Robust inference: Eicker-White and wild bootstrap

library(ivx)

Demetrescu, Georgiev, Rodrigues & Taylor (2023, DGRT) extend the IVX tests of Kostakis et al. (2015) in two directions implemented here: validity under general heteroskedasticity through Eicker–White standard errors (ivx(robust = TRUE), see vignette("ivx")), and bootstrap implementations that markedly improve finite-sample size, in particular for one-sided tests and with several predictors (ivx_boot()).

Why bootstrap

Even with a valid \(\chi^2\) limit, the IVX t-ratio is biased in finite samples when the predictor is strongly persistent and \(\delta\) is large in absolute value: the estimate of the intercept interacts with the instrument, and the leading higher-order term shifts the statistic (Hosseinkouchack & Demetrescu 2021). With \(T = 250\), \(\delta = -0.95\) and a unit-root predictor, DGRT report empirical sizes of 10–17% for the right-tailed 5% test.

The two schemes

Let \(\hat u_t\) be the OLS residuals of the predictive regression and \(R_t\) i.i.d. multipliers with mean zero and unit variance (Rademacher by default, dist = "normal" optional).

Fixed regressor wild bootstrap (FRWB), DGRT Algorithm 2. The bootstrap response imposes the null, \(y_t^* = R_t \hat u_t\); the predictors and the instrument are kept fixed. The IVX statistics are recomputed on \((y_t^*, x_{t-1}, z_{t-1})\).

Residual wild bootstrap (RWB), DGRT Algorithm 1 and Remark 22. Fit an autoregression in levels to the predictors (a VAR with several predictors),

\[ x_t = \hat m + \sum_{j=1}^{p+1} \hat a_j x_{t-j} + \hat v_t, \]

with the order chosen by BIC (Remark 24). Draw \((u_t^*, v_t^*) = (R_t \hat u_t, R_t \hat v_t)\) with the same \(R_t\) so the endogeneity correlation is preserved, rebuild \(x_t^* = \sum_j \hat a_j x_{t-j}^* + v_t^*\) from zero initial conditions and \(z_t^*\) from \(\Delta x_t^*\) with the original \(\rho_n\), and recompute the statistics on \((y_t^*, x_{t-1}^*, z_{t-1}^*)\). RWB is the scheme that controls size under strong persistence; FRWB is simpler and needs no lag choice.

The p-value is \(1 - G^*(\text{stat})\), the bootstrap tail probability, for the joint and individual Wald statistics and for the t-ratios against \(\beta < 0\), \(\beta > 0\) and \(\beta \ne 0\).

mod <- ivx(Ret ~ DP + TBL, data = kms)
ivx_boot(mod, B = 499, type = "rwb", seed = 1)
#> 
#> Call:
#> ivx(formula = Ret ~ DP + TBL, data = kms, horizon = 1)
#> 
#> Residual wild bootstrap, B = 499
#> 
#> Coefficients (bootstrap p-values):
#>      Estimate t value Wald Ind Pr(> chi) Pr(t < 0) Pr(t > 0)
#> DP   0.006145   1.349    1.819    0.3768    0.6693     0.331
#> TBL -0.080717  -1.399    1.957    0.2084    0.1002     0.900
#> 
#> Joint Wald statistic: 3.644, bootstrap p-value 0.3186

The asymptotic joint p-value is 0.162; the bootstrap moves it up, the direction DGRT document for persistent, endogenous predictors such as the dividend-price ratio.

Use cores to parallelise (the VAR recursion runs in C++, so a replication costs a few milliseconds; \(B = 9999\) takes seconds on a workstation). With cores > 1 reproducibility for a given seed relies on the L’Ecuyer streams of the parallel package, so results differ from the serial run.

Replication of DGRT (2023), Table 4

Table 4, Panel A, tests each Welch–Goyal predictor for the monthly equity premium, January 1927–December 2020, with asymptotic Eicker–White p-values and RWB p-values (\(B = 9999\)). The replication (in the research/replication folder of the source repository, with the data) reproduces the two-sided p-values for 12 of 14 predictors within bootstrap noise once the ex-dividend S&P return is used as the response; the table lists a selection.

predictor	EW paper	EW ivx	RWB paper	RWB ivx_boot
dp	0.510	0.536	0.612	0.641
ep	0.172	0.167	0.272	0.269
bm	0.594	0.529	0.453	0.425
tbl	0.073	0.068	0.087	0.091
lty	0.054	0.061	0.070	0.079
ntis	0.234	0.208	0.161	0.147
dfy	0.996	0.985	0.994	0.979
dfr	0.394	0.398	0.374	0.423

Two caveats from that exercise. First, the paper’s text describes the response as the return including dividends, but the published values for the three strongly endogenous predictors (dp, ep, bm) are only reproduced with the ex-dividend series; dy behaves the other way round. Second, the Goyal–Welch data are revised over time (inflation and net issuance in particular), so exact agreement is not expected. An independent re-implementation of the Eicker–White statistic in plain R reproduces the package’s C++ values exactly, so the statistics themselves are verified.

Caveats

• Bootstrap requires a plain ivx fit with horizon = 1 for the subsample tests, but ivx_boot() itself accepts any horizon: the null is imposed on the one-step response and the long-horizon statistic is recomputed.
• Weighted fits and the ivx_ar, ivx_ra, ivx_qr classes are not supported.
• RWB estimates the predictor’s autoregression; under a near-unit root the local-to-unity parameter is not consistently estimable, which DGRT show does not affect first-order validity but explains why the two schemes differ in finite samples.
• On Windows, cores > 1 starts PSOCK workers that need the package installed.

References

• 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.
• Hosseinkouchack, M., & Demetrescu, M. (2021). Finite-sample size control of IVX-based tests in predictive regressions. Econometric Theory, 37(4), 769–793.
• Welch, I., & Goyal, A. (2008). A comprehensive look at the empirical performance of equity premium prediction. Review of Financial Studies, 21(4), 1455–1508.