# How to decide between different robust standard errors?

Specifying my model I ran into some very mild heteroscedasticity problems. Given its superior small-sample properties (my dataset contains 79 observations) I used the HC3 specification of the White heteroscedasticity robust standard errors. However it renders almost all my estimates non-significant. Playing around with the data it showed that the significance levels relied on the type of the robust standard error used (HC0 to HC3) to a large extent. Also I couldn't use HC3 on all my models, apparently because its performance relies on the size of the hat values.

Now I'm slightly confused. How can I decide between these different standard errors? I don't actually have many high leverage points so is HC3 even appropriate?

Some more information on the data: The sample size is fixed, as I'm working within comparative political science. My dependent variable is a left skewed 0-100 index. Most of the independent variables follow a normal distribution, those that don't were logged. Further a region-dummy is included

structure(list(ID = c("Afghanistan", "Albania", "Algeria", "Angola",
"Azerbaijan", "Bahrain", "Bangladesh", "Belarus", "Benin", "Bolivia",
"Burkina Faso", "Burma/Myanmar", "Burundi", "Cambodia", "Cameroon",
"Central African Republic", "Chad", "China", "Cuba", "Democratic Republic of the Congo",
"Djibouti", "Egypt", "Eritrea", "Eswatini", "Ethiopia", "Fiji",
"Gabon", "Guinea", "Haiti", "Honduras", "Hungary", "Iran", "Iraq",
"Jordan", "Kazakhstan", "Kenya", "Kuwait", "Kyrgyzstan", "Laos",
"Lebanon", "Madagascar", "Malawi", "Malaysia", "Mali", "Mauritania",
"Morocco", "Mozambique", "Nicaragua", "Niger", "Nigeria", "Oman",
"Pakistan", "Papua New Guinea", "Philippines", "Qatar", "Republic of the Congo",
"Russia", "Rwanda", "Saudi Arabia", "Serbia", "Singapore", "Somalia",
"South Sudan", "Sudan", "Syria", "Tajikistan", "Tanzania", "Thailand",
"Togo", "Turkey", "Turkmenistan", "Uganda", "Ukraine", "United Arab Emirates",
"Uzbekistan", "Venezuela", "Vietnam", "Zambia", "Zimbabwe"),
Dep_Var = c(84.26, 89.81, 92.13, 90.74, 96.3, 78.7, 93.52,
19.44, 70.83, 96.3, 89.81, 86.11, 33.33, 68.52, 71.3, 75.93,
88.89, 81.94, 100, 80.56, 100, 84.26, 93.52, 89.81, 80.56,
88.89, 81.48, 78.7, 87.04, 100, 76.85, 64.35, 96.3, 100,
89.35, 93.52, 100, 92.13, 96.3, 85.19, 95.37, 57.41, 75,
72.22, 77.78, 93.52, 80.56, 16.67, 61.11, 85.65, 100, 96.3,
83.8, 100, 86.11, 97.22, 87.04, 90.74, 94.44, 100, 85.19,
60.19, 86.11, 91.67, 86.11, 73.15, 50, 82.41, 73.15, 77.78,
58.33, 93.52, 92.59, 89.81, 96.3, 87.04, 96.3, 70.83, 87.96
), Var1 = c(4.127, 5.251, 5.296, 6.541, 6.672, 6.414, 6.78,
6.062, 7.056, 5.78, 5.786, 4.552, 3.537, 5.527, 5.608, 6.275,
3.147, 5.291, 6.55, 4.069, 4.996, 4.586, 4.605, 6.525, 5.409,
6.302, 6.324, 5.558, 3.996, 6.325, 5.951, 4.481, 3.772, 4.786,
6.746, 5.645, 6.613, 6.603, 6.306, 3.814, 5.482, 6.14, 6.521,
4.924, 5.121, 5.355, 5.101, 7.214, 6.554, 6.047, 5.636, 7.175,
4.735, 5.151, 6.116, 4.799, 5.495, 6.855, 6.197, 6.398, 7.468,
3.565, 3.891, 5.121, 3.319, 6.826, 5.235, 5.737, 5.703, 4.748,
5.693, 6.474, 5.836, 6.91, 5.616, 5.979, 5.604, 6.875, 6.48
), Var2 = c(7.71, 7.85, 6.22, 6.93, 5.95, 5.98, 7.46, 1.26,
3.65, 3.42, 5.84, 3.31, 8.72, 7.88, 6.46, 8.05, 8.86, 5.62,
4.89, 6.42, 7.13, 5.63, 7.67, 2.84, 4.66, 4.57, 5.63, 6.62,
8.63, 8.64, 5.01, 3.43, 8.81, 5.72, 6.67, 5.12, 7.09, 6.05,
3.51, 7.46, 6.88, 5.64, 4.48, 5.68, 7.64, 4.56, 5.4, 7.27,
3.74, 7.67, 5.2, 7.83, 8.17, 7.09, 8.66, 9.11, 4.12, 2.87,
6.84, 6.36, 1.83, 8.71, 7.95, 6.59, 7.53, 6.92, 3.77, 6.72,
5.77, 4.83, 8.15, 5.81, 4.23, 4.3, 5.46, 8.92, 1.94, 5.48,
7.55), Var3 = c(1.51522335168878, 4.15325778828397, 2.94875838540978,
1.00628430938346, 25.4230503312376, 8.18998374332489, 1.04538591988069,
1.25351605643943, 1.03389499888613, 1.14765788536537, 1.26573001102583,
1.07401182190831, 1.02601777187271, 1.03639332007889, 1.62605356699817,
1.92725573397033, 1.01254164138481, 1.03505721838858, 3.46173219653658,
1.08295832426121, 57.2882616377008, 1.20022298864129, NA,
1.43549075453128, 1.01873682901296, 1.44946193787762, 1.64439543970553,
1.00783008956213, 1.14205709505493, 1.08208397251952, 2.40225911107622,
22.5235306849493, 1.36123323295883, 1.14848994633969, 1.27546825552435,
1.15597071197187, 48.3011823156329, 1.24779693041552, 1,
2.15232361681418, 1.10011380715122, 1.05904852323133, 1.67293114440204,
1, 1.02209604887292, 1.26595913129412, 1.57959491321159,
1.04583300104408, 1.00857974540892, 1.16022802219023, 206.447814325508,
1.3232281172556, 1.01139457132996, 1.10081706683103, 3.18921374388388,
1.0557568170143, 1.23896346376779, 1.03959784429336, 17.1956183236117,
1.80633839574657, 10.3099601319805, 1.02590186237628, 1.02711959279389,
3.88462180786233, 1.03514910690513, 6.23547964396164, 1.00861987775186,
1.87180599797356, 1.19796183444303, 24.2411476428364, NA,
1.02032998088937, 1.91697325378271, 5.22699733044137, 2.96243967880312,
3.06551946990564, 1.07671406220387, 1.1343707501751, 1.01365610166913
), Var4 = c(5.02115486913067, 53.5285741103671, 39.4834327892571,
29.7359115979868, 47.9358701966989, 235.03977126673, 18.5573982408046,
66.6329529282582, 12.1943267185875, 35.520687621416, 7.748396902346,
14.0781314341043, 2.61247472515742, 16.4312138876475, 14.9790917560296,
4.67907440636335, 7.09540310138533, 102.616791283744, 88.2181889125004,
5.45216212311342, 34.0884625440838, 30.2003134971543, NA,
38.3702699384472, 8.57501351042922, 62.2004642089247, 76.6736686126914,
10.6413123736656, 7.5458791774842, 25.7491219067443, 164.757418389402,
55.2031078932382, 59.5510901036816, 43.3032934593257, 97.3114520688729,
18.165469164389, 320.319801025988, 13.0939299165512, 25.3489827724281,
77.8431685692976, 5.2221980923047, 4.11552340423602, 114.148376618668,
8.90737285506382, 16.7791925268633, 32.0409500313298, 4.91804723063735,
19.1290374537869, 5.54600968697425, 22.2985869624463, 154.740324753635,
12.8470204090039, 28.4518018574842, 34.8508421836621, 647.817331974169,
20.1107239887663, 115.849953826104, 8.01656186978807, 231.397986561214,
74.0235455862041, 652.332824392302, NA, NA, 4.41505603374484,
NA, 8.70787589323223, 11.2212181047515, 78.0819291629949,
6.75542213340634, 90.4249298288882, 69.6663541063077, 7.76768575885308,
36.5903131229487, 431.033230583165, 17.248411344137, NA,
27.1527603640721, 12.9134335737364, 14.6398591018054), Var5 = c(2.086,
5.23, 4.112, 2.896, 4.8, 5.36, 3.504, 4.398, 3.868, 4.356,
3.844, 2.862, 2.16, 3.862, 3.4, 1.56, 1.942, 5.954, 4.51,
1.89, 3.202, 3.83, 1.544, 3.68, 3.786, 5.52, 3.376, 3.058,
1.182, 3.758, 5.974, 4.14, 2.358, 5.226, 5.044, 4.178, 4.824,
3.776, 3.654, 3.72, 2.694, 3.542, 7.15, 2.99, 3.546, 4.582,
3.252, 3.396, 3.468, 2.954, 5.374, 3.732, 3.646, 5.1, 6.262,
2.566, 4.878, 5.414, 5.646, 5.224, 9.462, 0.618, 0.102, 1.76,
1.658, 2.804, 3.488, 5.7, 2.884, 5.012, 2.912, 3.788, 4.17,
7.862, 3.906, 1.836, 4.994, 3.882, 2.602), Var6 = c(43.2,
10.8, 7.5, 2.7, 21.5, 7.4, 27.4, 63.1, 1.5, 3.6, 1.1, 20,
1.4, 18.9, 27.7, 1.9, 0.8, 38.3, 32.4, 18.8, 3.2, 5.6, 1.6,
5, 42.2, 8.2, 14.5, 0.7, 1.6, 2.8, 41.4, 32, 5.6, 37.4, 42.8,
20.7, 39.9, 17.2, 21, 37.4, 0.6, 19, 26.6, 0.5, 1.1, 20.5,
18.1, 46.2, 17.1, 2.8, 26.9, 3.4, 22.7, 20.9, 52.6, 3.5,
47.3, 20.1, 56.3, 19.4, 56.6, 1.5, 27.1, 2.7, 7.8, 16.6,
1.4, 48.1, 1.3, 10.7, 22.2, 1.4, 28.2, 32.9, 22.7, 6.4, 24.1,
21, 3.9), Var7 = c(0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,
0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
1, 0, 1, 0, 1, 0, 0, 0, 0), Var8 = c(0, 0, 0, 1, 0, 0, 0,
0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0,
1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1,
0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1,
0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1), Var9 = c(0,
0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0,
1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0,
1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0,
0, 0), Var10 = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 1, 0, 0, 0), Var11 = c(0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
1, 0, 0, 0, 0, NA, 0, 0, 0, 0, 0, 0, 0, 0), Var12 = c(1L,
1L, 1L, 1L, 1L, 0L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 0L, 0L, 1L, 1L, 1L, 0L, 0L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 0L, 1L, 1L, 0L, 1L, 0L, 1L, 1L, 1L, 1L, 1L, 1L, 0L,
1L, 1L, 1L, 1L, 0L, 1L, 1L, 1L, 0L, 1L, 1L, 1L, 0L, 1L, 1L,
0L, 0L, 0L, 0L, 1L, 1L, 0L, 1L, 1L, 1L, 1L, 1L, 0L, 0L, 1L,
0L, 1L, 1L)), row.names = c(1L, 2L, 3L, 5L, 11L, 12L, 13L,
15L, 18L, 21L, 28L, 29L, 30L, 31L, 32L, 36L, 37L, 39L, 43L, 46L,
48L, 52L, 54L, 56L, 57L, 59L, 62L, 71L, 73L, 74L, 76L, 80L, 81L,
88L, 89L, 90L, 92L, 93L, 94L, 96L, 103L, 104L, 105L, 106L, 107L,
113L, 114L, 119L, 120L, 121L, 123L, 124L, 127L, 130L, 135L, 136L,
138L, 139L, 141L, 143L, 146L, 150L, 153L, 156L, 160L, 162L, 163L,
164L, 167L, 170L, 171L, 173L, 174L, 175L, 179L, 181L, 182L, 184L,
185L), class = "data.frame")

model <- lm(formula = Dep_var ~ Var1 + Var2 + log(Var3) +
Var4 + Var5 + Var6 + Var7 + Var8 +
Var9 + Var10 + Var11 + Var12,
data = subset(df, ID !="Nicaragua" &
ID != "Belarus" &
ID != "Burundi"),
na.action = na.exclude)

coeftest(model, vcov = vcovHC(model, method = "arellano", type = "HC3"))

• What do you mean by "I couldn't use HC3 on all my models"? You can probably compute it in all models, but you have doubts about it's validity, right? – Lewian Sep 3 '20 at 11:51
• When different procedures give quite different results the possible actions range from choosing the most appropriate procedure to deciding that the data don't allow firm conclusions. I would choose a middle path here: if you're worried about heteroscedasticity the implication may well be that you need a different model. Also, when a model fit is fragile in any sense the implication may be that you need a simpler model. You aren't telling us anything about the data used except that you have 79 observations, which for your purposes may be anywhere from adequate to dangerously small, depending. – Nick Cox Sep 3 '20 at 12:07
• Possible problem areas include working with the data as they come rather than using appropriate transformations or link functions; using too many predictors. – Nick Cox Sep 3 '20 at 12:08
• @philipp.kn_98: "R returned 'INF' for all estimates and standard errors. This happens once two outliers are excluded or treated with an MM estimator" - this looks very strange to me. There's nothing in the definition of HC3 that requires that you have outliers or leverage points. I suspect you got something else wrong there. Your region dummy is a single 0-1 variable that has both enough zeroes and ones? – Lewian Sep 3 '20 at 14:14
• Thanks for showing the data. That format is fine for R people but pretty hard work for people using other software. I note that with missings (NAs) and your exclusion of 3 countries on whatever other grounds, you're down to 70 observations. I fitted a generalised linear model with logit link and binomial family to the dependent variable divided by 100, which at least matches the bounds. With your plain regression model, some predictions are indeed above 100. I'd put most emphasis on using logit link and choosing a simpler model. Naturally, you need robust SEs for a non-binary outcome. – Nick Cox Sep 4 '20 at 15:02