## FixedEffectModels.jl

Fast Estimation of Linear Models with IV and High Dimensional Categorical Variables
Popularity
99 Stars
Updated Last
10 Months Ago
Started In
June 2015

This package estimates linear models with high dimensional categorical variables and/or instrumental variables.

Its objective is similar to the Stata command `reghdfe` and the R function `felm`. The package tends to be much faster than these two options. ## Installation

The package is registered in the `General` registry and so can be installed at the REPL with `] add FixedEffectModels`.

## Syntax

```using DataFrames, RDatasets, FixedEffectModels
df = dataset("plm", "Cigar")
reg(df, @formula(Sales ~ NDI + fe(State) + fe(Year)), Vcov.cluster(:State), weights = :Pop)
# =====================================================================
# Number of obs:               1380   Degrees of freedom:            32
# R2:                         0.803   R2 Adjusted:                0.798
# F Statistic:              13.3382   p-value:                    0.001
# R2 within:                  0.139   Iterations:                     6
# Converged:                   true
# =====================================================================
#         Estimate  Std.Error  t value Pr(>|t|)   Lower 95%   Upper 95%
# ---------------------------------------------------------------------
# NDI  -0.00526264 0.00144097 -3.65216    0.000 -0.00808942 -0.00243586
# =====================================================================```
• A typical formula is composed of one dependent variable, exogeneous variables, endogeneous variables, instrumental variables, and a set of high-dimensional fixed effects.

`dependent variable ~ exogenous variables + (endogenous variables ~ instrumental variables) + fe(fixedeffect variable)`

High-dimensional 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, use `fe(State) + fe(Year)`. To add state-year fixed effects, use `fe(State)&fe(Year)`. To add state specific slopes for year, use `fe(State)&Year`. To add both state fixed-effects and state specific slopes for year use `fe(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 programatically, use

`reg(df, Term(:Sales) ~ Term(:NDI) + fe(Term(:State)) + fe(Term(:Year))`
• Standard errors are indicated with the prefix `Vcov`.

``` Vcov.robust()
Vcov.cluster(:State)
Vcov.cluster(:State, :Year)```
• The option `weights` specifies a variable for weights

` weights = :Pop`
• The option `subset` specifies a subset of the data

` subset = df.State .>= 30`
• The option `save` can be set to one of the following: `:residuals` to save residuals, `:fe` to save fixed effects, `true` to save both. You can access the result with `residuals()` and `fe()`

• The option `method` can be set to one of the following: `:cpu`, `:gpu` (see Performances below).

• The option `contrasts` specifies particular contrasts for categorical variables in the formula, e.g.

``` df.YearC = categorical(df.Year)
reg(df, @formula(Sales ~ YearC); contrasts = Dict(:YearC => DummyCoding(base = 80)))```

## Output

`reg` returns a light object. It is composed of

• the vector of coefficients & the covariance matrix (use `coef`, `coefnames`, `vcov` on the output of `reg`)
• a boolean vector reporting rows used in the estimation
• a set of scalars (number of observations, the degree of freedoms, r2, etc)
• with the option `save = true`, a dataframe aligned with the initial dataframe with residuals and, if the model contains high dimensional fixed effects, fixed effects estimates (use `residuals` or `fe` on the output of `reg`)

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 publication-quality regression tables.

## Performances

#### GPU

The package has support for GPUs (Nvidia) (thanks to Paul Schrimpf). This can make the package an order of magnitude faster for complicated problems.

First make sure that `using CuArrays` works without issue. Then, estimate a model with `method = :gpu`.

When working on the GPU, it is encouraged to set the floating point precision to `Float32` with `double_precision = false`, since it is usually much faster.

```using FixedEffectModels
df = dataset("plm", "Cigar")
reg(df, @formula(Sales ~ NDI + fe(State) + fe(Year)), method = :gpu, double_precision = false)```

## 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:

1. `y, X` are regressed on `D` using the package FixedEffects.jl
2. Estimates for `β`, along with their standard errors, are obtained by regressing the projected `y` on the projected `X` (an application of the Frisch Waugh-Lovell Theorem)
3. 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 on `D` using the package FixedEffects.jl

# References

Baum, C. and Schaffer, M. (2013) AVAR: Stata module to perform asymptotic covariance estimation for iid and non-iid data robust to heteroskedasticity, autocorrelation, 1- and 2-way clustering, and common cross-panel autocorrelated disturbances. Statistical Software Components, Boston College Department of Economics.

Correia, S. (2014) REGHDFE: Stata module to perform linear or instrumental-variable regression absorbing any number of high-dimensional fixed effects. Statistical Software Components, Boston College Department of Economics.

Fong, DC. and Saunders, M. (2011) LSMR: An Iterative Algorithm for Sparse Least-Squares Problems. SIAM Journal on Scientific Computing

Gaure, S. (2013) OLS with Multiple High Dimensional Category Variables. Computational Statistics and Data Analysis

Kleibergen, F, and Paap, R. (2006) Generalized reduced rank tests using the singular value decomposition. Journal of econometrics

Kleibergen, F. and Schaffer, M. (2007) RANKTEST: Stata module to test the rank of a matrix using the Kleibergen-Paap rk statistic. Statistical Software Components, Boston College Department of Economics.