Decide p, q values based on acf and pacf graphs and identify parameters of SARIMA based on the graphs

Question

I need to know the way how to calculate/decide the p and q value for ARIMA model based on the acf and the pacf graph. Kindly help

Answer 1

ACF (Autocorrelation Factor)

It is the correlation between the observations at the current time spot and observations at the previous time spots.

PACF (Partial Auto-correlation Factor)

The correlation between the observations at two time spots given that we consider both observations are correlated to the observations at the other time spots. For example, today’s stock price can be correlated to the day before yesterday, and yesterday can also be correlated to the day before yesterday. Then, PACF of the yesterday is the real correlation between today and yesterday after taking out the influence of the day before yesterday.

Procedure for determining ACF and PACF

· De-trending the data

The foremost step which we need to perform is to identify whether a presence of trend is visible in the data and if so, we need to detrend the data for the smooth calculations. We discussed about the various methods to eliminate trend from the data in Part-2 of the article. Usually, one-lag differencing is used here.

· Identifying the significant terms

Use PACF to determine the significant terms used in AR model. The numbers of terms determine the order of the model. For example, if the PACF of yesterday’s stock price is significant and all PACF of all other days are not significant. Then yesterday’s stock price will be used to predict today’s stock price. This type of AR model is called first order AR model. It is represented as AR (1).

Similar procedure can be used for identifying significant terms for MA model. We will be using ACF to determine the significant terms used in MA model.

· Choosing which model to be used

We should initially calculate the significant terms for both AR and MA using PACF and ACF respectively. Then the next step is to determine which one have more simpler terms that could work well for the implementation.

According to the above diagram,

· Number significant terms in ACF = 6

· Number significant terms in PACF = 8

Obviously we are going to use MA in this model since ACF < PACF. ACF = 6 signifies that if we are using MA model, we should use observations of 6 previous time spots which means MA (6). PACF = 8 signifies that if we are using AR model, we should use observations of 8 previous time spots which means AR (8). The minimal order out of AR and MA is chosen in order to reduce the complexity of the model. We would have chosen AR instead of MA if the order of PACF is less than ACF. ARIMA Algorithm ARIMA stands for Auto-regressive integrated moving average. It is nothing but the integration of both AR and MA in order to produce more sophisticated and accurate model. In ARIMA,”I” stands for integrated. It represents differencing used to handle non-stationary data.

For the above shown diagram,

If we took 1 level differencing to detrend the data, the integration factor will be 1.Then we can represent the model combining both AR and MA as ARIMA (8, 1, 6). If we took 2 level differencing to detrend the data, the integration factor will be 2.Then we can represent the model combining both AR and MA as ARIMA (8, 2, 6).

Mathematically, It is represented as ARIMA(p,d,q)

Here,

p = number of significant terms in PACF for trend

d = Order of differencing for trend

q= number of significant terms in ACF for trend

You can refer my article for more details here - URL