What is the Difference Between AR and MA Time Series Models?

AR stands for Auto-Regressive and MA stands for Moving Average.

Auto Regressive (AR)

It is a linear model where the current value is the sum of past values multiplied by a coefficient. AR model works well with stationary data.

As the AR model depends on past data, it needs time to adjust from past residuals. MA models come here for the rescue. Let’s see how moving average handles this issue.

Moving Average (MA)

Moving average models are the self-correcting models that take into account the past residual. They have the capacity to adjust past shocks easily as the predictions are corrected immediately following a big error.

For example, Random walk data is always adjusting from the errors of the previous period. Moving average models performs exceptionally well on random walk data.

AR Model Equation

y(t)= C + φy(t-1) + ε(t)

where,

• y(t-1) = value of x during the previous period.
• C = constant.
• φ = coefficient by which we multiply the previous period value, it can be interpreted as the part of the previous value that remains in the future. 
• The value of φ should always be between -1 and 1, else with time, it will blow up immeasurably.
• ε(t)= residual, the difference between the predicted value and the correct value at time t.

MA Model Equation

y(t)= C + θε(t-1) + ε(t)

where,

• y = time series variable.
• θ = numerical coefficient.
• ε(t-1) = residual for lag 1.
• ε(t) = residual for the current period.
• The value of θ should always be between -1 and 1, else with time, it will blow up immeasurably.

Comparing Equation of AR and MA Model

The equation of the AR and MA model looks very similar. The only difference is that the AR model uses the value of the variable whereas the MA model uses the value of the residual to calculate the current value.