MA(q) Process Basic Concepts

Introduction

A q-order moving average process, denoted MA(q), takes the form

Thinking of the subscripts i as representing time, we see that the value of y at time i is a linear function of past errors. We assume that the error terms are independently distributed with a normal distribution with mean zero and a constant variance σ². Thus

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

An MA(q) process can also be expressed as

where z_i = y_i – μ. Thus, we can often simplify our analyses by restricting ourselves to the case where the mean is zero.

Using the lag operator, we can express a zero mean MA(q) process as

where

Mean and variance

Property 1: The mean of an MA(q) process is μ.

Property 2: The variance of an MA(q) process is

Autocorrelation

Property 3: The autocorrelation function of an MA(1) process is

Property 4: The autocorrelation function of an MA(2) process is

We now generalize these properties to all MA(q).

Property 5: The autocorrelation function of an MA(q) process is

for h ≤ q and ρ_h = 0 for h > q

Observation: The proofs of Property 1 – 5 are given in Moving Average Proofs.

Property 6: The PACF of an MA(1) process is

where 1 ≤ j < n.

If the process is invertible (see Invertible MA(q) Processes) then

MA(1) Example

Example 1: Simulate a sample of size 199 from the MA(1) process y_i = 4 + ε_i + .5ε_i-1 where ε_i ∼ N(0,2).

Thus μ = 4, θ₁ = .5 and σ = 2. We simulate the independent ε_i by using the Excel formula =NORM.INV(RAND(),0,2) in column B of Figure 1 (only the first 20 of 199 values is shown). The y_i values are calculated by placing the formula =4+B5+0.5*B4 in cell C5, highlighting the range C5:C203 and pressing Ctrl-D. The graph of the y values is shown on the right side of Figure 1. As you can see, no particular pattern is visible.

Figure 1 – Simulated MA(1) data

By Properties 1 and 2, the theoretical values for the mean and variance are μ = 4 and var(y_i) = σ²(1+) = 2²(1+.5²) = 5. These compare to the actual time series values of y̅ = AVERAGE(C6:C204) = 4.358 and s² = VAR.P(C6:C204) = 4.401.

The ACF values are shown for lags 1 through 15 in Figure 2. These are calculated from the y values as in Example 1 of AR(p) Process Basic Concepts. Note that the ACF value at lag 1 is .301285. Based on Property 3, the population ACF value at lag 1 is

MA(1) ACF

The ACF values for lags h > 1 vary from about -.18 to .16, compared to the theoretical value of ρ_h = 0 (per Property 3). As you can see from Figure 2, the sample values can be quite different from the theoretical values.

Figure 2 – ACF for MA(1) process

Observation: By Property 3

But note that

and so the reciprocal of θ₁ yields the same ACF. Thus, if we are seeking a coefficient θ₁ that yields a particular ρ₁ value, we can always choose a coefficient whose absolute value is at most 1. In fact, it turns out that there is always a unique such θ₁ whose absolute value is less than 1.

This is also true for any MA(q) process, and so we will restrict our MA(q) processes to those where |θ_j| < 1 for all j. This also ensures that the MA(q) process is invertible (see Invertible MA(q) Processes).

MA(1) PACF

Example 2: Chart PACF for the data in Example 1.

The approach is as described in Example 1 of Partial Autocorrelation Function. The chart is shown in Figure 3.

Figure 3 – Graph of PACF for MA(1) Process

The theoretical PACF values are calculated using Property 6. In particular, we insert the following formula in cell G5, highlight the range G5:G19 and press Ctrl-D.

=-((-0.5)^E5*(1-0.5^2)/(1-0.5^(2*E5+2)))

Note that we couldn’t use the following formula

=-(-0.5)^E5*(1-0.5^2)/(1-0.5^(2*E5+2))

This is because Excel incorrectly evaluates any expressions of the form -a^n as if they were (-a)^n. Thus –1^2 and –(–1)^2 are evaluated as 1 instead of -1.

Observation: We can see from Figure 3 that the absolute value of the PACF values tends towards zero as the lag increases. This is generally true for an MA(q) model.

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