This article applies gw_test(), the Giacomini and White
(2006) test of conditional equal predictive ability (CEPA), to the
Survey of Professional Forecasters’ mean CPI inflation forecasts. The
benchmark is a naive random walk in inflation (next quarter’s forecast
equals this quarter’s realised inflation). The alternative is the SPF
mean at horizons
quarters. The bundled gw2006 dataset extends GW’s quarterly
sample to the present using the same Philadelphia Fed SPF source.
library(forecastdom)
data(gw2006)
str(gw2006)
#> 'data.frame': 180 obs. of 10 variables:
#> $ date : Date, format: "1981-09-01" "1981-12-01" ...
#> $ year : int 1981 1981 1982 1982 1982 1982 1983 1983 1983 1983 ...
#> $ quarter : int 3 4 1 2 3 4 1 2 3 4 ...
#> $ infl : num 11.61 6.66 3.6 5.91 7.13 ...
#> $ infl_lag: num NA 11.61 6.66 3.6 5.91 ...
#> $ spf_h0 : num 7.71 10.73 6.36 3.54 5.33 ...
#> $ spf_h1 : num NA 9.22 8.95 5.51 3.69 ...
#> $ spf_h2 : num NA NA 7.93 7.76 6.08 ...
#> $ spf_h3 : num NA NA NA 7.76 7.56 ...
#> $ spf_h4 : num NA NA NA NA 7.61 ...CEPA across forecast horizons
run_h <- function(h) {
spf_col <- paste0("spf_h", h)
ok <- complete.cases(gw2006[, c("infl", spf_col, "infl_lag")])
d <- gw2006[ok, ]
e_spf <- d$infl - d[[spf_col]] # SPF errors
e_rw <- d$infl - d$infl_lag # random-walk errors
r <- gw_test(e_spf, e_rw) # default: constant + lagged loss diff
data.frame(h = h, n = nrow(d),
mse_spf = mean(e_spf^2),
mse_rw = mean(e_rw^2),
wald = unname(r$statistic),
pvalue = unname(r$pvalue),
reject = unname(r$pvalue) < 0.05)
}
tab <- do.call(rbind, lapply(0:4, run_h))
knitr::kable(
tab, digits = 3, row.names = FALSE,
col.names = c("$h$", "$n$",
"$MSE_{SPF}$", "$MSE_{RW}$",
"Wald", "$p$-value", "Reject"))| Wald | -value | Reject | ||||
|---|---|---|---|---|---|---|
| 0 | 176 | 5.657 | 5.565 | 4.510 | 0.105 | FALSE |
| 1 | 176 | 4.906 | 5.565 | 1.206 | 0.547 | FALSE |
| 2 | 175 | 4.631 | 5.456 | 4.177 | 0.124 | FALSE |
| 3 | 174 | 4.708 | 5.434 | 3.581 | 0.167 | FALSE |
| 4 | 173 | 4.884 | 5.435 | 2.908 | 0.234 | FALSE |
At every horizon SPF has a lower mean squared error than the random-walk benchmark, yet the GW test fails to reject conditional equal predictive ability at the 5% level. The pattern matches Atkeson and Ohanian (2001): once the autocorrelation of the loss differential is taken into account, sophisticated inflation forecasts are hard to tell apart from a “no-change” benchmark in a formal test.
Choice of instruments
gw_test() uses two instruments by default: a constant
and the lagged loss differential. Different instruments target different
features of the conditioning information set. Below we replace the lag
with the lagged inflation level (a regime indicator that captures high-
and low-inflation environments) at horizon
.
h <- 1L
ok <- complete.cases(gw2006[, c("infl", paste0("spf_h", h), "infl_lag")])
d <- gw2006[ok, ]
e_spf <- d$infl - d[[paste0("spf_h", h)]]
e_rw <- d$infl - d$infl_lag
# Default: constant + lagged loss differential
r_default <- gw_test(e_spf, e_rw)
# Custom: constant + lagged inflation level (padded with NA at t = 1
# so that nrow(W) matches length(e1)).
W_infl <- cbind(1, c(NA, head(d$infl_lag, -1)))
keep <- !is.na(W_infl[, 2])
r_infl <- gw_test(e_spf[keep], e_rw[keep],
instruments = W_infl[keep, , drop = FALSE])
# Absolute-error loss instead of squared-error
r_ae <- gw_test(e_spf, e_rw, loss = "AE")
tab2 <- data.frame(
spec = c("Default (const + lagged Δloss)",
"Const + lagged inflation",
"Absolute-error loss"),
wald = c(r_default$statistic, r_infl$statistic, r_ae$statistic),
df = c(r_default$df, r_infl$df, r_ae$df),
pvalue = c(r_default$pvalue, r_infl$pvalue, r_ae$pvalue)
)
knitr::kable(
tab2, digits = 3, row.names = FALSE,
col.names = c("Specification", "Wald", "df", "$p$-value"))| Specification | Wald | df | -value |
|---|---|---|---|
| Default (const + lagged Δloss) | 1.206 | 2 | 0.547 |
| Const + lagged inflation | 10.738 | 2 | 0.005 |
| Absolute-error loss | 1.957 | 2 | 0.376 |
The decision is robust across all three specifications: SPF and the random walk cannot be statistically distinguished at the 5% level regardless of which conditioning instruments or loss function we choose.
Note on the original paper
Giacomini and White (2006, Section 4) compare SPF nowcasts to the Greenbook (the Federal Reserve staff’s internal inflation forecast) using a richer instrument set. They find that the test does reject equal conditional predictive ability between the two sophisticated forecasts in some conditioning states. Replicating that result needs Greenbook data with its five-year embargo. This article instead demonstrates the test mechanics against the simpler random-walk benchmark, which is freely available.