---
title: "Replicating Rapach et al. (2016)"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Replicating Rapach et al. (2016)}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment  = "#>",
  fig.width  = 7,
  fig.height = 4
)
```

This article replicates **Table A2** of the online appendix to Rapach, Ringgenberg, and Zhou (2016), *"Short interest and aggregate stock returns"* (*Journal of Financial Economics*, 121, 46-65), using two functions from **forecastdom**:

* `ivx_wald()`: the Kostakis, Magdalinos, and Stamatogiannis (2015) IVX-Wald test for return predictability with persistent regressors.
* `qll_hat()`: the Elliott and Müller (2006) $\widehat{qLL}$ test for time-varying coefficients.

The predictive regression is

$$r_{t:t+h} = \alpha + \beta\,\mathrm{SII}_t + \varepsilon_{t:t+h}, \qquad r_{t:t+h} = \frac{1}{h}\sum_{j=1}^{h} r_{t+j},$$

at horizons $h \in \{1, 3, 6, 12\}$. IVX-Wald tests $H_0: \beta = 0$. The $\widehat{qLL}$ statistic tests $H_0: \beta_t = \beta$ for all $t$.

```{r setup, message = FALSE}
library(forecastdom)
library(ggplot2)
data(rrz2016)
```

## The data

Monthly U.S. log excess return on the S&P 500 and the standardised linearly-detrended log of the equal-weighted short interest index (EWSI), 1973-01 to 2014-12 (504 observations).

```{r plot, fig.height = 3.5}
ggplot(rrz2016, aes(date, SII)) +
  geom_hline(yintercept = 0, linetype = "dashed", colour = "grey60") +
  geom_line(colour = "#47A5C5", linewidth = 0.6) +
  labs(x = NULL, y = "SII",
       title = "Short interest index, 1973-2014") +
  theme_minimal()
```

## Replicating Table A2

The original MATLAB program (`Compute_IVX_Wald.m` in the JFE data archive) calls the IVX-Wald with `beta = 0.99, M_n = 0` and the **negated** SII. RRZ hypothesise that SII negatively predicts returns, and the sign does not affect the Wald statistic. The $\widehat{qLL}$ test is called with Newey-West truncation `L = h`. We pass the same arguments to `ivx_wald()` and `qll_hat()`.

```{r table-a2}
horizons <- c(1, 3, 6, 12)

results <- do.call(rbind, lapply(horizons, function(h) {

  ivx <- ivx_wald(rrz2016$r, matrix(-rrz2016$SII, ncol = 1),
                  K = h, M_n = 0L, beta = 0.99)

  T_ <- nrow(rrz2016)
  P  <- T_ - h
  y_h <- sapply(seq_len(P), function(t) mean(rrz2016$r[(t + 1):(t + h)]))
  X_h <- matrix(rrz2016$SII[1:P], ncol = 1)
  Z_h <- matrix(1, P, 1)
  qll <- qll_hat(y_h, X_h, Z = Z_h, L = h)

  data.frame(h = h, IVX_Wald = ivx$statistic, qLL = qll$statistic)
  
}))

knitr::kable(results, digits = 3, row.names = FALSE,
             col.names = c("$h$", "IVX-Wald", "$\\widehat{qLL}$"))
```

Critical values (from RRZ 2016, online appendix):

* IVX-Wald: 10% = 2.71, 5% = 3.84, 1% = 6.64.
* $\widehat{qLL}$: 10% = $-7.14$, 5% = $-8.36$, 1% = $-11.05$ (reject for small values).

For comparison, the paper reports IVX-Wald = 3.38\*, 4.51\*\*, 4.60\*\*, 3.67\* and qLL = $-3.72$, $-4.86$, $-4.91$, $-5.02$. Our values match to two decimal places at every horizon. Conclusion: SII predicts the equity premium at all horizons (significant IVX-Wald) and constant $\beta$ is not rejected.

## References

- Elliott, G. and Müller, U. K. (2006). Efficient tests for general
  persistent time variation in regression coefficients. *Review of
  Economic Studies*, 73(4), 907-940.
- Kostakis, A., Magdalinos, T. and Stamatogiannis, M. P. (2015).
  Robust econometric inference for stock return predictability.
  *Review of Financial Studies*, 28(5), 1506-1553.
- Rapach, D. E., Ringgenberg, M. C. and Zhou, G. (2016). Short
  interest and aggregate stock returns. *Journal of Financial
  Economics*, 121(1), 46-65.