MOVING AVERAGE PROCESS

It is a process in function of time, and expressible through the summation of the linear combination of independent identically distributed, with zero mean and variance σ2 random variables

                                                                              

In a first-order moving-average process written as MA(1), the current value of the random variable rt is determined with the following relationship:

                                                            

where φ is the weight coefficient of the previous value ε. In this process we know the mean, variance and auto-covariance, in fact, respectively, we find:

                                              

                 

                   

Note that the last except at s=1, is always zero, because the ε are independent identically distributed with zero mean and variance σ2 random variables. So in a MA(1) we find:

                           

and consequently the autocorrelation function for a MA(1):

                                               

Therefore MA(1) process, but also generic process MA(q), have not time-dependent variance and auto covariance. This aspect implies that the process is stationary, regardless of the value of φ.

 

Editor: Giuliano DI TOMMASO