---
title: "IVX quantile predictive regression"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{IVX quantile predictive regression}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
has_qr <- requireNamespace("quantreg", quietly = TRUE)
```

```{r setup, message = FALSE}
library(ivx)
```

`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()`.

```{r, eval = has_qr}
m <- ivx_qr(Ret ~ DP, data = kms, tau = 0.5)
summary(m)
```

Several quantiles at once return a list:

```{r, eval = has_qr}
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))))
```

Predictability 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 $\beta = 0.8$:

```{r, eval = has_qr}
ivx_qr(Ret ~ DP, data = kms, tau = 0.5, beta = 0.8)$Wald_Joint
```

## Block 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.

```{r}
m <- ivx_qr(Ret ~ DP + TBL, data = kms, tau = 0.1)
ivx_qr_boot(m, B = 499, seed = 1)
```

In 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; use `ivx_qr_boot()`.
* The `rq` fit 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.