Impact of residuals on forecast I'm working with ARMA models right now and I was wondering about the following case:
If we have late significant lags in the residuals ACF and the rest of the earlier residual lags weren't significant, does this imply that a one or two period forecast is less likely to be impacted by these residuals? Thank you for your help in advance!

In theory, it absolutely could be. Autocorrelated residuals (regardless at which lags) suggest autocorrelated errors. Most estimation techniques assume uncorrelated errors. When the assumption is violated, coefficient estimators will generally be inefficient while their standard errors will generally be biased. Forecasts will also be suboptimal due to (1) poor coefficient estimates and (2) neglected autocorrelation in errors which could be extrapolated to yield more accurate forecasts. This applies to both long and short forecast horizons alike.
However, at 95% confidence level about 5% of lags in ACF and PACF should be statistically significant even if all autocorrelations and partial autocorrelations are actually zero in population. Thus autocorrelated residuals do not guarantee autocorrelated errors, and your model and estimator might still be adequate.
Also, people tend to pay less attention to high-order lags in practice, as processes with high-order autocorrelations (in population) are relatively less widespread than processes with low order autocorrelations (in population). Roughly speaking, people tend to ascribe high-order lags to chance due to sampling variation while low-order lags to genuine patterns in population. But these are only coarse observations. In certain applications you may expect genuine high-order autocorrelations, and thus these heuristics need not apply to all situations.
