Root Inference: How Fast Is the Bubble Growing

library(exuber)

A different question from “is there a bubble”

radf()/datestamp() (and the dating_*()/_test()/monitor()/monitor_*() families) all answer some version of “is there a bubble, and when did it happen.” None of them say anything about its magnitude: once an explosive episode is dated, how fast is the underlying autoregressive root actually growing? rootstamp() (Phillips & Magdalinos 2007; Guo, Sun & Wang 2019) answers exactly that, as a follow-up step after detection and dating, not a replacement for either.

It fits a no-intercept AR(1), y_t = rho * y_{t-1} + e_t, over a given sub-sample and reports rho’s estimate together with a confidence interval and an implied doubling time (log(2) / log(rho): how many periods until the bubble doubles in size at the estimated growth rate). Two methods, for two different starting points – a plain numeric sub-sample (fit once, one CI), or a radf_obj plus its datestamp() result (fit every episode at once, no manual loop). Neither method’s return value carries the radf_obj class itself, so rootstamp() doesn’t plug into summary()/tidy()/autoplot() – see vignette("naming-and-analysis").

Detect, date, then estimate the root

A unit-root run-up followed by a genuine explosive regime with rho = 1.04:

set.seed(2026)
burn <- cumsum(rnorm(60))
bubble <- burn[length(burn)] * 1.04^(1:40) + cumsum(rnorm(40, sd = 0.5))
y <- c(burn, bubble)

Detect and date it first, the ordinary way:

r <- radf(y, minw = 20)
cv <- radf_mc_cv(length(y), minw = 20, nrep = 300, seed = 4)
ds <- datestamp(r, cv = cv, min_duration = 3)
ds
#> 
#> ── Datestamp (min_duration = 3) ───────────────────────────────── Monte Carlo ──
#> 
#> series1 :
#>   Start Peak End Duration   Signal Ongoing
#> 1    83  100 100       18 negative    TRUE

Then estimate the root over the detected episode – the default method takes the sub-sample directly, sliced by the episode’s own Start/End:

ep <- ds[["series1"]]
rootstamp(y[ep$Start[1]:ep$End[1]]) # normal-t interval (Guo, Sun & Wang 2019), true rho = 1.04
#> 
#> ── rootstamp (n = 17, level = 95%, type = normal) ──────────────────────────────
#> 
#>     rho        se  t_stat  rho_lower  rho_upper  doubling_time  dt_lower
#>   1.042  0.006318   6.639       1.03      1.054          16.87      13.1
#>   dt_upper
#>      23.79
rootstamp(y[ep$Start[1]:ep$End[1]], type = "cauchy") # fixed-root Cauchy interval (Phillips & Magdalinos 2007)
#> 
#> ── rootstamp (n = 17, level = 95%, type = cauchy) ──────────────────────────────
#> 
#>     rho        se  t_stat  rho_lower  rho_upper  doubling_time  dt_lower
#>   1.042  0.006318   6.639     0.5007      1.583          16.87     1.509
#>   dt_upper
#>     -1.002

rho recovers something close to the true 1.04, alongside rho_ci and the implied doubling_time/doubling_time_ci. The two interval types answer slightly different questions. type = "normal" (the default) is the safer choice under drift/weak dependence and gives a noticeably tighter interval here; type = "cauchy" assumes a genuinely fixed, non-drifting root and, because a Cauchy distribution has much fatter tails than a normal, produces a visibly wider interval even at the same nominal level.

Every episode at once

For a datestamp() result with more than one episode, the radf_obj method runs the default method on each one without a manual loop – pass the original radf() result and the datestamp() result together:

rootstamp(r, ds)
#> 
#> ── rootstamp (level = 95%, type = normal) ──────────────────────────────────────
#> 
#> series1 :
#>   Start End   rho rho_lower rho_upper doubling_time doubling_time_lower
#> 1    83 100 1.042      1.03     1.054         16.87                13.1
#>   doubling_time_upper
#> 1               23.79

Root inference on a very short episode is close to meaningless (too few points to estimate an AR(1) coefficient precisely) – filter with datestamp(..., min_duration = ...) before piping in, rather than expecting this method to second-guess what counts as “too short.”