ivx_qr() implements the IVX-QR predictability test of
Lee (2016): a quantile regression version of the IVX test that is valid
for stationary, mildly integrated, near-unit-root and mildly explosive
predictors alike. It needs the quantreg package.
Setting
The conditional \(\tau\)-quantile of the return is modelled as
\[ Q_{y_t}(\tau \mid x_{t-1}) = \beta_{0,\tau} + \beta_\tau' x_{t-1}. \]
Ordinary quantile regression on \(x_{t-1}\) suffers from the same problem as OLS: with persistent, endogenous predictors the QR t-ratio has a non-standard limit that depends on the local-to-unity parameter and on the quantile-specific endogeneity \(\rho(\tau) = -\mathrm{corr}\big(1\{u_{0t} < 0\}, u_{xt}\big)\).
The test
Lee’s practical procedure (Section 3.3, Proposition 3.2) exploits that the IVX instrument \(z_{t-1}\) is “close” to \(x_{t-1}\): run the quantile regression of \(y_t\) on an intercept and the demeaned instrument,
\[ \hat\beta^{IVXQR}_\tau = \arg\min_{\beta_0, \beta} \sum_t \rho_\tau\big( y_t - \beta_0 - \beta' \tilde z_{t-1} \big), \]
and use the self-normalised statistic
\[ \frac{\hat f_u(0)^2}{\tau(1-\tau)} \; \hat\beta_\tau' \Big(\sum_t \tilde z_{t-1} \tilde z_{t-1}'\Big) \hat\beta_\tau \;\to\; \chi^2_K \quad \text{under } H_0: \beta_\tau = 0, \]
where \(\hat f_u(0)\) is the density
of the QR residuals at zero, estimated with a Gaussian kernel and
Silverman’s bandwidth (the paper’s footnote 4). The implied covariance
\(\tau(1-\tau)\hat f_u(0)^{-2}(\tilde
Z'\tilde Z)^{-1}\) gives the standard errors in
summary().
m <- ivx_qr(Ret ~ DP, data = kms, tau = 0.5)
summary(m)
#>
#> Call:
#> ivx_qr(formula = Ret ~ DP, data = kms, tau = 0.5)
#>
#> IVX-QR at tau = 0.5
#>
#> Coefficients:
#> Estimate Std. Error t value Wald Ind Pr(> chi)
#> DP 0.003571 0.005516 0.647 0.419 0.517
#>
#> Joint Wald statistic: 0.4191 on 1 DF, p-value 0.5174
#> QR endogeneity rho(tau): DP -0.669Several quantiles at once return a list:
ms <- ivx_qr(Ret ~ DP + TBL, data = kms, tau = c(0.1, 0.25, 0.5, 0.75, 0.9))
t(sapply(ms, function(f) c(tau = f$tau, coef(f), Wald = round(f$Wald_Joint, 2),
p = round(1 - pchisq(f$Wald_Joint, 2), 3))))
#> tau DP TBL Wald p
#> 0.1 0.10 -0.028305755 0.18938487 6.67 0.036
#> 0.25 0.25 -0.001707079 -0.06790951 0.75 0.688
#> 0.5 0.50 0.008204012 -0.17132111 7.54 0.023
#> 0.75 0.75 0.024622728 -0.19952197 21.90 0.000
#> 0.9 0.90 0.026739798 -0.43020894 34.59 0.000Predictability that is absent at the median can appear in the tails, which is the empirical point of the paper.
Tuning
Lee normalises \(c_z = 5\) (the
package default for ivx_qr(); ivx() uses 1)
and chooses the exponent \(\beta\) from
a look-up table indexed by the estimated endogeneity \(\hat\rho(\tau)\), so that the asymptotic
size of the nominal 5% test stays below 7.5%: the larger \(|\hat\rho(\tau)|\), the smaller \(\beta\). The table is in the paper’s
supplement rather than the paper, so the package reports \(\hat\rho(\tau)\) (rho_tau,
also printed by summary()) and leaves beta to
the user, defaulting to the Kostakis et al. (2015) value 0.95. In a
Monte Carlo with a unit-root predictor and \(\rho = -0.95\) the median test with the
default has empirical size of about 8% at \(n
= 250\), which is exactly the situation the rule is meant to
correct; with \(\beta = 0.8\):
ivx_qr(Ret ~ DP, data = kms, tau = 0.5, beta = 0.8)$Wald_Joint
#> [1] 0.005426976Block bootstrap under conditional heteroskedasticity
The asymptotic test needs the sparsity \(f_u(0)\) and, with conditionally
heteroskedastic errors, further nuisance parameters that enter the limit
(Fan & Lee, 2019, Theorem 3.2); their estimation error is what
distorts the test in the tails. ivx_qr_boot() implements
the paper’s moving block bootstrap: blocks of the pairs \((y_t, \tilde z_{t-1})\) of length \(\lceil n^{1/4} \rceil\) are resampled, the
quantile regression is refitted, and percentile intervals and p-values
are reported. No sparsity estimate is needed.
m <- ivx_qr(Ret ~ DP + TBL, data = kms, tau = 0.1)
ivx_qr_boot(m, B = 499, seed = 1)
#>
#> Call:
#> ivx_qr(formula = Ret ~ DP + TBL, data = kms, tau = 0.1)
#>
#> IVX-QR at tau = 0.1, moving block bootstrap, B = 499, block length 6
#>
#> Coefficients (percentile intervals and p-values):
#> Estimate 2.5% 97.5% Pr(|b| > 0)
#> DP -0.02831 -0.05947 0.01104 0.212
#> TBL 0.18938 -0.13693 0.55822 0.257In the paper’s ARCH(1) design (\(\alpha_1 = 0.9\), \(\rho = -0.9\), \(c = 0\), \(n = 200\), \(\tau = 0.1\)) the asymptotic test rejects a true null 18% of the time in 300 replications; the block bootstrap 7%.
Caveats
- Tail quantiles need long samples for an accurate density estimate; the paper uses \(n = 700\) to study the 5% quantile.
- The test in
summary()is the QR-on-instrument test of Proposition 3.2, not the full IVX-QR estimator of the paper’s equation (3.6), whose non-convex objective the paper itself avoids for testing \(\beta_\tau = 0\). The coefficients are therefore those of the regression on \(\tilde z_{t-1}\). - Short horizon only.
ivx_boot()does not apply; useivx_qr_boot(). - The
rqfit is stored in the result ($rq) for further quantreg methods.
References
Fan, R., & Lee, J. H. (2019). Predictive quantile regressions under persistence and conditional heteroskedasticity. Journal of Econometrics, 213(1), 261–280.
Lee, J. H. (2016). Predictive quantile regression with persistent covariates: IVX-QR approach. Journal of Econometrics, 192(1), 105–118.
Koenker, R. (2005). Quantile Regression. Cambridge University Press.