Package 'xtife'

Dataset on US Cigarette Demand Panel

Description

Balanced panel of cigarette sales and prices across 46 US states for 30 years (1963–1992). Originally used in Baltagi (1995) and widely used as a benchmark dataset for panel estimators.

Usage

cigar

Format

A data frame with 1,380 rows and 9 variables:

state

US state identifier (integer, 1–46)

year

year (integer, 1963–1992)

price

cigarette price index

pop

state population

pop16

population aged 16 and over

cpi

consumer price index

ndi

per-capita disposable income

sales

per-capita cigarette sales (packs per person per year)

pimin

minimum cigarette price in adjoining states

Source

Baltagi, B.H. (1995) Econometric Analysis of Panel Data. Wiley. Distributed with the plm R package (Croissant and Millo 2008).

References

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

Croissant, Y. and Millo, G. (2008). Panel data econometrics in R: the plm package. Journal of Statistical Software, 27(2), 1–43. doi:10.18637/jss.v027.i02

---

Estimate Interactive Fixed Effects Model (Bai 2009)

Description

Fits the panel model

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

for balanced panel data with analytical standard errors.

Usage

ife(
  formula,
  data,
  index,
  r = 1L,
  force = "two-way",
  se = "standard",
  bias_corr = FALSE,
  method = "static",
  M1 = 1L,
  tol = 1e-09,
  max_iter = 10000L
)

Arguments

formula	R formula: outcome ~ covariate1 + covariate2 + ...
data	data.frame in long format (one row per unit-time observation)
index	character(2): c("unit_id_column", "time_id_column")
r	integer >= 0, number of interactive factors (default 1)
force	additive FE specification: "none" | "unit" | "time" | "two-way" (default "two-way"). Additive unit effects $\alpha_i$ and time effects $\xi_t$ are removed via the standard within transformation (iterative demeaning) before the SVD algorithm runs, following Bai (2009) Section 3. Bai (2009, p.1) shows that two-way additive effects are a special case of the interactive structure with $r = 2$ (setting $F_t = (1, \xi_t)'$ and $\lambda_i = (\alpha_i, 1)'$), so the IFE estimator remains consistent when additive effects are present regardless of the force choice, but pre-demeaning improves efficiency.
se	SE type: "standard" | "robust" | "cluster" (default "standard"; "cluster" clusters by unit id)
bias_corr	logical; if TRUE apply bias correction. For method = "static" uses the two-term Bai (2009) Sec. 7 correction (B/N + C/T). For method = "dynamic" uses the three-term Moon and Weidner (2017) correction (B1/T + B2/N + B3/T). Requires r > 0 and at least one covariate. (default FALSE)
method	"static" (default) for Bai (2009) strictly-exogenous regressors; "dynamic" for Moon and Weidner (2017) predetermined regressors (e.g. lagged dependent variable). The dynamic estimator uses double projection M_Lambda M_F on X in the SVD loop.
M1	integer; lag bandwidth for the B1 dynamic bias term (default 1L). Only used when method = "dynamic" and bias_corr = TRUE.
tol	convergence tolerance (default 1e-9)
max_iter	maximum iterations (default 10000L)

Value

An S3 object of class "ife" with the following components:

• coef – named p-vector of estimated coefficients

• vcov – p x p variance-covariance matrix

• se – named p-vector of standard errors

• tstat – named p-vector of t-statistics

• pval – named p-vector of two-sided p-values

• ci – p x 2 matrix of 95% confidence intervals (CI.lower, CI.upper)

• table – data.frame coefficient table (Estimate, Std.Error, t.value, Pr.t, CI.lower, CI.upper)

• F_hat – T x r estimated factor matrix

• Lambda_hat – N x r estimated loading matrix

• residuals – T x N residual matrix (full model)

• sigma2 – estimated error variance

• df – residual degrees of freedom

• n_iter – iterations to convergence

• converged – logical

• N, T, r, force, se_type – model dimensions and options

• call – matched call

References

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

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

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

Examples

data(cigar, package = "xtife")
fit <- ife(sales ~ price, data = cigar, index = c("state", "year"),
           r = 2, force = "two-way", se = "standard")
print(fit)

---

Select the Number of Factors via Information Criteria

Description

Fits the IFE model for r = 0, 1, ..., r_max and evaluates five information criteria at each value of r. Returns IC1, IC2, and IC3 from Bai and Ng (2002) Proposition 1, applied to IFE residuals per Bai (2009) Section 9.4, plus a BIC-style penalty (IC_bic) and a small-sample-corrected prediction criterion (PC) from Bai (2009). The criterion-minimising r for each IC is flagged with "*" in the printed table, and a data-driven recommendation (favouring IC_bic when the Bai-Ng criteria decrease monotonically) is displayed.

Usage

ife_select_r(
  formula,
  data,
  index,
  r_max = NULL,
  force = "two-way",
  verbose = TRUE,
  tol = 1e-09,
  max_iter = 10000L
)

Arguments

formula	R formula passed to ife()
data	long-format data.frame
index	character(2): c("unit_id", "time_id")
r_max	maximum r to consider (default: min(8, floor(min(N,T)/2)))
force	additive FE type (default "two-way")
verbose	logical; if TRUE (default) print progress and results table to the console. Set to FALSE for silent operation.
tol	convergence tolerance (default 1e-9)
max_iter	maximum iterations (default 10000L)

Value

(invisibly) a data.frame with columns r, V_r, IC1, IC2, IC3, IC_bic, PC, converged, and attribute "suggested" (named integer vector giving the IC-minimising r for each criterion).

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

Examples

data(cigar, package = "xtife")
  sel <- ife_select_r(sales ~ price, data = cigar,
                      index = c("state", "year"), r_max = 4)

---

Factor Number Selection for Unbalanced Panel IFE via SVT

Description

Estimates the number of interactive factors in an unbalanced panel by the singular value thresholding (SVT) rule of Su, Wang and Wang (2025), applied to the nuclear-norm-regularised (soft-imputed) matrix — a missing-data counterpart of the information-criterion rules of Bai and Ng (2002).

Usage

ife_select_r_unb(formula, data, index, c_f = 0.6, nu_NT = NULL, verbose = TRUE)

Arguments

formula	R formula: outcome ~ covariate1 + covariate2 + ...
data	Data frame in long format.
index	Character vector of length 2: c("unit_col", "time_col").
c_f	SVT threshold constant (default 0.6).
nu_NT	Optional scalar or vector of NNR penalty values. If NULL (default), selects over c(0.01, 0.1, 1, 10) * max(N, TT), the grid of Su, Wang and Wang (2025, footnote 12), by minimising the observed-cell residual sum of squares of the rank-r fit.
verbose	Logical; print result table. Default TRUE.

Value

Invisibly returns a list with components r_hat, sv (normalised singular values), threshold, c_f, c_NT, and nu_used.

References

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

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

Examples

data(cigar, package = "xtife")
  set.seed(42)
  cigar_unb <- cigar[sample(nrow(cigar), 1200L), ]
  ife_select_r_unb(sales ~ price, data = cigar_unb,
                   index = c("state", "year"))

---

Unbalanced Panel Interactive Fixed Effects Estimator

Description

Fits the interactive fixed effects model

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

for unbalanced panels (units observed at different sets of time periods), where the additive unit effects $\alpha_i$ and time effects $\xi_t$ are controlled by force. The estimation and inference theory follows Su, Wang and Wang (2025). Estimation uses an alternating outer loop that updates $\hat\beta$ and the structure $(\hat\alpha, \hat\xi, \hat\lambda, \hat F)$, with an expectation-maximisation (EM) inner loop — in the spirit of Bai (2009, Appendix B) and the missing-data factor / matrix-completion framework of Bai and Ng (2021) — that imputes the unobserved cells from the current structure and re-estimates the additive and interactive components on the completed panel. Following the two-step procedure of Su, Wang and Wang (2025), the default initialisation (init = "nnr") is their nuclear-norm-regularised consistent initial estimator, computed here by the soft-impute algorithm of Mazumder, Hastie and Tibshirani (2010) (the same convex objective they solve by ADMM); a faster pooled-OLS warm start is available via init = "ols".

Usage

ife_unbalanced(
  formula,
  data,
  index,
  r = 1L,
  force = "none",
  se = "standard",
  init = "nnr",
  bias_corr = FALSE,
  exog = "strict",
  L_T = NULL,
  c_f = 0.6,
  nu_NT = NULL,
  tol = 1e-09,
  max_iter = 10000L,
  tol_em = 1e-07,
  max_iter_em = 500L
)

Arguments

formula	R formula: outcome ~ covariate1 + covariate2 + ...
data	Data frame in long format (one row per observed unit-time pair).
index	Character vector of length 2: c("unit_col", "time_col").
r	Positive integer. Number of interactive factors (default 1).
force	Additive fixed effects to remove jointly with the factors: "none" (default; intercept-free interactive model), "unit", "time", or "two-way". Additive FE are estimated jointly with the factors via EM on the imputed (completed) panel, which is robust to informative (factor-correlated) missingness; use force = "two-way" for data with level/trend structure (matching the balanced ife() default). Note: with strong, near-collinear common trends, "two-way" can converge slowly.
se	SE type: "standard" (homoskedastic), "robust" (HC1), "cluster" (cluster-robust by unit), or "hac" (HAC with a Bartlett kernel, for serially correlated errors). Default "standard".
init	Initialisation method: "nnr" (default, nuclear-norm-regularised initial estimator — the first step of Su, Wang and Wang (2025), solved by soft-impute) or "ols" (pooled-OLS warm start; faster, but not covered by the consistency theory for the initial estimator).
bias_corr	Logical. Apply the analytical incidental-parameter bias correction. Supports both strictly and weakly exogenous regressors (controlled by exog). Default FALSE.
exog	Exogeneity assumption: "strict" (default, regressors uncorrelated with past and future errors) or "weak" (weakly exogenous, e.g., lagged dependent variable $x_{it} = y_{i,t-1}$). When "weak" and bias_corr = TRUE, an additional dynamic bias term $\hat{b}_2$ is included.
L_T	Bartlett kernel bandwidth for HAC standard errors (se = "hac") and the dynamic bias term $\hat{b}_2$ (exog = "weak", bias_corr = TRUE). If NULL (default), set to $\lfloor 2 T^{1/5} \rfloor$ after the panel dimensions are known.
c_f	Singular-value-thresholding constant (default 0.6) used for factor-number selection. Used only when init = "nnr".
nu_NT	NNR penalty grid. If NULL (default), selects over c * max(N, TT) for c in c(0.01, 0.1, 1, 10), the grid of Su, Wang and Wang (2025, footnote 12).
tol	Outer-loop convergence tolerance on $\max|\hat\beta^{new} - \hat\beta^{old}|$. Default 1e-9.
max_iter	Maximum outer-loop iterations. Default 10000L.
tol_em	Inner EM convergence tolerance. Default 1e-7.
max_iter_em	Maximum inner EM iterations per outer step. Default 500L.

Details

Inference. Standard errors use a sandwich estimator on factor-projected regressors (the unbalanced analogue of the balanced formula of Bai 2009), following Su, Wang and Wang (2025), with heteroskedasticity-robust, cluster-robust (Arellano 1987; Cameron, Gelbach and Miller 2011) and HAC (Newey and West 1987) variants. The optional analytical bias correction of Su, Wang and Wang (2025) removes the leading incidental-parameter bias, extending the corrections of Bai (2009) and Moon and Weidner (2017) to the unbalanced and predetermined-regressor case.

Additive fixed effects. With force = "none" (default) the model is intercept-free and all heterogeneity is carried by the interactive factors. Setting force = "unit", "time" or "two-way" estimates the additive effects jointly with the factors by demeaning the imputed (completed) panel inside the EM loop, which is robust to informative missingness; the degrees-of-freedom adjustment is propagated to the standard errors.

Value

An S3 object of class "ife_unb" with components:

coef

Named p-vector of estimated coefficients $\hat\beta$ (bias-corrected when bias_corr = TRUE).

coef_raw

Named p-vector of uncorrected coefficients (only when bias_corr = TRUE).

vcov

p x p variance-covariance matrix.

se

Named p-vector of standard errors.

tstat

Named p-vector of t-statistics.

pval

Named p-vector of two-sided p-values.

ci

p x 2 matrix of 95 percent confidence intervals.

table

Data frame coefficient table.

F_hat

TT x r estimated factor matrix (normalised F'F/TT = I_r).

Lambda_hat

N x r estimated loading matrix.

residuals

n_obs numeric vector of full-model residuals at observed cells.

sigma2

Estimated error variance ($sum(u^2)/df$).

df

Residual degrees of freedom.

n_obs

Number of observed unit-time cells.

n_iter

Outer-loop iterations to convergence.

converged

Logical.

N, TT, r, se_type

Model dimensions and options.

init, bias_corr, exog, L_T

Options used.

b_hat, b2, b3, b4, b5, b6

Bias components (only when bias_corr = TRUE). b2 is a zero vector when exog = "strict".

y_name, x_names, id_col, time_col

Variable names.

unit_vals, time_vals

Unique unit and time identifiers.

unit_idx, time_idx

Integer index vectors for residuals.

call

Matched call.

References

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

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

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

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

Examples

data(cigar, package = "xtife")
# Drop ~10 % of rows to create an unbalanced panel
set.seed(1)
cigar_unb <- cigar[sample(nrow(cigar), 1200L), ]
fit <- ife_unbalanced(sales ~ price, data = cigar_unb,
                      index = c("state", "year"), r = 2L)
print(fit)

---

Print an IFE Model Summary

Description

Prints a formatted summary of an object of class "ife", including panel dimensions, number of factors, additive fixed effect specification, SE type, and a coefficient table with standard errors, t-statistics, p-values, and 95% confidence intervals. If bias correction was applied, bias terms are also reported. Information criteria are printed when the object contains them (i.e., when called from ife_select_r()).

Usage

## S3 method for class 'ife'
print(x, digits = 4, ...)

Arguments

x	an object of class "ife"
digits	number of significant digits (default 4)
...	unused

Value

x invisibly.

Examples

data(cigar, package = "xtife")
fit <- ife(sales ~ price, data = cigar, index = c("state", "year"),
           r = 2, force = "two-way", se = "standard")
print(fit)

---