I have a quadratic regression model with three statistically significant coefficients: intercept, time, and time^2. The residuals of the model show significant autocorrelation. I understand that you can find an ARMA process for the residuals simply by using the ar or arima function and getting the coefficients [i.e. res.ar=ar(resid(fit),method='mle')]… but how do you refit the regression model with the autocorrelated residuals (for purposes in forecasting)? I'm thinking of terms of auto.arima
Solved – How to fit an ARMA process to residuals in R
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The Actual , Fit and Forecast 
suggest a Mean Model where the forecast would be based upon the following equation 
. Note that there are 12 anomalous data points yielding a robust mean of 1.0378 . The visual definitely supports the unusual data points. Good time series analysis detecting the underlying signal ( a mean model ) while also detecting any exceptional data points rendering a cleansed/robust mean of 1.0378 as compared to a simple mean of 1.26 . Now that the anomalous points have been identified , one needs to ask what they have in common as a possible explanatory variable. Additionally the ACF of the errors from this model indicate randomness. Furthermore there is no evidence that the expected value is systematic with the error variance or error standard deviation suggesting that a power transform is not warranted. Finally there is no evidence of a structural shift in the robust mean over time suggesting that the parameter(s) of the model are invariant.
(1) Have you correctly fitted an ARIMA with some coefficients forced to zero? - Yes. But did you heed the Warnmeldung? If you didn't, & want to make sure your model is causal & invertible, check the roots of the AR & MA polynomials (abs(polyroot(your.polynomial)) is convenient in R).
(2) Is your model-building approach sensible? - I don't think it's the most sensible. Setting various coefficients to zero just because they're "not significant" is no more principled in ARIMA than in multiple regression - see the 'model selection' & 'variable selection' tags - & can lead to the same sorts of problems. The estimate for one parameter depends on all the others that are in the model, so by removing a whole bunch at once you can badly degrade the model. Stepwise selection is a little better, but after doing lots of tests you can't really justify the result overall.
Approaches vary, & depend on the goals of modelling, but I would suggest confining your attention to a smallish set of plausible models (is it really likely that an observation's affected by what happened four & five observations ago but by nothing in the intervening time?), picking a model using AIC, & validating it by out-of-sample forecasts.
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Best Answer
You are correct that it makes sense to estimate the "main part" of the model simultaneously allowing for the error term to have an ARMA structure (as opposed to doing this in two steps, which would be less efficient). Simply add the main regressors (
timeandtime^2) via the argumentxregin the functionarimaorauto.arima. It will estimate a regression with ARMA errors. You can read more about this and related techniques in an enlightening blog post "The ARIMAX model muddle" by Rob J. Hyndman.