Pkg.add("InteractiveFixedEffectModels")

This package implements a novel, fast and robust algorithm to estimate interactive fixed effect models (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

• The first argument of fit is an object of type InteractiveFixedEffectModel. Such an object can be constructed by specifying the id variable, the time variable, and the rank of the factor model (r in the model above). Both the id and time variable must be of type PooledDataVector.

   using RDatasets, DataFrames, InteractiveFixedEffectModels
   df = dataset("plm", "Cigar")
   # create PooledDataVector
   df[:pState] =  pool(df[:State])
   df[:pYear] =  pool(df[:Year])
   # create InteractiveFixedEffectModel in state, year, and rank 2
   factormodel = InteractiveFixedEffectModel(:pState, :pYear, 2)

• The second argument of fit is a formula

  • When the only regressor is 0, fit fits a factor model on the left hand side variable

     fit(InteractiveFixedEffectModel(:pState, :pYear, 2), Sales ~ 0, df)

    You can pre-demean the variable using |> as in the package FixedEffectModels.jl. Use only the variables specified in the factor model.

     fit(InteractiveFixedEffectModel(:pState, :pYear, 2), Sales ~ 1 |> pState, df)
     fit(InteractiveFixedEffectModel(:pState, :pYear, 2), Sales ~ 1 |> pYear, df)
     fit(InteractiveFixedEffectModel(:pState, :pYear, 2), Sales ~ 1 |> pState + pYear, df)

  • With multiple regressors, fit fits a linear model with interactive fixed effects (Bai (2009))

     fit(InteractiveFixedEffectModel(:pState, :pYear, 2), Sales ~ Price, df)

    Similarly, you may add id or time fixed effects

     fit(InteractiveFixedEffectModel(:pState, :pYear, 2), Sales ~ Price |> pState, df)

• Two minimization methods are available:

  • :levenberg_marquardt
  • :dogleg

• Compute robust standard errors by constructing an object of type AbstractVcovMethod. For now, VcovSimple() (default), VcovWhite() and VcovCluster(cols) are implemented.

   fit(InteractiveFixedEffectModel(:pState, :pYear, 2), Sales ~ Price, df, VcovCluster(:pState))

• 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 with NA).

The general syntax is

fit(pfm::InteractiveFixedEffectModel,
	f::Formula, 
    df::AbstractDataFrame, 
    vcov_method::AbstractVcovMethod = VcovSimple();
 	method::Symbol = :dogleg
    weight::Union{Symbol, Void} = nothing, 
    subset::Union{AbstractVector{Bool}, Void} = nothing,
    save::Bool = true, 
    maxiter::Int64 = 10000, tol::Float64 = 1e-8
    )

The algorithm can estimate models with missing observations per id x time, multiple observations per id x time, and weights (see below).

With multiple observations per id x time, or with weights non constant within id or time, the optimization problem may have local minima. The algorithm tries to catch these cases, and, when this happens, the optimization algorithm is restarted on a random starting point. However I'm not sure all cases are caught.

Some litterature using this estimation procedure::

• Eberhardt, Helmers, Strauss (2013) Do spillovers matter when estimating private returns to R&D?
• Hagedorn, Karahan, Movskii (2015) Unemployment Benefits and Unemployment in the Great Recession: The Role of Macro Effects
• Hagedorn, Karahan, Movskii (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

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).

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.

• Bai, Jushan. Panel data models with interactive fixed effects. (2009) Econometrica
• Ilin, Alexander, and Tapani Raiko. Practical approaches to principal component analysis in the presence of missing values. (2010) The Journal of Machine Learning Research 11
• Koren, Yehuda. Factorization meets the neighborhood: a multifaceted collaborative filtering model. (2008) Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining.
• Raiko, Tapani, Alexander Ilin, and Juha Karhunen. Principal component analysis for sparse high-dimensional data. (2008) Neural Information Processing.
• Srebro, Nathan, and Tommi Jaakkola. Weighted low-rank approximations (2010) The Journal of Machine Learning Research 11
• Nocedal, Jorge and Stephen Wright An Inexact Levenberg-Marquardt method for Large Sparse Nonlinear Least Squares (1985) The Journal of the Australian Mathematical Society

• 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