# Replicating Rapach & Zhou (2013)

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.

    library(forecastdom)
    data(rz2013)

## Helpers

    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.

    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"))

Predictor | \(R^2_{OS}\) (%) | CW stat | \(p\)-value
--- | --- | --- | ---
DP | 0.129 | 1.621 | 0.053
EP | -1.452 | 1.469 | 0.071
NTIS | -0.761 | 0.311 | 0.378
TBL | -0.043 | 1.308 | 0.095
INFL_lag | -0.086 | 0.047 | 0.481

## 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.