MA(1) process

Summary

• A stochastic process \(\{ Y_t \}\) is called an moving average process of order 1 or an MA(1) process if

\[Y_t=μ+ε_t+αε_{t-1}\]

• where \(μ\) and \(α\) are parameters and \(\{ ε_t \}\) is white noise.
• An MA(1) process is always stationarity .
• Result: The expected value of an MA(1) process \(Y_t=μ+ε_t+αε_{t-1}\) :

\[E\left( Y_t \right)=μ\]

• If \(Y_t\) follows an MA(1) process, \(Y_t=μ+ε_t+αε_{t-1}\) , then \(y_t=Y_t-E\left( Y_t \right)\) will follow an MA(1) process with intercept 0

\[y_t=ε_t+αε_{t-1}\]

• This is notationally much more convenient. We have

\[E\left( y_t \right)=0\]

• Result: The variance of an MA(1) process \(y_t=ε_t+αε_{t-1}\) :

\[Var\left( y_t \right)=(1+α^2)σ^2\]

• Result: The covariances of an MA(1) process \(y_t=ε_t+αε_{t-1}\) :

\[Cov\left( y_t,y_{t-1} \right)=ασ^2\]

• and

\[Cov\left( y_t,y_{t-k} \right)=0\]

• if \(k>1\) .
• Result: The correlations (or autocorrelations) of an MA(1) process \(y_t=ε_t+αε_{t-1}\) :

\[corr\left( y_t,y_{t-1} \right)= \frac{α}{1+α^2}\]

• \(corr\left( y_t,y_{t-k} \right)=0\) if \(k>1\) .