Confidence Interval and Doubling Time for an Explosive Root

Fits a no-intercept AR(1) regression \(y_t = \rho y_{t-1} + \epsilon_t\) (Phillips & Magdalinos 2007; no intercept, following their eq. 58, "to exclude the presence of a deterministically explosive component") and reports \(\hat\rho\) together with a confidence interval and implied doubling time \(\log(2)/\log(\hat\rho)\) – the number of periods for the series to double in magnitude at the estimated growth rate. Guo, Sun & Wang (2019) show that – unlike the classical (stationary or unit-root) case – the ordinary t-statistic for \(\hat\rho\), estimated by OLS with no intercept, is asymptotically standard normal under i.i.d. errors (and under weakly dependent errors, with a HAC standard error). This means an ordinary-looking Wald interval, \(\hat\rho \pm z_{\alpha/2}\cdot se(\hat\rho)\), is asymptotically valid here even though \(\hat\rho > 1\) – despite looking identical in form to a classical (invalid, for an explosive root) normal-theory interval, the justification is different (Guo, Sun & Wang's explosive-root CLT, not the classical stationary one).

Usage

rootstamp(object, ...)

# Default S3 method
rootstamp(object, level = 0.95, type = c("normal", "cauchy"), ...)

# S3 method for class 'radf_obj'
rootstamp(object, ds, level = 0.95, type = c("normal", "cauchy"), ...)

Arguments

object: For the default method, a numeric vector (the sub-sample to fit – already sliced to the episode of interest). For the radf_obj method, the radf_obj that ds was computed on.
...: further arguments passed to methods.
level: Confidence level (default 0.95).
type: "normal" (default) for Guo, Sun & Wang's normal-t interval, or "cauchy" for the Phillips-Magdalinos fixed-root Cauchy interval.
ds: (radf_obj method only) A datestamp result computed on object. Root inference on a very short episode is statistically meaningless – set min_duration in that datestamp call to exclude episodes too short for reliable root inference, rather than expecting this method to second-guess what counts as "too short".

Value

The default method returns a rootstamp_est object (a list with rho, se, t_stat, n, rho_ci, doubling_time, doubling_time_ci, with its own print() method).

The radf_obj method returns a rootstamp_episodes object (a named list, one element per series in ds; the panel sieve-bootstrap case, whose ds entry is named "panel" and has no single corresponding series, is dropped with a warning), each a data frame with one row per datestamped episode: Start, End, rho, rho_lower, rho_upper, doubling_time, doubling_time_lower, doubling_time_upper – also with its own print() method.

Details

type = "cauchy" instead uses the Phillips & Magdalinos (2007) fixed-root result (their eq. 27, restating White 1958): for a genuinely explosive, non-drifting root, \(\frac{\rho^n}{\rho^2-1}(\hat\rho-\rho)\) converges to a standard Cauchy variate. Plugging in \(\hat\rho\) for the unknown \(\rho\) in the normalization (the usual practice for this kind of self-normalized pivot) gives \(\hat\rho \pm q_{\alpha/2}\cdot (\hat\rho^2-1)/\hat\rho^n\), with \(q_{\alpha/2}\) a standard-Cauchy quantile. This interval assumes a fixed explosive root (no drift, no unknown localizing rate); the default "normal" type is the safer choice when that assumption is in doubt, since Guo, Sun & Wang's result allows drift and weak dependence.

Two methods, for two different starting points:

Default: object is a numeric vector – the sub-sample to fit (e.g. an episode already sliced out by hand, or by position from a datestamp result: y[from:to]). Fits once and returns one CI.
radf_obj: object is the radf_obj a datestamp result ds was computed on. Runs the default method once per datestamped episode, per series, slicing object's own data itself – no manual loop needed.

Note

Neither method's return value carries the radf_obj class – even the radf_obj method, which dispatches on that class for its input, returns its own rootstamp_episodes class – so rootstamp() does not plug into summary()/\link{datestamp}/tidy/autoplot. See vignette("naming-and-analysis", package = "exuber") for the full picture of which functions do and don't fit that pipeline.

Status

References

Guo, G., Sun, Y., & Wang, S. (2019). Testing for moderate explosiveness. The Econometrics Journal, 22(3), 279-303.

Phillips, P. C. B., & Magdalinos, T. (2007). Limit theory for moderate deviations from a unit root. Journal of Econometrics, 136(1), 115-130.

Examples

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)

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)

# default method: one episode, sliced by hand
rootstamp(y[ds[["series1"]]$Start[1]:ds[["series1"]]$End[1]]) # 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[ds[["series1"]]$Start[1]:ds[["series1"]]$End[1]], type = "cauchy")
#> 
#> ── 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
#> 


# radf_obj method: every datestamped episode at once
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
#> 
#>