Real-time monitoring for bubbles

library(exuber)

Monitoring and testing

radf(), datestamp() and the dating_*() family (see the dating and root inference page) all work on a finished sample. They answer the question of whether there was a bubble, and when, after every observation is already in hand. The monitor_*() family answers a real-time question instead. We fix a training window [1, T*] that we believe is free of exuberance and calibrate a boundary on it. We then watch each new observation T*+1, T*+2, ... and raise an alarm the first time the boundary is crossed. All four functions share this r_star, alarm and alarm_date structure. They differ in the statistic they monitor and in how they calibrate the boundary.

Function	Statistic monitored	Boundary	Static, full-sample counterpart
`monitor()`	`radf()`’s own `badf`/`bsadf` recursion	`"bootstrap"` (Phillips & Shi 2020 wild-bootstrap quantile), `"kurozumi"` (closed-form, Kurozumi 2020), or `"fluc"` (closed-form, Homm & Breitung 2012)	`radf()` and `datestamp()`. It is not a `_test()` function, but it uses the same recursive ADF core
`monitor_cusum()`	A CUSUM of the training-window-standardized series	Homm & Breitung (2012)‘s asymptotic (or finite-sample) constant	None. The CUSUM boundary of Homm & Breitung is a training and monitoring construction by design and has no full-sample form
`monitor_lbi()`	Breitung & Diegel (2025)‘s locally-best-invariant CUSUM (`mCUSUM`/`wCUSUM`, via `c_bar`)	Their Table 1 constant	`lbi_test()`, the static version of the same statistic
`monitor_quantile()`	A recursive quantile regression at `tau`	A simulated first-crossing boundary (Wu, Shi & Wu 2025)	`quantile_test()`, the static version of the same statistic

monitor() reuses badf and bsadf directly, which is the same recursive ADF core as in the radf_*() family. It is named after what it does and not after that internal detail (see the results, tidying and plotting page for the reasoning). Two of its three siblings, monitor_lbi() and monitor_quantile(), are the sequential extension of an existing static test with the same name minus the monitor_ prefix. monitor_cusum() has no such counterpart, because its source paper (Homm & Breitung 2012) proposed the CUSUM only as a monitoring detector.

One bubble, five monitors

The series starts with a training window of pure random walk (T* = 100) and continues as a random walk until t = 150. From then until the end of the sample it follows an explosive regime (rho = 1.04). We generate it with sim_psy1(), placing the bubble after the training window and leaving out the collapse:

y <- sim_psy1(n = 200, te = 150, tf = 200, c = 0.04, alpha = 0, seed = 7)

monitor_lbi(y, r_star = 100)
#> 
#> ── monitor_lbi (T* = 100 / 200, c_bar = 0, b_alpha = 1.95) ─────────────────────
#> 
#>    series  alarm  alarm_date
#>   series1    156         156
monitor_cusum(y, r_star = 0.5)
#> 
#> ── monitor_cusum (T* = 100 / 200, b_alpha = 4.6) ───────────────────────────────
#> 
#>    series  alarm  alarm_date
#>   series1    161         161
monitor_quantile(y, tau = 0.5, nrep = 200, seed = 1)
#> 
#> ── monitor_quantile (QPWY, n = 200, minw = 27, tau = 0.5, sig_lvl = 95%) ───────
#> 
#>    series  delta  boundary  alarm  alarm_date
#>   series1   0.64     1.909    161         161
monitor(y, r_star = 0.5, nboot = 200, seed = 1)
#> 
#> ── monitor (T* = 100 / 200, minw = 27, sig_lvl = 95%, boundary = bootstrap) ────
#> 
#>    series  boundary  alarm  alarm_date
#>   series1      2.17    156         156
monitor(y, r_star = 0.5, boundary = "kurozumi")
#> 
#> ── monitor (T* = 100 / 200, minw = 27, sig_lvl = 95%, boundary = kurozumi) ─────
#> 
#>    series  boundary  alarm  alarm_date
#>   series1     1.038    159         159

monitor_lbi() and monitor_quantile() each have a static, full-sample counterpart that asks the retrospective version of the same question. It runs on the whole series and does not wait for a first crossing:

lbi_test(y)
#> 
#> ── lbi_test (n = 200, sig_lvl = 95%) ───────────────────────────────────────────
#> 
#>    series   stat   crit  detected
#>   series1  6.502  1.645      TRUE
quantile_test(y, tau = 0.5)
#> 
#> ── quantile_test (n = 200, sig_lvl = 95%) ──────────────────────────────────────
#> 
#>    series  tau  tstat    crit  delta  detected
#>   series1  0.5  20.25  0.5041   0.64      TRUE

Every monitor alarms within about 15 observations of the true bubble start (150), and none alarms before it. Each function’s own test suite checks the absence of alarms before T* (or the true start) under the null. The timing of the alarm differs by design. The ADF-family statistics in monitor() tend to detect bubbles in the middle of the sample fastest, as the literature finds (for example Kurozumi 2020, 2021). The CUSUM-type detectors (monitor_cusum() and monitor_lbi()) are typically slower, but they are computationally simpler and need no bootstrap.

Which to reach for

If you want the fastest detection and can afford a wild bootstrap on each call, use monitor(boundary = "bootstrap"), which is the default.
If you want the same statistic without a bootstrap, using a published constant instead, use monitor(boundary = "kurozumi") or monitor(boundary = "fluc").
If you want a simpler CUSUM-based alternative with its own closed-form boundary, use monitor_cusum(). For the locally best invariant version of Breitung & Diegel, use monitor_lbi(), where c_bar > 0 gives up a little size in exchange for power against slowly building bubbles.
If you want to monitor a specific quantile of the distribution instead of the mean, use monitor_quantile().