Compute Bias-Corrected IFE Coefficients (Bai 2009)

Description

Applies the two-term asymptotic bias correction from Bai (2009) Theorems 7.1 and 7.2 to the raw IFE coefficient vector:

\hat{\beta}^\dagger = \hat{\beta} - \hat{B}/N - \hat{C}/T

where \hat{B} corrects for cross-sectional heteroskedasticity (Equation 17) and \hat{C} corrects for time-varying heteroskedasticity (Equation 19). Both terms require T/N^2 \to 0 and N/T^2 \to 0 respectively (Theorem 7.2). For panels of the scale used in the package examples (N \approx 50, T \approx 30) both conditions hold approximately.

Usage

.bias_correct(beta, F_hat, Lambda_hat, X_dm_arr, X_tilde, e_mat, N, TT, p, r)

Arguments

Value

list with elements: beta_bc p-vector: bias-corrected coefficients B_hat p-vector: estimated cross-section bias term (pre-scaling by 1/N) C_hat p-vector: estimated time-heteroskedasticity bias term (pre-scaling by 1/T)

Compute Bias-Corrected Coefficients for the Dynamic IFE Estimator

Description

Applies the three-term asymptotic bias correction from Moon and Weidner (2017) Corollary 4.5 to the raw dynamic IFE coefficient vector:

\hat{\beta}^* = \hat{\beta} + W^{-1}\!\left(\hat{B}_1/T + \hat{B}_2/N + \hat{B}_3/T\right)

where \hat{B}_1 corrects for the Nickell-type bias arising from predetermined (lagged) regressors using a lag-truncation bandwidth M1, and \hat{B}_2, \hat{B}_3 correct for cross-sectional and time-series heteroskedasticity respectively. The latter two terms are algebraically equivalent to the Bai (2009) \hat{B} and \hat{C} terms.

Usage

.bias_correct_mw(
  beta,
  F_hat,
  Lambda_hat,
  X_dm_arr,
  X_tilde,
  e_mat,
  N,
  TT,
  p,
  r,
  M1 = 1L
)

Arguments

Value

list: beta_bc p-vector: bias-corrected coefficients B1_hat p-vector: dynamic bias term (pre-multiplied by D0_inv; scaled by 1/(NT)) B2_hat p-vector: cross-section heteroscedasticity bias (same as Bai B_hat) B3_hat p-vector: time heteroscedasticity bias (same as Bai C_hat)

Compute Information Criteria for a Given Number of Factors (Internal)

Description

Evaluates five information criteria for a fitted IFE model with r factors, given the mean squared residual V_r. Returns IC1, IC2, and IC3 from Bai and Ng (2002) Proposition 1 (ICp1/ICp2/ICp3), applied to IFE residuals as suggested by Bai (2009) Section 9.4, plus a BIC-style criterion (IC_bic) and a small-sample-corrected prediction criterion (PC) as implemented in the fect package (C++ source, lines 196–224). Called once per candidate r inside ife_select_r().

Usage

.compute_ic(V_r, r, N, TT, p, force)

Arguments

Value

named list: IC1, IC2, IC3 (Bai & Ng 2002), IC_bic and PC (Bai 2009/fect)

Demean Panel Matrices for Additive Fixed Effects

Description

Removes additive unit and/or time fixed effects from a T x N outcome matrix and a T x N x p covariate array using the within-group transformation. Supports four specifications: no demeaning ("none"), unit-only ("unit"), time-only ("time"), and two-way ("two-way").

Usage

.demean_panel(Y_mat, X_arr, force)

Arguments

Value

list with: Y_dm T x N demeaned outcome X_dm T x N x p demeaned covariate array (or NULL) mu_Y grand mean of Y alpha_Y N x 1 unit means of Y (after grand mean removal) xi_Y T x 1 time means of Y (after grand mean removal)

Core IFE Estimation via SVD-Based Alternating Projections

Description

Estimates the Interactive Fixed Effects model by iterating between an OLS step for the regression coefficients (given current factor estimates) and an SVD step for the factor matrix (given current coefficients). Convergence is declared when the maximum absolute change in the coefficient vector falls below tol. Handles the degenerate case r = 0 (standard OLS on demeaned data) as a special case. Supports both the strictly-exogenous ("static", Bai 2009) and the predetermined-regressor ("dynamic", Moon and Weidner 2017) projection schemes.

Usage

.ife_fit(Y_dm, X_dm, r, tol = 1e-09, max_iter = 10000L, method = "static")

Arguments

Value

list: beta p x 1 coefficient vector (numeric(0) if p = 0) F_hat T x r factor matrix (normalized F'F/T = I_r) Lambda_hat N x r loading matrix X_tilde T x N x p factor-projected covariates (for SE computation) e_mat T x N residual matrix E = Y_dm - X_dm * beta - F Lambda' n_iter number of iterations used converged logical

Compute Standard Errors for the IFE Estimator

Description

Constructs the sandwich variance-covariance matrix for the IFE coefficient vector using the Frisch-Waugh-Lovell principle. The effective regressors are the factor-projected covariates \tilde{X}_{it} (i.e., demeaned X after removing the factor space). Three estimators are supported: homoskedastic ("standard"), HC1 heteroskedasticity-robust ("robust"), and cluster-robust by unit ("cluster"), following Cameron, Gelbach and Miller (2011). Degrees of freedom account for regression coefficients, interactive FE parameters, and additive FE parameters.

Usage

.ife_se(beta, X_tilde, u_mat, N, TT, r, force, se_type)

Arguments

Value

list: vcov_mat p x p estimated variance-covariance matrix df residual degrees of freedom

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

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

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)

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

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)