While plotting the PACF of the sample, the PACF values become insignificant post the second lag, then significant again post the 8th lag and so on. Basically, there's cyclicality in the partial autocorrelation. How then should I determine the order of the AR(p) model?
This is probably due to untreated deterministic (NOT STOCHASTIC) effects like level shifts or local time trends in the original series and is not representative of a recurring autoprojective (arima) process. If you post your actual data I will try to help further. Please describe your data as that often is useful in the analysis.