Installation
The package is registered in the General
registry and so can be installed at the REPL with
] add InteractiveFixedEffectModels
.
Motivation
This package implements a novel, fast and robust algorithm to estimate interactive fixed effect models.
The definition of interactive fixed effects follows Bai (2009).Formally, denote T(i)
and I(i))
the two categorical dimensions associated with observation i
(typically time and id). This package estimates the set of coefficients β
, of factors (f1, .., fr)
and of loadings (λ1, ..., λr)
in the model
using DataFrames, RDatasets, InteractiveFixedEffectModels
df = dataset("plm", "Cigar")
regife(df, @formula(Sales ~ Price + ife(State, Year, 2) + fe(State)))
# Interactive Fixed Effect Model
# ================================================================
# Number of obs: 1380 Degree of freedom: 47
# R2: 0.976 R2 within: 0.435
# Iterations: 436 Converged: true
# ================================================================
# Estimate Std.Error t value Pr(>t) Lower 95% Upper 95%
# 
# Price 0.425372 0.0132871 32.0139 0.000 0.451438 0.399306
# ================================================================
Syntax

Formula

Interactive fixed effects are indicated with the function
ife
. For instance, to specify a factor model with id variableState
, time variableYear
, and rank 2, useife(State, Year, 2)
. 
Highdimensional Fixed effects can be used, as in
fe(State)
but only for the variables specified in the factor model. See FixedEffectModels.jl for more informationregife(df, @formula(Sales ~ Price + ife(State, Year, 2))) regife(df, @formula(Sales ~ Price + ife(State, Year, 2) + fe(State)))
To construct formula programatically, use
regife(df, Term(:Sales) ~ Term(:Price) + ife(Term(:State), Term(:Year), 2) + fe(Term(:State)))


Standard errors are indicated as follows
Vcov.robust() Vcov.cluster(:State) Vcov.cluster(:State, :Year)

The option
weights
can add weightsweights = :Pop

The option
method
can be used to choose between two algorithms::levenberg_marquardt
:dogleg

The option
save = true
saves a new dataframe storing residuals, factors, loadings and the eventual fixed effects. Importantly, the returned dataframe is aligned with the initial dataframe (rows not used in the estimation are simply filled withmissing
s).
FAQ
Local minimum vs global minimum
The algorithm can estimate models with missing observations per id x time, multiple observations per id x time, and weights.
However, in these cases, the optimization problem may have local minima. The algorithm tries to catch these cases, and, if need be, restart the optimization until the global minimum is reached. However I am not sure that all the cases are caught.
Does the package estimate PCA / factor models?
Yes. Factor models are a particular case of interactive fixed effect models.
To estimate a factor model without any demeaning
using DataFrames, RDatasets, InteractiveFixedEffectModels
df = dataset("plm", "Cigar")
regife(df, @formula(Sales ~ 0 + ife(State, Year, 2)), save = true)
To demean with respect to one dimension, use
using DataFrames, RDatasets, InteractiveFixedEffectModels
df = dataset("plm", "Cigar")
regife(df, @formula(Sales ~ ife(State, Year, 2) + fe(State)), save = true)
The algorithm used in this package allows one to estimate models with multiple (or missing) observations per id x time.
When should one use interactive fixed effects models?
Some litterature using this estimation procedure::
 Eberhardt, Helmers, Strauss (2013) Do spillovers matter when estimating private returns to R&D?
 Hagedorn, Karahan, Manovskii (2015) Unemployment Benefits and Unemployment in the Great Recession: The Role of Macro Effects
 Hagedorn, Karahan, Manovskii (2015) The impact of unemployment benefit extensions on employment: the 2014 employment miracle?
 Totty (2015) The Effect of Minimum Wages on Employment: A Factor Model Approach
How are standard errors computed?
Errors are obtained by regressing y on x and covariates of the form i.id#c.year
and i.year#c.id
. This way of computing standard errors is hinted in section 6 of of Bai (2009).
Does this command implement the bias correction term in Bai (2009)?
In presence of cross or time correlation beyond the factor structure, the estimate for beta is consistent but biased (see Theorem 3 in Bai 2009, which derives the correction term in special cases). However, this package does not implement any correction. You may want to check that your residuals are approximately i.i.d (in which case there is no need for bias correction).
Related Packages
 https://github.com/joidegn/FactorModels.jl : fits and predict factor models on matrices
 https://github.com/madeleineudell/LowRankModels.jl : fits general low rank approximations on matrices
 https://github.com/aaw/IncrementalSVD.jl: implementation of the backpropagation algorithm