**ARMA(p,q)** model in time series forecasting is a combination of Autoregressive Process also known as AR Process and Moving Average (MA) Process where **p** corresponds to the autoregressive part and **q **corresponds to the moving average part.

**Autoregressive Process (AR) :- **When the value of Y_{t} in a time series data is regressed over its own past value then it is called an autoregressive process where p is the order of lag into consideration.

Where,

Y_{t} = observation which we need to find out.

α_{1}= parameter of an autoregressive model

Y_{t-1}= observation in the previous period

u_{t}= error term

The equation above follows the first order of autoregressive process or AR(1) and the value of p is 1. Hence the value of Y_{t} in the period ‘t’ depends upon its previous year value and a random term.

**Moving Average (MA) Process :- **When the value of Y_{t} of order q in a time series data depends on the weighted sum of current and the q recent errors i.e. a linear combination of error terms then it is called a moving average process which can be written as :-

y_{t} = observation which we need to find out

α= constant term

β_{ut-q}= error over the period q .

**ARMA (Autoregressive Moving Average) Process :-**

The above equation shows that value of Y in time period ‘t’ can be derived by taking into consideration the order of lag p which in the above case is 1 i.e. previous year’s observation and the weighted average of the error term over a period of time q which in case of the above equation is 1.

**How to decide the value of p and q?**

Two of the most important methods to obtain the best possible values of p and q are ACF and PACF plots.

**ACF (Auto-correlation function) :- **This function calculates the auto-correlation of the complete data on the basis of lagged values which when plotted helps us choose the value of q that is to be considered to find the value of Y_{t}. In simple words how many years residual can help us predict the value of Y_{t} can obtained with the help of ACF, if the value of correlation is above a certain point then that amount of lagged values can be used to predict Y_{t}.

Using the stock price of tesla between the years 2012 and 2017 we can use the .acf() method in python to obtain the value of p.

.DataReader() method is used to extract the data from web.

The above graph shows that beyond the lag 350 the correlation moved towards 0 and then negative.

**PACF (Partial auto-correlation function) :- **Pacf helps find the direct effect of the past lag by removing the residual effect of the lags in between. Pacf helps in obtaining the value of AR where as acf helps in obtaining the value of MA i.e. q. Both the methods together can be use find the optimum value of p and q in a time series data set.

Lets check out how to apply pacf in python.

As you can see in the above graph after the second lag the line moved within the confidence band therefore the value of p will be 2.

So, with that we come to the end of the discussion on the ARMA Model. Hopefully it helped you understand the topic, for more information you can also watch the video tutorial attached down this blog. **The blog is designed and prepared by Niharika Rai, Analytics Consultant, DexLab Analytics** DexLab Analytics offers **machine learning courses in Gurgaon**. To keep on learning more, follow DexLab Analytics blog.

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