This package estimates linear models with high dimensional categorical variables and/or instrumental variables.
Installation
The package is registered in the General
registry and so can be installed at the REPL with ] add FixedEffectModels
.
Benchmarks
The objective of the package is similar to the Stata command reghdfe
and the R packages lfe
and fixest
. The package is much faster than reghdfe
or lfe
. It also tends to be a bit faster than the more recent fixest
(depending on the exact command). For complicated models, FixedEffectModels
can also run on Nvidia GPUs for even faster performances (see below)
Syntax
using DataFrames, RDatasets, FixedEffectModels
df = dataset("plm", "Cigar")
reg(df, @formula(Sales ~ NDI + fe(State) + fe(Year)), Vcov.cluster(:State), weights = :Pop)
# FixedEffectModel
# =========================================================================
# Number of obs: 1380 Converged: true
# dof (model): 1 dof (residuals): 45
# R²: 0.803 R² adjusted: 0.798
# Fstatistic: 13.3382 Pvalue: 0.001
# R² within: 0.139 Iterations: 5
# =========================================================================
# Estimate Std. Error tstat Pr(>t) Lower 95% Upper 95%
# ─────────────────────────────────────────────────────────────────────────
# NDI 0.00526264 0.00144097 3.65216 0.0007 0.0081649 0.00236038
# =========================================================================

A typical formula is composed of one dependent variable, exogenous variables, endogenous variables, instrumental variables, and a set of highdimensional fixed effects.
dependent variable ~ exogenous variables + (endogenous variables ~ instrumental variables) + fe(fixedeffect variable)
Highdimensional fixed effect variables are indicated with the function
fe
. You can add an arbitrary number of high dimensional fixed effects, separated with+
. You can also interact fixed effects using&
or*
.For instance, to add state fixed effects use
fe(State)
. To add both state and year fixed effects, usefe(State) + fe(Year)
. To add stateyear fixed effects, usefe(State)&fe(Year)
. To add state specific slopes for year, usefe(State)&Year
. To add both state fixedeffects and state specific slopes for year usefe(State)*Year
.reg(df, @formula(Sales ~ Price + fe(State) + fe(Year))) reg(df, @formula(Sales ~ NDI + fe(State) + fe(State)&Year)) reg(df, @formula(Sales ~ NDI + fe(State)&fe(Year))) # for illustration only (this will not run here) reg(df, @formula(Sales ~ (Price ~ Pimin)))
To construct formula programmatically, use
reg(df, term(:Sales) ~ term(:NDI) + fe(:State) + fe(:Year))

The option
contrasts
specifies that a column should be understood as a set of dummy variables:reg(df, @formula(Sales ~ Price + Year); contrasts = Dict(:Year => DummyCoding()))
You can specify different base levels
reg(df, @formula(Sales ~ Price + Year); contrasts = Dict(:Year => DummyCoding(base = 80)))

The option
weights
specifies a variable for weightsweights = :Pop

Standard errors are indicated with the prefix
Vcov
(with the package Vcov)Vcov.robust() Vcov.cluster(:State) Vcov.cluster(:State, :Year)

The option
save
can be set to one of the following::none
(default) to save nothing,:residuals
to save residuals,:fe
to save fixed effects, and:all
to save both. Once saved, they can then be accessed usingresiduals(m)
orfe(m)
wherem
is the estimated model (the object returned by the functionreg
). Both residuals and fixed effects are aligned with the original dataframe used to estimate the model. 
The option
method
can be set to one of the following::cpu
,:CUDA
, or:Metal
(see Performances below).
Output
reg
returns a light object. It is composed of
 the vector of coefficients & the covariance matrix (use
coef
,coefnames
,vcov
on the output ofreg
)  a boolean vector reporting rows used in the estimation
 a set of scalars (number of observations, the degree of freedoms, r2, etc)
Methods such as predict
, residuals
are still defined but require to specify a dataframe as a second argument. The problematic size of lm
and glm
models in R or Julia is discussed here, here, here here (and for absurd consequences, here and there).
You may use RegressionTables.jl to get publicationquality regression tables.
Performances
MultiThreads
FixedEffectModels
is multithreaded. Use the option nthreads
to select the number of threads to use in the estimation (defaults to Threads.nthreads()
).
GPUs
The package has an experimental support for GPUs. This can make the package an order of magnitude faster for complicated problems.
If you have a Nvidia GPU, run using CUDA
before using FixedEffectModels
. Then, estimate a model with method = :CUDA
.
using CUDA, FixedEffectModels
@assert CUDA.functional()
df = dataset("plm", "Cigar")
reg(df, @formula(Sales ~ NDI + fe(State) + fe(Year)), method = :CUDA)
The package also supports Apple GPUs with Metal.jl
, although I could not find a way to get better performance
using Metal, FixedEffectModels
@assert Metal.functional()
df = dataset("plm", "Cigar")
reg(df, @formula(Sales ~ NDI + fe(State) + fe(Year)), method = :Metal)
Solution Method
Denote the model y = X β + D θ + e
where X is a matrix with few columns and D is the design matrix from categorical variables. Estimates for β
, along with their standard errors, are obtained in two steps:
y, X
are regressed onD
using the package FixedEffects.jl Estimates for
β
, along with their standard errors, are obtained by regressing the projectedy
on the projectedX
(an application of the Frisch WaughLovell Theorem)  With the option
save = true
, estimates for the high dimensional fixed effects are obtained after regressing the residuals of the full model minus the residuals of the partialed out models onD
using the package FixedEffects.jl
References
Baum, C. and Schaffer, M. (2013) AVAR: Stata module to perform asymptotic covariance estimation for iid and noniid data robust to heteroskedasticity, autocorrelation, 1 and 2way clustering, and common crosspanel autocorrelated disturbances. Statistical Software Components, Boston College Department of Economics.
Correia, S. (2014) REGHDFE: Stata module to perform linear or instrumentalvariable regression absorbing any number of highdimensional fixed effects. Statistical Software Components, Boston College Department of Economics.
Fong, DC. and Saunders, M. (2011) LSMR: An Iterative Algorithm for Sparse LeastSquares Problems. SIAM Journal on Scientific Computing
Gaure, S. (2013) OLS with Multiple High Dimensional Category Variables. Computational Statistics and Data Analysis