27190 4 31 or the nite sample correction h QLB m 2 Z fann j jzl proposed by Ljung and Box 1978 For 1 lt a lt 2 Runde 1997 considers using 2x h A gm 5 to test the iid hypothesis versus ARMA alternatives This test assumes 3 Simu lations indicate that the test performs poorly for small sample sizes especially for a close to 2 This is probably due to the apparent slow convergence of the norm al ized sample covariance and correlation functions to their asymptotic distributions When the underlying process has an in nite variance This slow convergence rate was observed in Phillips and Loretan 1990 and Adler Feldman and Gallagher 1999 as well D R A F T October 5 2001 319pm D R A F T 4 c GALLAGHER We consider using the covariation function 709 EX k 0 211 212 1 h Where 6 signX and its sample version A n1 2 539 for k gt 0 709 k y 71 1 X151 for k g 0 to test the above hypothesis For a process satisfying 1 709 AkEXk so that If the process is Gaussian Mk 2 M k 6 The ergodic theorem implies that 09 is a strongly consistent estimator of Gallagher 2000 gives the asymptotic joint distribution of the sample covariation for ARMA processes with iid innovation sequences If the innovation sequence satis es 3 for some a E 1 2 7614 W 709 gt Sm Where S has a symm etric astable distribution 7 It is shown in Gallagher Feldm an and Okuyama 2001 that for an iid sequence the univariate distribution function of the sample covariation converges at the same rate given in central limit theorems for iid processes For example if lt D R A F T October 5 2001 319pm D R A F T FINITE MEAN INDEPENDENCE TEST 5 00 the BerryEssen inequality applies The rate of convergence for the sample covariation function seems to be faster than that corresponding to the covariance In fact ifX is an iid symmetric stable Gaussian process 71 k1 1 quot fk has an exact stable Gaussian distribution We therefore expect hypothesis tests based on the sample covariation may perform well for small sample sizes In general 709 55 T k but if the process is iid with zero mean 709 T k 0 To test for independence we can de ne a symmetric covariation T0 TV TlkDZ which can be estimate with 21105909 k In this paper we consider two test statistics The rst is based on the rst It positive lag covariations When the process has an in nite variance we recommend the statistic h QM dd WM 2 8 jzl where of is any consistent estimate of the scale 1 If EX 2 02 lt 00 we can improve the small sample power by using the rst It symmetric covariations h A Qi if 63r1nZT3m 9 31 where f is any consistent estimator for Elej In the in nite variance case the null convergence of Q3 properly normalized seems to be quite slow Theorems 11 and 12 below give the asymptotic distributions of the statistics under the D R A F T October 5 2001 319pm D R A F T 6 c GALLAGHER null hypothesis Under the alternative hypothesis the test statistics diverge like g 2 as n 00 so that the tests have classical g 2 consistency The tests should have higher power as a 2 THEOREM 11 Let X1 X2 be a sequence of independent identically dis tributed symmetric random variables which belong to the normal domain of at traction ofan iistable law with or E O 2 Leth be an integer then as n 00 h 22 1n2 2a 220 gt kl QxylAy jzi where k Zh l Y is a vector ofiid symmetric iistable random variables with scale 1 1 and A is an idempotent matrix of rank h Remark 1 I If Elin 2 02 lt 00 and 62 is any consistent estimator 716 2 221 f2 j converges weakly to a chisquare distribution with h degrees of freedom Remark 1 2 If Y is not symmetric the sequence X2 2 Y2 Y21 can be used to test for independence Under the null hypothesis X21 is lid with a symmetric density If Y is an ARMA process Y 2 230 jZj with Z iid then X2 Z2 21 j Viv 0222 THEOREM 12 Let X1 X2 be a sequence of independent identically dis tributed symmetric random variables with EX 2 02 lt 00 Leth be an integer D R A F T October 5 2001 319pm D R A F T FINITE MEAN INDEPENDENCE TEST 7 then asn 00 h A if grlmzmi gt xi jzl where xi denotes a chisquare random variable with h degrees affreedam Remark 1 3 For a lt 2 Til 1quot 221 T2 can be shown to converge to a constant multiple of Y BY where B is an idempotent matrix of rank h and Y 2h 1 is a vector of 2 iid symmetric stable variables however initial simulations indicate the convergence under the null is quite slow For a lt 2 Theorem 11 is impractical as it stands In this case the limiting distribution has no known closed form expression In Section 2 we describe an algorithm for nding quantiles of the limiting distributions given above and give quantiles for h 3 5 and 10 for various a s In Section 3 we investigate the performance of the proposed tests on simulated data We simulate power under ARl alternatives In Section 31 we compare size and power of the test based on QCW to the Runde Portmanteau statistic using simulated data from stable normal distributions We rst assume the scale pa ram eter is known and simulate symmetric stable data for a lt 2 and see that under the null hypothesis QCW does not seem to be plagued with the slow distributional convergence of the sample covariance and correlation functions in the in nite vari ance case In Section 32 we simulate from the normal and tdistribution with 3 degrees of freedom to compare empirical size and power for QCW and Q3 to that of the traditional statistic 4 D R A F T October 5 2001 319pm D R A F T 8 c GALLAGHER The proof of our main results appear in the appendix Although Theorem 11 follows via the continuous mapping theorem from the limiting distribution given in Gallagher 2000 we present a complete proof 2 QUANTILES FOR a lt 2 In this section we consider estimating quantiles of the limiting distribution in Theorem 11 for a lt 2 We give an algorithm for simulating random variates from the asymptotic distribution To simulate quantiles we used this algorithm to simulate 500000 observations from the limiting distribution for each a 141516171819 and h 35 10 TABLE 1 Quantiles of ki QO39Y39AY h3 h5 hv10 a 90 95 90 95 90 95 14 3157 7803 5233 12998 10335 25991 15 2497 5540 4080 9190 7874 18071 16 2043 4099 3278 6597 6327 12972 17 1697 2969 2670 4750 5039 9218 18 1482 2264 2242 3459 4073 6438 19 1341 1810 1990 2630 3496 4576 20 1250 1563 1847 2214 1599 1831 A symmetric stable variable with 2 1 has variance 2 The last row ofthe table is 2 xiuz where xiuz is the appropriate quantile of the chisquare distribution with 1 degrees of freedom D R A F T October 5 2001 319pm D R A F T FINTTE MEAN INDEPENDENCE TEST 9 The algorithm is easily implemented in Splus Table 1 gives the 90 and 95 quantiles of Isl 2quot Y AY which were found via simulation using the S plus routine rstab to simulate symmetric stable variables The code is available at httpwwwresclemsoneducgallag In order to give our algorithm we must rst describe the matrix A Let 6 162 6h This vector takes k 2 2h 1 equally likely possible values Create the matrix M by letting each column correspond to one of these outcomes For example ifh 2 M The joint limiting distribution of 1quot 2 H1 jh39 is described in Proposition 21 below PROPOSITION 21 Let X be it39d and satisfy 3 IfX1 has a symmetric density the vector nl We gt k l MS where Y is a vector ofk it39d symmetric stable random variables each ofwhich has scale d It is easy to see that M has orthogonal rows It follows that A k lM M 10 D R A F T October 5 2001 319pm D R A F T 10 c GALLAGHER is an idempotent matrix Using standard results from multivariate analysis the quadratic form Y AY Y PP39Y where the columns of P are the h eigenvectors of A with nonzero eigenvalues This suggests the following algorithm for simulating a sample of size n from Y AY ALGORITHM l i Create M This can easily be done recursively in h 2 Find the eigenvalues of k lM M and create P 3 Foreachi12n i Simulate k symmetric stable random variables with index 1 ii Set 3 kHQWY39PP39Y 3 EMPIRICAL RESULTS In this section we report the results of a simulation study In the in nite variance case we simulated symmetric stable data In the nite variance case we considered the I distribution with 3 degrees of freedom and the normal distribution Under the alternative hypothesis we simulated from 2 where Z is an iid sequence with common symmetric astable or I density and gt 04 or 07 In the in nite variance 1 lt 2 case we will see that the proposed test statistics perform better as a approaches 2 The problem with the tests for smaller 1 is that D R A F T October 5 2001 319pm D R A F T FINITE MEAN INDEPENDENCE TEST 1 1 for small sample sizes the tests have low power This is probably due to the slow divergence ml 2quot under the alternative hypothesis For a g 15 Runde s test is performing quite well simulated type I error is essentially 0 and the test has good power properties However the covariation tests considered below seem to have better size and power properties than Runde s Portmanteau test for a gt 16 and n g 200 In the nite variance case the proposed test has the same asymptotic properties as the Portmanteau test However our simulations seem to indicate that the null convergence of Q3 is faster than the convergence for Q given by 4 The two tests seem to have comparable small sample power properties 31 Empirical size and power for symmetric stable data Here we compare the size and power of tests using QCW to that of the Portman teau and Runde tests on simulated symmetric stable data We will consider two different scale estimators 21 and of given below TABLE 2 Emperical size gtlt 100 for 005 tests when d is known h3 h5 h10 a n50 n100 n50 n100 n50 n100 15 4 92 449 489 477 4 56 4 74 17 5 04 514 482 4 8 449 446 19 4 53 4 84 482 479 4 29 5 24 D R A F T October 5 2001 319pm D R A F T 12 c GALLAGHER 311 Null convergence anCW for symmetric stable data To investigate the null convergence of our test statistic in the in nite variance case we simulated iid data from the symmetric stable distribution for various 1 values Since the sample mean can converge quite slowly in this case we used mediancorrected data Because distributional convergence can be quite slow for in nite variance sequences we rst investigate the null convergence of 221 Tj2 for a lt 2 when the scale parameter is known For symmetric stable data we consider two consistent estimators of d and compare the proposed test to the Portm anteau test from Runde 1997 When 1 is known simulations indicate that the empirical size does not signi cantly depend on a In the case of unkown scale for one of the scale estimators considered the distributional convergence seems to depend on a slowing as 1 decreases For 1 known we report results for 10000 tests for each n 50 100 and a 15 17 and 19 in table 2 We see thatmost of the numbers in the table are within 196 standard deviations of 005 It appears that the test will be slightly conservative since most of the empirical sizes are below 005 In the general case 1 must be estimated using some robust to moment assump tions estimator However if the data is coming from a stable distribution there are a number of estimators available We consider two which should behave fairly well under the alternative hypothesis If X1 has a symmetric stable distribution with scale 1 EX1 d2f1 1a7r D R A F T October 5 2001 319pm D R A F T FINITE MEAN INDEPENDENCE TEST 13 If the process generating the data sequence is stationary and ergodic with a nite mean the moment estimator A KHO 11 m 1 is a strongly consistent estimator of 1 Under the null hypothesis W0 Eerl Owl 4 which becomes increasingly slow as a approaches 1 Thus we expect the test statistic using this moment estimator to perform better for a close to 2 With this estimator of d the test statistic becomes h QM nH zm 1a7r Z 1209 12 331 where M We note here that if 1 is satis ed Rog Mk where the convergence is in the almost sure sense and A is given by 6 We also consider the quantile estimator of McCulloch 1986 This estimator was originally proposed for iid data and is a consistent estimator in this case Since the estimator uses the sample quantiles we expect it to work under the alternative hypothesis as well lndeedAdler Feldman and Gallagher 1998 give simulation D R A F T October 5 2001 319pm D R A F T 14 c GALLAGHER evidence indicating that the quantile estimator of the tail decay parameter 1 works well for small order moving average and autoregressive processes For this scale estimator we denote the test statistic by 13 where 22 denotes McCulloch s quantile estimator of 1 Table 3 contains simulated size for the tests based on Qwv Qwv and Q For each n and a we simulated 1000 data sets under the null hypothesis with d 3 TABLE 3 Emperical size gtlt 100 for 005 tests when d is unknown n50 n100 n200 a QM 2m Q QM 2m Q QM 2m Q 15 16 55 0 14 47 0 24 53 01 16 12 65 0 21 6 0 26 55 0 17 14 52 0 27 49 0 31 55 0 18 22 53 0 35 63 0 38 53 0 19 33 54 0 37 51 0 36 53 0 20 39 68 51 51 66 46 55 57 6 For a lt 2 the statistic QCW seems to perform best since all but 1 of its empirical sizes are within 05 1 196 95 xlt 051000 The test based on QCW and Runde s Portmanteau test appear to be conservative in the sense that the empirical size is systematically below its nominal value The Portmanteau test essentially had 0 D R A F T October 5 2001 319pm D R A F T FINITE MEAN INDEPENDENCE TEST 15 type I error for the in nite variance data while QCW has simulated sizes which seem to be approaching 005 as a and 71 increase 312 Empiricalpawerfar symmetric stable data We conducteda small simulation study to assess the nite sample power proper tiesof J J tet 394 4L L 39 39 ahnveand compared the performance to that of the test proposed in Runde 1997 For com parison purposes we assumed a was known We only had quantiles for Runde s test for speci c 1 values In practice a can be estimated using McCulloch s quantile method or any of the numerous methods available in the literature For each a and n 50 100E200 we simulated 1000 data sets from each of the two autoregressive models mentioned above We compare empirical power for Qwv QCW and Q given by 12 13 and 5 respectively Quantiles for the limiting distribution of 5 are given in Runde 1997 for various 1 E 1 2 and h 2 3y4and5 As mentioned above the proposed tests give better results for a closer to 2 Our simulations indicate that the covariation tests seem to have a higher power than Runde s test for small to moderate 71 and a gt 16 with a signi cantly higher power for a E 17 2 For a g 16 Runde s test seems to have a higher power This phenomenon can be seen by results given in Table 4 It is not clear which scale estimator is performing better It is not too surprising that for normal data a 2 the Portmanteau test has higher empirical power for all 71 considered 32 Simulations for nite variance data D R A F T October 5 2001 319pm D R A F T C GALLAGHER TABLE 4 Emperical power of 005 test when d is unknown n50 n100 n200 Model a Q Q60 Q Q Q60 Q Q Q60 Q o 07 15 99 121 598 362 223 986 797 534 999 a 04 14 64 10 16 66 135 68 97 851 o 07 16 257 192 338 656 489 951 976 936 998 a 04 28 65 01 54 85 37 157 154 459 o 07 17 535 414 214 891 847 828 999 999 999 a 04 82 111 0 193 198 05 483 441 173 o 07 18 711 645 57 981 973 632 1 1 995 a 04 142 155 0 387 369 01 779 734 29 o 07 19 860 821 016 996 991 389 1 1 973 a 04 304 290 0 591 582 0 933 925 05 o 07 2 0 919 905 980 999 999 1 1 1 1 a 04 383 388 550 726 729 894 980 984 999 For n 50 we took I 3 and for n 100 200 we used I 5 the normal distribution and Q given by 4 a 2 corresponds to To assess the small sample size and power properties of the tests for some nite variance distributions we perform ed a small simulation study We compared the Portmanteau test statistic 4 to QCW given by 8 as well as to Q3 given by 9 For Q3 we used simple moment estimators of 2E X1 and 02 while we usedthe D R A F T October 5 2001 319pm D R A F T FINITE MEAN INDEPENDENCE TEST 17 sample variance in place of i in 8 Since the test statistic Q3 had better empirical size and power than Qwv for brevity we report results only for this statistic To compare empirical sizes for the Portmanteau and covariation tests we sim ulated under the null from the normal distribution with variance 9 as well as from the Idistribution with 3 degrees of freedom For each n 10 20 50 and 100 we simulated 10000 data sets from each distribution and counted the number of rejections from each test statistic for h 5 The results appear below in Table 5 We see that the simulated sizes for Q3 are closer to 005 for all 71 considered for both normal as well as I data TABLE 5 Emperical size gtlt 100 0f005 tests for normal and t densities n10 n20 n50 n100 de SilY Q Qs Q Qs Q Qs Q Qs normal 048 108 218 285 330 399 457 470 t 0 29 103 130 2 97 273 4 08 3 63 4 49 Table 6 contains empirical power for 1000 data sets simulated under the two autoregressive processes considered above We see that not only do the two tests have the same asymptotic power properties the small sample empirical power is quite similar for both statistics The proposed test may work slightly better for smaller I quotthePnrtmanteaute t t 39 quot highclpuwcl r J for moderate sample sizes D R A F T October 5 2001 319pm D R A F T C GALLAGHER TABLE 6 Percent rejected power for 005 tests n10 n20 n50 n100 de SilY Model Q Qs Q Qs Q Qs t C3 07 0 10 401 408 966 952 10 10 C3 04 0 02 73 116 457 454 867 82 normal I 07 02 11 385 383 965 925 10 10 d 04 02 06 90 96 453 397 867 774 For all n We used I 5 APPENDIX PROOF OF THE MAIN RESULTS We derive the weak limit of QCW and Q3 under the null hypothesis The limiting distributions under ARMA alternatives follow from Theorem 22 in Gallagher 2000 via the continuous mapping theorem We will deal with the nite and in nite variance cases seperately A01 Finite variance case For or 2 Theorem 11 is an immediate consequence of Proposition Al PROPOSITION A1 Let X be an iid symmetric mean zero process with EX lt 00 For integerh n gtzE where Z is a vector ofiid normal random variables each ofwhich has variance mg D R A F T October 5 2001 319pm D R A F T FINITE MEAN INDEPENDENCE TEST 19 Proof Let a1 any 6 3 2quot and de ne Y1 1X1 ah6h 1161 1 Stationarity of Y follows from the fact that the process X is stationary Using the symmetry and independence assumptions we see Y is an hdependent process with covariance function EX2 f a3 k0 706 124 3 k 1h Using the central limit theorem for stationary mdependent processes eg see Brockwell and Davis 1991 we see that 71 71 12 3 gt N 1 h Where N has a normal distribution with variance E X 231 1 Theorem 12 follows directly from Proposition AZ PROPOSITION A121 Let X be an iid symmetric mean zero process with EX 2 02 lt 00 andc2 EX12 For integerh let T39 ltTlt1gtTlthgt The vector T f9 Z where Z is a vector ofiid normal random variables each ofwhich has variance 02 cg2 D R A F T October 5 2001 319pm D R A F T 20 c GALLAGHER Proof Let a1 any 6 3 2quot and de ne Y1 Xzz ahSI h Sum 1151 1 5141 Using the symmetry and independence assumptions we see Y is an hdependent process with covariance function yw ZL k0 away4 k 1 Using the central limit theorem for stationary mdependent processes eg see Brockwell and Davis 1991 we see that 71 7112 3 gt N 11 Where N has a normal distribution with variance Uh 2 EX 2 h 9 a 2111 A02 in nite variance data For a lt 2 Theorem 11 is an immediate consequence of Proposition 21 Proof Proposition 21 For a a1 any 6 3quot de ne 1 1 1leah5z h 115141 Y is an hdependent symmetric process with nP Yl gt Tilail cam 0 as 17 00 D R A F T October 5 2001 319pm D R A F T FINITE MEAN INDEPENDENCE TEST 21 where ca is a constant given below To describe 6 let the vector 6 6150 Then 6 Eja 6hj x The sequence 3 satis es all three conditions of Davis 1983 Condition D is implied by mixing and condition D follows from hdependence To see that D holds note that I I I 2 711303 gt Til m gt nl w g 71 131le gt owlm where C1 Z jaj Applying Theorems 2 and 3 in Davis 1983 we see that the partial sums 71 71 1quot 3 gt W2 11 where W has a symmetric stable distribution with scale ca The vector 6 has 2quot unique possible outcomes Meaning that ja 6 has at most k 2 2h 1 unique equally likely outcomes Each of these outcomes corresponds to a column of M Using well known properties of stable random variables e g see Samm orodnitsky and Taqqu 1994 it follows that 71 MW 2 Y gt 19 1quot a MY 11 where Y is a vector consisting of k iid symm etric stable random variables with scale D R A F T October 5 2001 319pm D R A F T 22 c GALLAGHER Remark1 I The limiting multivariate stable vector MY in Proposition 21 has spectral measure concentrated on k symmetric pairs of points on the unit circle in ERquot Unlike the nite variance case this limiting vector does not have independent components REFEREN C ES 1 R JAdler R Feldman and C Gallagher Analysing stable time series in A Practical Guide to Heavy Tails R J Adler R E Feldman andM S Taqqu eds Birkhauser Boston 1998 2 P J Brockwell and R A Davis Time Series Theory and Methodx Springer New York 1991 3 R A Davis Stable limits for partial sums of dependent random variables Ann Probab Vol 11 1983 2627269 4 W Feller An Introduction to Probability Theory and itsApplications Wiley New York 1971 5 C Gallagher Estimating the autocovariation from stationary heavytailed data with applications to time series modeling Preprint 2000 6 C Gallagher R E Feldman and T Okuyama Some differences in the rates of convergence of the sample covariance and covariation functions available at http10szcesclemsaneduNcgallag Working paper 2001 7 G M Ljung and G E P Box On ameasure of lack of t in time series models Biometrika Vol 65 1978 2977303 8 JH McCulloch Simple consistent estimators of stable distribution parameters Communications in Statinics 7 Computation and Simulation Vol 15 1986 110971136 D R A F T October 5 2001 319pm D R A F T FINITE MEAN INDEPENDENCE TEST 23 9 P C B Phillips and M Loretan The DurbinWatson ratio under in nitevariance errors 1 of Econometrics Vol 47 1991 857114 10 R Runde The asymptotic null distribution of the BoxPierce Qstatistic for random variables With in nite variance 1 ofEconomem39cs 3978 1997 2057216 11 G Sarnorodnitsky andM Taqqu Stable NonGaussian Random Processes ChapmanHall New York 1994 D R A F T October 5 2001 319pm D R A F T

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