Infinite Moving Average Processes

An infinite-order moving average process, denoted MA(∞), takes the form

image154z where the following infinite series is finite (i.e. converges to a real value)

image155z

andimage137z

We can express a MA(∞) process as image156z

where it is assumed that ψ0 = 1.

Observation: That \sum_{j=0}^\infty|ψj|converges ensures that the yi take finite values and that \sum_{j=0}^\infty \psi_j^2 converges.

Example 1: Show that the AR(1) process from Example 1 of Autoregressive Processes Basic Concepts can be represented by an MA(∞) process.

image062z

By Property 1 of Autoregressive Processes Basic Concepts

image159z

Now defineimage160z

Then the original AR(1) process can be transformed into the process

image161z

which isimage162z

But thenimage163z

and soimage164z

which means thatimage165z

Similarlyimage166z

which results in

image167z

Continuing in this way, we get

image168z

and soimage169z

is the desired MA(∞) process.

Property 1: Any stationary AR(1) process can be expressed as an MA(∞) process. In fact

image170zProof: Using the same approach as in Example 1, we find that the AR(1) process

image049z

can be expressed asimage171z

whereimage172z

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

image124z

whereimage173z

as well asimage174z

whereimage175z

Substituting the first equation inside the second, we get

image176z

i.e.image177z

image178zHere we recall that φ0 = 1. Equating the coefficients, we see that for all j > 0

image179z

Thus

image180z

image181zetc.

and so we thatimage182z

It follows thatimage183z

We also observed above thatimage184z

and so ψ(L) is the inverse of φ(L)image185z

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.

image001c By Property 1 of Autoregressive Processes Basic Concepts, the mean is

Now defineimage003c

Then the original AR(2) process can be transformed into the process

image004c But thenimage005cimage006cimage007cand so

image008cimage009c

image010c

etc. Thus the first few terms of the MA(∞) process are

image011c

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

image012cimage013cimage014cimage015c

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.

Psi coefficients AR model

Figure 1 – Convert AR(2) model into an MA(∞) model

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