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Rolling IVX tests for bubble detection

Source: vignettes/rolling-ivx.Rmd
rolling-ivx.Rmd

Pavlidis, Paya & Peel (2017) detect periodically collapsing bubbles by running the Fama (1984) regression inside a rolling window and drawing inference with IVX. Everything the procedure needs is already in ivx(), so this vignette shows how to build it in a few lines rather than through a dedicated function.

The procedure is:

  1. Fix a window of w observations. The paper uses w = max(25, r0 * T), where r0 * T is the minimum window rule of Phillips, Shi & Yu (2015): r0 = 0.01 + 1.8 / sqrt(T).
  2. For every window ending at t = w, ..., T, fit the predictive regression with ivx() and keep the IVX t-statistic of the slope. Under the null the statistic is standard normal.
  3. Compare the sequence against a constant critical value. For the overall test of “any bubble in the sample” the paper applies a Bonferroni correction for the T - w + 1 hypotheses, qnorm(1 - alpha / (T - w + 1)). For dating the episodes it compares each statistic with the plain qnorm(1 - alpha).
library(ivx)

ivx_roll <- function(formula, data, window, horizon = 1, alpha = 0.05, ...) {
  n <- nrow(data)
  ends <- window:n
  tstat <- vapply(ends, function(e) {
    fit <- ivx(formula, data[(e - window + 1):e, ], horizon = horizon, ...)
    fit$tstat[1]
  }, numeric(1))
  data.frame(
    end = ends,
    tstat = tstat,
    cv = qnorm(1 - alpha),
    cv_bonf = qnorm(1 - alpha / length(ends))
  )
}

fit$tstat is coef / se, i.e. the IVX t-ratio of the first regressor; the one-sided alternative in the paper is a slope above its efficient-market value. With the KMS monthly data:

n <- nrow(kms)
w <- max(25, floor((0.01 + 1.8 / sqrt(n)) * n))
r <- ivx_roll(Ret ~ LTY, kms, window = w)

max(r$tstat)
#> [1] 2.327991
r$cv_bonf[1]
#> [1] 3.882191

The overall test rejects if the maximum statistic exceeds the Bonferroni critical value. Episodes are dated by the periods where the statistic sits above the standard normal critical value:

plot(r$end, r$tstat, type = "l", xlab = "window end", ylab = "IVX t-statistic")
abline(h = r$cv[1], col = "red", lty = 2)
abline(h = r$cv_bonf[1], col = "red")

range(r$end[r$tstat > r$cv])
#> [1] 203 986

For a long-horizon Fama regression pass horizon = n and the overlapping regressand; ivx() then uses the Phillips & Lee (2013) long-horizon estimator. Bootstrap critical values for each window can be obtained by replacing the qnorm() constants with ivx_boot() quantiles, at the cost of B extra fits per window.

References

Pavlidis, E. G., Paya, I., & Peel, D. A. (2017). Testing for speculative bubbles using spot and forward prices. International Economic Review, 58(4), 1191-1226.

Phillips, P. C. B., Shi, S., & Yu, J. (2015). Testing for multiple bubbles: Historical episodes of exuberance and collapse in the S&P 500. International Economic Review, 56(4), 1043-1078.