An infinite-order moving average process, denoted MA(∞), takes the form
where the following infinite series is finite (i.e. converges to a real value)
We can express a MA(∞) process as
where it is assumed that ψ0 = 1.
Observation: That |ψj|converges ensures that the yi take finite values and that converges.
Example 1: Show that the AR(1) process from Example 1 of Autoregressive Processes Basic Concepts can be represented by an MA(∞) process.
By Property 1 of Autoregressive Processes Basic Concepts
Then the original AR(1) process can be transformed into the process
which results in
Continuing in this way, we get
is the desired MA(∞) process.
Property 1: Any stationary AR(1) process can be expressed as an MA(∞) process. In fact
Proof: Using the same approach as in Example 1, we find that the AR(1) process
Since the original process is a stationary AR(1), |φ1| < 1 and the εi have the desired properties.
Observation: Another way to see this is to use the lag operator, namely that an AR(1) process (with zero mean) can be expressed as
Substituting the first equation inside the second, we get
Here we recall that φ0 = 1. Equating the coefficients, we see that for all j > 0
Thus
and so ψ(L) is the inverse of φ(L)
Property 2: Any stationary AR(p) process can be expressed as an MA(∞) process.
Proof: The proof is similar to that of Property 1.
Example 2: Show that the following AR(2) process can be represented by an MA(∞) process.
By Property 1 of Autoregressive Processes Basic Concepts, the mean is
Then the original AR(2) process can be transformed into the process
etc. Thus the first few terms of the MA(∞) process are
Property 3 (Wold’s Decomposition Theorem): Any stationary process can be represented as a MA(∞) process
Property 4: The following are true for any MA(∞) process
Proof: See Moving Average Proofs
Real Statistics Function: The Real Statistics Resource Pack provides the following array function where R1 is a column range consisting of phi coefficients and R2 is a column range consisting of theta coefficients.
PSICoeff(R1, R2, k, rev): returns a k × 1 range containing the first k psi coefficients (starting with ψ0 = 1) for the ARMA model with the coefficients in R1 and R2.
If k is omitted (default) then k is set equal to the number of rows in the highlighted range. If rev = TRUE (default), then the phi and theta coefficients are listed in reverse order φp, φp-1, …, φ1 and order θq, θq-1, …, θ1.
Since both phi and theta coefficients can be present, this function can also handle ARMA processes as described in ARMA Processes.
We can use the PSICoeff function to find the psi coefficients for Example 2 as shown in range I10:I14 of Figure 1.
Figure 1 – Convert AR(2) model into an MA(∞) model
How can I calculate the covariance and autocorrelation of infinite moving average process
Ligawa,
Sorry, but I don’t know the answer to your question.
Charles
if y(t)=e(t)+5 e(t-1)+10e(t-2), e(t)~N(0,16), what’s the inverse of that MA model?
Hello Assma,
See the following webpage about the inverse of an MA(q) model.
https://www.real-statistics.com/time-series-analysis/moving-average-processes/invertibility-ma-processes/
Charles
Very resourceful and useful to me. Thank you