Comparing heteroscedasticity tests in R

Question

I've seen multiple explanations of comparisons of heteroscedasticity tests, but am still confused. (Part of the problem is that some of it is quite technical and has lost me.)

I've collected a few heteroscedasticity test functions in R and compared the results, some of which baffle me.

Here's a graph of a linear regression: To my untrained eye, the data look heteroscedastic.

I've run a few tests for confirmation:

> reg <- lm(est~emp_app, temp)
> lmtest::bptest(reg,studentize=TRUE)
    studentized Breusch-Pagan test

data:  reg
BP = 174.53, df = 1, p-value < 2.2e-16

> lmtest::bptest(reg,studentize=FALSE)
    Breusch-Pagan test

data:  reg
BP = 4163.8, df = 1, p-value < 2.2e-16

> car::ncvTest(reg) #original Breusch-Pagan
Non-constant Variance Score Test 
Variance formula: ~ fitted.values 
Chisquare = 4163.766    Df = 1     p = 0 

> res <- gvlma::gvlma(reg)
> gvlma::display.gvlmatests(res)

ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
Level of Significance =  0.05 

Call:
 gvlma::gvlma(x = reg) 

                       Value p-value                   Decision
Global Stat        3.378e+04  0.0000 Assumptions NOT satisfied!
Skewness           4.505e+02  0.0000 Assumptions NOT satisfied!
Kurtosis           3.318e+04  0.0000 Assumptions NOT satisfied!
Link Function      1.504e+02  0.0000 Assumptions NOT satisfied!
Heteroscedasticity 2.137e-01  0.6439    Assumptions acceptable.

As I understand, the first three tests report heteroscedasticity with very low p-values, but the fourth rejects it quite squarely. How can the tests disagree so strongly?

Here's my data:

temp<-structure(list(cbsa_code = c("10180", "10420", "10500", "10540", 
"10580", "10740", "10780", "10900", "11020", "11100", "11180", 
"11260", "11460", "11500", "11540", "11700", "12020", "12060", 
"12100", "12220", "12260", "12420", "12540", "12580", "12620", 
"12700", "12940", "12980", "13020", "13140", "13220", "13380", 
"13460", "13740", "13780", "13820", "13900", "13980", "14010", 
"14020", "14100", "14260", "14460", "14500", "14540", "14740", 
"14860", "15180", "15260", "15380", "15500", "15540", "15680", 
"15940", "15980", "16020", "16060", "16180", "16220", "16300", 
"16540", "16580", "16620", "16700", "16740", "16820", "16860", 
"16940", "16980", "17020", "17140", "17300", "17420", "17460", 
"17660", "17780", "17820", "17860", "17900", "17980", "18020", 
"18140", "18580", "18700", "18880", "19060", "19100", "19140", 
"19180", "19300", "19340", "19380", "19460", "19500", "19660", 
"19740", "19780", "19820", "20020", "20100", "20220", "20260", 
"20500", "20700", "20740", "20940", "21060", "21140", "21300", 
"21340", "21500", "21660", "21780", "21820", "22020", "22140", 
"22180", "22220", "22380", "22420", "22500", "22520", "22540", 
"22660", "22900", "23060", "23420", "23460", "23540", "23580", 
"23900", "24020", "24140", "24220", "24260", "24300", "24340", 
"24420", "24500", "24540", "24580", "24660", "24780", "24860", 
"25060", "25180", "25220", "25260", "25420", "25500", "25540", 
"25620", "25860", "25940", "25980", "26140", "26300", "26380", 
"26420", "26580", "26620", "26820", "26900", "26980", "27060", 
"27100", "27140", "27180", "27260", "27340", "27500", "27620", 
"27740", "27780", "27860", "27900", "27980", "28020", "28100", 
"28140", "28420", "28660", "28700", "28740", "28940", "29020", 
"29100", "29180", "29200", "29340", "29420", "29460", "29540", 
"29620", "29700", "29740", "29820", "29940", "30020", "30140", 
"30300", "30340", "30460", "30620", "30700", "30780", "30860", 
"30980", "31020", "31080", "31140", "31180", "31340", "31420", 
"31460", "31540", "31700", "31740", "31860", "31900", "32580", 
"32780", "32820", "32900", "33100", "33140", "33220", "33260", 
"33340", "33460", "33540", "33660", "33700", "33740", "33780", 
"33860", "34060", "34100", "34580", "34620", "34740", "34820", 
"34900", "34940", "34980", "35100", "35300", "35380", "35620", 
"35660", "35840", "35980", "36100", "36140", "36220", "36260", 
"36420", "36500", "36540", "36740", "36780", "36980", "37100", 
"37340", "37460", "37620", "37860", "37900", "37980", "38060", 
"38220", "38300", "38340", "38540", "38860", "38900", "38940", 
"39140", "39300", "39340", "39380", "39460", "39540", "39580", 
"39660", "39740", "39820", "39900", "40060", "40140", "40220", 
"40340", "40380", "40420", "40580", "40660", "40900", "40980", 
"41060", "41100", "41140", "41180", "41420", "41500", "41540", 
"41620", "41660", "41700", "41740", "41860", "41940", "42020", 
"42100", "42140", "42200", "42220", "42340", "42540", "42660", 
"42680", "42700", "43100", "43300", "43340", "43420", "43580", 
"43620", "43780", "43900", "44060", "44100", "44140", "44180", 
"44220", "44300", "44420", "44700", "44940", "45060", "45220", 
"45300", "45460", "45500", "45540", "45780", "45820", "45940", 
"46060", "46140", "46220", "46340", "46520", "46540", "46660", 
"46700", "47020", "47220", "47260", "47300", "47380", "47460", 
"47580", "47900", "47940", "48060", "48140", "48260", "48300", 
"48540", "48620", "48660", "48700", "48900", "49020", "49180", 
"49340", "49420", "49620", "49660", "49700", "49740"), emp_app = c(59998, 
294127, 45814, 35251, 344504, 293984, 49098, 313182, 54469, 95485, 
32074, 166243, 148265, 35910, 112362, 159295, 59932, 2241972, 
103524, 41553, 186878, 773940, 194840, 1139580, 57350, 74140, 
337264, 53395, 30703, 134579, 34931, 73743, 60309, 72287, 79217, 
445349, 59192, 51175, 82343, 52204, 36838, 242687, 2421578, 150110, 
57912, 57374, 421295, 104560, 32901, 476709, 56328, 102068, 28799, 
147570, 201523, 40052, 40758, 21200, 36096, 129661, 49464, 75399, 
89688, 262866, 988186, 82306, 215347, 35530, 4120166, 59475, 
907677, 68582, 39401, 923418, 47532, 70268, 232291, 75408, 284984, 
95823, 46808, 836802, 163715, 25919, 78046, 31188, 2954801, 56250, 
24983, 58868, 159359, 318802, 44878, 45585, 158021, 1211011, 
307055, 1709983, 49354, 51662, 55142, 109624, 241065, 44970, 
72853, 33153, 40291, 123927, 31597, 229609, 114164, 123022, 140182, 
27673, 120979, 39736, 97042, 199864, 50051, 117246, 74128, 46700, 
42920, 120470, 93234, 191745, 249702, 29603, 92134, 72187, 28508, 
41666, 33461, 43601, 35966, 51843, 474938, 21420, 30802, 85157, 
156091, 318528, 60323, 322404, 127159, 83253, 35571, 23393, 271533, 
51459, 533971, 50122, 127573, 60115, 12898, 27791, 32412, 81983, 
2576412, 111891, 172187, 50584, 856951, 67841, 49525, 48974, 
216677, 56192, 534828, 35491, 56646, 52435, 63657, 46531, 44177, 
69998, 61999, 121603, 36979, 923782, 79661, 103572, 104648, 45604, 
324448, 32084, 65354, 195811, 69714, 75839, 41520, 174572, 226286, 
160786, 75850, 50155, 824659, 40240, 34425, 44109, 21824, 45347, 
219269, 46122, 139267, 275443, 43653, 85075, 30598, 5456991, 
560894, 113641, 97143, 82500, 26580, 316916, 179115, 29258, 48648, 
44133, 187842, 70228, 527509, 42036, 2122149, 34473, 34633, 91487, 
775911, 1778005, 49149, 148864, 135488, 66903, 36847, 129973, 
51192, 38306, 39772, 39382, 53027, 126884, 62275, 118795, 797655, 
30768, 335845, 496807, 8123112, 53404, 232638, 102909, 80011, 
26513, 65095, 177240, 504306, 68684, 414460, 991947, 82979, 46639, 
257011, 169860, 65829, 32257, 126500, 165198, 2563343, 1619025, 
22919, 1094529, 53467, 24406, 231298, 978967, 111852, 57307, 
616068, 192435, 48828, 37485, 69285, 481122, 54861, 154975, 49163, 
179215, 523449, 1118924, 138842, 108519, 443427, 133966, 48602, 
34336, 697430, 79117, 97391, 45178, 48349, 1223383, 114637, 106652, 
123924, 582047, 41413, 825174, 1239334, 2045647, 1010970, 91540, 
75572, 46233, 146504, 166604, 147128, 236587, 1626232, 42560, 
19861, 54832, 39777, 156849, 26092, 77546, 136675, 126814, 130867, 
189825, 86462, 227956, 179168, 41116, 44884, 39796, 173611, 32393, 
258000, 110022, 1080271, 56795, 46653, 20144, 271481, 85504, 
188908, 310979, 403144, 80890, 91548, 355607, 99886, 43020, 107550, 
36273, 44519, 603621, 94155, 103218, 20122, 45576, 2559666, 81913, 
29527, 64477, 35986, 35650, 56778, 265128, 47202, 46211, 99208, 
52110, 231599, 319287, 66044, 164072, 197478, 30114, 41516), 
    est = c(3899, 16486, 3123, 2511, 21298, 18592, 3325, 18293, 
    3191, 6282, 2045, 10887, 8096, 2300, 5898, 12163, 4592, 137077, 
    6329, 2658, 10395, 48893, 12642, 66489, 4109, 8488, 18098, 
    2614, 2193, 7953, 2587, 6401, 6530, 5994, 5045, 25544, 3938, 
    3438, 4006, 3305, 1936, 17296, 127170, 12073, 3539, 5676, 
    27190, 6381, 2855, 27251, 3153, 6803, 1909, 8681, 17459, 
    2764, 2953, 1960, 3006, 6472, 3048, 4900, 5343, 17916, 57660, 
    6043, 11213, 3139, 243420, 4663, 45916, 4328, 2133, 51551, 
    4513, 4744, 17221, 4649, 16921, 5663, 1821, 41361, 9490, 
    2111, 7243, 2005, 156111, 2545, 1426, 5127, 8883, 16576, 
    2951, 2461, 14380, 80560, 15629, 98561, 3402, 3323, 2762, 
    7071, 12362, 3368, 4274, 2494, 2932, 4866, 1806, 14208, 6163, 
    9696, 7583, 2459, 6725, 2721, 6046, 11416, 3527, 7737, 4197, 
    3221, 2361, 10291, 5818, 10377, 16350, 1980, 6285, 4100, 
    1942, 3346, 2174, 2703, 2570, 4404, 23939, 1907, 2433, 5830, 
    7673, 17428, 3523, 19162, 7132, 4993, 2317, 1585, 13478, 
    2972, 29338, 3294, 7370, 5629, 872, 2715, 2697, 4766, 135923, 
    6967, 9527, 3920, 45981, 3909, 2369, 2892, 13128, 2971, 36198, 
    2763, 3275, 3623, 3742, 3194, 2808, 3978, 4546, 6822, 2339, 
    52739, 5732, 6038, 6004, 4746, 17783, 1751, 3458, 13273, 
    4042, 4597, 3677, 11218, 12594, 9453, 5259, 3570, 43396, 
    2698, 2280, 2684, 1568, 2706, 12282, 2404, 8759, 17801, 3531, 
    5445, 2099, 357910, 29101, 7233, 5908, 5112, 1929, 16765, 
    10938, 2189, 2587, 2618, 11866, 6091, 25206, 2969, 188379, 
    2303, 2167, 5299, 37970, 94806, 4296, 8600, 8744, 4549, 2254, 
    7671, 2860, 1962, 3417, 2393, 3137, 10873, 4160, 11293, 41609, 
    2517, 19566, 30498, 572361, 3543, 21885, 5805, 6912, 3810, 
    3717, 13364, 35224, 6039, 23116, 60881, 3499, 2612, 20602, 
    13648, 4868, 2095, 9339, 8422, 145816, 92265, 1605, 59858, 
    3868, 1987, 17640, 66947, 10681, 5741, 40821, 12438, 3020, 
    3815, 4004, 31493, 4737, 8404, 4141, 11989, 31020, 70200, 
    8063, 5172, 24580, 7342, 2745, 1940, 46889, 4300, 5249, 4557, 
    2996, 75922, 9244, 8557, 10493, 31772, 2860, 44267, 81710, 
    127015, 48731, 8164, 6961, 4696, 11455, 13746, 8884, 13185, 
    101754, 4156, 1876, 2651, 2511, 9994, 2188, 4415, 7463, 6473, 
    6560, 13750, 5203, 13176, 11904, 2281, 3313, 2774, 11025, 
    1755, 15352, 8788, 74726, 3514, 3069, 1383, 13265, 5135, 
    9664, 20152, 24241, 4490, 5776, 21167, 6047, 3050, 6855, 
    2547, 2832, 37174, 6259, 5277, 1483, 3130, 149805, 4125, 
    2435, 3273, 2231, 3215, 3346, 14695, 3458, 2817, 8073, 3149, 
    13114, 19860, 4680, 8675, 12313, 2529, 2942)), .Names = c("cbsa_code", 
"emp_app", "est"), row.names = c(13L, 16L, 45L, 53L, 63L, 85L, 
97L, 117L, 132L, 136L, 163L, 178L, 195L, 199L, 214L, 232L, 255L, 
257L, 278L, 287L, 301L, 316L, 341L, 346L, 363L, 389L, 401L, 411L, 
428L, 440L, 462L, 468L, 491L, 510L, 511L, 527L, 541L, 564L, 573L, 
592L, 610L, 622L, 641L, 660L, 667L, 676L, 692L, 713L, 726L, 739L, 
759L, 777L, 788L, 798L, 823L, 833L, 851L, 861L, 884L, 891L, 912L, 
926L, 944L, 955L, 963L, 985L, 993L, 1017L, 1031L, 1049L, 1065L, 
1078L, 1086L, 1102L, 1125L, 1140L, 1153L, 1156L, 1171L, 1198L, 
1208L, 1228L, 1231L, 1251L, 1268L, 1277L, 1305L, 1311L, 1332L, 
1342L, 1357L, 1377L, 1385L, 1399L, 1415L, 1433L, 1448L, 1461L, 
1485L, 1497L, 1515L, 1530L, 1533L, 1558L, 1564L, 1589L, 1603L, 
1614L, 1632L, 1650L, 1655L, 1674L, 1686L, 1698L, 1717L, 1728L, 
1747L, 1756L, 1779L, 1791L, 1811L, 1830L, 1845L, 1856L, 1866L, 
1883L, 1902L, 1913L, 1926L, 1942L, 1963L, 1966L, 1989L, 2006L, 
2014L, 2028L, 2043L, 2066L, 2077L, 2090L, 2103L, 2120L, 2132L, 
2150L, 2161L, 2189L, 2204L, 2218L, 2231L, 2238L, 2257L, 2270L, 
2286L, 2304L, 2316L, 2335L, 2344L, 2361L, 2376L, 2391L, 2406L, 
2430L, 2432L, 2452L, 2465L, 2477L, 2491L, 2509L, 2535L, 2545L, 
2564L, 2580L, 2588L, 2610L, 2625L, 2639L, 2654L, 2668L, 2685L, 
2700L, 2705L, 2718L, 2731L, 2748L, 2771L, 2786L, 2798L, 2806L, 
2826L, 2845L, 2859L, 2876L, 2890L, 2908L, 2924L, 2939L, 2943L, 
2968L, 2973L, 2997L, 3003L, 3016L, 3033L, 3046L, 3066L, 3081L, 
3105L, 3108L, 3125L, 3139L, 3153L, 3176L, 3187L, 3205L, 3213L, 
3231L, 3245L, 3259L, 3276L, 3287L, 3315L, 3322L, 3331L, 3349L, 
3369L, 3379L, 3396L, 3414L, 3427L, 3441L, 3463L, 3478L, 3494L, 
3506L, 3511L, 3539L, 3551L, 3569L, 3585L, 3600L, 3606L, 3619L, 
3643L, 3647L, 3667L, 3687L, 3694L, 3712L, 3723L, 3748L, 3755L, 
3769L, 3784L, 3810L, 3820L, 3840L, 3843L, 3865L, 3881L, 3891L, 
3915L, 3923L, 3936L, 3950L, 3970L, 3980L, 4003L, 4006L, 4033L, 
4047L, 4062L, 4073L, 4088L, 4097L, 4120L, 4138L, 4149L, 4167L, 
4180L, 4196L, 4210L, 4229L, 4238L, 4252L, 4261L, 4277L, 4305L, 
4307L, 4322L, 4349L, 4351L, 4368L, 4394L, 4403L, 4412L, 4440L, 
4447L, 4469L, 4475L, 4496L, 4511L, 4526L, 4537L, 4559L, 4571L, 
4583L, 4595L, 4606L, 4626L, 4648L, 4658L, 4671L, 4691L, 4700L, 
4713L, 4733L, 4755L, 4768L, 4780L, 4797L, 4811L, 4830L, 4833L, 
4859L, 4861L, 4885L, 4897L, 4912L, 4931L, 4939L, 4952L, 4968L, 
4985L, 5004L, 5012L, 5032L, 5047L, 5063L, 5076L, 5092L, 5112L, 
5120L, 5140L, 5149L, 5175L, 5188L, 5193L, 5210L, 5230L, 5246L, 
5261L, 5276L, 5285L, 5300L, 5311L, 5339L, 5346L, 5358L, 5380L, 
5386L, 5410L, 5420L, 5443L, 5454L, 5470L, 5486L, 5494L, 5514L, 
5531L, 5539L, 5552L, 5572L, 5591L, 5604L, 5625L, 5634L, 5641L, 
5666L, 5671L, 5686L, 5713L), class = "data.frame")

Answer 1

As Pena and Slate (2006) write in their paper that is the basis of the gvlma function:

The difficulty with these tests [meaning tests like, e.g., Durbin-Watson or Cook-Weisberg/Breusch-Pagan] is that each is designed to detect departures from one assumption, and the impact of violations of other assumptions on this test, as well as its sensitivity against these violations are not apparent. Hence, when a specific test indicates a violation, it might be due to the violation of another assumption which affects this test.

Judging from the plot, your fit is more an example of a skewed residual distribution than of heteroscedasticity. The other tests might be giving false positive results.