I am currently estimating a cointegrating regression (DOLS), where my residuals have autocorrelation. Sometimes it is just in one or two lags, but sometimes it is more. My question is: Can I apply HAC and carry on with my estimation? I.e. would HAC standard errors compensate for the autocorrelation?
The motivation for including leads/lags within a DOLS is to control for bias in small samples. Actually, to be more specific the lags\leads are included to ensure that the OLS estimates are normally distributed. More importantly, lag/leads are not related to correcting for autocorrelation. These issue are first discussed in Banerjee 1986 et al, who shows that bias in introduced as a results of the strict exogeneity assumption not holding. Saikkonen 1991 is the first to suggest using lag/lead difference to control for bias.
More importantly for your application is that it is actually essential you use HAC variance. As a matter of fact the seminal paper on DOLS, Stock and Watson (1993), discusses this in-depth. Read these Zivot Notes, since they contain the essential information. For a application with Newey-West errors read this.