Episodic predictability: subsample IVX tests

library(ivx)

Full-sample tests find little return predictability; a growing literature argues that predictability comes and goes in “pockets”. ivx_episodic() implements the subsample IVX tests of Demetrescu, Georgiev, Rodrigues & Taylor (2023, Section 3.2), which formalise the rolling and recursive approaches of Demetrescu et al. (2022) and, for one-sided tests, of Pavlidis, Paya & Peel (2017) (see vignette("rolling-ivx") for the latter).

Subsample statistic

For a window \(t = \lfloor \tau_1 T \rfloor + 1, \dots, \lfloor \tau_2 T \rfloor\) the IVX statistic is computed from the window’s observations but with the full-sample instrument (DGRT eqs 15–17):

\[ \hat\beta_{zx}(\tau_1, \tau_2) = \frac{\sum_{t} z_{t-1}\,(y_t - \bar y(\tau_1,\tau_2))} {\sum_{t} z_{t-1}\,(x_{t-1} - \bar x_{-1}(\tau_1,\tau_2))}, \qquad t_{zx}(\tau_1, \tau_2) = \frac{\hat\beta_{zx}(\tau_1,\tau_2)}{\mathrm{s.e.}(\hat\beta_{zx}(\tau_1,\tau_2))}, \]

with \(\hat\sigma_u^2\) from the window’s OLS residuals (or Eicker–White weights with robust = TRUE). Three agnostic sequences are considered:

  • forward recursive, \(\{t_{zx}(0, \tau)\}\) for \(\tau \in [\tau_L, 1]\) — pockets that start at the beginning of the sample;
  • backward recursive, \(\{t_{zx}(\tau, 1)\}\) for \(\tau \in [0, \tau_U]\) — end-of-sample pockets;
  • rolling, \(\{t_{zx}(\tau, \tau + \Delta\tau)\}\) for a fixed window fraction \(\Delta\tau\).

The tests are the maximum (right-tailed, \(H_1: \beta > 0\)), minimum (left-tailed) and maximum squared (two-sided) of the sequence; with several predictors, the maximum of the subsample Wald statistics (Remark 11). Their limits are functionals of Brownian motions, so critical values come from the wild bootstrap of vignette("robust-inference"): the same functional is computed on each bootstrap sample. The default is the fixed regressor wild bootstrap used by Demetrescu et al. (2022); type = "rwb" is available.

mod <- ivx(Ret ~ DP, data = kms)
e <- ivx_episodic(mod, scheme = "rolling", window = 0.2, B = 499, seed = 1)
e
#> 
#> Call:
#> ivx(formula = Ret ~ DP, data = kms, horizon = 1)
#> 
#> Subsample IVX tests, rolling scheme (window = 0.2), 827 windows
#> Fixed regressor wild bootstrap, B = 499
#> 
#>                         statistic bootstrap p
#> sup t   (H1: beta > 0)      2.921      0.1363
#> inf t   (H1: beta < 0)    -0.4166      0.9980
#> sup t^2 (H1: beta != 0)     8.533      0.2846

The sequence itself is returned, with the window bounds as row indices of the regression sample:

head(e$sequence, 3)
#>   start end      Wald      t_DP
#> 1     2 207 0.3000535 0.5477714
#> 2     3 208 0.2980866 0.5459731
#> 3     4 209 0.3001673 0.5478752
plot(kms$Date[e$sequence$end], e$sequence$t_DP, type = "l",
     xlab = "window end", ylab = "subsample IVX t-ratio", main = "Rolling windows, 20% of the sample")
abline(h = c(-1.96, 1.96), lty = 2)

Forward and backward recursive sequences:

ivx_episodic(mod, scheme = "forward", window = 0.2, B = 199, seed = 1)$p.value
#>       sup       inf    sup_sq 
#> 0.2462312 0.7788945 0.4371859
ivx_episodic(mod, scheme = "backward", window = 0.8, B = 199, seed = 1)$p.value
#>        sup        inf     sup_sq 
#> 0.03015075 0.98994975 0.05025126

Interpretation and caveats

  • A rejection says that predictability of the given sign exists in at least one window; the plot of the sequence locates it. The window(s) above the pointwise \(\pm 1.96\) line are not individually significant at 5% — that ignores the multiplicity the sup-test accounts for.
  • window is a fraction of the regression sample; the paper uses window fractions between 0.1 and 0.3 for rolling tests and warm-in fractions of 0.1–0.25 for recursive ones. Very short windows give noisy statistics and low power.
  • The instrument is built once from the full sample. Refitting ivx() on each window (as in vignette("rolling-ivx")) rebuilds the instrument inside the window, which is a different statistic; the Bonferroni approach used there is conservative, the bootstrap sup-test is not.
  • Short horizon only; the fit must be a plain ivx object without weights.

References

  • Demetrescu, M., Georgiev, I., Rodrigues, P. M. M., & Taylor, A. M. R. (2022). Testing for episodic predictability in stock returns. Journal of Econometrics, 227(1), 85–113.
  • Demetrescu, M., Georgiev, I., Rodrigues, P. M. M., & Taylor, A. M. R. (2023). Extensions to IVX methods of inference for return predictability. Journal of Econometrics, 237(2), 105271.
  • 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.