SBZ Weighted Least Squares Bubble Test with Union-of-Rejections

radf_sbz_union performs the HLST (2016) wild bootstrap – the same algorithm as radf_wb_cv – jointly on the classic sup-ADF statistic (supDF, i.e. radf()'s sadf) and the WLS/kernel-volatility statistic supBZ of Harvey, Leybourne & Zu (2019), and combines them into the paper's union-of-rejections statistic U. supBZ can have substantially higher power than supDF under many time-varying-volatility patterns, at the cost of lower power under others (e.g. upward volatility trends); U is designed to capture whichever of the two is more powerful for a given series.

Usage

radf_sbz_union(
  data,
  minw = NULL,
  nboot = 499L,
  kernel = c("gaussian", "uniform"),
  h = NULL,
  seed = NULL
)

# S3 method for class 'radf_sbz_union'
autoplot(object, sig_lvl = 95, ...)

Arguments

data: A univariate or multivariate numeric time series object, a numeric vector or matrix, or a data.frame. A column may have leading and/or trailing NA values (an uneven/unbalanced panel where series enter or exit the sample at different times) – those periods are filled with NA in badf/bsadf and excluded from that series' adf/sadf/ gsadf. Interior NA values (a gap in the middle of a series) are not supported. When any series is padded this way, the panel statistic (bsadf_panel/gsadf_panel) is not available and is returned as NA, with a warning.
minw: A positive integer. The minimum window size (default = \((0.01 + 1.8/\sqrt(T))T\), where T denotes the sample size).
nboot: A positive integer. Number of bootstraps (default = 500L).
kernel: Kernel for the spot-volatility estimator (eq. 6), "gaussian" (default, as in the paper) or "uniform".
h: Bandwidth for the spot-volatility estimator. Default: leave-one-out cross-validation over the paper's own search range.
seed: An object specifying if and how the random number generator (rng) should be initialized. Either NULL or an integer will be used in a call to set.seed before simulation. If set, the value is saved as "seed" attribute of the returned value. The default, NULL, will not change rng state, and return .Random.seed as the "seed" attribute. Results are reproducible across the parallel and non-parallel option when the same seed is used.
object: A radf_sbz_union object.
sig_lvl: Significance level to plot the critical value at, one of 90, 95 (default), 99.
...: Further arguments passed to methods. Not used.

Value

A list with bootstrap p-values (p_supDF, p_supBZ, p_U) and critical values (supDF_cv, supBZ_cv, U_cv) for each series.

Details

U's value itself – not just its significance – is defined using a bootstrap-calibrated scaling ratio between supDF and supBZ's own 95\ the joint bootstrap needs supDF/supBZ computed from the same resampled series each replicate for the union's size guarantee (the paper's Theorem 3) to hold. That coupling is why this stays one bundled function rather than splitting into a statistic and a critical-value function the way most of exuber does – unlike U, supBZ alone has no such coupling, so it does split that way: see radf_sbz/radf_sbz_cv for the supBZ-only route, with the usual summary()/datestamp/ tidy/autoplot pipeline.

Note

This function bundles the statistic and its critical values in a single call – there is no separate un-cv'd statistic function and no other critical-value function to pair it with, unlike radf()/ radf_wb_cv() (U's own value structurally requires the bootstrap, see Details).

Returns its own class (not radf_obj), so it does not plug into summary()/\link{datestamp}/tidy/autoplot – prints its own statistic/critical-value summary (bundles the test statistic and its critical value in one object), but has its own autoplot method (a per-series comparison of supDF/supBZ/U against their critical values) – see vignette("naming-and-analysis", package = "exuber") for the full picture of which functions do and don't fit the shared pipeline.

Status

References

Harvey, D. I., Leybourne, S. J., & Zu, Y. (2019). Testing explosive bubbles with time-varying volatility. Econometric Reviews, 38(10), 1131-1151.

Examples

# \donttest{
res <- radf_sbz_union(sim_data, nboot = 200)
print(res)
#> 
#> ── radf_sbz_union (minw = 19, nboot = 200) ─────────────────────────────────────
#> 
#>   series  supDF   supBZ      U  p_supDF  p_supBZ    p_U
#>     psy1  1.946  0.2802  1.946    0.055    0.675  0.105
#>     psy2  7.880  1.5349  7.880    0.000    0.175  0.000
#>    evans  5.283  1.9138  5.283    0.120    0.285  0.190
#>      div  1.113  2.2607  1.338    0.090    0.100  0.130
#>     blan  3.930  1.4008  3.930    0.080    0.240  0.130
#> 
autoplot(res)

# }