Conditional vs. Exact Maximum Likelihood Estimation I am reading through the Hamilton book on Time Series Analysis. There are numerous references to estimation of models using both exact log-likelihood estimation and conditional estimation. 
From what I've gathered so far exact likelihood estimation has to be done with any number of numerical optimization techniques. However, the log of the conditional likelihood function can be estimated via OLS on a constant and $p$ of its own lagged values.
It seems to me that the conditional log likelihood estimation is computationally easier than the optimization method, and it only requires $t - p$ values of $y$ instead of $t$. However, since the exact likelihood is well...exact I would think it would be the preferred method, especially considering how fast optimization algorithms are on a modern computer.
Which should I prefer when implementing MLE for a time series model? The book doesn't exactly make this clear. It does however specify that the exact and conditional method of estimation have the same large-sample distribution making this question  more interesting.
 A: Even though optimization algorithms are fast on modern computers, the likelihood function of an ARMA model can be sufficiently complicated to make the computing time noticeable. Try fitting an ARMA(5,0) model on a 1000-long time series a thousand times, and you will find yourself bored waiting (takes 60 seconds on an i7-6600U Skylake processor or 97 seconds on a i5-2410M Sandy Bridge processor). Try it yourself:
set.seed(1); x=rnorm(1000)  
Sys.time(); for(i in 1:1000) m=arima(x,order=c(5,0,0),method="ML"); Sys.time()

When you need to run this for tens of thousands or millions of time series (instead of just a thousand), even supercomputers might sweat. 
The same estimations with CSS are some 8-9 times faster (7 seconds on i7 Skylake and 11 seconds on i5 Sandy Bridge). (Replace "ML" with "CSS" in the code above.)
If computational time is not a problem, you could prefer exact MLE to CSS for small or medium samples as the effect of the initial conditions are often noticeable (that can be simulated, too). In really large samples this might not matter much.
A: But why would you use arima() for a pure AR model? Use ar.burg() instead.
Burg's algorithm gives max entropy estimates equal to exact likelihood, right?
And no explicit optimization technique. Try it in your example to check speed.
