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 | (%) | CW stat | -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 — 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.