ACF and PACF of residuals to determine ARIMA model I'm having trouble interpreting an ACF/PACF plot of the residuals of a regression to determine what the corresponding ARIMA model would be for the error term. This is the plot of the ACF/PACF of the regression. Since the ACF trails off at a lag of 4 and the PACF cuts off after a lag of 2, I believe it would be an ARIMA(4,0,2) model, but when I run the model the p-values are very low. When I run an ARIMA(4,0,0) model, the p values increase to a satisfactory amount. Am I interpreting these models correctly? 

 A: The thread Terms "cut off" and "tail off" about ACF, PACF functions on this forum will come in handy, Benjamin!
In looking at your plots, I see that the PACF cuts off after 2 lags and the ACF 'decays' towards zero. 
As per the above thread, that would suggest an AR(2) process for the residuals from your initial regression model. In general, the 'decay' in the ACF can look like what you have in your plot (i.e., exponential decay) or have some sort of sinusoidal flavour, as seen in What does my ACF graph tell me about my data?, for instance.
If you work in R, you could try to fit an ARIMA process to your residuals using the auto.arima function from the forecast package just to see how 'close' your guess (or mine) that the residuals follow an AR(p) process - where p = 4 in your guess or p = 2 in mine - would be to what auto.arima comes with after automated selection of a time series model from the ARIMA class for the model residuals. Just use something like this:
install.packages("forecast")
require(forecast)

auto.arima(residuals(initialreg))

and see what comes out. Another thing to keep in mind is the length of your time series of regression residuals - if that series is not too long, you'll know upfront you won't be able to fit an AR model with too many parameters to it. In particular, you might be able yo fit an AR(1) or AR(2) model, but not an AR(4) or AR(5). The shorter the series, the less complex the AR model that it can support. 
Of course, after you fit your AR(2) model to the regression residuals, you have to look at diagnostic plots of the AR(2) model residuals to make sure they look fine (i.e., like white noise). 
A: To review , auto.arima in a brute force list-based procedure that tries a fixed set of models and selects the calculated AIC based upon estimated parameters. The AIC should be calculated from residuals using models that control for intervention administration, otherwise the intervention effects are taken to be Gaussian noise, underestimating the actual model's autoregressive effect and thus miscalculating the model parameters which leads directly to an incorrect error sum of squares and ultimately an incorrect AIC and bad model identification. Most SE responders do not point out this assumption when they promote the auto.arima tool as they are unaware of this subtlety.
Modern/Correct/Advanced ARIMA time series analysis is conducted by identifying a starting model and then iterating to refine the initially suggested model as implied by @isabella-ghement and then carefully examining the residuals for the existence of structure BOTH arima AND deterministic structure like pulses, level.step shifts, seasonal pulses and.or local time trends.
As an example of a very bad model identification using auto.arima see https://www.omicsonline.org/open-access/an-implementation-of-the-mycielski-algorithm-as-a-predictor-in-r-2090-4541-1000195.php?aid=65324 .
Furthermore your model residuals may still have an impact from lags of your predictor. Thus is often the case when your combined regression model + ARIMA structure + Identified Interventions has insufficient lag X structures.
If you wish to post your data I will try and help further to give better guidance.
