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## MODEL FOR LARGE VOLATILITY JUMPS

Volatility jumps are defined by daily difference of $v(t) = \log(r_t^2)$.  When we take the threshold $\delta=1.0$ for large volatility jumps, one pleasant surprise is that the kurtosis becomes negligible which suggests that we can model the jump distribution with a Gaussian.  The mean is in the order -0.05 but over 47 stocks we have a regularity that the standard deviation is 2.83 +/- 0.27 and the kurtosis is in the range (-1,1) so small enough that Gaussian model makes sense for the jumps.  Now the fits of the Mittag-Leffler distribution for waiting times uniformly have R^2 at 0.95+ and the $\alpha \sim 0.85-0.95$ and the ARMA(5,0) model fit the series with Durbin-Watson statistics almost uniformly at 2.0.

So we have some solid basis for our model for large jumps.  Fit details follow.

stock= LRN  theta= [ 0.99996892  0.88161969]  R2-proxy= 0.937850327889
jmean= 0.0294315518205  jstd= 2.80054291004  jkurt= 0.314458587651
ARMA(5,0) params =  [ 0.06086052 -0.53998629 -0.33441736 -0.11587049 -0.05408026  0.09652985]
conf interval = [[-0.17681571  0.29853676]
[-0.72543442 -0.35453816]
[-0.54684029 -0.12199444]
[-0.34048325  0.10874227]
[-0.27257009  0.16440958]
[-0.09183989  0.28489959]]
Durbin-Watson = 1.94909509277
stock= PRE  theta= [ 0.9999513   0.85421703]  R2-proxy= 0.960269554899
jmean= 0.0621828700621  jstd= 2.84739548654  jkurt= -0.237051558082
ARMA(5,0) params =  [ 0.04734985 -0.44788097 -0.1445403  -0.14781883 -0.1600812  -0.05612818]
conf interval = [[-0.2221646   0.3168643 ]
[-0.64672537 -0.24903657]
[-0.36202446  0.07294385]
[-0.36515048  0.06951283]
[-0.37863494  0.05847254]
[-0.26013844  0.14788208]]
Durbin-Watson = 2.01069838791
stock= IRBT  theta= [ 0.99996069  0.89310215]  R2-proxy= 0.980616974737
jmean= 0.0508959377504  jstd= 2.88440846345  jkurt= 0.646118633237
ARMA(5,0) params =  [ 0.03852739 -0.54581742 -0.3243831  -0.25562566 -0.17016157 -0.04757723]
conf interval = [[-0.09513529  0.17219008]
[-0.66714377 -0.42449106]
[-0.46137678 -0.18738943]
[-0.39456483 -0.11668649]
[-0.30672635 -0.03359679]
[-0.16854604  0.07339158]]
Durbin-Watson = 2.01008853309
stock= G  theta= [ 0.99998258  0.99815473]  R2-proxy= 0.990137522673
jmean= 0.0400095758891  jstd= 2.93975204693  jkurt= 0.641294312775
ARMA(5,0) params =  [ 0.01945534 -0.55995793 -0.43905131 -0.34876499 -0.18904465 -0.08170556]
conf interval = [[-0.08710931  0.12601998]
[-0.66663689 -0.45327897]
[-0.56003835 -0.31806428]
[-0.47328308 -0.22424691]
[-0.30992269 -0.06816661]
[-0.18865203  0.02524091]]
Durbin-Watson = 2.00646979156
stock= ICUI  theta= [ 0.90412203  0.96234142]  R2-proxy= 0.992505464322
jmean= 0.0185884811961  jstd= 3.01252541619  jkurt= 0.495061052979
ARMA(5,0) params =  [-0.09300529 -0.49001522 -0.21573804 -0.11289468 -0.05761088 -0.00252118]
conf interval = [[-0.297063    0.11105242]
[-0.65827169 -0.32175875]
[-0.40329493 -0.02818115]
[-0.30451523  0.07872587]
[-0.24678255  0.13156078]
[-0.17172824  0.16668589]]
Durbin-Watson = 2.00186355625
stock= IGT  theta= [ 1.1305236   0.85017439]  R2-proxy= 0.97695118233
jmean= -0.0893489899464  jstd= 2.52075469457  jkurt= -0.348287841095
ARMA(5,0) params =  [ 0.01339653 -0.5288701  -0.2898832  -0.23884346 -0.24415113 -0.18933466]
conf interval = [[-0.13036234  0.15715539]
[-0.67747824 -0.38026195]
[-0.45582481 -0.1239416 ]
[-0.40849428 -0.06919264]
[-0.41400887 -0.07429338]
[-0.3405554  -0.03811392]]
Durbin-Watson = 2.01953854508
stock= SIAL  theta= [ 0.72051876  0.90319602]  R2-proxy= 0.981637054059
jmean= 0.00901449242706  jstd= 2.70572065159  jkurt= 0.00494812999156
ARMA(5,0) params =  [ 0.15874632 -0.52403816 -0.33154891 -0.18080352 -0.17386892  0.00269378]
conf interval = [[-0.20295765  0.52045028]
[-0.80693043 -0.24114588]
[-0.65082615 -0.01227167]
[-0.51763103  0.156024  ]
[-0.49965701  0.15191917]
[-0.28889166  0.29427923]]
Durbin-Watson = 1.96147435554
stock= KND  theta= [ 0.1391246   1.60682853]  R2-proxy= 0.966362077675
jmean= 0.214073615483  jstd= 3.1949526767  jkurt= 0.368830304527
ARMA(5,0) params =  [ 0.02777949 -0.3894543  -0.21170628 -0.17133642 -0.17274839 -0.0855376 ]
conf interval = [[-0.21989724  0.27545623]
[-0.59135154 -0.18755707]
[-0.42613532  0.00272277]
[-0.38690275  0.04422991]
[-0.38654793  0.04105115]
[-0.28640224  0.11532704]]
Durbin-Watson = 2.01935586981
stock= ASH  theta= [ 0.9861872   0.95091635]  R2-proxy= 0.992134910176
jmean= 0.0121124607631  jstd= 2.65609837172  jkurt= -0.17520355874
ARMA(5,0) params =  [-0.16918392 -0.47037578 -0.32130846 -0.25507023 -0.26929911 -0.16348391]
conf interval = [[-0.32059221 -0.01777563]
[-0.62436362 -0.31638795]
[-0.48731056 -0.15530636]
[-0.4234842  -0.08665626]
[-0.43512809 -0.10347013]
[-0.31746562 -0.0095022 ]]
Durbin-Watson = 2.02273974971
stock= ABMD  theta= [ 0.70484455  0.97523926]  R2-proxy= 0.994663890429
jmean= -0.17335521284  jstd= 2.69586505875  jkurt= 0.563975504556
ARMA(5,0) params =  [-0.06967265 -0.37927941 -0.06836187 -0.15830212 -0.02806978 -0.01129251]
conf interval = [[-0.28212083  0.14277553]
[-0.56395246 -0.19460635]
[-0.26554736  0.12882362]
[-0.35321611  0.03661186]
[-0.2248097   0.16867013]
[-0.19478599  0.17220096]]
Durbin-Watson = 2.00028399329
stock= WSM  theta= [ 0.90417518  0.88528278]  R2-proxy= 0.979200676615
jmean= -0.0758359125838  jstd= 2.04587883004  jkurt= -0.738194945686
ARMA(5,0) params =  [-0.07759654 -0.63412656 -0.35847934 -0.40687518 -0.32489902 -0.10054766]
conf interval = [[-0.2467333   0.09154021]
[-0.83893454 -0.42931858]
[-0.5913581  -0.12560058]
[-0.64394773 -0.16980262]
[-0.55909516 -0.09070288]
[-0.32723388  0.12613856]]
Durbin-Watson = 1.98019003093
stock= RTEC  theta= [ 0.99997471  0.95095665]  R2-proxy= 0.98738055187
jmean= -0.0726923603272  jstd= 2.85105509943  jkurt= 0.409323456797
ARMA(5,0) params =  [-0.01990631 -0.61781906 -0.44662155 -0.22899039 -0.14689118 -0.06684741]
conf interval = [[-0.15727368  0.11746106]
[-0.74547294 -0.49016518]
[-0.59619733 -0.29704577]
[-0.38715019 -0.07083059]
[-0.2978158   0.00403345]
[-0.19641206  0.06271725]]
Durbin-Watson = 2.00482883223
stock= LXU  theta= [ 0.9999005   0.95224293]  R2-proxy= 0.988483231896
jmean= -0.0231412032909  jstd= 3.17393327474  jkurt= -0.449650364269
ARMA(5,0) params =  [ 0.03932204 -0.56342612 -0.4081642  -0.31210742 -0.10649639 -0.04643003]
conf interval = [[-0.07899715  0.15764124]
[-0.67016519 -0.45668706]
[-0.53116632 -0.28516208]
[-0.43821326 -0.18600158]
[-0.2294076   0.01641482]
[-0.15411345  0.06125339]]
Durbin-Watson = 2.00907209179
stock= ELNK  theta= [ 0.999968   0.9213232]  R2-proxy= 0.986000669409
jmean= 0.0413440491317  jstd= 3.11474759639  jkurt= 0.0991784878528
ARMA(5,0) params =  [ 0.0012655  -0.38802911 -0.57528908 -0.31638771 -0.13320647 -0.24019667]
conf interval = [[-0.28873359  0.29126458]
[-0.71332644 -0.06273178]
[-0.93010682 -0.22047135]
[-0.70038708  0.06761165]
[-0.48499478  0.21858185]
[-0.5622117   0.08181836]]
Durbin-Watson = 2.07253489333
stock= CVO  theta= [ 0.85541401  0.84612859]  R2-proxy= 0.9670393023
jmean= -0.000375969042805  jstd= 2.65795083161  jkurt= -0.0647402701212
ARMA(5,0) params =  [-0.1406462  -0.42909725 -0.2882266  -0.25570435 -0.23604482  0.0161032 ]
conf interval = [[-0.38082953  0.09953712]
[-0.7008704  -0.15732411]
[-0.57769461  0.00124141]
[-0.54050336  0.02909465]
[-0.51867059  0.04658094]
[-0.2533382   0.2855446 ]]
Durbin-Watson = 1.984188253
stock= GOV  theta= [ 0.95575066  0.99817121]  R2-proxy= 0.971826381963
jmean= -0.163095015318  jstd= 2.14094258343  jkurt= -0.30008383327
ARMA(5,0) params =  [ 0.10027954 -0.55416268 -0.14634413 -0.12178947 -0.17429381 -0.0978434 ]
conf interval = [[-0.18159692  0.38215601]
[-0.80183432 -0.30649104]
[-0.42579448  0.13310621]
[-0.39819059  0.15461165]
[-0.45358058  0.10499296]
[-0.36338928  0.16770248]]
Durbin-Watson = 2.02098051411
stock= RRGB  theta= [ 0.65652263  0.98604002]  R2-proxy= 0.988128633248
jmean= 0.0915109500596  jstd= 2.73387451511  jkurt= 1.25316497823
ARMA(5,0) params =  [ 0.02048212 -0.37193339 -0.20100143 -0.07148445 -0.09553108 -0.12803281]
conf interval = [[-0.22549535  0.2664596 ]
[-0.54737647 -0.19649031]
[-0.38814289 -0.01385996]
[-0.26228454  0.11931564]
[-0.28287456  0.0918124 ]
[-0.30696347  0.05089785]]
Durbin-Watson = 2.01457163572
stock= SIRO  theta= [ 0.91913269  0.90214069]  R2-proxy= 0.960775810547
jmean= 0.0219361865932  jstd= 2.76774120343  jkurt= 0.116456260397
ARMA(5,0) params =  [ 0.03579251 -0.45195314 -0.18457566 -0.16853396 -0.06132811 -0.19767589]
conf interval = [[-0.16198369  0.2335687 ]
[-0.63049507 -0.2734112 ]
[-0.3820811   0.01292979]
[-0.36572004  0.02865212]
[-0.25941037  0.13675414]
[-0.37816536 -0.01718643]]
Durbin-Watson = 2.01391674468
stock= FFIN  theta= [ 0.75637651  0.84692317]  R2-proxy= 0.974699980779
jmean= 0.0246738667589  jstd= 2.52586246113  jkurt= -0.375911219342
ARMA(5,0) params =  [-0.1327562  -0.69290713 -0.39474342 -0.32534413 -0.25512602 -0.14183444]
conf interval = [[-0.30459975  0.03908734]
[-0.86192119 -0.52389308]
[-0.59646105 -0.19302579]
[-0.52964049 -0.12104778]
[-0.45510115 -0.05515088]
[-0.30870983  0.02504095]]
Durbin-Watson = 2.02535220516
stock= PBCT  theta= [ 0.99998252  1.05584298]  R2-proxy= 0.819130524377
jmean= -0.140674883209  jstd= 3.4369916645  jkurt= 0.257548368561
ARMA(5,0) params =  [ 0.11303233 -0.64678227 -0.35096918 -0.25418904 -0.18593906 -0.14135883]
conf interval = [[-0.09721903  0.32328369]
[-0.86463886 -0.42892567]
[-0.60961467 -0.0923237 ]
[-0.51871161  0.01033354]
[-0.44757253  0.07569441]
[-0.36171418  0.07899651]]
Durbin-Watson = 1.98526752154
stock= MM  theta= [ 3.71649539  0.87487951]  R2-proxy= 0.901219968427
jmean= 0.120326715594  jstd= 2.92274201481  jkurt= -0.736208470008
ARMA(5,0) params =  [-0.03012048 -0.49317388 -0.32768766 -0.15968667 -0.17800506 -0.01961474]
conf interval = [[-0.25764933  0.19740837]
[-0.67075437 -0.31559338]
[-0.52307616 -0.13229916]
[-0.36255305  0.04317971]
[-0.37287938  0.01686925]
[-0.19639737  0.15716789]]
Durbin-Watson = 1.99717708939
stock= SWKS  theta= [ 0.999925    0.93874668]  R2-proxy= 0.987113589413
jmean= -0.0436152260188  jstd= 3.14062479357  jkurt= 0.530733492819
ARMA(5,0) params =  [-0.01001171 -0.5902579  -0.29387853 -0.16038168 -0.1203014  -0.15294941]
conf interval = [[-0.10243313  0.0824097 ]
[-0.67659876 -0.50391705]
[-0.39401974 -0.19373732]
[-0.26284498 -0.05791837]
[-0.2204709  -0.02013189]
[-0.23922494 -0.06667388]]
Durbin-Watson = 2.04473537563
stock= WABC  theta= [ 0.99989841  0.96155137]  R2-proxy= 0.986527649263
jmean= -0.00782053808363  jstd= 2.8736925083  jkurt= -0.076849848246
ARMA(5,0) params =  [ 0.06378981 -0.56362914 -0.38215818 -0.29907376 -0.25818181 -0.03560436]
conf interval = [[-0.05587069  0.18345032]
[-0.68805943 -0.43919885]
[-0.52142168 -0.24289467]
[-0.44103789 -0.15710963]
[-0.39679347 -0.11957015]
[-0.15947341  0.08826469]]
Durbin-Watson = 1.99587323171
stock= ITRI  theta= [ 0.99999045  0.96846545]  R2-proxy= 0.990423332284
jmean= 0.0615601374759  jstd= 2.83441739931  jkurt= 0.711771540735
ARMA(5,0) params =  [-0.03780849 -0.56476071 -0.45264726 -0.19573311 -0.1036705  -0.14044692]
conf interval = [[-0.23572947  0.16011248]
[-0.74007688 -0.38944455]
[-0.65849974 -0.24679478]
[-0.41894185  0.02747563]
[-0.31293834  0.10559734]
[-0.32345428  0.04256043]]
Durbin-Watson = 2.01714220619
stock= APH  theta= [ 0.99997087  0.90554231]  R2-proxy= 0.986838941448
jmean= -0.0468059384601  jstd= 3.2256552151  jkurt= 0.402811158044
ARMA(5,0) params =  [-0.1382487  -0.43370062 -0.48237105 -0.21822605 -0.08130685  0.1097229 ]
conf interval = [[-0.47598674  0.19948933]
[-0.72356968 -0.14383155]
[-0.79591614 -0.16882596]
[-0.56433186  0.12787977]
[-0.40336965  0.24075595]
[-0.22513336  0.44457915]]
Durbin-Watson = 1.93700372835
stock= DF  theta= [ 1.52158777  0.85590622]  R2-proxy= 0.975013901
jmean= -0.062612213295  jstd= 2.78316583175  jkurt= 1.88873401075
ARMA(5,0) params =  [-0.17384638 -0.26603544 -0.26434288 -0.15845732 -0.12736092 -0.10798378]
conf interval = [[-0.54297038  0.19527761]
[-0.56915491  0.03708404]
[-0.57239929  0.04371353]
[-0.47140243  0.1544878 ]
[-0.43036599  0.17564415]
[-0.40933649  0.19336892]]
Durbin-Watson = 2.0184672955
stock= HTZ  theta= [ 0.82925885  0.8408738 ]  R2-proxy= 0.966439185707
jmean= -0.163601869967  jstd= 2.41179427204  jkurt= 1.13116333592
ARMA(5,0) params =  [-0.04686149 -0.57943133 -0.5593799  -0.45323373 -0.26790802 -0.28918723]
conf interval = [[-0.26191092  0.16818794]
[-0.82467987 -0.33418278]
[-0.83794539 -0.2808144 ]
[-0.74836003 -0.15810743]
[-0.55075361  0.01493757]
[-0.5356335  -0.04274096]]
Durbin-Watson = 1.94479749179
stock= ADSK  theta= [ 0.99990461  0.87975262]  R2-proxy= 0.966511655279
jmean= -0.0608812470196  jstd= 3.11985008957  jkurt= 1.59626290008
ARMA(5,0) params =  [-0.00673632 -0.54673592 -0.40667261 -0.17887949  0.02444398 -0.08980181]
conf interval = [[-0.15691     0.14343736]
[-0.68364732 -0.40982453]
[-0.56492959 -0.24841564]
[-0.34636048 -0.01139851]
[-0.13421696  0.18310493]
[-0.22907946  0.04947584]]
Durbin-Watson = 2.0227822874
stock= KNL  theta= [ 0.95427927  0.99804168]  R2-proxy= 0.995543290108
jmean= -0.00713138895544  jstd= 2.80825219332  jkurt= 0.32656384927
ARMA(5,0) params =  [ 0.08619826 -0.49359831 -0.31064723 -0.25699899 -0.182569   -0.02008305]
conf interval = [[-0.08676613  0.25916265]
[-0.67134807 -0.31584856]
[-0.51097792 -0.11031654]
[-0.46406261 -0.04993537]
[-0.38812317  0.02298517]
[-0.20431723  0.16415114]]
Durbin-Watson = 1.99190602202
stock= ACGL  theta= [ 0.99989168  0.8814233 ]  R2-proxy= 0.967120911123
jmean= 0.0938852337025  jstd= 2.44941820204  jkurt= 0.137137823914
ARMA(5,0) params =  [-0.03421737 -0.58401673 -0.41565273 -0.22129808 -0.15664703 -0.0030519 ]
conf interval = [[-0.2126274   0.14419266]
[-0.76367013 -0.40436333]
[-0.62194742 -0.20935803]
[-0.4366325  -0.00596366]
[-0.36277743  0.04948337]
[-0.18185003  0.17574622]]
Durbin-Watson = 1.98975419161
stock= BKD  theta= [ 0.99999168  0.94905234]  R2-proxy= 0.984059650429
jmean= -0.036348243495  jstd= 2.74922067025  jkurt= -0.422663921316
ARMA(5,0) params =  [ 0.08990865 -0.62800203 -0.44498539 -0.40243617 -0.20670902  0.0143907 ]
conf interval = [[-0.09772067  0.27753798]
[-0.8189747  -0.43702936]
[-0.67158485 -0.21838594]
[-0.63326316 -0.17160919]
[-0.43350717  0.02008913]
[-0.18041147  0.20919288]]
Durbin-Watson = 1.94445921826
stock= TITN  theta= [ 0.88026953  0.91809454]  R2-proxy= 0.949216107532
jmean= 0.0752816585322  jstd= 3.15815645718  jkurt= 0.0779119339733
ARMA(5,0) params =  [-0.0612254  -0.58965975 -0.32520984 -0.32666317 -0.35209366 -0.2650505 ]
conf interval = [[-0.2095061   0.0870553 ]
[-0.75322799 -0.42609151]
[-0.5107132  -0.13970647]
[-0.51794601 -0.13538034]
[-0.54360565 -0.16058168]
[-0.43541752 -0.09468349]]
Durbin-Watson = 1.96074031549
stock= FNSR  theta= [ 0.99997566  0.92195708]  R2-proxy= 0.981508363376
jmean= -0.0812974127143  jstd= 2.97538920587  jkurt= -0.119269543595
ARMA(5,0) params =  [-0.02252418 -0.44930442 -0.34837653 -0.2484514  -0.12215994 -0.12952747]
conf interval = [[-0.16515118  0.12010283]
[-0.5692527  -0.32935614]
[-0.47966866 -0.21708441]
[-0.38375695 -0.11314586]
[-0.25413184  0.00981196]
[-0.25071525 -0.00833969]]
Durbin-Watson = 2.03930865536
stock= CR  theta= [ 0.99996777  0.95592438]  R2-proxy= 0.991360130799
jmean= -0.0152596950666  jstd= 3.00546588651  jkurt= 0.706067872065
ARMA(5,0) params =  [-0.01755189 -0.53094024 -0.42382986 -0.21436448 -0.26592312 -0.07076629]
conf interval = [[-0.1286674   0.09356361]
[-0.64517809 -0.41670239]
[-0.54960726 -0.29805245]
[-0.34698798 -0.08174099]
[-0.39173496 -0.14011128]
[-0.18510913  0.04357654]]
Durbin-Watson = 2.0212206639
stock= UEIC  theta= [ 0.99998695  1.00037433]  R2-proxy= 0.964813942523
jmean= -0.0167775620858  jstd= 2.81951571272  jkurt= -0.26297173277
ARMA(5,0) params =  [ 0.05958843 -0.53431779 -0.1813146  -0.12467358 -0.20001362 -0.1037217 ]
conf interval = [[-0.14166165  0.2608385 ]
[-0.72255218 -0.34608339]
[-0.39115506  0.02852586]
[-0.33572192  0.08637477]
[-0.40944815  0.00942091]
[-0.29163296  0.08418957]]
Durbin-Watson = 1.87204339943
stock= VOD  theta= [ 0.99988511  0.9684429 ]  R2-proxy= 0.988574706045
jmean= 0.0341689907168  jstd= 2.68428560669  jkurt= 0.0760904997802
ARMA(5,0) params =  [-0.05122744 -0.48501565 -0.41501785 -0.24404891 -0.14194714 -0.10856455]
conf interval = [[-0.16909409  0.06663921]
[-0.59802878 -0.37200252]
[-0.54040584 -0.28962986]
[-0.3747499  -0.11334792]
[-0.26713812 -0.01675616]
[-0.22283651  0.00570741]]
Durbin-Watson = 2.00102539335
stock= CNA  theta= [ 0.71826137  0.97461694]  R2-proxy= 0.995891078916
jmean= -0.0492465190465  jstd= 2.7931347235  jkurt= -0.0348922231121
ARMA(5,0) params =  [-0.05369399 -0.59573017 -0.39671404 -0.27418054 -0.17190969 -0.12382285]
conf interval = [[-0.123026    0.01563803]
[-0.66902867 -0.52243167]
[-0.48140455 -0.31202353]
[-0.36154725 -0.18681383]
[-0.25660664 -0.08721274]
[-0.19720788 -0.05043781]]
Durbin-Watson = 2.00888138987
stock= SYY  theta= [ 2.76419304  0.87336548]  R2-proxy= 0.937049618714
jmean= -0.0507207006549  jstd= 2.80811566284  jkurt= -0.218892394808
ARMA(5,0) params =  [ 0.02119349 -0.53940883 -0.34433548 -0.2873687  -0.0492023  -0.03619517]
conf interval = [[-0.0932662   0.13565318]
[-0.64775365 -0.431064  ]
[-0.46736608 -0.22130487]
[-0.41203292 -0.16270448]
[-0.17208607  0.07368146]
[-0.14423561  0.07184526]]
Durbin-Watson = 2.00398692379
stock= NJR  theta= [ 0.99997756  0.89954243]  R2-proxy= 0.966276881647
jmean= 0.023756193789  jstd= 2.74137033882  jkurt= -0.446214916682
ARMA(5,0) params =  [-0.25152044 -0.43653422 -0.63326935 -0.22509904 -0.34268369  0.01211151]
conf interval = [[-0.55543805  0.05239716]
[-0.77396888 -0.09909956]
[-0.98026562 -0.28627309]
[-0.64983747  0.19963939]
[-0.70686152  0.02149413]
[-0.35422376  0.37844677]]
Durbin-Watson = 2.05802814232
stock= VIAB  theta= [ 2.00838757  0.87315402]  R2-proxy= 0.967646301708
jmean= -0.231294144342  jstd= 2.89656118  jkurt= -0.67614828937
ARMA(5,0) params =  [ 0.03564024 -0.5580667  -0.43847423 -0.25067058  0.04581651  0.26278659]
conf interval = [[ -2.02252714e-01   2.73533191e-01]
[ -7.58833424e-01  -3.57299983e-01]
[ -6.75134566e-01  -2.01813894e-01]
[ -5.01691461e-01   3.50292822e-04]
[ -1.96534906e-01   2.88167917e-01]
[  5.52303001e-02   4.70342885e-01]]
Durbin-Watson = 1.9477401723
stock= EXLS  theta= [ 0.73685341  0.8285745 ]  R2-proxy= 0.931674210449
jmean= 0.0558825528152  jstd= 2.93067910202  jkurt= 0.0095287770534
ARMA(5,0) params =  [-0.04420292 -0.2484656  -0.34699447 -0.13913302 -0.17605756 -0.29131132]
conf interval = [[-0.26828527  0.17987943]
[-0.44461544 -0.05231577]
[-0.54755916 -0.14642978]
[-0.36695784  0.0886918 ]
[-0.39430374  0.04218861]
[-0.50202161 -0.08060103]]
Durbin-Watson = 1.99899094517
stock= YRCW  theta= [ 0.99996333  0.84023746]  R2-proxy= 0.983817595569
jmean= -0.0570336638377  jstd= 2.64894345043  jkurt= 0.440215848544
ARMA(5,0) params =  [-0.06597955 -0.66169291 -0.35974653 -0.08207264  0.00604032  0.00075826]
conf interval = [[-0.2344301   0.10247101]
[-0.80405751 -0.51932831]
[-0.53135033 -0.18814272]
[-0.26184341  0.09769813]
[-0.16630391  0.17838454]
[-0.14140864  0.14292517]]
Durbin-Watson = 1.99308530888
stock= AFAM  theta= [ 0.6088053   0.99710295]  R2-proxy= 0.995338473687
jmean= -0.0750479717568  jstd= 2.99430216076  jkurt= 0.784948129538
ARMA(5,0) params =  [ 0.06809011 -0.42836933 -0.24515901 -0.22646454 -0.01462457 -0.10068529]
conf interval = [[-0.1653122   0.30149243]
[-0.64272086 -0.2140178 ]
[-0.48007286 -0.01024515]
[-0.46179564  0.00886656]
[-0.24724669  0.21799755]
[-0.31277882  0.11140825]]
Durbin-Watson = 2.01180692655
stock= CDR  theta= [ 0.99998816  0.78248214]  R2-proxy= 0.962179159197
jmean= 0.0678235846154  jstd= 2.40588710639  jkurt= 0.519379787365
ARMA(5,0) params =  [-0.05272832 -0.58995994 -0.40761636 -0.23785661 -0.31847617 -0.08155858]
conf interval = [[-0.21297232  0.10751568]
[-0.76975458 -0.41016531]
[-0.61096603 -0.2042667 ]
[-0.45071244 -0.02500078]
[-0.52113403 -0.11581832]
[-0.26070203  0.09758487]]
Durbin-Watson = 2.01911924946
stock= PSEM  theta= [ 0.85137043  0.9974492 ]  R2-proxy= 0.98893167627
jmean= -0.0575733921264  jstd= 2.78570433969  jkurt= -0.1865601175
ARMA(5,0) params =  [-0.14254734 -0.630148   -0.41103835 -0.2297352   0.06516916  0.09030555]
conf interval = [[-0.43399439  0.1488997 ]
[-0.88638481 -0.3739112 ]
[-0.71316482 -0.10891188]
[-0.5429047   0.08343429]
[-0.23493227  0.3652706 ]
[-0.1608401   0.34145119]]
Durbin-Watson = 2.01208455331
stock= PHH  theta= [ 0.99993928  0.90744927]  R2-proxy= 0.972040977034
jmean= -0.113406976097  jstd= 2.87095587282  jkurt= 0.202135711007
/usr/lib/python2.7/dist-packages/scipy/integrate/quadpack.py:321: IntegrationWarning: The maximum number of subdivisions (50) has been achieved.
If increasing the limit yields no improvement it is advised to analyze
the integrand in order to determine the difficulties.  If the position of a
local difficulty can be determined (singularity, discontinuity) one will
probably gain from splitting up the interval and calling the integrator
on the subranges.  Perhaps a special-purpose integrator should be used.
warnings.warn(msg, IntegrationWarning)
ARMA(5,0) params =  [-0.05178284 -0.66324716 -0.46699886 -0.30719121 -0.20250933 -0.04684167]
conf interval = [[-0.13893775  0.03537208]
[-0.75872653 -0.56776779]
[-0.58034894 -0.35364877]
[-0.42548749 -0.18889492]
[-0.31586398 -0.08915468]
[-0.14222426  0.04854091]]
Durbin-Watson = 1.99312766937
stock= PVA  theta= [ 0.98068119  1.00003109]  R2-proxy= 0.967399052158
jmean= -0.0543065432594  jstd= 2.95063094724  jkurt= 0.120339706885
ARMA(5,0) params =  [ 0.04962918 -0.64038529 -0.50157434 -0.42551256 -0.23581859 -0.15677098]
conf interval = [[-0.1220842   0.22134256]
[-0.83199875 -0.44877183]
[-0.72550141 -0.27764728]
[-0.65346111 -0.19756402]
[-0.45721838 -0.01441879]
[-0.34558449  0.03204254]]
Durbin-Watson = 2.02911883261
stock= WY  theta= [ 5.07933269  0.81997644]  R2-proxy= 0.966626549696
jmean= 0.0324613955858  jstd= 3.11252435011  jkurt= -0.161639159355

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