Package 'xtbhst'

Plot method for xtbhst objects

Description

Produces diagnostic plots for the bootstrap slope heterogeneity test.

Usage

## S3 method for class 'xtbhst'
plot(x, which = c(1L, 2L), ask = NULL, ...)

Arguments

x	An object of class "xtbhst".
which	Integer vector specifying which plots to produce: 1 = Bootstrap distribution of the statistic S, 2 = Bootstrap distribution of S on the Delta scale (the fixed rescaling sqrt(N) (S/N - K)/sqrt(2K) of S and S*), 3+ = Individual coefficient distributions (plot 3 is the first regressor, plot 4 the second, and so on). Default is c(1, 2).
ask	Logical. If TRUE, prompt before each plot (default: TRUE if multiple plots and interactive session).
...	Additional arguments passed to plotting functions.

Value

Invisibly returns NULL.

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Print method for xtbhst objects

Description

Print method for xtbhst objects

Usage

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

Arguments

x	An object of class "xtbhst".
digits	Number of digits to display (default: 4).
...	Additional arguments (ignored).

Value

Invisibly returns the input object.

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Summary method for xtbhst objects

Description

Summary method for xtbhst objects

Usage

## S3 method for class 'xtbhst'
summary(object, digits = 4L, ...)

Arguments

object	An object of class "xtbhst".
digits	Number of digits to display (default: 4).
...	Additional arguments (ignored).

Value

Invisibly returns a list with summary statistics.

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Bootstrap Slope Heterogeneity Test for Panel Data

Description

Implements the panel bootstrap test of slope homogeneity of Blomquist and Westerlund (2016). The null hypothesis is that the slope coefficients are equal across all cross-sectional units.

Usage

xtbhst(
  formula,
  data,
  id,
  time,
  reps = 999L,
  blocklength = NULL,
  variance = c("bw", "py"),
  partial = NULL,
  csa = NULL,
  csa_lags = 0L,
  constant = TRUE,
  seed = NULL
)

Arguments

formula	A formula of the form y ~ x1 + x2 + ... specifying the dependent variable and regressors.
data	A data frame containing the panel data.
id	A character string specifying the name of the cross-sectional identifier variable.
time	A character string specifying the name of the time variable.
reps	Integer. Number of bootstrap replications (default: 999).
blocklength	Integer. Block length for the block bootstrap. If NULL (default), set to round(2 * T^(1/3)), the deterministic rule of Blomquist and Westerlund (2016, Section 4.2), where T is the number of time periods in the estimation sample. Must lie between 1 and T.
variance	Character. Which unit-specific error variance estimate to use in the weights of the weighted fixed-effects estimator and in the test statistic. "bw" (default) is the estimator of Blomquist and Westerlund (2016, Section 3.1); "py" is the estimator of Pesaran and Yamagata (2008). See Details.
partial	Optional formula specifying variables to be partialled out unit by unit. For example, ~ z1 + z2.
csa	Optional formula specifying variables for which cross-sectional averages should be computed and partialled out, in the spirit of Pesaran (2006). See Details.
csa_lags	Integer. Number of lags of the cross-sectional averages to include (default: 0).
constant	Logical. If TRUE (default), a unit-specific constant is partialled out.
seed	Optional integer seed for reproducibility.

Details

Let $\hat\beta_i$ be the least-squares estimate of the slopes of unit $i$ after partialling out the constant (and any further variables given in partial and csa), let $M$ denote the corresponding projection matrix, and let

$\hat\beta_{WFE} = \left(\sum_{i=1}^N \frac{x_i' M x_i}{\hat\sigma_i^2}\right)^{-1} \sum_{i=1}^N \frac{x_i' M y_i}{\hat\sigma_i^2}$

be the weighted fixed-effects estimator. The test statistic of Blomquist and Westerlund (2016, Section 3.1) is

$S = \sum_{i=1}^N (\hat\beta_i - \hat\beta_{WFE})' \frac{x_i' M x_i}{\hat\sigma_i^2} (\hat\beta_i - \hat\beta_{WFE}).$

Two estimators of $\sigma_i^2$ are available:

variance = "bw"

$\hat\sigma_i^2 = T^{-1} \sum_t \hat\varepsilon_{i,t}^2$, where $\hat\varepsilon_{i,t}$ are the residuals of the unit-specific least-squares regression. This is the estimator used by Blomquist and Westerlund (2016).

variance = "py"

$\tilde\sigma_i^2 = (T - K - 1)^{-1} (y_i - x_i \hat\beta_{FE})' M (y_i - x_i \hat\beta_{FE})$, where $\hat\beta_{FE}$ is the pooled fixed-effects estimator. This is the estimator used by Pesaran and Yamagata (2008). When further variables are partialled out, the degrees of freedom are $T - K - K_{partial}$, which reduces to $T - K - 1$ in the standard case.

The same choice is used for the weights of $\hat\beta_{WFE}$, in $S$ and in every bootstrap statistic $S^*$.

The bootstrap follows Algorithm BOOT of Blomquist and Westerlund (2016): the unit-specific least-squares residuals are resampled in blocks of length blocklength along the time dimension (keeping the cross-section intact), pseudo-data are generated under the null hypothesis with slopes $\hat\beta_{WFE}$, and $S^*$ is computed exactly as $S$. The reported bootstrap p-value is the proportion of $S^*$ that are at least as large as $S$.

For reference, the standardized statistics of Pesaran and Yamagata (2008) are also reported with their asymptotic standard normal p-values:

$\Delta = \sqrt{N} \frac{N^{-1} S - K}{\sqrt{2K}}, \qquad \Delta_{adj} = \sqrt{N} \frac{N^{-1} S - K}{\sqrt{2K (T - K - 1)/(T + 1)}}.$

These are always computed from $S$ evaluated with the Pesaran and Yamagata (2008) variance estimator $\tilde\sigma_i^2$ (returned as S_py), whatever the value of variance, so that they are the statistics of that paper with their asymptotic null distribution; variance governs only $S$ and the bootstrap. Because $\Delta$ and $\Delta_{adj}$ are fixed rescalings of S_py, a bootstrap p-value for them would coincide with the bootstrap p-value of $S$ when variance = "py"; only the asymptotic p-values are therefore reported for them. $\Delta_{adj}$ is NA when $T - K - 1 \le 0$.

Rows of data with missing values in any of the variables used (the model, partial, csa, id and time) are dropped before the panel structure is checked. The remaining panel must be strongly balanced: every unit must be observed in exactly the same set of time periods, and every (id, time) pair must occur once.

Partialling out cross-sectional averages (csa, csa_lags) is an extension not covered by Blomquist and Westerlund (2016). With csa_lags > 0 the first csa_lags periods are dropped for all units, so that the estimation sample is a common sample of T - csa_lags periods. If the dependent variable appears in csa, its cross-sectional averages are recomputed from the bootstrap pseudo-data in each replication (observed values are used for lagged averages that fall before the estimation sample). The dependent variable must then be a plain column of data.

Value

An object of class "xtbhst" containing:

S

The Swamy-type statistic $S$ that is bootstrapped, computed with the variance estimator selected by variance.

S_py

The same statistic computed with the variance estimator of Pesaran and Yamagata (2008); it underlies delta and delta_adj. Equal to S when variance = "py".

pval

Bootstrap p-value: the proportion of bootstrap statistics $S^*$ that are at least as large as $S$.

delta

The Delta statistic of Pesaran and Yamagata (2008), computed from S_py whatever the value of variance.

delta_adj

The adjusted Delta statistic of Pesaran and Yamagata (2008), or NA if $T - K - 1 \le 0$.

pval_delta_asy

Asymptotic (standard normal, upper tail) p-value of delta.

pval_delta_adj_asy

Asymptotic (standard normal, upper tail) p-value of delta_adj.

S_stars

Vector of bootstrap statistics $S^*$.

delta_stars

S_stars on the Delta scale, that is sqrt(N) (S_stars/N - K)/sqrt(2K).

blocklength

The block length used in the bootstrap.

reps

Number of bootstrap replications.

variance

The variance estimator used ("bw" or "py").

N

Number of cross-sectional units.

T

Number of time periods in the estimation sample (after dropping the first csa_lags periods).

K

Number of regressors.

Kpartial

Number of partialled-out variables (including the constant).

sigma2

Vector of unit-specific error variance estimates.

beta_i

Matrix of individual slope estimates (N x K).

beta_fe

Vector of weighted fixed-effects (pooled) estimates.

n_dropped

Number of rows of data dropped because of missing values.

call

The matched call.

References

Blomquist, J. and Westerlund, J. (2016). Panel bootstrap tests of slope homogeneity. Empirical Economics, 50(4), 1359-1381. doi:10.1007/s00181-015-0978-z

Pesaran, M. H. and Yamagata, T. (2008). Testing slope homogeneity in large panels. Journal of Econometrics, 142(1), 50-93. doi:10.1016/j.jeconom.2007.05.010

Pesaran, M. H. (2006). Estimation and inference in large heterogeneous panels with a multifactor error structure. Econometrica, 74(4), 967-1012. doi:10.1111/j.1468-0262.2006.00692.x

Examples

# Generate example panel data
set.seed(123)
N <- 20  # cross-sectional units
T_periods <- 30  # time periods

# Homogeneous slopes (H0 is true)
data_hom <- data.frame(
  id = rep(1:N, each = T_periods),
  time = rep(1:T_periods, N),
  x = rnorm(N * T_periods)
)
data_hom$y <- 1 + 0.5 * data_hom$x + rnorm(N * T_periods)

# Test for slope heterogeneity
result <- xtbhst(y ~ x, data = data_hom, id = "id", time = "time",
                 reps = 199, seed = 42)
print(result)
summary(result)

# Pesaran and Yamagata (2008) variance estimator
result_py <- xtbhst(y ~ x, data = data_hom, id = "id", time = "time",
                    reps = 199, seed = 42, variance = "py")

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