Package 'xtfifevd'

Between/Within SD Ratio for Time-Invariant Variables

Description

Computes the between-panel and within-panel standard deviations for specified variables, along with their ratio. This diagnostic helps assess whether FEVD/FEF methods may improve upon standard FE estimation.

Usage

bw_ratio(data, variables, id)

Arguments

data	A data frame containing the panel data.
variables	A character vector of variable names to analyze.
id	Character string naming the panel identifier variable.

Details

For truly time-invariant variables, the within-panel SD should be zero (or near-zero due to numerical precision), giving an infinite ratio. Plumper and Troeger (2007, p. 134) define the b/w ratio as the between SD of a variable divided by its within SD, which is what this function reports.

Plumper and Troeger (2007, Section 6) compare the root mean squared error of the FE and FEVD estimators of a rarely changing variable in a Monte Carlo design with N = 30 units and T = 20 periods (their footnote 13 reports that the findings for rarely changing variables were confirmed for N in 15 to 100 and T in 20 to 100). The b/w ratio above which FEVD has the lower RMSE depends on the (unobservable) correlation between the rarely changing variable and the unit effects: about 0.2 when the correlation is 0, about 1.7 at a correlation of 0.3, about 2.8 at 0.5 and close to 3.8 at 0.8 (their Fig. 4, pp. 135-136). The often quoted threshold of 1.7 is therefore a result for corr(z, u) = 0.3 from their Fig. 4 (N = 30, T = 20), and Plumper and Troeger themselves write that they "cannot offer a simple rule of thumb", adding (p. 136) that "the odds are that at a b/w ratio of at least 2.8, the variable is better included into the stage 2 estimation". This function prints these thresholds for orientation only.

Two caveats from Plumper and Troeger apply. First, the FEVD (and FEF) coefficient on a time-invariant or rarely changing variable is biased whenever that variable is correlated with the unobserved unit effects (p. 129); the bias is the usual omitted variable bias and does not vanish with a high b/w ratio. Second, this correlation cannot be observed or tested, because the unit effects are unobservable (p. 135); the trade-off between the bias of FEVD and the inefficiency of FE for rarely changing variables therefore rests on an untestable assumption.

Value

A data frame with columns:

variable

Variable name

sd_between

Between-panel standard deviation

sd_within

Within-panel standard deviation

bw_ratio

Ratio of between to within SD

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

Examples

# Create example data
set.seed(42)
N <- 50
T <- 5
id <- rep(1:N, each = T)
z_invariant <- rep(rnorm(N), each = T)  # Truly time-invariant
z_slow <- rep(rnorm(N), each = T) + rnorm(N * T, sd = 0.1)  # Slowly varying
x_varying <- rnorm(N * T)  # Time-varying

data <- data.frame(id = id, z_inv = z_invariant,
                   z_slow = z_slow, x = x_varying)

bw_ratio(data, c("z_inv", "z_slow", "x"), id = "id")

---

Panel Fixed Effects Estimation for Time-Invariant Variables

Description

Estimates panel models with time-invariant regressors using FEVD, FEF, or FEF-IV methods. Standard fixed effects estimation cannot identify coefficients on time-invariant variables; these methods decompose or filter the unit effects to recover these coefficients.

Usage

xtfifevd(
  formula,
  data,
  id,
  time,
  method = c("fevd", "fef", "fef_iv"),
  instruments = NULL,
  vcov_beta = c("robust", "classical"),
  na.action = na.omit
)

fevd(
  formula,
  data,
  id,
  time,
  vcov_beta = c("robust", "classical"),
  na.action = na.omit
)

fef(
  formula,
  data,
  id,
  time,
  vcov_beta = c("robust", "classical"),
  na.action = na.omit
)

fef_iv(
  formula,
  data,
  id,
  time,
  instruments,
  vcov_beta = c("robust", "classical"),
  na.action = na.omit
)

Arguments

formula	A formula of the form y ~ x1 + x2 | z1 + z2 where terms before | are time-varying and terms after | are time-invariant. Both parts are processed with stats::model.matrix(), so transformations (log(x), I(x^2), x:w, log(y) on the left-hand side) and factors (with their contrasts) are handled as in stats::lm(). Intercept columns are removed; the model intercept is estimated in stage 2.
data	A data frame containing the variables.
id	Character string naming the panel (individual) identifier variable.
time	Character string naming the time identifier variable.
method	Estimation method: "fevd" (default), "fef", or "fef_iv".
instruments	For method = "fef_iv", a one-sided formula specifying instrumental variables, e.g., ~ iv1 + iv2.
vcov_beta	Covariance estimator for the stage 1 (within) coefficients beta: "robust" (default) is the Arellano-type panel-robust matrix of Pesaran and Zhou (2018, eq. 18), valid under heteroskedasticity and serial correlation within panels; "classical" is the homoskedastic $\hat\sigma_e^2 (X'MX)^{-1}$. The chosen matrix is also used inside the Pesaran and Zhou variance of gamma and of the intercept.
na.action	How to handle missing values. Default is na.omit.

Details

Model

The panel model is:

$y_{it} = \alpha + \alpha_i + x_{it}'\beta + z_i'\gamma + \varepsilon_{it}$

where $x_{it}$ are time-varying regressors, $z_i$ are time-invariant regressors, and $\alpha_i$ are individual effects that may be correlated with $x_{it}$.

Stage 1 (all methods)

Within (fixed effects) regression of $y_{it}$ on $x_{it}$ yields $\hat{\beta}$ and the time-averaged FE residuals $\bar u_i = \bar y_i - \bar x_i'\hat\beta$ (Pesaran and Zhou 2018, eq. 3).

Stage 2

• FEF: unit-level OLS of $\bar u_i$ on an intercept and $z_i$ (Pesaran and Zhou eq. 4 and 5).

• FEF-IV: unit-level 2SLS using instruments $r_i$ (Pesaran and Zhou eq. 48).

• FEVD: as FEF; the residuals $h_i$ are the unexplained part of the unit effect (Plumper and Troeger 2007, eq. 6).

Stage 3 (FEVD only)

Pooled OLS of $y_{it}$ on an intercept, $x_{it}$, $z_i$ and $h_i$ (Plumper and Troeger 2007, eq. 7). Plumper and Troeger's stage 2 equation (5) is printed without an intercept, but their $\hat u_i$ (eq. 4) contains the model constant and their stage 3 equation (7) has an intercept; the package therefore includes an intercept in stage 2, which makes the stage 3 estimates identical to FEF and the coefficient $\delta$ on $h_i$ identically equal to 1 (Pesaran and Zhou 2018, Proposition 3). The identity holds in balanced and unbalanced panels: with $(a, \hat\beta, \hat\gamma, 1)$ the stage 3 residuals are the within residuals, which sum to zero within every unit and are orthogonal to $x_{it}$, so they satisfy the stage 3 normal equations exactly. Without the stage 2 intercept the FEVD estimator is in general biased (Pesaran and Zhou 2018, Section 3.4). Plumper and Troeger are silent on unbalanced panels; the package runs stage 2 at the unit level without weights, one observation per panel unit (Pesaran and Zhou's FEF), rather than at the observation level, where units would be weighted by $T_i$.

The naive stage 3 OLS standard errors of Plumper and Troeger are too small for the time-invariant coefficients because they ignore that $h_i$ is a generated regressor (Breusch, Ward, Nguyen and Kompas 2010, Theorem 3; Greene 2011; Pesaran and Zhou 2018). In a Monte Carlo check by the package author (400 replications, N = 200, T = 8, two time-varying and two time-invariant regressors, AR(1) errors with coefficient 0.8 and heteroskedastic across units, x correlated with the unit effects) their 95 percent coverage for $\gamma$ was 35 to 40 percent, against 92 to 95 percent for the Pesaran and Zhou standard errors. They are returned in stage3$se_naive for reference only and are never used for inference.

Rarely changing variables

If a variable after | varies within panels, a warning is issued and its unit (panel) mean is used as $z_i$ in all stages. This is the package's choice; Plumper and Troeger (2007) do not specify how rarely changing variables enter stage 2.

Variance estimation

The gamma block of vcov is Pesaran and Zhou (2018) equation 17 (FEF and FEVD) or equation 51 (FEF-IV), which account for the estimation uncertainty of $\hat\beta$ through the matrix vcov_beta. The beta block is vcov_beta itself (robust eq. 18 by default). The remaining blocks follow from Pesaran and Zhou eq. (A.11),

$\hat\gamma - \gamma = Q_{zz}^{-1}\left[N^{-1}\sum_i (z_i - \bar z) v_i - Q_{z\bar x}(\hat\beta - \beta)\right],$

so that $Cov(\hat\gamma, \hat\beta) = -Q_{zz}^{-1} Q_{z\bar x} Var(\hat\beta)$, and from eq. (5), $\hat\alpha = \bar u - \bar z'\hat\gamma$, by the delta method with $c = \bar x - Q_{z\bar x}' Q_{zz}^{-1} \bar z$: $Var(\hat\alpha) = N^{-2}\sum_i w_i^2 \hat v_i^2 + c' Var(\hat\beta) c$, $w_i = 1 - \bar z' Q_{zz}^{-1}(z_i - \bar z)$, with the corresponding covariances. For FEF-IV the same derivation is used with $Q_{zz}^{-1}(z_i - \bar z)$ replaced by $H_{zr}(r_i - \bar r)$ and $Q_{z\bar x}$ by $Q_{r\bar x}$. As in Pesaran and Zhou's eq. (17), the cross term between the unit-level scores and $\hat\beta$ (the term defined in their eq. 15, negligible under their condition 16) is dropped in $Var(\hat\alpha)$ and in $Cov(\hat\gamma, \hat\beta)$.

Value

An object of class "xtfifevd" containing:

coefficients

Named vector of all coefficients (beta, gamma, ⁠_cons⁠). For method = "fevd" these are the stage 3 pooled OLS estimates, which equal the FEF estimates (see Details).

vcov

Full variance-covariance matrix of coefficients from the Pesaran and Zhou (2018) derivation, including the covariances between beta, gamma and the intercept (see Details).

beta

Coefficients on time-varying variables

gamma

Coefficients on time-invariant variables

intercept

Overall intercept

delta

(FEVD only) Stage 3 coefficient on the unexplained unit effect $h_i$; equal to 1 by construction.

stage3

(FEVD only) List with the full stage 3 pooled OLS coefficient vector (coefficients, including h), the naive OLS covariance matrix vcov_naive and standard errors se_naive. These naive standard errors are known to be too small for the time-invariant coefficients and are returned for reference only.

fef

(FEVD only) The stage 1 and 2 (FEF) coefficients.

residuals

Idiosyncratic (within) residuals from stage 1

fitted.values

Fitted values $x_{it}'\hat\beta + \bar z_i'\hat\gamma + \hat\alpha$ (unit effects excluded)

sigma2_e

Variance of the idiosyncratic error

sigma2_u

Variance of the unexplained unit effect: the stage 2 residual variance minus $\hat\sigma_e^2 \, \mathrm{mean}(1/T_i)$, truncated at zero

vcov_beta

The vcov_beta choice used

V_beta_robust, V_beta_classical

Both covariance matrices of beta

V_gamma_pz

The gamma block of vcov (Pesaran and Zhou eq. 17 or 51)

stage2_residuals

Unit-level stage 2 residuals

N

Total number of observations

N_g

Number of groups (panels)

T_bar

Average time periods per panel

balanced

Logical, whether the panel is balanced

method

Estimation method used

call

The matched call

Functions

• fevd(): FEVD estimation (3-stage, Plumper and Troeger 2007, with Pesaran and Zhou 2018 standard errors)

• fef(): FEF estimation (2-stage, Pesaran and Zhou 2018)

• fef_iv(): FEF-IV estimation with instruments

References

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

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

See Also

fevd(), fef(), fef_iv(), bw_ratio()

Examples

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

# Generate data
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
x <- rnorm(n)                        # Time-varying
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)

# Estimate with different methods
fit_fevd <- xtfifevd(y ~ x | z, data = data, id = "id", time = "time")
summary(fit_fevd)
fit_fevd$delta   # equals 1 by construction

fit_fef <- xtfifevd(y ~ x | z, data = data, id = "id", time = "time",
                    method = "fef")
summary(fit_fef)

# Transformations in the formula are allowed
fit_log <- fef(y ~ x + I(x^2) | z, data = data, id = "id", time = "time")
coef(fit_log)

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