$title GARCH -- Generalized Autoregressive Conditional Heteroskedasticity $ontext Example of restricted maximum likelihood estimation. Erwin Kalvelagen, 2001 References: Jon Kierkegaard, Estimation of Nonlinear Stochastic Processes, MSc Thesis, Technical University of Denmark, report IMM-EKS-2000-16, April 2000. Bollerslev, T. (1986), "Generalized Autoregressive Conditional Heteroskedasticity", Journal of Econometrics 31, 307-327. Bollerslev, T., Chou, R.Y. and Kroner, K.F. (1992), "ARCH Modeling in Finance: A Review of the Theory and Empirical Evidence," Journal of Econometrics 52, 5-59. Data set from: Stephen F. Gray, GARCH Code, http://www.duke.edu/~sg12/software/garch/garch.htm $offtext *------- order of the GARCH(p,q) model ---------- $set p 1 $set q 1 *------------------------------------------------ set t 'dates yymmdd' / 920101, 920102, 920103, 920106, 920107, 920108, 920109, 920110, 920113 920114, 920115, 920116, 920117, 920120, 920121, 920122, 920123, 920124 920127, 920128, 920129, 920130, 920131, 920203, 920204, 920205, 920206 920207, 920210, 920211, 920212, 920213, 920214, 920217, 920218, 920219 920220, 920221, 920224, 920225, 920226, 920227, 920228, 920302, 920303 920304, 920305, 920306, 920309, 920310, 920311, 920312, 920313, 920316 920317, 920318, 920319, 920320, 920323, 920324, 920325, 920326, 920327 920330, 920331, 920401, 920402, 920403, 920406, 920407, 920408, 920409 920410, 920413, 920414, 920415, 920416, 920417, 920420, 920421, 920422 920423, 920424, 920427, 920428, 920429, 920430, 920501, 920504, 920505 920506, 920507, 920508, 920511, 920512, 920513, 920514, 920515, 920518 920519, 920520, 920521, 920522, 920525, 920526, 920527, 920528, 920529 920601, 920602, 920603, 920604, 920605, 920608, 920609, 920610, 920611 920612, 920615, 920616, 920617, 920618, 920619, 920622, 920623, 920624 920625, 920626, 920629, 920630, 920701, 920702, 920703, 920706, 920707 920708, 920709, 920710, 920713, 920714, 920715, 920716, 920717, 920720 920721, 920722, 920723, 920724, 920727, 920728, 920729, 920730, 920731 920803, 920804, 920805, 920806, 920807, 920810, 920811, 920812, 920813 920814, 920817, 920818, 920819, 920820, 920821, 920824, 920825, 920826 920827, 920828, 920831, 920901, 920902, 920903, 920904, 920907, 920908 920909, 920910, 920911, 920914, 920915, 920916, 920917, 920918, 920921 920922, 920923, 920924, 920925, 920928, 920929, 920930, 921001, 921002 921005, 921006, 921007, 921008, 921009, 921012, 921013, 921014, 921015 921016, 921019, 921020, 921021, 921022, 921023, 921026, 921027, 921028 921029, 921030, 921102, 921103, 921104, 921105, 921106, 921109, 921110 921111, 921112, 921113, 921116, 921117, 921118, 921119, 921120, 921123 921124, 921125, 921126, 921127, 921130, 921201, 921202, 921203, 921204 921207, 921208, 921209, 921210, 921211, 921214, 921215, 921216, 921217 921218, 921221, 921222, 921223, 921224, 921225, 921228, 921229, 921230 921231, 930101, 930104, 930105, 930106, 930107, 930108, 930111, 930112 930113, 930114, 930115, 930118, 930119, 930120, 930121, 930122, 930125 930126, 930127, 930128, 930129, 930201, 930202, 930203, 930204, 930205 930208, 930209, 930210, 930211, 930212, 930215, 930216, 930217, 930218 930219, 930222, 930223, 930224, 930225, 930226, 930301, 930302, 930303 930304, 930305, 930308, 930309, 930310, 930311, 930312, 930315, 930316 930317, 930318, 930319, 930322, 930323, 930324, 930325, 930326, 930329 930330, 930331, 930401, 930402, 930405, 930406, 930407, 930408, 930409 930412, 930413, 930414, 930415, 930416, 930419, 930420, 930421, 930422 930423, 930426, 930427, 930428, 930429, 930430, 930503, 930504, 930505 930506, 930507, 930510, 930511, 930512, 930513, 930514, 930517, 930518 930519, 930520, 930521, 930524, 930525, 930526, 930527, 930528, 930531 930601, 930602, 930603, 930604, 930607, 930608, 930609, 930610, 930611 930614, 930615, 930616, 930617, 930618, 930621, 930622, 930623, 930624 930625, 930628, 930629, 930630, 930701, 930702, 930705, 930706, 930707 930708, 930709, 930712, 930713, 930714, 930715, 930716, 930719, 930720 930721, 930722, 930723, 930726, 930727, 930728, 930729, 930730, 930802 930803, 930804, 930805, 930806, 930809, 930810, 930811, 930812, 930813 930816, 930817, 930818, 930819, 930820, 930823, 930824, 930825, 930826 930827, 930830, 930831, 930901, 930902, 930903, 930906, 930907, 930908 930909, 930910, 930913, 930914, 930915, 930916, 930917, 930920, 930921 930922, 930923, 930924, 930927, 930928, 930929, 930930, 931001, 931004 931005, 931006, 931007, 931008, 931011, 931012, 931013, 931014, 931015 931018, 931019, 931020, 931021, 931022, 931025, 931026, 931027, 931028 931029, 931101, 931102, 931103, 931104, 931105, 931108, 931109, 931110 931111, 931112, 931115, 931116, 931117, 931118, 931119, 931122, 931123 931124, 931125, 931126, 931129, 931130, 931201, 931202, 931203, 931206 931207, 931208, 931209, 931210, 931213, 931214, 931215, 931216, 931217 931220, 931221, 931222, 931223, 931224, 931227, 931228, 931229, 931230 931231, 940103, 940104, 940105, 940106, 940107, 940110, 940111, 940112 940113, 940114, 940117, 940118, 940119, 940120, 940121, 940124, 940125 940126, 940127, 940128, 940131, 940201, 940202, 940203, 940204, 940207 940208, 940209, 940210, 940211, 940214, 940215, 940216, 940217, 940218 940221, 940222, 940223, 940224, 940225, 940228, 940301, 940302, 940303 940304, 940307, 940308, 940309, 940310, 940311, 940314, 940315, 940316 940317, 940318, 940321, 940322, 940323, 940324, 940325, 940328, 940329 940330, 940331, 940401, 940404, 940405, 940406, 940407, 940408, 940411 940412, 940413, 940414, 940415, 940418, 940419, 940420, 940421, 940422 940425, 940426, 940427, 940428, 940429, 940502, 940503, 940504, 940505 940506, 940509, 940510, 940511, 940512, 940513, 940516, 940517, 940518 940519, 940520, 940523, 940524, 940525, 940526, 940527, 940530, 940531 940601, 940602, 940603, 940606, 940607, 940608, 940609, 940610, 940613 940614, 940615, 940616, 940617, 940620, 940621, 940622, 940623, 940624 940627, 940628, 940629, 940630, 940701, 940704, 940705, 940706, 940707 940708, 940711, 940712, 940713, 940714, 940715, 940718, 940719, 940720 940721, 940722, 940725, 940726, 940727, 940728, 940729, 940801, 940802 940803, 940804, 940805, 940808, 940809, 940810, 940811, 940812, 940815 940816, 940817, 940818, 940819, 940822, 940823, 940824, 940825, 940826 940829, 940830, 940831, 940901, 940902, 940905, 940906, 940907, 940908 940909, 940912, 940913, 940914, 940915, 940916, 940919, 940920, 940921 940922, 940923, 940926, 940927, 940928, 940929, 940930, 941003, 941004 941005, 941006, 941007, 941010, 941011, 941012, 941013, 941014, 941017 941018, 941019, 941020, 941021, 941024, 941025, 941026, 941027, 941028 941031, 941101, 941102, 941103, 941104, 941107, 941108, 941109, 941110 941111, 941114, 941115, 941116, 941117, 941118, 941121, 941122, 941123 941124, 941125, 941128, 941129, 941130, 941201, 941202, 941205, 941206 941207, 941208, 941209, 941212, 941213, 941214, 941215, 941216, 941219 941220, 941221, 941222, 941223, 941226, 941227, 941228, 941229, 941230 950102, 950103, 950104, 950105, 950106, 950109, 950110, 950111, 950112 950113, 950116, 950117, 950118, 950119, 950120, 950123, 950124, 950125 950126, 950127, 950130, 950131, 950201, 950202, 950203, 950206, 950207 950208, 950209, 950210, 950213, 950214, 950215, 950216, 950217, 950220 950221, 950222, 950223, 950224, 950227, 950228, 950301, 950302, 950303 950306, 950307, 950308, 950309, 950310, 950313, 950314, 950315, 950316 950317, 950320, 950321, 950322, 950323, 950324, 950327, 950328, 950329 950330, 950331, 950403, 950404, 950405, 950406, 950407, 950410, 950411 950412, 950413, 950414, 950417, 950418, 950419, 950420, 950421, 950424 950425, 950426, 950427, 950428, 950501, 950502, 950503, 950504, 950505 950508, 950509, 950510, 950511, 950512, 950515, 950516, 950517, 950518 950519, 950522, 950523, 950524, 950525, 950526, 950529, 950530, 950531 950601, 950602, 950605, 950606, 950607, 950608, 950609, 950612, 950613 950614, 950615, 950616, 950619, 950620, 950621, 950622, 950623, 950626 950627, 950628, 950629, 950630, 950703, 950704, 950705, 950706, 950707 950710, 950711, 950712, 950713, 950714, 950717, 950718, 950719, 950720 950721, 950724, 950725, 950726, 950727, 950728, 950731, 950801, 950802 950803, 950804, 950807, 950808, 950809, 950810, 950811, 950814, 950815 950816, 950817, 950818, 950821, 950822, 950823, 950824, 950825, 950828 950829, 950830, 950831, 950901, 950904, 950905, 950906, 950907, 950908 950911, 950912, 950913, 950914, 950915, 950918, 950919, 950920, 950921 950922, 950925, 950926, 950927, 950928, 950929, 951002, 951003, 951004 951005, 951006, 951009, 951010, 951011, 951012, 951013, 951016, 951017 951018, 951019, 951020, 951023, 951024, 951025, 951026, 951027, 951030 951031, 951101, 951102, 951103, 951106, 951107, 951108, 951109, 951110 951113, 951114, 951115, 951116, 951117, 951120, 951121, 951122, 951123 951124, 951127, 951128, 951129, 951130, 951201, 951204, 951205, 951206 951207, 951208, 951211, 951212, 951213, 951214, 951215, 951218, 951219 951220, 951221, 951222, 951225, 951226, 951227, 951228, 951229 /; parameter s(t) 'daily series US$/DM exchange rate' / 920101 0.6601, 920102 0.6532, 920103 0.6484, 920106 0.6597, 920107 0.6623 920108 0.6636, 920109 0.641 , 920110 0.6321, 920113 0.63441, 920114 0.6266 920115 0.6135, 920116 0.6174, 920117 0.6277, 920120 0.6277, 920121 0.6302 920122 0.6289, 920123 0.6225, 920124 0.6268, 920127 0.6203, 920128 0.6246 920129 0.6238, 920130 0.6192, 920131 0.62 , 920203 0.6244, 920204 0.6289 920205 0.6323, 920206 0.6337, 920207 0.641 , 920210 0.6339, 920211 0.6268 920212 0.6219, 920213 0.6161, 920214 0.6165, 920217 0.6165, 920218 0.6072 920219 0.6085, 920220 0.6075, 920221 0.6044, 920224 0.6051, 920225 0.6028 920226 0.6076, 920227 0.6112, 920228 0.6101, 920302 0.6086, 920303 0.6026 920304 0.599 , 920305 0.5979, 920306 0.5996, 920309 0.6014, 920310 0.5988 920311 0.599 , 920312 0.599 , 920313 0.5999, 920316 0.5999, 920317 0.607 920318 0.6048, 920319 0.5984, 920320 0.5986, 920323 0.5995, 920324 0.6014 920325 0.6053, 920326 0.6017, 920327 0.6101, 920330 0.6083, 920331 0.6085 920401 0.6053, 920402 0.6079, 920403 0.6154, 920406 0.6155, 920407 0.616 920408 0.6142, 920409 0.6187, 920410 0.6107, 920413 0.6042, 920414 0.6042 920415 0.6017, 920416 0.6102, 920417 0.5988, 920420 0.6006, 920421 0.5992 920422 0.6015, 920423 0.6019, 920424 0.6062, 920427 0.6053, 920428 0.6046 920429 0.6031, 920430 0.6064, 920501 0.6098, 920504 0.6073, 920505 0.6112 920506 0.6143, 920507 0.6105, 920508 0.6086, 920511 0.6103, 920512 0.616 920513 0.6211, 920514 0.6209, 920515 0.6207, 920518 0.6268, 920519 0.6287 920520 0.6244, 920521 0.619 , 920522 0.6179, 920525 0.6179, 920526 0.6205 920527 0.6102, 920528 0.6147, 920529 0.6221, 920601 0.6234, 920602 0.6205 920603 0.624 , 920604 0.6249, 920605 0.6289, 920608 0.6297, 920609 0.6283 920610 0.6275, 920611 0.6341, 920612 0.6337, 920615 0.6369, 920616 0.6396 920617 0.6346, 920618 0.6384, 920619 0.6357, 920622 0.6376, 920623 0.6388 920624 0.6452, 920625 0.6496, 920626 0.6515, 920629 0.6589, 920630 0.6561 920701 0.6572, 920702 0.6618, 920703 0.6596, 920706 0.6616, 920707 0.6709 920708 0.672 , 920709 0.6614, 920710 0.6687, 920713 0.6776, 920714 0.6741 920715 0.6752, 920716 0.6782, 920717 0.6859, 920720 0.6705, 920721 0.6716 920722 0.6727, 920723 0.6736, 920724 0.6691, 920727 0.6754, 920728 0.6791 920729 0.675 , 920730 0.6748, 920731 0.6781, 920803 0.678 , 920804 0.6771 920805 0.6748, 920806 0.6767, 920807 0.6814, 920810 0.6828, 920811 0.6812 920812 0.6832, 920813 0.6873, 920814 0.6824, 920817 0.6826, 920818 0.6863 920819 0.6863, 920820 0.6911, 920821 0.6995, 920824 0.7128, 920825 0.7141 920826 0.7114, 920827 0.7101, 920828 0.7102, 920831 0.7133, 920901 0.7194 920902 0.7184, 920903 0.7077, 920904 0.713 , 920907 0.713 , 920908 0.7179 920909 0.708 , 920910 0.7046, 920911 0.6916, 920914 0.6725, 920915 0.6707 920916 0.6601, 920917 0.6757, 920918 0.666 , 920921 0.6736, 920922 0.6687 920923 0.6673, 920924 0.6741, 920925 0.6757, 920928 0.6875, 920929 0.708 920930 0.707 , 921001 0.7025, 921002 0.7097, 921005 0.7052, 921006 0.7008 921007 0.6906, 921008 0.6745, 921009 0.6727, 921012 0.6727, 921013 0.6821 921014 0.6845, 921015 0.6885, 921016 0.6768, 921019 0.6678, 921020 0.6594 921021 0.6596, 921022 0.6633, 921023 0.6542, 921026 0.6513, 921027 0.6547 921028 0.6467, 921029 0.6498, 921030 0.6489, 921102 0.6371, 921103 0.6376 921104 0.6345, 921105 0.6321, 921106 0.6262, 921109 0.6254, 921110 0.6256 921111 0.6256, 921112 0.6331, 921113 0.6373, 921116 0.6275, 921117 0.6275 921118 0.6293, 921119 0.6357, 921120 0.6229, 921123 0.6262, 921124 0.6264 921125 0.6302, 921126 0.6302, 921127 0.6254, 921130 0.6283, 921201 0.6355 921202 0.6352, 921203 0.6313, 921204 0.6276, 921207 0.6416, 921208 0.6425 921209 0.6367, 921210 0.6329, 921211 0.6335, 921214 0.6369, 921215 0.6378 921216 0.642 , 921217 0.6416, 921218 0.639 , 921221 0.6376, 921222 0.6283 921223 0.6268, 921224 0.6254, 921225 0.6254, 921228 0.6173, 921229 0.619 921230 0.6188, 921231 0.6177, 930101 0.6177, 930104 0.61 , 930105 0.6155 930106 0.6111, 930107 0.6107, 930108 0.6079, 930111 0.613 , 930112 0.6131 930113 0.6156, 930114 0.6156, 930115 0.612 , 930118 0.612 , 930119 0.6217 930120 0.6231, 930121 0.6194, 930122 0.6285, 930125 0.6341, 930126 0.6337 930127 0.6307, 930128 0.6305, 930129 0.6207, 930201 0.6114, 930202 0.6094 930203 0.6079, 930204 0.6048, 930205 0.6031, 930208 0.6037, 930209 0.6051 930210 0.6029, 930211 0.6033, 930212 0.6031, 930215 0.6031, 930216 0.6141 930217 0.6161, 930218 0.6127, 930219 0.6099, 930222 0.6161, 930223 0.6172 930224 0.6146, 930225 0.6112, 930226 0.6112, 930301 0.604 , 930302 0.6095 930303 0.607 , 930304 0.6099, 930305 0.5993, 930308 0.6015, 930309 0.6004 930310 0.6003, 930311 0.602 , 930312 0.601 , 930315 0.6015, 930316 0.6015 930317 0.6017, 930318 0.6088, 930319 0.6111, 930322 0.6105, 930323 0.6132 930324 0.6109, 930325 0.6098, 930326 0.6123, 930329 0.6148, 930330 0.6183 930331 0.6221, 930401 0.6287, 930402 0.6246, 930405 0.627 , 930406 0.6203 930407 0.618 , 930408 0.6221, 930409 0.6221, 930412 0.627 , 930413 0.6325 930414 0.6272, 930415 0.6227, 930416 0.6184, 930419 0.6258, 930420 0.6274 930421 0.6248, 930422 0.6252, 930423 0.6319, 930426 0.6384, 930427 0.6329 930428 0.6327, 930429 0.6335, 930430 0.6306, 930503 0.6327, 930504 0.6347 930505 0.6332, 930506 0.6349, 930507 0.632 , 930510 0.6217, 930511 0.6219 930512 0.6202, 930513 0.619, 930514 0.6205, 930517 0.6197, 930518 0.6159 930519 0.6152, 930520 0.6198, 930521 0.6154, 930524 0.6114, 930525 0.6143 930526 0.6141, 930527 0.6223, 930528 0.6305, 930531 0.6305, 930601 0.6287 930602 0.6254, 930603 0.6248, 930604 0.6148, 930607 0.6167, 930608 0.6153 930609 0.6116, 930610 0.6135, 930611 0.6148, 930614 0.6139, 930615 0.6072 930616 0.6028, 930617 0.6028, 930618 0.5952, 930621 0.5922, 930622 0.5896 930623 0.5912, 930624 0.5857, 930625 0.587 , 930628 0.5888, 930629 0.5925 930630 0.5857, 930701 0.5901, 930702 0.5896, 930705 0.5896, 930706 0.5868 930707 0.5854, 930708 0.5872, 930709 0.5879, 930712 0.5785, 930713 0.5819 930714 0.5828, 930715 0.5802, 930716 0.5824, 930719 0.586 , 930720 0.5886 930721 0.5893, 930722 0.5865, 930723 0.5819, 930726 0.5799, 930727 0.5791 930728 0.5819, 930729 0.5745, 930730 0.5745, 930802 0.5851, 930803 0.586 930804 0.5847, 930805 0.5841, 930806 0.5898, 930809 0.5892, 930810 0.5825 930811 0.5805, 930812 0.5829, 930813 0.5844, 930816 0.5934, 930817 0.5906 930818 0.5954, 930819 0.5926, 930820 0.5972, 930823 0.5933, 930824 0.5954 930825 0.5934, 930826 0.6005, 930827 0.6008, 930830 0.5984, 930831 0.5964 930901 0.6026, 930902 0.6085, 930903 0.6165, 930906 0.6165, 930907 0.6211 930908 0.6192, 930909 0.6254, 930910 0.6266, 930913 0.6209, 930914 0.6213 930915 0.6268, 930916 0.6238, 930917 0.62 , 930920 0.6207, 930921 0.6086 930922 0.6141, 930923 0.6085, 930924 0.6103, 930927 0.6146, 930928 0.6194 930929 0.6198, 930930 0.6127, 931001 0.6131, 931004 0.616 , 931005 0.6156 931006 0.6165, 931007 0.6161, 931008 0.6236, 931011 0.6236, 931012 0.6274 931013 0.624 , 931014 0.6202, 931015 0.6187, 931018 0.6106, 931019 0.609 931020 0.6094, 931021 0.6002, 931022 0.5967, 931025 0.5935, 931026 0.5945 931027 0.5952, 931028 0.5974, 931029 0.5931, 931101 0.5904, 931102 0.5869 931103 0.5903, 931104 0.5895, 931105 0.5903, 931108 0.5938, 931109 0.5911 931110 0.5919, 931111 0.5919, 931112 0.594 , 931115 0.5909, 931116 0.5867 931117 0.5879, 931118 0.5839, 931119 0.5824, 931122 0.5868, 931123 0.5877 931124 0.5869, 931125 0.5869, 931126 0.584 , 931129 0.5851, 931130 0.5828 931201 0.5807, 931202 0.5805, 931203 0.5811, 931206 0.5881, 931207 0.5875 931208 0.5867, 931209 0.5863, 931210 0.5896, 931213 0.5857, 931214 0.5831 931215 0.5819, 931216 0.5847, 931217 0.586 , 931220 0.5843, 931221 0.5851 931222 0.5879, 931223 0.5901, 931224 0.5901, 931227 0.5877, 931228 0.5888 931229 0.5785, 931230 0.5765, 931231 0.5759, 940103 0.5732, 940104 0.5759 940105 0.5752, 940106 0.5731, 940107 0.5789, 940110 0.5765, 940111 0.5741 940112 0.5767, 940113 0.5712, 940114 0.5716, 940117 0.5716, 940118 0.5727 940119 0.5727, 940120 0.5755, 940121 0.5708, 940124 0.5716, 940125 0.5713 940126 0.5732, 940127 0.5779, 940128 0.5741, 940131 0.5772, 940201 0.5767 940202 0.577 , 940203 0.5745, 940204 0.5679, 940207 0.5682, 940208 0.5666 940209 0.5683, 940210 0.5706, 940211 0.571 , 940214 0.5794, 940215 0.5799 940216 0.5802, 940217 0.5794, 940218 0.5841, 940221 0.5841, 940222 0.5798 940223 0.5784, 940224 0.5818, 940225 0.5851, 940228 0.587 , 940301 0.5855 940302 0.5866, 940303 0.5847, 940304 0.5818, 940307 0.5812, 940308 0.5831 940309 0.5858, 940310 0.5955, 940311 0.5938, 940314 0.5918, 940315 0.5893 940316 0.5927, 940317 0.592 , 940318 0.59 , 940321 0.5915, 940322 0.5924 940323 0.5944, 940324 0.5996, 940325 0.6003, 940328 0.5976, 940329 0.5964 940330 0.5968, 940331 0.5983, 940401 0.59 , 940404 0.59 , 940405 0.5822 940406 0.5833, 940407 0.5826, 940408 0.5843, 940411 0.5838, 940412 0.5818 940413 0.5851, 940414 0.5847, 940415 0.5835, 940418 0.5855, 940419 0.5879 940420 0.5912, 940421 0.5913, 940422 0.5917, 940425 0.5947, 940426 0.5965 940427 0.599 , 940428 0.6019, 940429 0.6051, 940502 0.6066, 940503 0.6109 940504 0.6048, 940505 0.6001, 940506 0.6019, 940509 0.6051, 940510 0.5976 940511 0.599 , 940512 0.5998, 940513 0.5984, 940516 0.5976, 940517 0.5979 940518 0.6033, 940519 0.6035, 940520 0.6077, 940523 0.6092, 940524 0.6046 940525 0.6075, 940526 0.6068, 940527 0.6086, 940530 0.6086, 940531 0.6072 940601 0.6077, 940602 0.6046, 940603 0.5992, 940606 0.5984, 940607 0.6002 940608 0.5986, 940609 0.5993, 940610 0.6002, 940613 0.6077, 940614 0.6079 940615 0.6116, 940616 0.6131, 940617 0.6209, 940620 0.6209, 940621 0.6279 940622 0.6226, 940623 0.6234, 940624 0.6309, 940627 0.6319, 940628 0.6327 940629 0.6311, 940630 0.6303, 940701 0.6266, 940704 0.6266, 940705 0.6327 940706 0.6347, 940707 0.6361, 940708 0.6397, 940711 0.6542, 940712 0.6545 940713 0.65 , 940714 0.6427, 940715 0.6433, 940718 0.6466, 940719 0.638 940720 0.6398, 940721 0.6289, 940722 0.6258, 940725 0.6292, 940726 0.6307 940727 0.6351, 940728 0.6281, 940729 0.6309, 940801 0.6335, 940802 0.6317 940803 0.6346, 940804 0.6299, 940805 0.6333, 940808 0.6315, 940809 0.6323 940810 0.6315, 940811 0.6408, 940812 0.6436, 940815 0.6441, 940816 0.6421 940817 0.6445, 940818 0.6482, 940819 0.6503, 940822 0.6547, 940823 0.6525 940824 0.647 , 940825 0.6485, 940826 0.6351, 940829 0.6339, 940830 0.6344 940831 0.6325, 940901 0.6347, 940902 0.6429, 940905 0.6429, 940906 0.6485 940907 0.6452, 940908 0.6431, 940909 0.6502, 940912 0.6479, 940913 0.6485 940914 0.6486, 940915 0.6466, 940916 0.6481, 940919 0.6457, 940920 0.6441 940921 0.6466, 940922 0.6466, 940923 0.6467, 940926 0.6437, 940927 0.6479 940928 0.6448, 940929 0.6463, 940930 0.6446, 941003 0.6435, 941004 0.6466 941005 0.6478, 941006 0.6485, 941007 0.646 , 941010 0.646 , 941011 0.647 941012 0.6491, 941013 0.6515, 941014 0.6579, 941017 0.6673, 941018 0.6656 941019 0.666 , 941020 0.67 , 941021 0.6671, 941024 0.6696, 941025 0.6692 941026 0.6702, 941027 0.668 , 941028 0.6623, 941031 0.6648, 941101 0.6683 941102 0.6599, 941103 0.6583, 941104 0.6596, 941107 0.659 , 941108 0.6627 941109 0.654 , 941110 0.6532, 941111 0.6532, 941114 0.6477, 941115 0.6472 941116 0.6447, 941117 0.6463, 941118 0.6435, 941121 0.642 , 941122 0.6425 941123 0.6433, 941124 0.6433, 941125 0.6417, 941128 0.6384, 941129 0.6367 941130 0.6397, 941201 0.6355, 941202 0.6333, 941205 0.636 , 941206 0.6359 941207 0.6376, 941208 0.6341, 941209 0.6343, 941212 0.6361, 941213 0.6365 941214 0.6372, 941215 0.6367, 941216 0.6362, 941219 0.6352, 941220 0.6369 941221 0.6334, 941222 0.6348, 941223 0.6336, 941226 0.6336, 941227 0.6347 941228 0.6484, 941229 0.6441, 941230 0.6453, 950102 0.6453, 950103 0.6428 950104 0.641 , 950105 0.6451, 950106 0.6402, 950109 0.6509, 950110 0.6517 950111 0.6521, 950112 0.6543, 950113 0.6522, 950116 0.6522, 950117 0.6528 950118 0.6516, 950119 0.6603, 950120 0.6614, 950123 0.6612, 950124 0.661 950125 0.6593, 950126 0.659 , 950127 0.66 , 950130 0.6646, 950131 0.6558 950201 0.6586, 950202 0.6586, 950203 0.6549, 950206 0.6531, 950207 0.6509 950208 0.6527, 950209 0.6542, 950210 0.6582, 950213 0.6575, 950214 0.6628 950215 0.6627, 950216 0.6713, 950217 0.6746, 950220 0.6746, 950221 0.6786 950222 0.6804, 950223 0.6815, 950224 0.6839, 950227 0.683 , 950228 0.6845 950301 0.6834, 950302 0.6931, 950303 0.7005, 950306 0.71286, 950307 0.7297 950308 0.7158, 950309 0.7166, 950310 0.7066, 950313 0.711 , 950314 0.7062 950315 0.7188, 950316 0.7161, 950317 0.7215, 950320 0.7126, 950321 0.7087 950322 0.713, 950323 0.7126, 950324 0.7051, 950327 0.711 , 950328 0.7207 950329 0.7233, 950330 0.7095, 950331 0.7271, 950403 0.7287, 950404 0.7257 950405 0.7282, 950406 0.7278, 950407 0.7263, 950410 0.7094, 950411 0.7136 950412 0.7143, 950413 0.7205, 950414 0.7186, 950417 0.7307, 950418 0.7391 950419 0.7297, 950420 0.7227, 950421 0.7303, 950424 0.7273, 950425 0.7299 950426 0.7306, 950427 0.7254, 950428 0.7215, 950501 0.7192, 950502 0.7258 950503 0.7279, 950504 0.7286, 950505 0.7278, 950508 0.731 , 950509 0.7241 950510 0.7205, 950511 0.6983, 950512 0.6908, 950515 0.6969, 950516 0.6925 950517 0.6913, 950518 0.694 , 950519 0.69252, 950522 0.694 , 950523 0.6925 950524 0.6943, 950525 0.714 , 950526 0.7265, 950529 0.7265, 950530 0.7202 950531 0.706 , 950601 0.7101, 950602 0.71 , 950605 0.709 , 950606 0.7089 950607 0.7075, 950608 0.7097, 950609 0.711 , 950612 0.7133, 950613 0.7102 950614 0.713 , 950615 0.7117, 950616 0.7134, 950619 0.7156, 950620 0.7186 950621 0.7231, 950622 0.7158, 950623 0.7218, 950626 0.7192, 950627 0.7213 950628 0.7162, 950629 0.7247, 950630 0.7236, 950703 0.7246, 950704 0.7246 950705 0.7245, 950706 0.7254, 950707 0.7181, 950710 0.7168, 950711 0.7117 950712 0.7128, 950713 0.7199, 950714 0.7188, 950717 0.7166, 950718 0.7205 950719 0.7273, 950720 0.7241, 950721 0.7218, 950724 0.7219, 950725 0.7176 950726 0.721 , 950727 0.7229, 950728 0.7241, 950731 0.7214, 950801 0.7267 950802 0.7153, 950803 0.7188, 950804 0.7158, 950807 0.71 , 950808 0.7105 950809 0.7111, 950810 0.7046, 950811 0.695 , 950814 0.6969, 950815 0.6775 950816 0.6766, 950817 0.6791, 950818 0.6775, 950821 0.6773, 950822 0.6729 950823 0.6742, 950824 0.6787, 950825 0.6793, 950828 0.6829, 950829 0.677 950830 0.677 , 950831 0.6819, 950901 0.6835, 950904 0.6835, 950905 0.6841 950906 0.6771, 950907 0.6762, 950908 0.6766, 950911 0.6789, 950912 0.6787 950913 0.6693, 950914 0.6725, 950915 0.672 , 950918 0.6745, 950919 0.6718 950920 0.6845, 950921 0.703 , 950922 0.7025, 950925 0.6974, 950926 0.6957 950927 0.7022, 950928 0.7045, 950929 0.7005, 951002 0.6999, 951003 0.6957 951004 0.697 , 951005 0.7047, 951006 0.7035, 951009 0.7035, 951010 0.705 951011 0.7013, 951012 0.705 , 951013 0.7009, 951016 0.7031, 951017 0.7075 951018 0.7027, 951019 0.7115, 951020 0.7156, 951023 0.7216, 951024 0.7174 951025 0.7186, 951026 0.719 , 951027 0.7116, 951030 0.7095, 951031 0.7107 951101 0.7058, 951102 0.7026, 951103 0.7058, 951106 0.7077, 951107 0.706 951108 0.7035, 951109 0.7085, 951110 0.706 , 951113 0.7055, 951114 0.7075 951115 0.7104, 951116 0.711 , 951117 0.7113, 951120 0.711 , 951121 0.7102 951122 0.7091, 951123 0.7091, 951124 0.7045, 951127 0.6961, 951128 0.6976 951129 0.6959, 951130 0.6916, 951201 0.6914, 951204 0.6946, 951205 0.6976 951206 0.6926, 951207 0.6915, 951208 0.6906, 951211 0.6923, 951212 0.6911 951213 0.6897, 951214 0.6949, 951215 0.6927, 951218 0.6992, 951219 0.694 951220 0.6961, 951221 0.695 , 951222 0.6962, 951225 0.6962, 951226 0.6991 951227 0.6974, 951228 0.6944, 951229 0.6961 /; set t1(t) 'all but first'; t1(t+1)=yes; parameter y(t) 'percentage returns'; y(t1(t)) = 100*log(s(t)/s(t-1)); display y; scalar mpq 'max(p,q)'; mpq = max(%p%,%q%); set lag(t) 'observations from t >= max(p,q)'; lag(t+(1+mpq)) = yes; *display lag set p /p1 * p%p%/ q /q1 * q%q%/ ; variables L 'log likelihood (constants taken out)' h(t) 'sigma(t)=sqrt(h(t))' e(t) 'residuals' alpha_0 'GARCH parameter (constant)' alpha(q) 'GARCH parameters (q)' beta(p) 'GARCH parameters (p)' constant 'information structure: y = constant + e' ; equations loglike 'transformed log likelihood function' def_h(t) 'h(t) structure' stat_garch 'stationarity condition (GARCH model)' stat_igarch 'stationarity condition (IGARCH model)' series(t) 'defines the time series' ; loglike.. L =e= sum(lag(t), log(h(t)) + sqr(e(t))/h(t) ); def_h(lag(t)).. h(t) =e= alpha_0 + sum(q, alpha(q)*sqr(e(t-ord(q)))) + sum(p, beta(p)*h(t-ord(p))) ; stat_garch.. sum(q,alpha(q)) + sum(p,beta(p)) =l= 1; stat_igarch.. sum(q,alpha(q)) + sum(p,beta(p)) =e= 1; series(lag(t)).. y(t) =e= constant + e(t); * * lower bounds * alpha_0.lo = 0; alpha.lo(q) = 0; beta.lo(p) = 0; h.lo(t) = 0.01; * * upper bounds * alpha_0.up = 100; alpha.up(q) = 100; beta.up(p) = 100; h.up(t) = 100; * * initial values * alpha_0.l = .1; alpha.l(q) = .1; beta.l(p) = .1; h.l(t) = .1; e.l(t) = .1; model garch /loglike, def_h, stat_garch, series/; model igarch /loglike, def_h, stat_igarch, series/; solve garch using nlp minimizing L; solve igarch using nlp minimizing L;