sandwich + mlogit: `Error in ef/X : non-conformable arrays` when using `vcovHC()` to compute robust/clustered standard errors

Question

I am trying to compute robust/cluster standard errors after using mlogit() to fit a Multinomial Logit (MNL) in a Discrete Choice problem. Unfortunately, I suspect I am having problems with it because I am using data in long format (this is a must in my case), and getting the error #Error in ef/X : non-conformable arrays after sandwich::vcovHC( , "HC0").

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The Data

For illustration, please gently consider the following data. It represents data from 5 individuals (id_ind ) that choose among 3 alternatives (altern). Each of the five individuals chose three times; hence we have 15 choice situations (id_choice). Each alternative is represented by two generic attributes (x1 and x2), and the choices are registered in y (1 if selected, 0 otherwise).

df <- read.table(header = TRUE, text = "
id_ind id_choice altern           x1          x2 y
1       1         1      1  1.586788801  0.11887832 1
2       1         1      2 -0.937965347  1.15742493 0
3       1         1      3 -0.511504401 -1.90667519 0
4       1         2      1  1.079365680 -0.37267925 0
5       1         2      2 -0.009203032  1.65150370 1
6       1         2      3  0.870474033 -0.82558651 0
7       1         3      1 -0.638604013 -0.09459502 0
8       1         3      2 -0.071679538  1.56879334 0
9       1         3      3  0.398263302  1.45735788 1
10      2         4      1  0.291413453 -0.09107974 0
11      2         4      2  1.632831160  0.92925495 0
12      2         4      3 -1.193272276  0.77092623 1
13      2         5      1  1.967624379 -0.16373709 1
14      2         5      2 -0.479859282 -0.67042130 0
15      2         5      3  1.109780885  0.60348187 0
16      2         6      1 -0.025834772 -0.44004183 0
17      2         6      2 -1.255129594  1.10928280 0
18      2         6      3  1.309493274  1.84247199 1
19      3         7      1  1.593558740 -0.08952151 0
20      3         7      2  1.778701074  1.44483791 1
21      3         7      3  0.643191170 -0.24761157 0
22      3         8      1  1.738820924 -0.96793288 0
23      3         8      2 -1.151429915 -0.08581901 0
24      3         8      3  0.606695064  1.06524268 1
25      3         9      1  0.673866953 -0.26136206 0
26      3         9      2  1.176959443  0.85005871 1
27      3         9      3 -1.568225496 -0.40002252 0
28      4        10      1  0.516456176 -1.02081089 1
29      4        10      2 -1.752854918 -1.71728381 0
30      4        10      3 -1.176101700 -1.60213536 0
31      4        11      1 -1.497779616 -1.66301234 0
32      4        11      2 -0.931117325  1.50128532 1
33      4        11      3 -0.455543630 -0.64370825 0
34      4        12      1  0.894843784 -0.69859139 0
35      4        12      2 -0.354902281  1.02834859 0
36      4        12      3  1.283785176 -1.18923098 1
37      5        13      1 -1.293772990 -0.73491317 0
38      5        13      2  0.748091387  0.07453705 1
39      5        13      3 -0.463585127  0.64802031 0
40      5        14      1 -1.946438667  1.35776140 0
41      5        14      2 -0.470448172 -0.61326604 1
42      5        14      3  1.478763383 -0.66490028 0
43      5        15      1  0.588240775  0.84448489 1
44      5        15      2  1.131731049 -1.51323232 0
45      5        15      3  0.212145247 -1.01804594 0
")

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The problem

Consequently, we can fit an MNL using mlogit() and extract their robust variance-covariance as follows:

library(mlogit)
library(sandwich)
mo <-  mlogit(formula = y ~ x1 + x2|0 , 
              method ="nr",  
              data =  df,
              idx  =  c("id_choice", "altern"))

sandwich::vcovHC(mo, "HC0")
#Error in ef/X : non-conformable arrays

As we can see there is an error produced by sandwich::vcovHC, which says that ef/X is non-conformable. Where X <- model.matrix(x) and ef <- estfun(x, ...). After looking through the source code on the mirror on GitHub I spot the problem which comes from the fact that, given that the data is in long format, ef has dimensions 15 x 2 and X has 45 x 2.

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My workaround

Given that the show must continue, I am computing the robust and cluster standard errors manually using some functions that I borrow from sandwich and I adjusted to accommodate the Stata's output.

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> Robust Standard Errors

These lines are inspired on the sandwich::meat() function.

psi<- estfun(mo)
k <- NCOL(psi)
n <- NROW(psi)
rval <-  (n/(n-1))* crossprod(as.matrix(psi))
vcov(mo) %*% rval %*% vcov(mo)

#            x1         x2
# x1 0.23050261 0.09840356
# x2 0.09840356 0.12765662

Stata Equivalent

qui clogit y x1 x2 ,group(id_choice) r
mat li e(V)
symmetric e(V)[2,2]
            y:         y:
            x1         x2
y:x1  .23050262
y:x2  .09840356  .12765662

---

> Clustered Standard Errors

Here, given that each individual answers 3 questions is highly likely that there is some degree of correlation among individuals; hence cluster corrections should be preferred in such situations. Below I compute the cluster correction in this case and I show the equivalence with the Stata output of clogit , cluster().

id_ind_collapsed <- df$id_ind[!duplicated(mo$model$idx$id_choice,)]
psi_2 <- rowsum(psi, group = id_ind_collapsed )

k_cluster <- NCOL(psi_2)
n_cluster <- NROW(psi_2)
rval_cluster <-  (n_cluster/(n_cluster-1))* crossprod(as.matrix(psi_2))
vcov(mo) %*% rval_cluster %*% vcov(mo)

#           x1        x2
# x1 0.1766707 0.1007703
# x2 0.1007703 0.1180004

Stata equivalent

qui clogit y x1 x2 ,group(id_choice) cluster(id_ind)
symmetric e(V)[2,2]
            y:         y:
            x1         x2
y:x1  .17667075
y:x2   .1007703  .11800038

The Question:

I would like to accommodate my computations within the sandwich ecosystem, meaning not computing the matrices manually but actually using the sandwich functions. Is it possible to make it work with models in long format like the one described here? For example, providing the meat and bread objects directly to perform the computations? Thanks in advance.

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PS: I noted that there is a dedicated bread function in sandwich for mlogit, but I could not spot something like meat for mlogit, but anyways I am probably missing something here...

Answer 1

Why vcovHC does not work for mlogit

The class of HC covariance estimators can just be applied in models with a single linear predictor where the score function aka estimating function is the product of so-called "working residuals" and a regressor matrix. This is explained in some detail in the Zeileis (2006) paper (see Equation 7), provided as vignette("sandwich-OOP", package = "sandwich") in the package. The ?vcovHC also pointed to this but did not explain it very well. I have improved this in the documentation at http://sandwich.R-Forge.R-project.org/reference/vcovHC.html now:

The function meatHC is the real work horse for estimating the meat of HC sandwich estimators - the default vcovHC method is a wrapper calling sandwich and bread. See Zeileis (2006) for more implementation details. The theoretical background, exemplified for the linear regression model, is described below and in Zeileis (2004). Analogous formulas are employed for other types of models, provided that they depend on a single linear predictor and the estimating functions can be represented as a product of “working residual” and regressor vector (Zeileis 2006, Equation 7).

This means that vcovHC() is not applicable to multinomial logit models as they generally use separate linear predictors for the separate response categories. Similarly, two-part or hurdle models etc. are not supported.

Basic "robust" sandwich covariance

Generally, for computing the basic Eicker-Huber-White sandwich covariance matrix estimator, the best strategy is to use the sandwich() function and not the vcovHC() function. The former works for any model with estfun() and bread() methods.

For linear models sandwich(..., adjust = FALSE) (default) and sandwich(..., adjust = TRUE) correspond to HC0 and HC1, respectively. In a model with n observations and k regression coefficients the former standardizes with 1/n and the latter with 1/(n-k).

Stata, however, divides by 1/(n-1) in logit models, see: Different Robust Standard Errors of Logit Regression in Stata and R. To the best of my knowledge there is no clear theoretical reason for using specifically one or the other adjustment. And already in moderately large samples, this makes no difference anyway.

Remark: The adjustment with 1/(n-1) is not directly available in sandwich() as an option. However, coincidentally, it is the default in vcovCL() without specifying a cluster variable (i.e., treating each observation as a separate cluster). So this is a convenient "trick" if you want to get exactly the same results as Stata.

Clustered covariance

This can be computed "as usual" via vcovCL(..., cluster = ...). For mlogit models you just have to consider that the cluster variable just needs to be provided once (as opposed to stacked several times in long format).

Replicating Stata results

With the data and model from your post:

vcovCL(mo)
##            x1         x2
## x1 0.23050261 0.09840356
## x2 0.09840356 0.12765662
vcovCL(mo, cluster = df$id_choice[1:15])
##           x1        x2
## x1 0.1766707 0.1007703
## x2 0.1007703 0.1180004