Model identification in time series - ARMA (p,q)

ARMA (p,q) model is commonly used in time series analysis. Therefore it is essential to be able to identify the appropriate ARMA (p,q) model for the particular series. Employing correlogram and partial correlogram to help in the identification process is common. Basically, correlogram and partial correlogram are plot diagrams of the autocorrelation function (ACF) and partial autocorrelation function (PACF) respectively.

The auto-correlation function (ACF) can be defined as a set of correlation coefficients between the series and lags of itself over time. While the definition of partial auto-correlation function (PACF) is the partial correlation coefficients between the series and lags of itself over time. 

1. Autoregressive (AR) model
The AR model is a model which includes lagged terms of the time series itself.
We conclude that the series is AR process if

• the PACF displays a sharp cutoff
• while the ACF decays more slowly (i.e., has significant spikes at higher lags) or oscillates (exponentially decays).

Moreover, the lag at which the PACF cuts off is the indicated number of AR terms.

2. Moving Average (MA) model
The MA model is a model which includes lagged terms on the noise or residuals.
The patterns for MA process as follows

• the ACF of the differenced series displays a sharp cutoff
• while the decays slowly or oscillates (exponentially decays).

The lag at which the ACF cuts off is the indicated number of MA terms

3. ARMA model
This is just a combination of MA and AR terms. Therefore the pattern shows combination of AR and MA process. 

Thus, we may head on to following table.