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,
  level = 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.

level

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).

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. radf_monitor for the OLS-based monitoring alternative.

Examples

# \donttest{
res <- monitor_quantile(sim_data$sim_psy1, tau = 0.5, nrep = 100, seed = 1)
#> Warning: Unknown or uninitialised column: `sim_psy1`.
#> Error: unsupported class
print(res)
#> Error: object 'res' not found
# }