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xtife: Interactive Fixed Effects Estimator for Panel Data

R CMD Check CRAN status License: GPL v2/v3

xtife provides a pure base-R implementation of the Interactive Fixed Effects (IFE) panel estimator for both balanced and unbalanced panels. It delivers full analytical standard errors, asymptotic bias correction, and factor number selection with no external dependencies beyond base R.

For a comprehensive review of interactive fixed effects, see Ditzen & Karavias (2025).

The Model

Standard two-way fixed effects (TWFE) assumes unobserved heterogeneity enters additively. IFE generalises this by allowing unobserved confounders to interact across units and time:

$$y_{it} = \alpha_i + \xi_t + X_{it}'\beta + \lambda_i'F_t + u_{it}$$

where $F_t \in \mathbb{R}^r$ are common factors and $\lambda_i \in \mathbb{R}^r$ are unit-specific loadings. Setting $r = 0$ reduces the model to standard TWFE. For unbalanced panels, ife_unbalanced() supports the same additive fixed effects via its force argument ("none", "unit", "time", "two-way"), estimated jointly with the factors by EM on the imputed panel — robust to informative (factor-correlated) missingness. Its default is force = "none" (the intercept-free interactive model); use force = "two-way" for data with level or trend structure.

Features

Feature Balanced (ife) Unbalanced (ife_unbalanced)
Estimator Bai (2009) SVD alternating projections Su, Wang & Wang (2025); EM with matrix completion (Bai & Ng 2021); NNR (default) or OLS init
Standard errors Homoskedastic · HC1 robust · Cluster Homoskedastic · HC1 robust · Cluster · HAC
Static bias correction Bai (2009) $\hat B/N + \hat C/T$ Analytical incidental-parameter correction
Dynamic bias correction Moon & Weidner (2017) Analytical, incl. predetermined-regressor term
Factor number selection IC1/2/3 · IC(BIC) · PC SVT rule (singular value thresholding)
Initialisation — Nuclear-norm regularisation (NNR, default) · OLS
Dependencies Base R only Base R only

Installation

# From CRAN
install.packages("xtife")

# Development version from GitHub
# install.packages("remotes")
remotes::install_github("Rickchen0910/xtife")

What's new in 0.1.5

The unbalanced-panel estimator is now fully aligned with the reference two-step procedure of Su, Wang & Wang (2025): the NNR penalty grid is $c \cdot \max(N,T)$, $c \in {0.01, 0.1, 1, 10}$ (their footnote 12), and ife_unbalanced() defaults to init = "nnr" (their nuclear-norm-regularised first-step estimator). Degenerate penalty candidates (which previously could make ife_select_r_unb() return $\hat r = \min(N,T)$) are now excluded. See NEWS.md for the full history.

Balanced Panel: Quick Start

library(xtife)
data(cigar)   # 46 US states x 30 years cigarette panel (Baltagi 1995)

# Fit IFE with r = 2 factors, two-way FE, cluster-robust SE
fit <- ife(sales ~ price, data = cigar,
           index  = c("state", "year"),
           r      = 2,
           force  = "two-way",
           se     = "cluster")
print(fit)
Interactive Fixed Effects (Bai 2009, Econometrica)
-------------------------------------------------------
Panel    : N = 46 units, T = 30 periods
Factors  : r = 2
Force    : two-way fixed effects
SE type  : cluster (by state)
Outcome  : sales
-------------------------------------------------------
      Estimate Std.Error t.value Pr(>|t|)             95% CI
price  -0.5242    0.0802 -6.5333   0.0000 [-0.6816, -0.3667] ***
---
Signif. codes: *** <0.01  ** <0.05  * <0.1
-------------------------------------------------------
sigma^2  : 18.456076 | df = 1157
Converged: YES | Iterations: 10

Standard error types

fit_std <- ife(sales ~ price, data = cigar,
               index = c("state", "year"), r = 2, se = "standard")
fit_rob <- ife(sales ~ price, data = cigar,
               index = c("state", "year"), r = 2, se = "robust")
fit_cl  <- ife(sales ~ price, data = cigar,
               index = c("state", "year"), r = 2, se = "cluster")
se = Assumption Typical use
"standard" Homoskedasticity Benchmark
"robust" HC1 sandwich Heteroskedasticity across cells
"cluster" Cluster-robust by unit Serial correlation within units (recommended)

Factor number selection

sel <- ife_select_r(sales ~ price, data = cigar,
                    index = c("state", "year"),
                    r_max = 6, force = "two-way")

Prints a table of IC1, IC2, IC3 (Bai & Ng 2002), IC(BIC), and PC (Bai 2009) for each candidate $r$. IC(BIC) is recommended for panels with $\min(N, T) &lt; 60$.

Asymptotic bias correction

# Static bias correction — Bai (2009)
fit_bc <- ife(sales ~ price, data = cigar,
              index = c("state", "year"), r = 2, bias_corr = TRUE)

# Dynamic bias correction — Moon & Weidner (2017)
# Use when regressors include lagged dependent variables
fit_dyn <- ife(sales ~ price, data = cigar,
               index = c("state", "year"), r = 2,
               method = "dynamic", bias_corr = TRUE, M1 = 1L)

For the cigar panel ($N = 46$, $T = 30$, $T/N \approx 0.65$):

Estimator Price coefficient
TWFE ($r = 0$) −1.0847
IFE ($r = 2$) −0.5242
IFE + Bai (2009) bias correction −0.5285
IFE dynamic + Moon & Weidner (2017) bias correction −0.5316

Comparison with TWFE

Setting r = 0 recovers the standard two-way FE estimator, identical to lm() with unit and time dummies at machine precision:

fit0 <- ife(sales ~ price, data = cigar,
            index = c("state", "year"), r = 0)
# Equivalent to plm(..., model = "within", effect = "twoways")

Unbalanced Panel: Quick Start

ife_unbalanced() fits the IFE model on genuinely unbalanced panels following the estimation and inference theory of Su, Wang & Wang (2025), which extends the interactive fixed effects estimator of Bai (2009) to the unbalanced case using the missing-data factor analysis / matrix completion of Bai & Ng (2021). The core algorithm is an alternating outer loop that updates β and the structure $(\hat\alpha, \hat\xi, \hat\lambda, \hat F)$ until convergence, with an expectation-maximisation inner loop that imputes the unobserved cells from the current structure. Initialisation defaults to the nuclear-norm-regularised (NNR) consistent initial estimator of Su, Wang & Wang (2025) (init = "nnr", solved by the soft-impute algorithm of Mazumder, Hastie & Tibshirani 2010 — the same convex objective they solve by ADMM); a faster pooled-OLS warm start is available via init = "ols". The resulting estimator is $\sqrt{NT}$-consistent and asymptotically normal.

# Simulate a 10% randomly missing panel
set.seed(42)
cigar_unb <- cigar[sample(nrow(cigar), size = floor(0.9 * nrow(cigar))), ]

fit_unb <- ife_unbalanced(sales ~ price,
                           data  = cigar_unb,
                           index = c("state", "year"),
                           r     = 2L,
                           se    = "cluster")
print(fit_unb)

Standard error types

se = Assumption When to use
"standard" Homoskedastic i.i.d. errors Benchmark
"robust" HC1 Cell-level heteroskedasticity
"cluster" Cluster-robust by unit Serial correlation within units
"hac" Bartlett kernel (bandwidth $L_T = \lfloor 2T^{1/5}\rfloor$) Serial correlation over time within units

Factor number selection

ife_select_r_unb() applies the singular value thresholding (SVT) rule of Su, Wang & Wang (2025) to the nuclear-norm regularised matrix $\hat\Theta^{(0)}$ — a missing-data counterpart of the Bai & Ng (2002) information criteria:

sel_unb <- ife_select_r_unb(sales ~ price, data = cigar_unb,
                              index = c("state", "year"))
# Returns: r_hat, singular values, SVT threshold, nu_used (NNR penalty from cross-validation)

Analytical bias correction

Setting bias_corr = TRUE applies the analytical incidental-parameter correction of Su, Wang & Wang (2025), $\hat\beta^{abc} = \hat\beta - (NT)^{-1/2}\hat W_X^{-1}\hat b$, where the bias vector $\hat b$ depends on the exogeneity assumption:

# Strictly exogenous regressors (default): b3 + b4 + b5 + b6
fit_bc <- ife_unbalanced(sales ~ price, data = cigar_unb,
                          index = c("state", "year"), r = 2,
                          se = "standard", bias_corr = TRUE,
                          exog = "strict")

# Weakly exogenous regressors (e.g. lagged dep. var.): b2 + b3 + ... + b6
fit_dyn_unb <- ife_unbalanced(y ~ lag_y,
                               data = df_dynamic, index = c("i", "t"),
                               r = 2, se = "hac", bias_corr = TRUE,
                               exog = "weak")
exog = Bias terms Typical application
"strict" (default) $\hat b_3 + \hat b_4 + \hat b_5 + \hat b_6$ Standard panel regressors
"weak" $\hat b_2 + \hat b_3 + \hat b_4 + \hat b_5 + \hat b_6$ Lagged dependent variable

Initialisation options

# Default: NNR initial estimator (Su, Wang & Wang 2025 first step)
fit_nnr <- ife_unbalanced(sales ~ price, data = cigar_unb,
                           index = c("state", "year"), r = 2, init = "nnr")

# Faster pooled-OLS warm start (not covered by the initial-estimator theory)
fit_ols <- ife_unbalanced(sales ~ price, data = cigar_unb,
                           index = c("state", "year"), r = 2, init = "ols")

Function Reference

Function Panel type Description
ife() Balanced Fit IFE model (Bai 2009); returns coefficients, SEs, factors, loadings
print.ife() Balanced Formatted coefficient table and model summary
ife_select_r() Balanced Fit IFE for $r = 0, \ldots, r_{\max}$; compare IC1/2/3, IC(BIC), PC
ife_unbalanced() Unbalanced Fit IFE via EM with matrix completion; NNR (default) or OLS initialisation; analytical SE and bias correction
print.ife_unb() Unbalanced Formatted coefficient table with bias components
ife_select_r_unb() Unbalanced SVT factor selection (singular value thresholding)

Key ife() arguments

Argument Default Description
formula — outcome ~ covariate1 + ...
data — Long-format data.frame (balanced)
index — c("unit_col", "time_col")
r 1 Number of interactive factors
force "two-way" Additive FE: "none", "unit", "time", "two-way"
se "standard" SE type: "standard", "robust", "cluster"
bias_corr FALSE Apply analytical bias correction
method "static" "static" (Bai 2009) or "dynamic" (Moon & Weidner 2017)
M1 1L Lag bandwidth for dynamic $\hat B_1$ bias term

Key ife_unbalanced() arguments

Argument Default Description
formula — outcome ~ covariate1 + ...
data — Long-format data.frame (balanced or unbalanced)
index — c("unit_col", "time_col")
r 1L Number of interactive factors
force "none" Additive FE: "none", "unit", "time", "two-way" (jointly estimated with the factors; default differs from ife() as the unbalanced model is intercept-free by default)
se "standard" SE type: "standard", "robust", "cluster", "hac"
init "nnr" Initialisation: "nnr" (nuclear-norm initial estimator, Su, Wang & Wang 2025 first step) or "ols" (faster warm start)
bias_corr FALSE Apply analytical incidental-parameter bias correction
exog "strict" Exogeneity: "strict" or "weak" (dynamic regressors)
L_T NULL HAC bandwidth ($\lfloor 2T^{1/5}\rfloor$ if NULL)

About

Author

Binzhi Chen (University of Essex)

Email: Binzhi.Chen9@gmail.com

Web: https://rickchen0910.github.io/

Citation

Please cite as follows:

Chen, B. (2026). xtife: Interactive Fixed Effects Estimator for Panel Data. R package version 0.1.5. https://CRAN.R-project.org/package=xtife.

@Manual{xtife,
  title  = {{xtife}: Interactive Fixed Effects Estimator for Panel Data},
  author = {Binzhi Chen},
  year   = {2026},
  note   = {R package version 0.1.5},
  url    = {https://CRAN.R-project.org/package=xtife},
}

References

Bai, J. (2009). Panel data models with interactive fixed effects. Econometrica, 77(4), 1229–1279. doi:10.3982/ECTA6135

Bai, J. and Ng, S. (2002). Determining the number of factors in approximate factor models. Econometrica, 70(1), 191–221. doi:10.1111/1468-0262.00273

Bai, J. and Ng, S. (2021). Matrix completion, counterfactuals, and factor analysis of missing data. Journal of the American Statistical Association, 116(536), 1746–1763. doi:10.1080/01621459.2021.1967163

Baltagi, B.H. (1995). Econometric Analysis of Panel Data. Wiley.

Cameron, A.C., Gelbach, J.B. and Miller, D.L. (2011). Robust inference with multiway clustering. Journal of Business & Economic Statistics, 29(2), 238–249. doi:10.1198/jbes.2010.07136

Ditzen, J. and Karavias, Y. (2025). Interactive, Grouped and Non-separable Fixed Effects: A Practitioner's Guide to the New Panel Data Econometrics. arXiv:2507.19099. doi:10.48550/arXiv.2507.19099

Mazumder, R., Hastie, T. and Tibshirani, R. (2010). Spectral regularization algorithms for learning large incomplete matrices. Journal of Machine Learning Research, 11, 2287–2322.

Moon, H.R. and Weidner, M. (2017). Dynamic linear panel regression models with interactive fixed effects. Econometric Theory, 33, 158–195. doi:10.1017/S0266466615000328

Newey, W.K. and West, K.D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55(3), 703–708. doi:10.2307/1913610

Su, L., Wang, F. and Wang, Y. (2025). Estimation and inference for interactive fixed effects panel data models with unbalanced panels. SSRN Working Paper No. 5177283. doi:10.2139/ssrn.5177283

License

GPL-2 | GPL-3 © 2026 Binzhi Chen

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Interactive Fixed Effects Estimator for Balanced Panel Data

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