Infinite Moving Average Processes

Basic Concepts

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

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

and

ε_i ∼ N(0, σ²)          cov(ε_i, ε_j) = 0 if i ≠ j

We can also express an MA(∞) process as

where it is assumed that ψ₀ = 1.

Observation: That |ψj|converges ensures that the y_i take finite values and that  converges.

AR(1) Process

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

Now define

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

which is

But then

and so

which means that

Similarly

which results in

Continuing in this way, we get

and so

which 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

can be expressed as

where

Since the original process is a stationary AR(1), |φ₁| < 1 and the ε_i 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

where

as well as

where

Substituting the first equation inside the second, we get

i.e.

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

Thus

etc.

and so we that

It follows that

We also observed above that

and so ψ(L) is the inverse of φ(L)

AR(p) and MA(∞) equivalence

Property 2: Any stationary AR(p) process can be expressed as an MA(∞) process.

Proof: The proof is similar to that of Property 1.

AR(2) Example

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

Now define

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

But thenand so

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

Key Properties

Property 3 (Wold’s Decomposition Theorem): Any stationary process can be represented as an MA(∞) process

Property 4: The following are true for any MA(∞) process

Proof: See Moving Average Proofs

Worksheet Function

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 φ_p, φ_p-1, …, φ₁ and order θ_q, θ_q-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

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

References

Greene, W. H. (2002) Econometric analysis. 5th Ed. Prentice-Hall
https://www.scirp.org/(S(351jmbntvnsjt1aadkposzje))/reference/referencespapers.aspx?referenceid=1243286

Gujarati, D. & Porter, D. (2009) Basic econometrics. 5th Ed. McGraw Hill
http://www.uop.edu.pk/ocontents/gujarati_book.pdf

Hamilton, J. D. (1994) Time series analysis. Princeton University Press
https://press.princeton.edu/books/hardcover/9780691042893/time-series-analysis

Wooldridge, J. M. (2009) Introductory econometrics, a modern approach. 5th Ed. South-Western, Cegage Learning
https://cbpbu.ac.in/userfiles/file/2020/STUDY_MAT/ECO/2.pdf

Wei, W. (2006) Time series analysis: univariate and multivariate methods, 2nd edition. Pearson Addison Wesley https://www.researchgate.net/publication/236651810_Time_Series_Analysis_Univariate_and_Multivariate_Methods_2nd_edition_2006