DisplayWesterlund: 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
)

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.

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.

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:

Compute observed raw statistics via WesterlundPlain.
Optionally compute a bootstrap distribution via WesterlundBootstrap.
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 )

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.

References

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