Different result using fixest and multiple fixed effects - r

Question

First of all, I have to apologize if my headline is misleading. I am not sure how to put it appropriately for my question.
I am currently working on the fixed-effect model. My data looks like the following table, although it is not actual data due to the information privacy.
state
district
year
grade
Y
X
id
AK
1001
2009
3
0.1
0.5
1001.3
AK
1001
2010
3
0.8
0.4
1001.3
AK
1001
2011
3
0.5
0.7
1001.3
AK
1001
2009
4
1.5
1.3
1001.4
AK
1001
2010
4
1.1
0.7
1001.4
AK
1001
2011
4
2.1
0.4
1001.4
...
...
...
..
..
..
...
WY
5606
2011
6
4.2
5.3
5606.6
I used the fixest package to run the fixed-effect model for this project. To get the unique observation in this dataset, I have to combine district, grade, and year. Note that I avoided using plm because there is no way to specify three fixed effects in the model unless you combine two identities (in my case, I generated id by combining district and grade). fixest seems to be able to solve this problem. However, I got different results when specifying three fixed effects (district, grade, and year) compared to two fixed effects (id and year). The following results and codes may clear up some confusion from my explanation.
# Two fixed effects (id and year)
df <- transform(df, id = apply(df[c("district", "grade")], 1, paste, collapse = "."))
fe = feols(y ~ x | id + year, df, se = "standard")
summary(fe)
OLS estimation, Dep. Var.: y
Observations: 499,112
Fixed-effects: id: 64,302, year: 10
Standard-errors: IID
Estimate Std. Error t value Pr(>|t|)
X 0.012672 0.003602 3.51804 0.00043478 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
RMSE: 0.589222 Adj. R2: 0.761891
Within R2: 2.846e-5
###########################################################################
# Three fixed effects (district, grade, and year)
fe = feols(y ~ x | district + grade + year, df, se = "standard")
summary(fe)
OLS estimation, Dep. Var.: y
Observations: 499,112
Fixed-effects: district: 11,097, grade: 6, year: 10
Standard-errors: IID
Estimate Std. Error t value Pr(>|t|)
X 0.014593 0.00401 3.63866 0.00027408 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
RMSE: 0.702543 Adj. R2: 0.698399
Within R2: 2.713e-5
Questions:
Why are the results different?
This is an equation I plan to use;. I am not sure which model is associated with this specification. To my feeling, it could be the second one. But if it is the case, why do many websites recommend combining two identities and running normal plm.
Thank you so much for reading my question. Any answers/ suggestions/ advice would be appreciated!

Answer 1

The answer is simply that you are estimating two different models.
Three fixed-effects (FEs):
Year + id FEs (I renamed id in to district_grade):
The first set of fixed-effects is strictly included in the set of FEs of the second estimation, which is more restrictive.
Here is a reproducible example in which we see that we obtain two different sets of coefficients.
data(trade)
est = fepois(Euros ~ log(dist_km) | sw(Origin + Product, Origin^Product) + Year, trade)
etable(est, vcov = "iid")
#> model 1 model 2
#> Dependent Var.: Euros Euros
#>
#> log(dist_km) -1.020*** (1.18e-6) -1.024*** (1.19e-6)
#> Fixed-Effects: ------------------- -------------------
#> Origin Yes No
#> Product Yes No
#> Year Yes Yes
#> Origin-Product No Yes
#> _______________ ___________________ ___________________
#> S.E. type IID IID
#> Observations 38,325 38,325
#> Squared Cor. 0.27817 0.35902
#> Pseudo R2 0.53802 0.64562
#> BIC 2.75e+12 2.11e+12
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
We can see that they have different FEs, which confirms that the models estimated are completely different:
summary(fixef(est$`Origin + Product + Year`))
#> Fixed_effects coefficients
#> Origin Product Year
#> Number of fixed-effects 15 20 10
#> Number of references 0 1 1
#> Mean 23.5 -0.012 0.157
#> Standard-deviation 1.15 1.35 0.113
#> COEFFICIENTS:
#> Origin: AT BE DE DK ES
#> 22.91 23.84 24.62 23.62 24.83 ... 10 remaining
#> -----
#> Product: 1 2 3 4 5
#> 0 1.381 0.624 1.414 -1.527 ... 15 remaining
#> -----
#> Year: 2007 2008 2009 2010 2011
#> 0 0.06986 0.006301 0.07463 0.163 ... 5 remaining
summary(fixef(est$`Origin^Product + Year`))
#> Fixed_effects coefficients
#> Origin^Product Year
#> Number of fixed-effects 300 10
#> Number of references 0 1
#> Mean 23.1 0.157
#> Standard-deviation 1.96 0.113
#> COEFFICIENTS:
#> Origin^Product: 101 102 103 104 105
#> 22.32 24.42 24.82 21.28 23.04 ... 295 remaining
#> -----
#> Year: 2007 2008 2009 2010 2011
#> 0 0.06962 0.006204 0.07454 0.1633 ... 5 remaining