README

Overview

xtfifevd implements fixed effects estimators for time-invariant variables in panel data models. Standard fixed effects (FE) estimation cannot identify coefficients on time-invariant regressors because they are collinear with the individual fixed effects. This package provides three methods to estimate these coefficients:

FEVD: Fixed Effects Vector Decomposition (Plumper and Troeger, 2007), three stages with an intercept in stage 2

FEF: Fixed Effects Filtered (Pesaran and Zhou, 2018)

FEF-IV: FEF with instrumental variables for endogenous regressors

All methods use the Pesaran and Zhou (2018) variance estimators, which account for generated regressor uncertainty (the naive FEVD stage 3 standard errors are too small for the time-invariant coefficients), and report the full covariance matrix of the time-varying coefficients, the time-invariant coefficients and the intercept.

Installation

# Install from CRAN (when available)
install.packages("xtfifevd")

# Or install development version from GitHub
# install.packages("remotes")

Usage

library(xtfifevd)

# Simulate panel data
set.seed(123)
N <- 100  # panels
T <- 10   # time periods
n <- N * T

id <- rep(1:N, each = T)
time <- rep(1:T, N)
alpha_i <- rep(rnorm(N), each = T)  # Fixed effects
z <- rep(rnorm(N), each = T)        # Time-invariant variable
x <- rnorm(n)                        # Time-varying variable
y <- 1 + 2 * x + 0.5 * z + alpha_i + rnorm(n, sd = 0.5)

data <- data.frame(id = id, time = time, y = y, x = x, z = z)

# Formula: y ~ time_varying_vars | time_invariant_vars
# Transformations and factors are allowed, e.g. log(y) ~ x + I(x^2) | z
fit <- xtfifevd(y ~ x | z, data = data, id = "id", time = "time")
summary(fit)
fit$delta   # FEVD stage 3 coefficient on h_i, equal to 1 by construction

====================================================================== FEVD Estimation Results xtfifevd 1.1.0 ====================================================================== Dep. variable: y Method: FEVD Variance: Pesaran and Zhou (2018), beta vcov: robust Observations: 1000 Groups: 100 T (average): 10.00 ---------------------------------------------------------------------- Estimate Std. Error z value Pr(>|z|) x 2.03092 0.01619 125.469 < 2e-16 *** z 0.42660 0.08702 4.902 9.48e-07 *** _cons 1.08528 0.09019 12.033 < 2e-16 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ---------------------------------------------------------------------- Time-varying (FE): x Time-invariant: z sigma_e: 0.4881 sigma_u (unexplained unit effect): 0.9046 FEVD stage 3 coefficient on h_i (delta): 1.000000 [equals 1 by construction] Naive stage 3 OLS SEs are too small for the time-invariant coefficients; see ?xtfifevd. ======================================================================

Methods Comparison

# All three methods
fit_fevd <- fevd(y ~ x | z, data, id = "id", time = "time")
fit_fef  <- fef(y ~ x | z, data, id = "id", time = "time")

# FEF and FEVD produce identical point estimates (Proposition 3)
all.equal(coef(fit_fevd), coef(fit_fef))
# [1] TRUE

# With instruments (when z may be endogenous)
data$iv <- data$z + rnorm(n, sd = 0.3)  # Instrument
fit_iv <- fef_iv(y ~ x | z, data, id = "id", time = "time",
                 instruments = ~ iv)

Diagnostics

# Between/Within SD ratio (Plumper and Troeger 2007 define it as the
# between SD divided by the within SD)
bw_ratio(data, c("z", "x"), id = "id")

Plumper and Troeger (2007, Fig. 4; N = 30, T = 20) find that FEVD has lower RMSE than FE for a rarely changing variable when its b/w ratio exceeds about 0.2 if corr(z, u) = 0, 1.7 at corr 0.3, 2.8 at 0.5 and 3.8 at 0.8. The correlation with the unit effects is not observable or testable, the FEVD and FEF coefficients are biased whenever it is non-zero, and Plumper and Troeger state that they “cannot offer a simple rule of thumb”.

Why Use This Package?

The Problem

Standard FE estimation “absorbs” time-invariant variables into the fixed effects, making their coefficients unidentified. Researchers often want to estimate effects of variables like:

References

Plumper, T. and Troeger, V. E. (2007). Efficient Estimation of Time-Invariant and Rarely Changing Variables in Finite Sample Panel Analyses with Unit Fixed Effects. Political Analysis, 15(2), 124-139. doi:10.1093/pan/mpm002

Pesaran, M. H. and Zhou, Q. (2018). Estimation of time-invariant effects in static panel data models. Econometric Reviews, 37(10), 1137-1171. doi:10.1080/07474938.2016.1222225

Breusch, T., Ward, M. B., Nguyen, H. and Kompas, T. (2010). On the fixed-effects vector decomposition. MPRA Paper No. 21452. https://mpra.ub.uni-muenchen.de/21452/

Greene, W. H. (2011). Fixed Effects Vector Decomposition: A Magical Solution to the Problem of Time-Invariant Variables in Fixed Effects Models? Political Analysis, 19(2), 135-146. doi:10.1093/pan/mpq034

xtfifevd