Guo, Sun & Wang (2019) show that – unlike the classical (stationary or unit-root) case – the ordinary t-statistic for the autoregressive root \(\hat\rho\) of a (moderately) explosive AR(1), 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
root_ci(x, level = 0.95, type = c("normal", "cauchy"))Arguments
- x
A list as returned by
explosive_root.- 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.
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.
root_ci also reports the implied doubling time
\(\log(2)/\log(\hat\rho)\): the number of periods for the bubble to
double in magnitude at the estimated growth rate, with its own interval
obtained by transforming the endpoints of the \(\hat\rho\) interval
(doubling time is strictly decreasing in \(\rho\), so the CI's lower and
upper doubling-time bounds come from the upper and lower \(\rho\) bounds,
respectively).
![[Experimental]](figures/lifecycle-experimental.svg)