# Diebold-Mariano test: robust standard errors for 1-day ahead forecasts?

Diebold & Mariano suggest using Newey-West standard errors to correct for autocorrelation and heteroscedasticity in the error terms when comparing forecast accuracy.

However, if I understand correctly, they also assume dependency only for (k-1) day ahead forecast errors.

As such, am I correct in assuming that I do not need to use Newey-West standard errors when looking at 1-day ahead forecasts?

• The implementation code of the test in R package "forecast" function dm.test is quite easy to read; there you see that the variance estimation accounts for autocorrelation up to $h-1$. Aug 12 '16 at 9:45
• I was going through my old answers and noticed this one was not accepted. Do you perhaps need further clarification? Mar 17 '17 at 11:47

Autocorrelation
$h$-step-ahead forecast errors for $h>1$ are autocorrelated by construction, because the forecasts cover overlapping time periods. This holds even for "optimal" forecasts. Meanwhile, 1-step-ahead forecast errors need not be autocorrelated, at least for "optimal" forecasts.

However, they still might be autocorrelated for "suboptimal" forecasts. E.g. if the forecast for $t+1$ is just the actual value as of $t-k$ for some $k>0$, you effectively get the same overlap and thus autocorrelation in forecast errors as with $h$-step-ahead forecasts. Of course, this need not be a sensible forecast, but it serves as an example.

Using autocorrelation-robust standard errors for 1-step-ahead forecasts in the Diebold-Mariano test is thus a conservative, "safe" choice.

Heteroskedasticity
Regarding heteroskedasticity, neither Diebold & Mariano (1995) nor Diebold (2015) mentions it, so probably there is nothing special about it in the context of the Diebold-Mariano test. If there is reason to believe that the forecast error distribution had a varying variance in your sample, you could use the heteroskedasticity-robust standard errors.

References:

• Now I started wondering myself what one should do with "suboptimal" forecasts. Does the fact that the 1-step-ahead forecast errors are autocorrelated effectively reduce the sample (as with optimal $h$-step-ahead forecasts) so that we have to use autocorrelation-robust variance estimator? Or do the errors carry independent information nevertheless and regular variance estimator should be used? Aug 12 '16 at 9:50