InteractiveFixedEffectModels.jl

InteractiveFixedEffectModels.jl estimates interactive fixed effect models of the Bai (2009) form:

The package is designed for panel settings with latent factors, optional high-dimensional fixed effects on the same panel dimensions, weights, clustering, and incomplete panels.

It integrates with the Julia StatsModels / FixedEffectModels ecosystem:

• formulas use @formula(...)
• high-dimensional fixed effects use fe(...)
• covariance estimators come from Vcov.jl

Installation

The package is registered in the General registry:

] add InteractiveFixedEffectModels

Current package compatibility requires Julia 1.9+.

Quick Start

using DataFrames, RDatasets, InteractiveFixedEffectModels

df = dataset("plm", "Cigar")

result = regife(
    df,
    @formula(Sales ~ Price + ife(State, Year, 2) + fe(State))
)

display(result)

Typical output:

Interactive Fixed Effect Model
Number of obs: ...
R²: ...
R² within: ...

Model Specification

Every regife formula must contain exactly one interactive fixed effect term:

ife(id, time, rank)

where:

• id is the unit dimension
• time is the time dimension
• rank is the number of latent factors

Example:

@formula(Sales ~ Price + ife(State, Year, 2))

High-dimensional fixed effects can also be added with fe(...), but only on the same panel dimensions used in ife(...):

@formula(Sales ~ Price + ife(State, Year, 2) + fe(State))
@formula(Sales ~ Price + ife(State, Year, 2) + fe(Year))
@formula(Sales ~ Price + ife(State, Year, 2) + fe(State) + fe(Year))

This is invalid:

@formula(Sales ~ Price + ife(State, Year, 2) + fe(Region))

unless Region is the same dimension as one of the ife(...) variables.

Main Options

Covariance Estimators

Pass a Vcov.jl covariance estimator as the third positional argument:

regife(df, @formula(y ~ x + ife(id, time, 2)), Vcov.simple())
regife(df, @formula(y ~ x + ife(id, time, 2)), Vcov.robust())
regife(df, @formula(y ~ x + ife(id, time, 2)), Vcov.cluster(:id))
regife(df, @formula(y ~ x + ife(id, time, 2)), Vcov.cluster(:id, :time))

Weights

Use a positive weight column:

regife(df, @formula(y ~ x + ife(id, time, 2)); weights = :w)

Subsetting

Restrict estimation to a subset while keeping saved outputs aligned with the original data:

regife(df, @formula(y ~ x + ife(id, time, 2)); subset = df.year .>= 1980)

Optimization Method

Available methods are:

• :dogleg
• :levenberg_marquardt
• :gauss_seidel

Example:

regife(df, @formula(y ~ x + ife(id, time, 2)); method = :dogleg)

Saved Outputs

With save = true, the returned object contains an augmentdf aligned with the original data. Depending on the specification, it can include:

• residuals
• factors1, factors2, ...
• loadings1, loadings2, ...
• absorbed fixed effects such as fe_State

Rows excluded from estimation are filled with missing.

Worked Examples

Interactive Fixed Effect Regression

using DataFrames, RDatasets, InteractiveFixedEffectModels

df = dataset("plm", "Cigar")

result = regife(
    df,
    @formula(Sales ~ Price + ife(State, Year, 2) + fe(State)),
    Vcov.cluster(:State);
    save = true
)

coef(result)
result.augmentdf

Factor Model / PCA-Like Use

Factor models are a special case with no observed regressors.

using DataFrames, RDatasets, InteractiveFixedEffectModels

df = dataset("plm", "Cigar")

result = regife(
    df,
    @formula(Sales ~ 0 + ife(State, Year, 2));
    save = true
)

To demean with respect to one panel dimension:

regife(df, @formula(Sales ~ ife(State, Year, 2) + fe(State)); save = true)

Programmatic Formula Construction

using StatsModels

regife(
    df,
    Term(:Sales) ~ Term(:Price) + ife(Term(:State), Term(:Year), 2) + fe(Term(:State))
)

Returned Objects

If the model includes regressors, regife returns an InteractiveFixedEffectModel, which behaves like a regression result and supports methods such as:

• coef
• coefnames
• vcov
• confint
• coeftable
• nobs
• r2
• adjr2

If the model contains no regressors, it returns a lighter FactorResult with estimation metadata and the optional saved dataframe.

Notes and Caveats

Local Minima

The package supports:

• missing observations within the id × time panel
• multiple observations per id × time
• weights

In those cases, the optimization problem can have local minima. The package includes a restart heuristic to reduce that risk, but it is still a numerical optimization problem rather than a closed-form estimator.

Standard Errors

The reported standard errors are based on regressing on covariates of the form:

• i.id # c.time
• i.time # c.id

This follows the discussion in Section 6 of Bai (2009).

Bias Correction

The package does not implement the Bai (2009) bias correction for settings with residual cross-sectional or serial dependence beyond the factor structure. In such cases, coefficient estimates can remain consistent but biased in finite samples.

When should one use interactive fixed effects models?

Some literature 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

Related Packages

• FixedEffectModels.jl: linear models with high-dimensional fixed effects
• FactorModels.jl: factor models on matrices
• LowRankModels.jl: general low-rank approximation models

Reference

• Bai, Jushan (2009), "Panel Data Models With Interactive Fixed Effects," Econometrica.