Multiple estimations can be a chore to set up. Varying left-hand-sides (LHS), right-hand-sides (RHS) or samples require either many lines of code, or loops with formula/data manipulation. The package fixest simplifies multiple estimations by providing an optimized procedure along with a clear and concise syntax.

On the one hand, stepwise functions facilitate the sequential inclusion of variables in the RHS or in the fixed-effects part of the formula. On the other hand, intuitive methods are introduced to manipulate the results from multiple estimations, making it easy to visualize or export any wanted set of results.

# First illustration

What does a multiple estimation look like? Let’s give it a try:

base = iris
names(base) = c("y1", "y2", "x1", "x2", "species")

res_multi = feols(c(y1, y2) ~ x1 + csw(x2, x2^2) | sw0(species), base, fsplit = ~species)

With the previous line of code (90 characters long), we have just performed 32 estimations: eight different functional forms on four different samples.

The previous code leads to the following results:

summary(res_multi, "compact", se = "hetero")
##         sample lhs   fixef               rhs      (Intercept)
## 1  Full sample  y1 1       x1 + x2           4.19*** (0.104)
## 2  Full sample  y1 1       x1 + x2 + I(x2^2) 4.27*** (0.101)
## 3  Full sample  y1 species x1 + x2
## 4  Full sample  y1 species x1 + x2 + I(x2^2)
## 5  Full sample  y2 1       x1 + x2           3.59*** (0.103)
## 6  Full sample  y2 1       x1 + x2 + I(x2^2) 3.68*** (0.0969)
## 7  Full sample  y2 species x1 + x2
## 8  Full sample  y2 species x1 + x2 + I(x2^2)
## 9  setosa       y1 1       x1 + x2           4.25*** (0.474)
## 10 setosa       y1 1       x1 + x2 + I(x2^2)    4*** (0.504)
## 11 setosa       y1 species x1 + x2
## 12 setosa       y1 species x1 + x2 + I(x2^2)
## 13 setosa       y2 1       x1 + x2           2.89*** (0.416)
## 14 setosa       y2 1       x1 + x2 + I(x2^2) 2.82*** (0.423)
## 15 setosa       y2 species x1 + x2
## 16 setosa       y2 species x1 + x2 + I(x2^2)
## 17 versicolor   y1 1       x1 + x2           2.38*** (0.423)
## 18 versicolor   y1 1       x1 + x2 + I(x2^2)   0.323 (1.44)
## 19 versicolor   y1 species x1 + x2
## 20 versicolor   y1 species x1 + x2 + I(x2^2)
## 21 versicolor   y2 1       x1 + x2           1.25*** (0.275)
## 22 versicolor   y2 1       x1 + x2 + I(x2^2)   0.097 (1.01)
## 23 versicolor   y2 species x1 + x2
## 24 versicolor   y2 species x1 + x2 + I(x2^2)
## 25 virginica    y1 1       x1 + x2             1.05. (0.539)
## 26 virginica    y1 1       x1 + x2 + I(x2^2)   -2.39 (2.04)
## 27 virginica    y1 species x1 + x2
## 28 virginica    y1 species x1 + x2 + I(x2^2)
## 29 virginica    y2 1       x1 + x2             1.06. (0.572)
## 30 virginica    y2 1       x1 + x2 + I(x2^2)     1.1 (1.76)
## 31 virginica    y2 species x1 + x2
## 32 virginica    y2 species x1 + x2 + I(x2^2)
##                    x1               x2           I(x2^2)
## 1   0.542*** (0.0761)   -0.32. (0.17)
## 2   0.719*** (0.0815) -1.52*** (0.307) 0.348*** (0.0748)
## 3   0.906*** (0.0759)   -0.006 (0.163)
## 4     0.9*** (0.0767)     0.29 (0.408)  -0.0879 (0.117)
## 5  -0.257*** (0.0664)   0.364* (0.142)
## 6    -0.0301 (0.0778) -1.18*** (0.313) 0.446*** (0.0737)
## 7     0.155* (0.0735) 0.623*** (0.114)
## 8     0.148. (0.0755)   0.951* (0.472)  -0.0973 (0.125)
## 9      0.399 (0.325)    0.712. (0.418)
## 10     0.405 (0.325)     2.51. (1.47)     -2.91 (2.1)
## 11     0.399 (0.325)    0.712. (0.418)
## 12     0.405 (0.325)     2.51. (1.47)     -2.91 (2.1)
## 13     0.247 (0.305)     0.702 (0.56)
## 14     0.248 (0.304)      1.27 (2.39)    -0.911 (3.28)
## 15     0.247 (0.305)     0.702 (0.56)
## 16     0.248 (0.304)      1.27 (2.39)    -0.911 (3.28)
## 17  0.934*** (0.166)     -0.32 (0.364)
## 18  0.901*** (0.164)      3.01 (2.31)     -1.24 (0.841)
## 19  0.934*** (0.166)     -0.32 (0.364)
## 20  0.901*** (0.164)      3.01 (2.31)     -1.24 (0.841)
## 21    0.0669 (0.0949) 0.929*** (0.244)
## 22     0.048 (0.0993)     2.8. (1.65)    -0.695 (0.583)
## 23    0.0669 (0.0949) 0.929*** (0.244)
## 24     0.048 (0.0993)     2.8. (1.65)    -0.695 (0.583)
## 25  0.995*** (0.0898)  0.00706 (0.205)
## 26  0.994*** (0.0881)     3.5. (2.09)     -0.87 (0.519)
## 27  0.995*** (0.0898)  0.00706 (0.205)
## 28  0.994*** (0.0881)     3.5. (2.09)     -0.87 (0.519)
## 29     0.149 (0.107)  0.535*** (0.122)
## 30     0.149 (0.108)     0.503 (1.56)   0.00798 (0.388)
## 31     0.149 (0.107)  0.535*** (0.122)
## 32     0.149 (0.108)     0.503 (1.56)   0.00798 (0.388)

This vignette now details how to perform multiple estimations for multiple: LHSs, RHSs, fixed-effects, or samples. It then comes to describe the various methods to access the results.

# Performing multiple estimations

#### Multiple LHS

To perform an estimation on multiple LHS, simply wrap the different LHS in c():

etable(feols(c(y1, y2) ~ x1 + x2, base))
##                            model 1             model 2
## Dependent Var.:                 y1                  y2
##
## (Intercept)      4.191*** (0.0970)   3.587*** (0.0937)
## x1              0.5418*** (0.0693) -0.2571*** (0.0669)
## x2               -0.3196* (0.1605)    0.3640* (0.1550)
## _______________ __________________ ___________________
## S.E. type                      IID                 IID
## Observations                   150                 150
## R2                         0.76626             0.21310
## Adj. R2                    0.76308             0.20240

#### Multiple RHS and fixed-effects: stepwise functions

To estimate multiple RHS (or fixed-effects), you need to use a specific set of functions: the stepwise functions. There are four of them: sw, sw0, csw, csw0.

• sw: this function is replaced sequentially by each of its arguments. For example, y ~ x1 + sw(x2, x3) leads to two estimations: y ~ x1 + x2 and y ~ x1 + x3.

• sw0: identical to sw but first adds the empty element. E.g. y ~ x1 + sw0(x2, x3) leads to three estimations: y ~ x1, y ~ x1 + x2 and y ~ x1 + x3.

• csw: it stands for cumulative stepwise. It adds to the formula each of its arguments sequentially. E.g. y ~ x1 + csw(x2, x3) will become y ~ x1 + x2 and y ~ x1 + x2 + x3.

• csw0: identical to csw but first adds the empty element. E.g. y ~ x1 + csw0(x2, x3) leads to three estimations: y ~ x1, y ~ x1 + x2 and y ~ x1 + x2 + x3.

The stepwise functions can be applied both to the linear part and the fixed-effects part of the formula. Note that at most one stepwise function can be applied per part.

Here is an example:

etable(feols(y1 ~ csw(x1, x2) | sw0(species), base, cluster = ~species))
##                           model 1          model 2           model 3
## Dependent Var.:                y1               y1                y1
##
## (Intercept)      4.307** (0.1917) 4.191** (0.2443)
## x1              0.4089** (0.0401) 0.5418* (0.1115) 0.9046** (0.0758)
## x2                                -0.3196 (0.1707)
## Fixed-Effects:  ----------------- ---------------- -----------------
## species                        No               No               Yes
## _______________ _________________ ________________ _________________
## S.E.: Clustered       by: species      by: species       by: species
## Observations                  150              150               150
## R2                        0.75995          0.76626           0.83672
## Within R2                      --               --           0.57178
##                           model 4
## Dependent Var.:                y1
##
## (Intercept)
## x1              0.9059** (0.0814)
## x2               -0.0060 (0.1260)
## Fixed-Effects:  -----------------
## species                       Yes
## _______________ _________________
## S.E.: Clustered       by: species
## Observations                  150
## R2                        0.83673
## Within R2                 0.57179

As you can see, if the stepwise functions are in both parts, there will be as many estimations as the cardinal product of the two parts.

#### Split sample estimations

To perform split sample estimations, use either the argument split or fsplit. The argument split accepts a variable that will be treated as a factor, and an estimation will be performed for each sub-sample defined by each level of this variable. The argument fsplit is identical but first adds the full sample.

etable(feols(y1 ~ x1 + x2, base, fsplit = ~species))
##                            model 1           model 2            model 3
## Dependent Var.:                 y1                y1                 y1
##
## (Intercept)      4.191*** (0.0970) 4.248*** (0.4114)  2.381*** (0.4493)
## x1              0.5418*** (0.0693)   0.3990 (0.2958) 0.9342*** (0.1693)
## x2               -0.3196* (0.1605)   0.7121 (0.4874)   -0.3200 (0.4024)
## _______________ __________________ _________________ __________________
## S.E. type                      IID               IID                IID
## Observations                   150                50                 50
## R2                         0.76626           0.11173            0.57432
## Adj. R2                    0.76308           0.07393            0.55620
##                            model 4
## Dependent Var.:                 y1
##
## (Intercept)        1.052* (0.5139)
## x1              0.9946*** (0.0893)
## x2                 0.0071 (0.1795)
## _______________ __________________
## S.E. type                      IID
## Observations                    50
## R2                         0.74689
## Adj. R2                    0.73612

#### Combining multiple estimations

You can combine multiple LHS to multiple RHS to multiple fixed-effects to multiple samples. The total number of estimations is always equal to the cardinal product of the total number of parts.

# Manipulation of multiple estimations

We’ve just seen how to perform multiple estimations, now let’s see how to manipulate them. First a multiple estimation is a fixest_multi object with its own set of methods. We can access its elements by using keys. There are five keys: sample, lhs, rhs, fixef, and iv.

### Basic access

res_multi[sample = 1] returns all the results for the first sample. res_multi[lhs = .N] returns all the results for the last dependent variable (the special variable .N can be used to refer to the last element). etc.

You can combine different keys: res_multi[sample = -1, lhs = 1] will select all results for all samples but the first, and for the first dependent variable.

Note that these arguments accept regular expressions, so res_multi[fixef = "spe"] returns all results for which the character string "spe" is contained in the fixed-effects part of the formula.

### Putting order

The results in a fixest_multi object have a specific order, organized in a tree. By default the order is $$sample \rightarrow lhs \rightarrow fixef \rightarrow rhs \rightarrow iv$$.

Changing the order of the results is important to organize/export them. By default, when one accesses fixest_multi objects the results are reordered according to the order of the arguments used.

For instance, res_mutli[rhs = 1:.N, fixef = 1:.N] will place the RHS at the root of the tree followed by the fixed-effects. Then the sample and the LHS will follow.

The arguments accept logical values: res_multi[fixef = TRUE, sample = FALSE] will keep all results but will place fixef as the root and sample as the last leaf.

This subsetting can then be used to easily obtain the appropriate set of results and ordering:

etable(res_multi[lhs = 1, fixef = 1, rhs = TRUE, sample = -1])
##                           model 1            model 2            model 3
## Dependent Var.:                y1                 y1                 y1
##
## (Intercept)     4.248*** (0.4114)  2.381*** (0.4493)    1.052* (0.5139)
## x1                0.3990 (0.2958) 0.9342*** (0.1693) 0.9946*** (0.0893)
## x2                0.7121 (0.4874)   -0.3200 (0.4024)    0.0071 (0.1795)
## x2 square
## _______________ _________________ __________________ __________________
## S.E. type                     IID                IID                IID
## Observations                   50                 50                 50
## R2                        0.11173            0.57432            0.74689
## Adj. R2                   0.07393            0.55620            0.73612
##                           model 4            model 5            model 6
## Dependent Var.:                y1                 y1                 y1
##
## (Intercept)     4.004*** (0.4892)     0.3234 (1.791)     -2.393 (2.173)
## x1                0.4048 (0.2963) 0.9006*** (0.1710) 0.9939*** (0.0878)
## x2                  2.511 (2.011)      3.015 (2.839)      3.504 (2.153)
## x2 square          -2.913 (3.157)     -1.236 (1.042)   -0.8703 (0.5340)
## _______________ _________________ __________________ __________________
## S.E. type                     IID                IID                IID
## Observations                   50                 50                 50
## R2                        0.12786            0.58696            0.76071
## Adj. R2                   0.07098            0.56002            0.74510

# Some notes

#### Note on standard-errors

Defining the standard-errors at estimation time, by using the arguments se or cluster, can be useful to obtain a coherent set of standard-errors across results, especially if the fixed-effects are modified (which will modify the default clustering of standard-errors across models).

#### Note on IVs

IV estimations return a regular fixest object. The summary applied to it however can return a fixest_multi object. This is the case when both the first and second stage regressions are requested using the argument stage = 1:2. You can then cherry-pick the results as before using, e.g. res[iv = 1]. Note, importantly, that the index refers to the order of the results and 1 here does not mean the first stage.

#### Note on memory usage

The objects returned by fixest estimations are large. They contain the necessary information to apply various methods without incurring additional computing costs. This is particularly true for clustering the standard-errors for instance. Stated differently speed is privileged over memory usage.

The problem when it comes to multiple estimations is that it is very easy to perform many many estimations leading to a ballooning of object size possibly getting out of control at some point. To circumvent this issue, here’s what to do:

1. use the argument vcov to get a summary of the results with the appropriate standard-errors at estimation time,
2. use the argument lean = TRUE.

This will perform the estimation with the appropriate standard errors (point 1) and clean any large object from the results (point 2).

The drawback of this is that you won’t be able to apply some methods to the results (like changing the type of standard-errors, predict, resid, etc). But the amount of memory saved can be considerable.