Lagged dependent variable or handling residual as AR process?

I have been confused with these two methodologies when doing regression.

Let's say I have a dependent variable (DV), which is auto-correlated. When I regress the dependent variable on a number of independent variables (IV), most likely, the residuals are auto-correlated as well.

After doing some search on the internet, my understanding is that, when such a situation emerges, roughly speaking, the methodologies to handle this fall into two categories:

1. Including lagged DV as independent variables

2. Treating the residuals as an AR process. It seems that there are a number of ways to do this:

a. Use Heteroskedasticity and Autocorrelation Consistent (HAC) robust standard errors while leaving the model specification unchanged.

b. Something like the Cochrane-Orcutt Procedure.

c. Using Maximum Likelihood Estimate assuming a ARIMA(p,d,q) structure for the residuals.

I would like to know if my above understanding is correct. If so, then how could one make a decision about which approach to use? Last, when using the first methodology, i.e., including lagged DV as independent variables, can Ordinary Least Squares (OLS) still be used to estimate the parameters and is OLS still consistent, unbiased, etc.?

• Relevant thread: stats.stackexchange.com/questions/110757/… – Jake Westfall Apr 15 '17 at 14:28
• 1 and 2c are similar when you write them out, but 2c is effectively a restricted form of 1 under some conditions (which I forgot), so then you can even test 1 against 2c using, e.g., a likelihood ratio test. OLS is fine for 1; the estimator is consistent, although biased. But most of the sensible estimators will be biased for 1 (if you want an unbiased estimator, you will get a large variance and thus a large expected squared error). – Richard Hardy Apr 15 '17 at 17:54
• Many thanks, but I don't quite understand why 1 and 2c are similar, since in 1 it is the DV that is lagged whereas in 2c it is the residual that is lagged. I did try 1 with OLS and 2c with ARIMA(1,0,0), but they gave quite different results. – user3692418 Apr 15 '17 at 21:48