---
title: "Replicating Neely et al. (2014)"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Replicating Neely et al. (2014)}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

This article applies `cw_test()` to the `nrtz2014` bundled dataset:
the 14 binary technical indicators used in Neely, Rapach, Tu and
Zhou (2014). The benchmark is the prevailing historical mean and
the alternative is a bivariate predictive regression on a single
technical indicator.

```{r setup, message = FALSE}
library(forecastdom)
data(nrtz2014)
```

## Helpers

```{r helper}
recursive_forecasts <- function(y, x, R) {

  P <- length(y) - R
  e1 <- e2 <- f1 <- f2 <- numeric(P)
  
  for (j in seq_len(P)) {

    ty <- y[1:(R + j - 1)]
    tx <- x[1:(R + j - 1)]
    f1[j] <- mean(ty)
    fit   <- lm(yt ~ xlag, data = data.frame(yt = ty[-1], xlag = tx[-length(tx)]))
    f2[j] <- as.numeric(predict(fit, newdata = data.frame(xlag = tx[length(tx)])))
    e1[j] <- y[R + j] - f1[j]
    e2[j] <- y[R + j] - f2[j]

  }

  list(e1 = e1, e2 = e2, f1 = f1, f2 = f2)

}

run_cw <- function(data, predictors, R) {

  do.call(rbind, lapply(predictors, function(p) {

    fc  <- recursive_forecasts(data$eq_prem, data[[p]], R = R)
    res <- cw_test(fc$e1, fc$e2, fc$f1, fc$f2)

    data.frame(predictor = p,
               R2OS_pct  = unname(res$r2os),
               CW_stat   = unname(res$statistic),
               p_value   = unname(res$pvalue))

  }))

}
```

## Technical indicators

Initial window of 181 months (1950-12 to 1965-12); out-of-sample
period 1966-01 to 2011-12.

```{r nrtz}
preds_nr <- c("MA_1_9", "MA_2_12", "MOM_9", "MOM_12", "VOL_2_12")

knitr::kable(
  run_cw(nrtz2014, preds_nr, R = 181), digits = 3, row.names = FALSE,
  col.names = c("Predictor", "$R^2_{OS}$ (%)", "CW stat", "$p$-value"))
```

## Takeaway

Technical indicators deliver positive $R^2_{OS}$ with Clark-West statistics that are significant at conventional levels. This is the central finding of Neely et al. (2014): technical signals carry information beyond what macro variables provide.

## References

- Clark, T. E. and West, K. D. (2007). Approximately normal tests
  for equal predictive accuracy in nested models. *Journal of
  Econometrics*, 138(1), 291-311.
- Neely, C. J., Rapach, D. E., Tu, J. and Zhou, G. (2014).
  Forecasting the equity risk premium: The role of technical
  indicators. *Management Science*, 60(7), 1772-1791.