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Posted: November 23rd, 2011, 3:24 am
I am using 4 years of data to pick co-integrated pairs. I am regressing S1 ~ S2 over 4 years of daily data. I used ADF test to test for co-integration and list of pairs which are co-integrated. I found that spread is mean reverting or stationary but the rate of mean reversion is very slow. It returns to mean 2-3 times in 4 years of data. How the mean reversion rate can be increased such that spread reverts to mean more than 20 times in 4 years of data? I want frequent reversion to mean to reduce the number of days I am in the position. Reducing the trade length will help me realizing the returns so quickly rather than waiting for so long.

Posted: November 23rd, 2011, 10:08 am
Try changing the time frame and reference point. Then look at your costs.

Posted: November 23rd, 2011, 10:41 am
I looked at different data frames and overall I found spread revert to its mean after around 20-30 days and this happens once in approximately 100 days.

Posted: November 24th, 2011, 12:54 pm
What time frames? Daily, hours, minutes, ticks? Usually these things swing in and out more frequently as you increase the granularity.

Posted: November 24th, 2011, 1:35 pm
I am using daily price only as I don't have intra day data. I used 4 years of daily data to get the co-integration coefficient. I have also found that, there are few pairs of stocks which are co integrated with 4 years of data but reverts to mean only once or twice. I want only pairs of stocks which return to mean frequently. Is there any technical way to do this?

Posted: November 24th, 2011, 4:18 pm
Ok, can I ask one thing ? Have you checked that the stock prices are actually I(1) before doing the cointegration ? ie are you unable to reject a unit root in the individual prices ? I find it strange that you are able to reject a unit root on the pair with so few crosses of the mean. I guess on 4 years worth of data we are talking like 1000 data points, so would have thought critical level at 1% on adf statistic would be something like ~ -3.5. That might be wrong as I don't have my adf stats with me at the mo. But I know it's pretty strict. Back when I did these sorts of analysis, sometimes I'd get a successful cointegration because one of the stocks had risen and then tanked. So the success / mean reversion etc was coming from that one name, and if you test it alone you couldn't say it was I(1), and hence it wouldn't actually make a good trade although the pair would pass the adf test.

Posted: November 25th, 2011, 4:24 am
Yes, I tested each time series i.e. individual stock prices are I(1). I used the adf.test to test for unit root in stocks price and statistics are greater than -2 => Non-stationarityI also checked their first difference d(t) = S(t) - s(t-1) and tested the difference which is I(0) i.e. stationaryI think that is valid point to say that Stock prices are I(1).For the case of spread I considered statistics value less than -3.43. Pairs of stocks whose spread has statistic less than -3.43, I considered the pair as co-integrated and good for pair trading. Below is the data set. You can plot and see. Does this spread make sense to trade in pair? Stock 1 Stock 2 Coefficient Spread726.5 72.05 15.99348228 -425.8303982726.7 72.8 15.99348228 -437.6255099730.35 73.15 15.99348228 -439.5732287720.25 76.55 15.99348228 -504.0510684726.1 77.3 15.99348228 -510.1961801752.45 73.3 15.99348228 -419.872251731.2 70.7 15.99348228 -399.5391971736.55 73.15 15.99348228 -433.3732287747.8 72.85 15.99348228 -417.325184753.75 74.75 15.99348228 -441.7628003743.85 77.75 15.99348228 -499.6432472743 79.9 15.99348228 -534.8792341742.1 79.85 15.99348228 -534.9795599735.9 78.65 15.99348228 -521.9873812735.1 77.3 15.99348228 -501.1961801753.3 75.85 15.99348228 -459.8056308766.5 75.35 15.99348228 -438.6088897764.8 76.9 15.99348228 -465.0987872766.35 75.5 15.99348228 -441.157912761.85 75.4 15.99348228 -444.0585638762.15 75.65 15.99348228 -447.7569344761.45 74.85 15.99348228 -435.6621486757.55 74.9 15.99348228 -440.3618227754.8 75.55 15.99348228 -453.5075861757.15 76.9 15.99348228 -472.7487872771.05 75.55 15.99348228 -437.2575861765 77 15.99348228 -466.4981355760 75.35 15.99348228 -445.1088897763.6 77.1 15.99348228 -469.4974837769.1 75.9 15.99348228 -444.8053049764.7 75.6 15.99348228 -444.4072603763.9 75.75 15.99348228 -447.6062826769.7 75.7 15.99348228 -441.0066085781.7 79.2 15.99348228 -484.9837965807.45 81.4 15.99348228 -494.4194575826.6 86.6 15.99348228 -558.4355653838.9 85.35 15.99348228 -526.1437125842.25 81.45 15.99348228 -460.4191316831.5 82.45 15.99348228 -487.1626139822 82.2 15.99348228 -492.6642433817.7 81.1 15.99348228 -479.3714128844.2 80.05 15.99348228 -436.0782564859.85 78.05 15.99348228 -388.4412918850.1 76.65 15.99348228 -375.8004167845.6 76.85 15.99348228 -383.4991131872.6 76.55 15.99348228 -351.7010684860.1 76.55 15.99348228 -364.2010684849.85 76.4 15.99348228 -372.0520461852 76 15.99348228 -363.5046532864.4 74.85 15.99348228 -332.7121486851.9 73.45 15.99348228 -322.8212734890.85 73.6 15.99348228 -286.2702957870.25 79.2 15.99348228 -396.4337965880.6 79.5 15.99348228 -390.8818411901.8 78.45 15.99348228 -352.8886848930.65 83.2 15.99348228 -400.0077256901.8 80.25 15.99348228 -381.6769529910.35 83.85 15.99348228 -430.7034891927.3 81.85 15.99348228 -381.7665245899.4 83.85 15.99348228 -441.6534891887.9 83.35 15.99348228 -445.1567479878.2 83.7 15.99348228 -460.4544667890.9 84.2 15.99348228 -455.7512079891 83.85 15.99348228 -450.0534891900.45 83.95 15.99348228 -442.2028373880.05 83.15 15.99348228 -449.8080515892.6 82.4 15.99348228 -425.2629398894.85 82.35 15.99348228 -422.2132656905.05 89.15 15.99348228 -520.7689451907.4 89.65 15.99348228 -526.4156863907.6 89.3 15.99348228 -520.6179675925.8 87.9 15.99348228 -480.0270923925.25 88.05 15.99348228 -482.9761146929.75 88.4 15.99348228 -484.0738334927.45 86.75 15.99348228 -459.9845877918.2 85.95 15.99348228 -456.4398018906 86.85 15.99348228 -483.0339359893.35 84.6 15.99348228 -459.6986008897.9 81.75 15.99348228 -409.5671763909.15 84.6 15.99348228 -443.8986008950.95 84.1 15.99348228 -394.1018596937.45 82.2 15.99348228 -377.2142433939.3 85.15 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562.77090541888.6 75.75 15.99348228 677.09371742066.55 79.3 15.99348228 798.26685532085.5 83.35 15.99348228 752.44325212140.8 86.9 15.99348228 750.966392376.4 94.6 15.99348228 863.41657642352.65 92.1 15.99348228 879.65028212214.8 90.95 15.99348228 760.19278682223.5 92.5 15.99348228 744.10288922108 83.35 15.99348228 774.94325212273.1 85.6 15.99348228 904.0579172277.35 83.7 15.99348228 938.69553332401.75 87.9 15.99348228 995.92290772373 91.35 15.99348228 911.99539392465 95.9 15.99348228 931.22504952391.35 99.7 15.99348228 796.79981682477.6 98.3 15.99348228 905.4406922516.1 95.65 15.99348228 986.32342012576.75 98.9 15.99348228 994.99460262466.75 97.4 15.99348228 908.98482612406.85 98.3 15.99348228 834.6906922288.35 93.45 15.99348228 793.75908112363.7 92.7 15.99348228 881.10419282284.65 90.95 15.99348228 830.04278682315.9 91.7 15.99348228 849.29767512207.85 89.05 15.99348228 783.63040312280.1 90.2 15.99348228 837.48789852364.75 91.2 15.99348228 906.14441622342 89.2 15.99348228 915.38138082296.4 87.45 15.99348228 897.76997472246.3 84.6 15.99348228 893.25139922343.5 90.3 15.99348228 899.28855022339.4 90.3 15.99348228 895.18855022447.6 97.9 15.99348228 881.83808492419.1 97.65 15.99348228 857.33645552281.85 95.5 15.99348228 754.47244242379.9 98.95 15.99348228 797.34492852379.8 95 15.99348228 860.41918352319.85 98 15.99348228 752.48873672261 101.05 15.99348228 644.85861572182 100.1 15.99348228 581.05242392203.5 94.55 15.99348228 691.31625062175.55 100.3 15.99348228 571.40372752168.15 102.75 15.99348228 524.81969592130.5 110.25 15.99348228 367.21857882317.35 108.75 15.99348228 578.05880222332.3 105.95 15.99348228 637.79055262197.25 105.05 15.99348228 517.13468662178.7 102.7 15.99348228 536.169372165.4 106.4 15.99348228 463.69348562091.15 97.35 15.99348228 534.18450022031.8 91.3 15.99348228 571.5950682143.95 92.2 15.99348228 669.35093392207.05 93.4 15.99348228 713.25875522079.55 93 15.99348228 592.15614811961.05 86.05 15.99348228 584.81084991952.15 87.8 15.99348228 547.92225591888.2 85.1 15.99348228 527.15465811721.8 87.35 15.99348228 324.7693231811.85 89.65 15.99348228 378.03431371833.4 93.05 15.99348228 345.2064741710.05 88.3 15.99348228 297.82551481804.75 94.9 15.99348228 286.96853181782.05 93.2 15.99348228 291.45745161883 94.95 15.99348228 364.41885761975.15 98.2 15.99348228 404.59004021904.95 98.2 15.99348228 334.39004021763.8 93.8 15.99348228 263.61136231575.05 80.95 15.99348228 280.37760951467.4 68.1 15.99348228 378.24385681541.65 76.15 15.99348228 323.74632451504.5 70.05 15.99348228 384.15656641764.7 73.5 15.99348228 589.17905251894.35 78.05 15.99348228 646.05870821918.95 83.5 15.99348228 583.49422971749.8 83 15.99348228 422.34097091657.3 80.35 15.99348228 372.22369891701.55 83.25 15.99348228 370.09260031776.5 83.8 15.99348228 436.24618511659.3 82.9 15.99348228 333.44031911636.15 83 15.99348228 308.69097091558.65 80.05 15.99348228 278.37174361497.3 75.3 15.99348228 292.99078441419.85 70 15.99348228 300.30624051366.15 67.05 15.99348228 293.78701321287.1 63.25 15.99348228 275.51224591396.05 62.35 15.99348228 398.85637991371.95 62.75 15.99348228 368.3589871350.6 64.05 15.99348228 326.21746011415.2 64 15.99348228 391.61713421469.45 62.35 15.99348228 472.25637991443.75 64.9 15.99348228 405.77300011387.4 64.65 15.99348228 353.42137071410.5 65.95 15.99348228 355.72984371511.75 67.85 15.99348228 426.59222741430.3 66.8 15.99348228 361.93538381511.95 65.55 15.99348228 463.57723661618.55 66.15 15.99348228 560.58114731606 66.35 15.99348228 544.83245081634.3 68.6 15.99348228 537.14711571612.75 70.2 15.99348228 490.0075441546.2 69.6 15.99348228 433.05363341538.3 66.05 15.99348228 481.93049551534.55 68.05 15.99348228 446.19353091523.8 69.3 15.99348228 415.45167811496.25 70.8 15.99348228 363.91145471480.05 69.2 15.99348228 373.30102631463.55 69.55 15.99348228 351.20330751449.05 69.45 15.99348228 338.30265581498.05 70.2 15.99348228 375.3075441532.65 71.15 15.99348228 394.71373591486.4 71.75 15.99348228 338.86764651505.35 73 15.99348228 337.82579371545.15 75.4 15.99348228 339.24143621630.15 77.85 15.99348228 385.05740461680.45 75.35 15.99348228 475.34111031629.25 69.9 15.99348228 511.30558871643.7 68.3 15.99348228 551.34516041616.8 67.95 15.99348228 530.04287921553.2 67.05 15.99348228 480.83701321586.35 68 15.99348228 498.79320511534.85 66.6 15.99348228 469.68408021565.5 67 15.99348228 493.93668731546.1 67.15 15.99348228 472.1376651485.8 65.55 15.99348228 437.42723661373.55 64.5 15.99348228 341.9703931403.25 63.55 15.99348228 386.86420121389.1 60.35 15.99348228 423.89334451445.6 60.8 15.99348228 473.19627751504.55 60.55 15.99348228 536.1446481530.2 61.05 15.99348228 553.79790691531.6 60.9 15.99348228 557.59692921421.5 58.85 15.99348228 480.28356791431.3 58.15 15.99348228 501.27900551438.55 58.7 15.99348228 499.73259021387.9 57.9 15.99348228 461.87737611412 58.3 15.99348228 479.57998321492.05 58.8 15.99348228 551.6332421510.2 58.95 15.99348228 567.38421971531.95 58.55 15.99348228 595.53161261525.05 58.45 15.99348228 590.23096081540.4 58.75 15.99348228 600.78291611498.55 58.35 15.99348228 565.3303091433.6 55.8 15.99348228 541.16368891362.35 53.1 15.99348228 513.0960911392.8 51.4 15.99348228 570.73501091352.4 50.1 15.99348228 551.12653781291.7 47.15 15.99348228 537.60731061254.75 47.3 15.99348228 498.25828821212.05 45.6 15.99348228 482.74720811274.5 46.35 15.99348228 533.20209641240.45 44.3 15.99348228 531.93873511201.3 42.95 15.99348228 514.37993611160.9 41 15.99348228 505.16722661151.25 40.45 15.99348228 504.31364181220.25 40.8 15.99348228 567.7159231255.8 38.2 15.99348228 644.8489771290.2 40.4 15.99348228 644.06331591377.5 43.3 15.99348228 684.98221731392.85 43.7 15.99348228 693.93482441364.6 42.2 15.99348228 689.67504781354 43.35 15.99348228 660.68254321387.95 44.15 15.99348228 681.83775741410.8 43.7 15.99348228 711.88482441527.9 45.55 15.99348228 799.39688221564.85 46 15.99348228 829.14981521594 45.7 15.99348228 863.09785991656.45 48.35 15.99348228 883.16513181585.4 47.9 15.99348228 819.31219891446.8 44.85 15.99348228 729.49231981412.2 45.6 15.99348228 682.89720811505.8 45.95 15.99348228 770.89948931576.9 48.75 15.99348228 797.21773891698.35 49.9 15.99348228 900.27523431708.5 50.45 15.99348228 901.6288191705.1 52.05 15.99348228 872.63924741688.65 58.85 15.99348228 747.43356791770.8 63.95 15.99348228 748.01680831754.3 60.5 15.99348228 786.69432211799.55 66.5 15.99348228 735.98342851751.7 70.15 15.99348228 629.75721821750.7 68.1 15.99348228 661.54385681742.15 65.05 15.99348228 701.77397781758 64.8 15.99348228 721.62234831805.4 68.1 15.99348228 716.24385681807.95 68.45 15.99348228 713.1961381665.05 64.7 15.99348228 630.27169661726.95 64.35 15.99348228 697.76941541965.05 67.9 15.99348228 879.09255331850.85 64 15.99348228 827.2671342

Posted: November 25th, 2011, 4:41 pm
Ok not plotted this, but if column 2 is the 2nd stock price, then it looks like you are looking at one illiquid name (stock 2) vs a liquid one (stock 1). Alot of the values in the 2nd row look repeated over many days so it looks like it trades only so often and you are just interpolating forward the values. This would not make a good candidate to trade as it seems the values for stock 2 are stale, and you are not looking at true prices synchronized on the same day.

Posted: November 26th, 2011, 5:43 am
Thanks for your help. I took other pair which is liquid 1. does this pair look stationary to you ? 2. Should the co-integrated pair's spread be stationary?Stock-1 Stock-2 Coefficient SPread726.5 72.05 15.99348228 -425.8303982726.7 72.8 15.99348228 -437.6255099730.35 73.15 15.99348228 -439.5732287720.25 76.55 15.99348228 -504.0510684726.1 77.3 15.99348228 -510.1961801752.45 73.3 15.99348228 -419.872251731.2 70.7 15.99348228 -399.5391971736.55 73.15 15.99348228 -433.3732287747.8 72.85 15.99348228 -417.325184753.75 74.75 15.99348228 -441.7628003743.85 77.75 15.99348228 -499.6432472743 79.9 15.99348228 -534.8792341742.1 79.85 15.99348228 -534.9795599735.9 78.65 15.99348228 -521.9873812735.1 77.3 15.99348228 -501.1961801753.3 75.85 15.99348228 -459.8056308766.5 75.35 15.99348228 -438.6088897764.8 76.9 15.99348228 -465.0987872766.35 75.5 15.99348228 -441.157912761.85 75.4 15.99348228 -444.0585638762.15 75.65 15.99348228 -447.7569344761.45 74.85 15.99348228 -435.6621486757.55 74.9 15.99348228 -440.3618227754.8 75.55 15.99348228 -453.5075861757.15 76.9 15.99348228 -472.7487872771.05 75.55 15.99348228 -437.2575861765 77 15.99348228 -466.4981355760 75.35 15.99348228 -445.1088897763.6 77.1 15.99348228 -469.4974837769.1 75.9 15.99348228 -444.8053049764.7 75.6 15.99348228 -444.4072603763.9 75.75 15.99348228 -447.6062826769.7 75.7 15.99348228 -441.0066085781.7 79.2 15.99348228 -484.9837965807.45 81.4 15.99348228 -494.4194575826.6 86.6 15.99348228 -558.4355653838.9 85.35 15.99348228 -526.1437125842.25 81.45 15.99348228 -460.4191316831.5 82.45 15.99348228 -487.1626139822 82.2 15.99348228 -492.6642433817.7 81.1 15.99348228 -479.3714128844.2 80.05 15.99348228 -436.0782564859.85 78.05 15.99348228 -388.4412918850.1 76.65 15.99348228 -375.8004167845.6 76.85 15.99348228 -383.4991131872.6 76.55 15.99348228 -351.7010684860.1 76.55 15.99348228 -364.2010684849.85 76.4 15.99348228 -372.0520461852 76 15.99348228 -363.5046532864.4 74.85 15.99348228 -332.7121486851.9 73.45 15.99348228 -322.8212734890.85 73.6 15.99348228 -286.2702957870.25 79.2 15.99348228 -396.4337965880.6 79.5 15.99348228 -390.8818411901.8 78.45 15.99348228 -352.8886848930.65 83.2 15.99348228 -400.0077256901.8 80.25 15.99348228 -381.6769529910.35 83.85 15.99348228 -430.7034891927.3 81.85 15.99348228 -381.7665245899.4 83.85 15.99348228 -441.6534891887.9 83.35 15.99348228 -445.1567479878.2 83.7 15.99348228 -460.4544667890.9 84.2 15.99348228 -455.7512079891 83.85 15.99348228 -450.0534891900.45 83.95 15.99348228 -442.2028373880.05 83.15 15.99348228 -449.8080515892.6 82.4 15.99348228 -425.2629398894.85 82.35 15.99348228 -422.2132656905.05 89.15 15.99348228 -520.7689451907.4 89.65 15.99348228 -526.4156863907.6 89.3 15.99348228 -520.6179675925.8 87.9 15.99348228 -480.0270923925.25 88.05 15.99348228 -482.9761146929.75 88.4 15.99348228 -484.0738334927.45 86.75 15.99348228 -459.9845877918.2 85.95 15.99348228 -456.4398018906 86.85 15.99348228 -483.0339359893.35 84.6 15.99348228 -459.6986008897.9 81.75 15.99348228 -409.5671763909.15 84.6 15.99348228 -443.8986008950.95 84.1 15.99348228 -394.1018596937.45 82.2 15.99348228 -377.2142433939.3 85.15 15.99348228 -422.545016939.75 89.25 15.99348228 -487.6682934927.6 88.15 15.99348228 -482.2254629927.6 87.8 15.99348228 -476.6277441913.5 86 15.99348228 -461.939476909.25 83.2 15.99348228 -421.4077256899.75 83.25 15.99348228 -431.7073997885.4 85.5 15.99348228 -482.0427348893.05 88.1 15.99348228 -515.9757887871.7 86.8 15.99348228 -516.5342618898.25 86.2 15.99348228 -480.3881724903.8 88.9 15.99348228 -518.0205746918.5 89.85 15.99348228 -518.5143827903.5 87.85 15.99348228 -501.5274182903.65 87.1 15.99348228 -489.3823065901.8 86.8 15.99348228 -486.4342618937.3 86.75 15.99348228 -450.1345877917.7 89.45 15.99348228 -512.9169898932.6 92.55 15.99348228 -547.5967849961.45 96.5 15.99348228 -581.9210399987.15 94.75 15.99348228 -528.23244591001.1 95.65 15.99348228 -528.67657991064.8 95.25 15.99348228 -458.5791871052.3 95.65 15.99348228 -477.47657991029.85 94.25 15.99348228 -477.5357048990.35 92.6 15.99348228 -490.646459966.75 88.25 15.99348228 -444.6748111955.85 86.85 15.99348228 -433.1839359988.95 90.25 15.99348228 -454.46177561001.75 92.35 15.99348228 -475.24808841037.85 91.9 15.99348228 -431.95102141019.75 91.5 15.99348228 -443.65362851039.4 93.5 15.99348228 -455.9905931018.45 92.45 15.99348228 -460.14743671023.55 93 15.99348228 -463.84385191030.4 92.85 15.99348228 -454.59482961000.9 92.2 15.99348228 -473.6990661994.55 90.2 15.99348228 -448.0621015983.8 91.25 15.99348228 -475.6052579991.9 95.2 15.99348228 -530.6795129981.6 92.55 15.99348228 -498.5967849985.85 91.65 15.99348228 -479.9526508979.4 90.6 15.99348228 -469.6094944983.25 91.05 15.99348228 -472.9565615973.9 88 15.99348228 -433.5264405950.85 86.25 15.99348228 -428.5878465923.7 85.4 15.99348228 -442.1433866912.9 86.85 15.99348228 -476.1339359910.6 88.15 15.99348228 -499.2254629926.8 89.75 15.99348228 -508.6150345944.35 91.45 15.99348228 -518.2539544938.95 91 15.99348228 -516.4568874967.9 90.95 15.99348228 -486.7072132969.65 90.8 15.99348228 -482.55819091000.3 89.95 15.99348228 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Posted: November 26th, 2011, 10:53 am
This series is more liquid so a better candidate. Both stocks rise and then drop, so you may be getting a false positive like I stated on my previous post due to this, but they do peak at similar times, so it's possible cointegration exists. Again I've not done the analysis or plotted, just looking at your numbers. A pair spread is stationary I(0), that's what you are looking for in a cointegrated pair. You can trade pairs based on other things and don't necessarily need a stationary spread, but for what you are doing with cointegration, the whole point is you are looking for a combination that gives you stationarity, with which you'll try to trade successfully.

Posted: November 26th, 2011, 3:43 pm
Hi Blade,I am sorry, in my previous blog's question no 2, I suppose to ask is co-integrated pair's spread normal? When I plot the spread, first half of the data remains below mean and second half of the data remains above the mean and spread is rising continuously. The problem with the spread is it does not fluctuate around mean. That raise concern that it is co-integrated spread but it just start from bottom ream mean and cross and never come back in 1000 points. Once you plot, you will get what I mean. I am not sure, how to use this spread in trading. I find strategy profitable if spread fluctuate around mean frequently.

Posted: November 26th, 2011, 4:58 pm
Ok on plotting, I think this might be ok and could be cointegrated. Try having a look at the spread of log prices though and fitting an model with trend. There are different statistics for adf with trend. If the trend is fairly constant then you can maybe play the spread around the trend. Don't think cointegrated spread has to be normal, but not 100% sure. Think as long as spread is stationary are there then you can call the spread cointegrated. Think in the case of non-normal all the adf methodology would work but the critical value may change.

Posted: November 27th, 2011, 1:47 pm
I worked out with the log of the series but looks like same. Still the spread is rising it implies mean is changing time to time. How to do the trend analysis? Could you guide me towards trend analysis? Should I first remove the trend in the price series and then calculate the spread or should I try to remove the trends directly from spread ? Could you guide me towards some material on that. I am new to time series analysis and don't know much about time series.

Posted: November 27th, 2011, 7:50 pm
Have a look at that. Should cover what you need to know, with prob sections 1.1 and 6.3 most relevant. http://www.yats.com/doc/cointegration-en.html