System of Equations Error (Near Singular Matrix) in a GMM(HAC) estimation

[blessingrugara]

Good day all,
Please help, I am new to to econometrics and Eviews, but I am doing a research paper on a fama and french asset pricing model applied on African stocks, I have 81 portfolios that I have setup as a system of equations (3 systems of 27 equations) and im trying to use the GMM(HAC)(my data is in times series format) method to regress them on the 5 fama and french factors.
So my issue is that I ran the model initially, it was fine, but i realised I had missed a single factor, when I added that factor back in, I got the Near Singular Matrix error. What is odd to me, is that if i switch to OLS estimation, the model works fine. But my research specifies the GMM method.
Also, I managed to estimate the equations individually, but I need them specified as a system so I can run a wald test to check that all the intercepts are jointly equal to 0, I've read on the forum that, that can only be done in a system of equations. I tried changing estimation settings, adding the identity weighting matrix works, but the p-values i get are 0 for the first 5 coefficients, then ~0.999 for the rest, which seems very wrong, based on the data I've seen in similar papers.
Can anyone please advise on where I'm going wrong or how to do this properly.
I setup my estimation setting (default), and picked instrumental variables based on youtube tutorials I saw of similar work, so I'm not actually sure what changing those settings would mean.

Please see below my equation specs (first set of 3):

P1_s_cma_w = c(1) + c(2)*(mkt_w) + c(3)*(smb_2x2_w) + c(4)*(hml_2x2_w) + c(5)*(rmw_2x2_w) + c(6)*(cma_2x2_w) @ P1_s_cma_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P2_s_cma_w = c(7) + c(8)*(mkt_w) + c(9)*(smb_2x2_w) + c(10)*(hml_2x2_w) + c(11)*(rmw_2x2_w) + c(12)*(cma_2x2_w) @ P2_s_cma_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P3_s_cma_w = c(13) + c(14)*(mkt_w) + c(15)*(smb_2x2_w) + c(16)*(hml_2x2_w) + c(17)*(rmw_2x2_w) + c(18)*(cma_2x2_w) @ P3_s_cma_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P4_s_cma_w = c(19) + c(20)*(mkt_w) + c(21)*(smb_2x2_w) + c(22)*(hml_2x2_w) + c(23)*(rmw_2x2_w) + c(24)*(cma_2x2_w) @ P4_s_cma_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P5_s_cma_w = c(25) + c(26)*(mkt_w) + c(27)*(smb_2x2_w) + c(28)*(hml_2x2_w) + c(29)*(rmw_2x2_w) + c(30)*(cma_2x2_w) @ P5_s_cma_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P6_s_cma_w = c(31) + c(32)*(mkt_w) + c(33)*(smb_2x2_w) + c(34)*(hml_2x2_w) + c(35)*(rmw_2x2_w) + c(36)*(cma_2x2_w) @ P6_s_cma_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P7_s_cma_w = c(37) + c(38)*(mkt_w) + c(39)*(smb_2x2_w) + c(40)*(hml_2x2_w) + c(41)*(rmw_2x2_w) + c(42)*(cma_2x2_w) @ P7_s_cma_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2(-1)
P8_s_cma_w = c(43) + c(44)*(mkt_w) + c(45)*(smb_2x2_w) + c(46)*(hml_2x2_w) + c(47)*(rmw_2x2_w) + c(48)*(cma_2x2_w) @ P8_s_cma_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2(-1)
P9_s_cma_w = c(49) + c(50)*(mkt_w) + c(51)*(smb_2x2_w) + c(52)*(hml_2x2_w) + c(53)*(rmw_2x2_w) + c(54)*(cma_2x2_w) @ P9_s_cma_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2(-1)
P1_s_hml_w = c(55) + c(56)*(mkt_w) + c(57)*(smb_2x2_w) + c(58)*(hml_2x2_w) + c(59)*(rmw_2x2_w) + c(60)*(cma_2x2_w) @ P1_s_hml_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P2_s_hml_w = c(61) + c(62)*(mkt_w) + c(63)*(smb_2x2_w) + c(64)*(hml_2x2_w) + c(65)*(rmw_2x2_w) + c(66)*(cma_2x2_w) @ P2_s_hml_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P3_s_hml_w = c(67) + c(68)*(mkt_w) + c(69)*(smb_2x2_w) + c(70)*(hml_2x2_w) + c(71)*(rmw_2x2_w) + c(72)*(cma_2x2_w) @ P3_s_hml_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P4_s_hml_w = c(73) + c(74)*(mkt_w) + c(75)*(smb_2x2_w) + c(76)*(hml_2x2_w) + c(77)*(rmw_2x2_w) + c(78)*(cma_2x2_w) @ P4_s_hml_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P5_s_hml_w = c(79) + c(80)*(mkt_w) + c(81)*(smb_2x2_w) + c(82)*(hml_2x2_w) + c(83)*(rmw_2x2_w) + c(84)*(cma_2x2_w) @ P5_s_hml_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P6_s_hml_w = c(85) + c(86)*(mkt_w) + c(87)*(smb_2x2_w) + c(88)*(hml_2x2_w) + c(89)*(rmw_2x2_w) + c(90)*(cma_2x2_w) @ P6_s_hml_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P7_s_hml_w = c(91) + c(92)*(mkt_w) + c(93)*(smb_2x2_w) + c(94)*(hml_2x2_w) + c(95)*(rmw_2x2_w) + c(96)*(cma_2x2_w) @ P7_s_hml_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P8_s_hml_w = c(97) + c(98)*(mkt_w) + c(99)*(smb_2x2_w) + c(100)*(hml_2x2_w) + c(101)*(rmw_2x2_w) + c(102)*(cma_2x2_w) @ P8_s_hml_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P9_s_hml_w = c(103) + c(104)*(mkt_w) + c(105)*(smb_2x2_w) + c(106)*(hml_2x2_w) + c(107)*(rmw_2x2_w) + c(108)*(cma_2x2_w) @ P9_s_hml_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P1_s_rmw_w = c(109) + c(110)*(mkt_w) + c(111)*(smb_2x2_w) + c(112)*(hml_2x2_w) + c(113)*(rmw_2x2_w) + c(114)*(cma_2x2_w) @ P1_s_rmw_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P2_s_rmw_w = c(115) + c(116)*(mkt_w) + c(117)*(smb_2x2_w) + c(118)*(hml_2x2_w) + c(119)*(rmw_2x2_w) + c(120)*(cma_2x2_w) @ P2_s_rmw_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P3_s_rmw_w = c(121) + c(122)*(mkt_w) + c(123)*(smb_2x2_w) + c(124)*(hml_2x2_w) + c(125)*(rmw_2x2_w) + c(126)*(cma_2x2_w) @ P3_s_rmw_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P4_s_rmw_w = c(127) + c(128)*(mkt_w) + c(129)*(smb_2x2_w) + c(130)*(hml_2x2_w) + c(131)*(rmw_2x2_w) + c(132)*(cma_2x2_w) @ P4_s_rmw_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P5_s_rmw_w = c(133) + c(134)*(mkt_w) + c(135)*(smb_2x2_w) + c(136)*(hml_2x2_w) + c(137)*(rmw_2x2_w) + c(138)*(cma_2x2_w) @ P5_s_rmw_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P6_s_rmw_w = c(139) + c(140)*(mkt_w) + c(141)*(smb_2x2_w) + c(142)*(hml_2x2_w) + c(143)*(rmw_2x2_w) + c(144)*(cma_2x2_w) @ P6_s_rmw_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P7_s_rmw_w = c(145) + c(146)*(mkt_w) + c(147)*(smb_2x2_w) + c(148)*(hml_2x2_w) + c(149)*(rmw_2x2_w) + c(150)*(cma_2x2_w) @ P7_s_rmw_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P8_s_rmw_w = c(151) + c(152)*(mkt_w) + c(153)*(smb_2x2_w) + c(154)*(hml_2x2_w) + c(155)*(rmw_2x2_w) + c(156)*(cma_2x2_w) @ P8_s_rmw_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)
P9_s_rmw_w = c(157) + c(158)*(mkt_w) + c(159)*(smb_2x2_w) + c(160)*(hml_2x2_w) + c(161)*(rmw_2x2_w) + c(162)*(cma_2x2_w) @ P9_s_rmw_w(-2) mkt_w(-1) smb_2x2_w(-1) hml_2x2_w(-1) rmw_2x2_w(-1) cma_2x2_w(-1)

I am running Eviews 13