QPWY Recursive Quantile Monitoring (Wu, Shi & Wu 2025)

monitor_quantile implements the QPWY real-time monitoring strategy of Wu, Shi & Wu (2025): a quantile-regression (QR) analogue of PWY's own recursive ADF t-statistic, testing at a chosen conditional quantile tau over an expanding window [1, r] (start fixed at the beginning of the sample, exactly radf's own badf convention) rather than quantile_test's single full-sample test.

Usage

monitor_quantile(
  data,
  tau = 0.5,
  minw = NULL,
  nrep = 500L,
  sig_lvl = 95,
  seed = NULL
)

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.

tau

Quantile to test at, in (0, 1) (fixed, unlike quantile_test's "optimal" grid search – WSW's own eq. 25 takes tau as a given parameter for the monitoring statistic, not re-selected at each recursion point).

minw

A positive integer. The minimum window size (default = \((0.01 + 1.8/\sqrt{T})T\), where T denotes the sample size).

nrep

Number of Monte Carlo replications for the boundary.

sig_lvl

Significance level, one of 90, 95, 99.

seed

Optional seed for the Monte Carlo draws.

Value

An object of class monitor_quantile_obj: a list with the statistic path stat, the (flat) boundary, the estimated delta, and alarm/alarm_date (the first breach, NA if none).

Details

Only QPWY (single recursion) is implemented, not the paper's own QPSY (double recursion, additionally optimizing over the window start): QPWY_r(tau) needs O(T) actual quantile -regression fits (no closed-form recursive update the way OLS has), tractable at the same cost order as radf()'s own badf; QPSY needs O(T^2) such fits, a substantially larger undertaking left unimplemented.

The critical value is simulated per call: QPWY_r(tau)'s limiting null distribution at each r is sqrt(1 - delta^2) * z + delta * Q_{0,r}, with z ~ N(0, 1), delta a data-estimated correlation coefficient (as in quantile_test), and Q_{0,r} exactly radf()'s own badf sequence under a simulated null path – reusing radf() directly for the simulation rather than new theory. A single flat boundary is used (not one value per r): controlling the first-crossing false-alarm rate requires calibrating against each simulated path's own supremum, exactly how radf_mc_cv's own sadf_cv is constructed, not a per-r marginal quantile (which would badly inflate the false-alarm rate).

Note

The critical value (boundary) is simulated internally on every call (via an unexported helper, qpwy_boundary_sim) – there is currently no reusable/exported cv counterpart for this function (a known, separately-tracked gap, not addressed here).

Returns its own class (not radf_obj), so it does not plug into summary()/\link{datestamp}/tidy; it has its own print() and autoplot() methods instead. Prints its own statistic/boundary/delta summary – see vignette("naming-and-analysis", package = "exuber") for the full picture of which functions do and don't fit that pipeline.

Status

References

Wu, R., Shi, S., & Wu, J. (2025). Quantile analysis for financial bubble detection and surveillance. Journal of Time Series Analysis, 46(5), 908-931.

See also

quantile_test for the static, full-sample version of this test. monitor for the OLS-based monitoring alternative.

Other monitoring: monitor(), monitor_cusum(), monitor_lbi()

Examples

# \donttest{
# Heavy-tailed (t3) innovations, explosive from t = 150 to the sample end
y <- sim_psy1(n = 200, te = 150, tf = 200, seed = 7,
  e = sim_innov(199, dist = "t", df = 3))
res <- monitor_quantile(y, tau = 0.5, nrep = 100, seed = 1)
print(res)
#> 
#> ── monitor_quantile (n = 200, minw = 27, tau = 0.5, sig_lvl = 95%) ─────────────
#> 
#>    series  delta  boundary  alarm  alarm_date
#>   series1  0.623      1.53    164         164
#> 
autoplot(res)
#> Warning: Removed 1 row containing missing values or values outside the scale range
#> (`geom_segment()`).


# Upper-quantile monitoring is typically more powerful for right-tailed bubbles
autoplot(monitor_quantile(y, tau = 0.9, nrep = 100, seed = 1))
#> Warning: Removed 1 row containing missing values or values outside the scale range
#> (`geom_segment()`).

# }