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?
 A: 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:


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*Diebold, Francis X., and Roberto S. Mariano. "Comparing Predictive Accuracy." Journal of Business & Economic Statistics 13.3 (1995): 253-263.

*Diebold, Francis X. "Comparing predictive accuracy, twenty years later: A personal perspective on the use and abuse of Diebold–Mariano tests." Journal of Business & Economic Statistics 33.1 (2015): 1-1.

