Confidence Interval and Doubling Time for an Explosive Root

rootstamp(object, ...)

rootstamp(object, sig_lvl = 95, type = c("normal", "cauchy"), ...)

rootstamp(object, ds, sig_lvl = 95, type = c("normal", "cauchy"), ...)

Fits a no-intercept AR(1) regression $y_t = \rho y_{t-1} + \epsilon_t$ (Phillips & Magdalinos 2007, who omit the intercept in their eq. 58 "to exclude the presence of a deterministically explosive component") and reports $\hat\rho$ together with a confidence interval and the implied doubling time $\log(2)/\log(\hat\rho)$, which is the number of periods the series needs to double in magnitude at the estimated growth rate.

Arguments

object	For the default method, a numeric vector with the sub-sample to fit, already sliced to the episode of interest. For the radf_obj method, the radf_obj on which ds was computed.
...	Further arguments passed to methods.
sig_lvl	Confidence level of the interval on the 0 to 100 scale used throughout the package (default 95). Any value in [50, 100) is allowed.
type	"normal" (default) for the normal-t interval of Guo, Sun & Wang, or "cauchy" for the fixed-root Cauchy interval of Phillips & Magdalinos.
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 that are too short for reliable root inference, because this method does not decide what counts as too short.

Value

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

The radf_obj method returns a rootstamp_episodes object, which is a named list with one element for each series in ds. Each element is a data frame with one row for each datestamped episode and the columns Start, End, rho, rho_lower, rho_upper, doubling_time, doubling_time_lower and doubling_time_upper. The object has its own print() method. The panel sieve-bootstrap case has a ds entry named "panel" that corresponds to no single series, and it is dropped with a warning.

Details

Guo, Sun & Wang (2019) show that the ordinary t-statistic for $\hat\rho$, estimated by OLS with no intercept, is asymptotically standard normal under i.i.d. errors, and also under weakly dependent errors when a HAC standard error is used. This differs from the classical stationary and unit-root cases. An ordinary-looking Wald interval, $\hat\rho \pm z_{\alpha/2}\cdot se(\hat\rho)$, is therefore asymptotically valid here even though $\hat\rho > 1$. The interval has the same form as a classical normal-theory interval, which would be invalid for an explosive root, but the justification is different: it rests on the explosive-root central limit theorem of Guo, Sun & Wang and not on the classical stationary one.

type = "cauchy" instead uses the fixed-root result of Phillips & Magdalinos (2007, their eq. 27, which restates White 1958). For an explosive root that does not drift, $\frac{\rho^n}{\rho^2-1}(\hat\rho-\rho)$ converges to a standard Cauchy variate. We replace the unknown $\rho$ in the normalization with $\hat\rho$, as is usual for this kind of self-normalized pivot, and obtain $\hat\rho \pm q_{\alpha/2}\cdot (\hat\rho^2-1)/\hat\rho^n$, where $q_{\alpha/2}$ is a standard-Cauchy quantile. This interval assumes a fixed explosive root, with no drift and no unknown localizing rate. When that assumption is in doubt, the default "normal" type is safer, because the result of Guo, Sun & Wang allows for drift and weak dependence.

There are two methods, for two different starting points:

• Default: object is a numeric vector, the sub-sample to fit. It can be an episode that you sliced out by hand or by position from a datestamp result (y[from:to]). The method fits the sub-sample once and returns one confidence interval.
• radf_obj: object is the radf_obj on which a datestamp result ds was computed. The method runs the default method once for each datestamped episode of each series and slices the data of object itself, so no manual loop is needed.

# The martingale-to-explosive process of sim_psy1(), explosive through the sample end
y <- sim_psy1(n = 100, te = 60, tf = 100, seed = 2026)

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
ep <- y[ds[["series1"]]$Start[1]:ds[["series1"]]$End[1]]
fit <- rootstamp(ep) # recovers the DGP's explosive AR coefficient
fit
#> 
#> ── rootstamp (n = 22, sig_lvl = 95%, type = normal) ────────────────────────────
#> 
#>     rho        se  t_stat  rho_lower  rho_upper  doubling_time  dt_lower
#>   1.059  0.005279   11.17      1.049      1.069           12.1     10.34
#>   dt_upper
#>       14.6
rootstamp(ep, type = "cauchy")
#> 
#> ── rootstamp (n = 22, sig_lvl = 95%, type = cauchy) ────────────────────────────
#> 
#>     rho        se  t_stat  rho_lower  rho_upper  doubling_time  dt_lower
#>   1.059  0.005279   11.17     0.6216      1.496           12.1      1.72
#>   dt_upper
#>     -1.458

# Plot the episode with the fitted explosive-root path overlaid
autoplot(fit)

# radf_obj method: every datestamped episode at once
res_all <- rootstamp(r, ds)
res_all
#> 
#> ── rootstamp (sig_lvl = 95%, type = normal) ────────────────────────────────────
#> 
#> series1 :
#>   Start End   rho rho_lower rho_upper doubling_time doubling_time_lower
#> 1    78 100 1.059     1.049     1.069          12.1               10.34
#>   doubling_time_upper
#> 1                14.6

# Plot the estimated rho (with its CI) for every episode
autoplot(res_all)