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Replicating Clark & McCracken (2001)

Source: vignettes/cm2001-enc-new.Rmd
cm2001-enc-new.Rmd

This article reproduces the Phillips-curve forecasting illustration of Clark and McCracken (2001, Section 5) on the bundled cm2001 dataset. The benchmark forecasts US unemployment from its own lags (univariate AR). The alternative adds lagged inflation (ARX). enc_new() tests whether the AR forecasts encompass those of the larger ARX model, i.e. whether lagged inflation adds any predictive content beyond what lagged unemployment already contains.

library(forecastdom)
data(cm2001)

# CM (2001) sample: 1957:01-1997:08.
cm <- subset(cm2001,
             date >= as.Date("1957-01-01") &
             date <= as.Date("1997-08-01"))

nrow(cm)
#> [1] 488

Recursive forecasts

Both models are estimated by OLS on an expanding window with p lags of each variable. At each origin we form a one-step-ahead forecast, then roll forward by one month.

make_lags <- function(z, p) {

  n <- length(z)
  sapply(seq_len(p), function(k) c(rep(NA, k), z[seq_len(n - k)]))

}

recursive_arx <- function(y, x, p, R) {

  n <- length(y)
  YL <- make_lags(y, p)
  XL <- make_lags(x, p)
  target <- y[(p + 1):n]
  YL <- YL[(p + 1):n, , drop = FALSE]
  XL <- XL[(p + 1):n, , drop = FALSE]
  P <- length(target) - R
  
  e_ar <- e_arx <- numeric(P)
  
  for (j in seq_len(P)) {

    idx     <- seq_len(R + j - 1)
    fit_ar  <- lm.fit(cbind(1, YL[idx, ]),         target[idx])
    fit_arx <- lm.fit(cbind(1, YL[idx, ], XL[idx, ]), target[idx])
    pred_ar  <- sum(coef(fit_ar)  * c(1, YL[R + j, ]))
    pred_arx <- sum(coef(fit_arx) * c(1, YL[R + j, ], XL[R + j, ]))
    e_ar[j]  <- target[R + j] - pred_ar
    e_arx[j] <- target[R + j] - pred_arx

  }
  
  list(e_ar = e_ar, e_arx = e_arx)

}

ENC-NEW across lag orders

CM (2001) consider lag orders p = 1, 3, 6, 12. The initial estimation window is R = 120 months (10 years); the remaining observations form the out-of-sample period.

R <- 120L

rows <- lapply(c(1L, 3L, 6L, 12L), function(p) {

  fc  <- recursive_arx(cm$unrate, cm$infl, p = p, R = R)
  enc <- enc_new(fc$e_ar, fc$e_arx)
  msfe1 <- mean(fc$e_ar ^ 2)
  msfe2 <- mean(fc$e_arx ^ 2)
  data.frame(p        = p,
             n_oos    = length(fc$e_ar),
             pi_ratio = round(length(fc$e_ar) / R, 2),
             MSFE_AR  = msfe1,
             MSFE_ARX = msfe2,
             R2OS_pct = 100 * (1 - msfe2 / msfe1),
             ENC_NEW  = unname(enc$statistic))
})

tab <- do.call(rbind, rows)

knitr::kable(
  tab, digits = 3, row.names = FALSE, format = "html",
  table.attr = "style='width:auto;'", escape = FALSE,
  col.names = c("\\(p\\)", "\\(T_{OOS}\\)", "\\(\\pi = P/R\\)",
                "\\(\\mathrm{MSFE}_{AR}\\)", "\\(\\mathrm{MSFE}_{ARX}\\)",
                "\\(R^2_{OS}\\) (%)", "ENC-NEW"))
pp TOOST_{OOS} π=P/R\pi = P/R MSFEAR\mathrm{MSFE}_{AR} MSFEARX\mathrm{MSFE}_{ARX} ROS2R^2_{OS} (%) ENC-NEW
1 367 3.06 0.035 0.034 2.968 13.930
3 365 3.04 0.033 0.032 2.184 11.412
6 362 3.02 0.031 0.031 1.833 14.754
12 356 2.97 0.032 0.033 -2.459 20.984

At every lag order the ARX model achieves a lower MSFE than the pure AR. That is, knowing past inflation reduces out-of-sample unemployment-forecast error. The ENC-NEW values are large relative to the Clark-McCracken (2001) Table 2 asymptotic 5% critical values, which are tabulated in k2k_2 (number of extra regressors, equal to pp) and π=P/R\pi = P/R. For k2=6k_2 = 6 and π≈2\pi \approx 2 the 5% critical value is about 8.6; for k2=12k_2 = 12 and π≈2\pi \approx 2 it is about 12.0. Our statistics exceed these benchmarks decisively at every specification, so the AR forecasts do not encompass the ARX: lagged inflation carries real predictive content for unemployment.

Comparison with Clark-West MSFE-adjusted

enc_new() answers the encompassing question. cw_test() answers the closely related “equal MSFE” question using a tt-statistic that is approximately N(0,1)N(0, 1) under the null. The two tests typically point the same way, and reporting both is informative.

rows2 <- lapply(c(1L, 3L, 6L, 12L), function(p) {

  fc <- recursive_arx(cm$unrate, cm$infl, p = p, R = R)
  f1 <- cm$unrate[(p + 1 + R):nrow(cm)] - fc$e_ar
  f2 <- cm$unrate[(p + 1 + R):nrow(cm)] - fc$e_arx
  cw <- cw_test(fc$e_ar, fc$e_arx, f1, f2)

  data.frame(p         = p,
             CW_stat   = unname(cw$statistic),
             CW_pvalue = unname(cw$pvalue))
             
})

knitr::kable(
  do.call(rbind, rows2), digits = 3, row.names = FALSE,
  col.names = c("$p$", "CW stat", "CW $p$-value"))
pp CW stat CW pp-value
1 2.531 0.006
3 2.295 0.011
6 2.522 0.006
12 2.489 0.006

CW p-values are far below 5% for every lag order, agreeing with the ENC-NEW decision: the ARX model has significantly lower out-of-sample MSFE than the pure AR.

References

  • Clark, T. E. and McCracken, M. W. (2001). Tests of equal forecast accuracy and encompassing for nested models. Journal of Econometrics, 105(1), 85-110.
  • 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.