---
title: "Replicating Rapach & Zhou (2013)"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Replicating Rapach & Zhou (2013)}
  %\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 `rz2013` bundled dataset:
the 14 Goyal-Welch macroeconomic predictors used in Rapach and
Zhou (2013) to forecast the U.S. equity premium. The benchmark is
the prevailing historical mean and the alternative is a bivariate
predictive regression on a single macro variable.

```{r setup, message = FALSE}
library(forecastdom)
data(rz2013)
```

## 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))

  }))

}
```

## Macro predictors

Initial estimation window of 241 months (1926-12 to 1946-12);
out-of-sample period 1947-01 to 2010-12.

```{r rz}
preds_rz <- c("DP", "EP", "NTIS", "TBL", "INFL_lag")

knitr::kable(
  run_cw(rz2013, preds_rz, R = 241), digits = 3, row.names = FALSE,
  col.names = c("Predictor", "$R^2_{OS}$ (%)", "CW stat", "$p$-value"))
```

## Takeaway

Out-of-sample gains over the historical mean are economically small across the Goyal-Welch macro predictors. This is the well-known Goyal-Welch puzzle that motivated Rapach and Zhou (2013).

## 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.
- Rapach, D. E. and Zhou, G. (2013). Forecasting stock returns.
  In G. Elliott and A. Timmermann (Eds.), *Handbook of Economic
  Forecasting*, Vol. 2A, pp. 328-383. Elsevier.
