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+1 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.
Topics
- Basic Concepts
- Infinite-order Moving Average
- Invertibility
- Finding Model Coefficients using ACF
- Finding Model Coefficients using Solver
The mathematical proofs of some of the properties of Moving Average Processes are provided in Moving Average Proofs
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
MA(q) is a linear function of past q terms of the errors.
My confusion is that where are these errors coming from? I have not yet developed a model.
To explain further:
When starting to work on an ARIMA model, I have only the time series data. How do I get the errors to identify q for the MA component of ARIMA.
Amar,
1. MA(q) is a process. The instantiation of the process is a time series. Sort of like the relationship between a population that follows a distribution and a sample from this population.
2. The errors don’t identify the q value. You can try to estimate q using other approaches (ACF and PACF), but in the end you may need to try several q values and see which one gives the best fit to the time series data that you have.
3. For any specific value of q, the errors are the difference between the values predicted by the MA(q) process and the time series data. This is similar to the differences between the observed y values and the y values predicted by a linear regression.
4. The added problem is what sort of errors are there for the first few elements in the time series since there are no time series terms prior to time = 1. Let’s look at the MA(1) process defined by y_i = ε_i – .4ε_i-1 with σ^2 = .25, as described in Example 1 of
https://real-statistics.com/time-series-analysis/moving-average-processes/calculating-ma-coefficients-solver/
What values should be used for ε_0? Once we determine what this value, finding the values for ε_i for i >= 1 is straightforward. As shown in Figure 1 of the above webpage, we arbitrarily set ε_0 = 0. This is not the only choice and other choices may have advantages. In any case, if you have a stationary process, then after some time, the initial choices for ε_0 (as well as ε_i for negative i) won’t have much of an impact of the fit.
Charles
I really want to understand where -0.694200914686689 (first residual term in MA(1 )) is from
It is calculated by the worksheet formula =NORM.INV(RAND(),0,2)
Charles
Thanks a lot.
It means that first residual term could be change every run
Mee,
Yes, it could change on every run.
Keep in mind, this was my way of creating a time series that came from a moving average process. For your actual time series you wouldn’t need to use a random first residual.
Charles
Charles,
I came across the following Q&A at https://stats.stackexchange.com/questions/58242/why-are-maq-time-series-models-called-moving-averages, which had illuminated the above issue.
In the meantime, I succeeded in gaining insight into the process of AR, MR, ARMA and SARIMA thanks to your step-by-step examples.
Thank you very much for your support!
Jaron Lee
Hello Charles,
According to the above equation, a MA is the linear function value of the past errors.
This is different from the commonly-known formula, which is an average of the past values.
Is there any connection between two or are they totally different from each other?
I really appreciate any help you can provide.
Thanks.
Jaron Lee
Hello Jaron,
The version you are referring to is described at
Simple Moving Average
I don’t know why they are both called moving average,
Charles