DisplayWesterlund: Standardize and Display Westerlund ECM Panel Cointegration...

In Westerlund: Panel Cointegration Tests Based on Westerlund (2007)

Standardize and Display Westerlund ECM Panel Cointegration Test Results

Description

Formats, standardizes, and prints the Westerlund (2007) ECM-based panel cointegration test results for the null hypothesis of no cointegration. Given raw statistics G_t, G_a, P_t, and P_a, the function computes standardized Z-statistics using tabulated (or hard-coded) asymptotic moments and reports left-tail asymptotic p-values. If a bootstrap distribution is supplied, it also computes bootstrap (robust) p-values for the raw statistics.

Usage

DisplayWesterlund(
  stats,
  bootstats = NULL,
  nobs,
  nox,
  constant = FALSE,
  trend = FALSE,
  meanlag = -1,
  meanlead = -1,
  realmeanlag = -1,
  realmeanlead = -1,
  auto = 0,
  westerlund = FALSE,
  aic = TRUE,
  verbose = FALSE
)

Arguments

stats	A numeric vector of length 4 or a 1x4 numeric matrix containing the raw test statistics in the order c(Gt, Ga, Pt, Pa).
bootstats	Optional. A numeric matrix with bootstrap replications of the raw statistics, with 4 columns in the order [Gt, Ga, Pt, Pa]. If provided, robust (bootstrap) p-values are computed and displayed.
nobs	Integer. Number of valid cross-sectional units (series) used in the test.
nox	Integer. Number of covariates in the long-run relationship (length of xvars in upstream calls). Used to select asymptotic moments for standardization in the non-westerlund case.
constant	Logical. Indicates whether a constant was included in the cointegrating relationship. Affects the deterministic-case row used for asymptotic moment lookup.
trend	Logical. Indicates whether a trend was included. If TRUE, constant is assumed to be TRUE and the deterministic-case row for moment lookup changes accordingly.
meanlag	Integer. Mean selected lag length (rounded down to integer) reported in the header when auto is non-zero.
meanlead	Integer. Mean selected lead length (rounded down to integer) reported in the header when auto is non-zero.
realmeanlag	Numeric. Unrounded mean selected lag length (for display when auto is non-zero).
realmeanlead	Numeric. Unrounded mean selected lead length (for display when auto is non-zero).
auto	Logical/integer. If non-zero, the header prints the average AIC-selected lag and lead lengths based on realmeanlag and realmeanlead.
westerlund	Logical. If TRUE, uses the hard-coded Westerlund-specific moment constants for standardization (separate constants for trend vs no-trend). If FALSE, uses lookup tables indexed by deterministic case and number of covariates.
aic	Logical. If TRUE, AIC was used for lag/lead selection; If FALSE, BIC was used.
verbose	Logical. If TRUE, prints additional output.

Details

What this function does. DisplayWesterlund() takes raw Westerlund ECM test statistics and produces:

• standardized Z-statistics for G_t, G_a, P_t, P_a,

• left-tail asymptotic p-values using pnorm(),

• optionally, bootstrap (robust) p-values when bootstats is provided,

• a console table summarizing the results.

Standardization and deterministic cases. The function selects a deterministic-case index:

• Row 1: no constant, no trend,

• Row 2: constant, no trend,

• Row 3: constant and trend,

implemented as row_idx = (as.integer(constant) + as.integer(trend)) + 1. In the non-westerlund case, asymptotic means and variances are taken from hard-coded lookup matrices indexed by row_idx and nox. Z-statistics are computed using:

Z = \frac{\sqrt{N}S - \sqrt{N}\mu}{\sqrt{\sigma^2}}

for mean-group statistics and analogous formulas for pooled statistics as implemented in the code, where N = nobs and S is the raw statistic.

In the westerlund=TRUE case, the function uses a separate set of hard-coded mean/variance constants, with different values depending on whether trend is included.

Asymptotic p-values. All asymptotic p-values are computed as left-tail probabilities: pnorm(Z).

Bootstrap p-values (robust p-values). If bootstats is supplied, bootstrap p-values are computed for each raw statistic using a left-tail empirical rule:

\hat{p} = \frac{r + 1}{B + 1}, \qquad r = \sum_{b=1}^{B} \mathbb{I}(S_b \le S_{\text{obs}}),

after dropping non-finite bootstrap draws. This is a common finite-sample correction used in Stata-style bootstrap code.

Return values. In addition to printing, the function returns a list containing the raw statistics, Z-statistics, asymptotic p-values, and (if applicable) bootstrap p-values.

Value

A list containing at minimum:

• gt, ga, pt, pa: raw test statistics,

• gt_z, ga_z, pt_z, pa_z: standardized Z-statistics,

• gt_pval, ga_pval, pt_pval, pa_pval: left-tail asymptotic p-values.

If bootstats is provided, the list also includes:

• gt_pvalboot, ga_pvalboot, pt_pvalboot, pa_pvalboot: bootstrap (robust) p-values for the raw statistics.

Vignette

This section explains how to use DisplayWesterlund() and how its output connects to the broader testing workflow.

Where does DisplayWesterlund() fit?

Typically, the workflow is:

1. Compute observed raw statistics via WesterlundPlain.

2. Optionally compute a bootstrap distribution via WesterlundBootstrap.

3. Call DisplayWesterlund() to standardize, print, and return p-values.

The user-facing westerlund_test wraps these steps and collects the returned scalars.

Input format for stats

stats can be either a numeric vector c(Gt, Ga, Pt, Pa) or a 1x4 matrix with [1,] corresponding to Gt, Ga, Pt, Pa.

Asymptotic standardization and left-tail p-values

The function converts raw statistics to Z-statistics using hard-coded asymptotic means and variances. P-values are computed as pnorm(Z), i.e., a left-tail test.

Bootstrap (robust) p-values

If bootstats is provided, robust p-values are computed by comparing the observed raw statistic to its bootstrap distribution, using a finite-sample correction: (r+1)/(B+1) where r is the number of bootstrap draws less than or equal to the observed statistic.

Examples

## Example 1: Asymptotic-only display (no bootstrap)
stats <- c(Gt = -2.1, Ga = -9.5, Pt = -1.8, Pa = -6.2)

res1 <- DisplayWesterlund(
  stats      = stats,
  bootstats = NULL,
  nobs       = 18,
  nox        = 2,
  constant   = TRUE,
  trend      = FALSE,
  auto       = 0,
  westerlund = FALSE
)

## Example 2: With bootstrap distribution (robust p-values)
set.seed(123)
bootstats <- cbind(
  rnorm(399, mean = -1.8, sd = 0.9),
  rnorm(399, mean = -8.0, sd = 3.0),
  rnorm(399, mean = -1.5, sd = 1.0),
  rnorm(399, mean = -5.0, sd = 3.5)
)

res2 <- DisplayWesterlund(
  stats      = stats,
  bootstats = bootstats,
  nobs       = 18,
  nox        = 2,
  constant   = TRUE,
  trend      = FALSE,
  auto       = 1,
  realmeanlag  = 1.35,
  realmeanlead = 0.40,
  westerlund = FALSE
)

References

Westerlund, J. (2007). Testing for error correction in panel data. Oxford Bulletin of Economics and Statistics, 69(6), 709–748.

See Also

westerlund_test, WesterlundPlain, WesterlundBootstrap

Westerlund documentation built on Feb. 7, 2026, 5:07 p.m.