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# P1.T2.20.23. Stationary Time Series: autoregressive moving average (ARMA) models

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Learning objectives: Explain mean reversion and calculate a mean-reverting level. Define and describe the properties of autoregressive moving average (ARMA) processes. Describe the application of AR, MA, and ARMA processes.

Questions:

20.23.1. Below are plotted the autocorrelation function (ACF) and partial autocorrelation function (PACF) for a simulated time-series process. Please note that the horizontal dashed blue lines represent the 95.0% confidence interval. Which of the following time-series models is most likely described by these ACF and PACF plots?

a. MA(1) with an MA coefficient of -0.67
b. MA(2) with MA coefficients of -0.45 and +1.83
c. AR(1) with an AR coefficient of 0.25
d. AR(2) with AR coefficients of 1.4 and -0.7

20.23.2. Below are plotted the autocorrelation function (ACF) and partial autocorrelation function (PACF) for a simulated time-series process. Please note that the horizontal dashed blue lines represent the 95.0% confidence interval. Which of the following time-series models is most likely described by these ACF and PACF plots?

a. MA(1) with a coefficient of 1.3
b. MA(3) with coefficients of 0.4, 0.6, and 0.8
c. AR(2) with coefficients of 1.5 and 0.8
d. AR(3) with coefficients of 0.3, -1.4 and 0.7

20.23.3. An ARMA(1,1) process evolves according to Y(t) = ẟ + ϕ*Y(t-1) + θ*ε(t-1) + ε(t). The plot below contains two 300-step ARMA(1,1) processes that differ only in their coefficients. Which of the following statements is TRUE about the two ARMA(1,1) simulations above?

a. Both of the ARMA(1,1) processes are likely to exhibit slow decay in both the ACF and PACF
b. Both of the ARMA(1,1) processes are likely to cut off sharply at lag 1 in both the ACF and PACF
c. The first ARMA(1,1) process (i.e., plotted with a red dashed line) cannot be covariance-stationary
d. The unconditional mean of both ARMA(1,1) processes must be zero because all ARMA(1,1) have a long-run mean of zero

Last edited by a moderator:

20.23.1 is (d)
20.23.2 is (b)
20.23.3 is (a)

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
20.23.1 is (d)
20.23.2 is (b)
20.23.3 is (a)
Hello @BIONICANURAG

The answers and explanations are provided to our paid members. If you would like access to the answers, along with access to the full paid forum where these questions are discussed in detail, you can view the study packages that we offer here: https://www.bionicturtle.com/shop-courses/.

#### z.jiejie

##### New Member
I thought Stationary Time Series is excluded from the FRM 2020 Oct exam? Anyone knows?

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
I thought Stationary Time Series is excluded from the FRM 2020 Oct exam? Anyone knows?
Hello @z.jiejie

Stationary time series has an entire chapter assigned under Quantitative Analysis (QA-10, Chapter 10). It was not removed from the curriculum. You will need to make sure to follow the GARP study guide to make sure that you are studying the learning objectives that are assigned. We also have a curriculum spreadsheet analysis here, which shows any learning objectives that have been added/removed/changed through the years: https://www.bionicturtle.com/forum/threads/2019-2020-curriculum-change-analysis-spreadsheets.22938/.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
If you are interested, the code and the plots for this questions set were generated in R (#rstats). To render attractive plots, I used ggplot which is part of the amazing tidyverse (https://www.tidyverse.org/). The forecast package (https://github.com/robjhyndman/forecast) by Rob Hyndman enables ggAcf() and ggPacf(); these functions produces a ggplot object of their equivalent base Acf, Pacf functions. If you would like to learn more about data science, see the following links:

#### kumar gopalakrishnan

##### New Member
Hello @BIONICANURAG

The answers and explanations are provided to our paid members. If you would like access to the answers, along with access to the full paid forum where these questions are discussed in detail, you can view the study packages that we offer here: https://www.bionicturtle.com/shop-courses/.
Hi There Mam,
Good day
Mam how to collect daily Questions in Pdf Form can you help out kindly

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Hi There Mam,
Good day
Mam how to collect daily Questions in Pdf Form can you help out kindly
Hello @kumar gopalakrishnan

These practice questions are published in PDF format in the study planner under the appropriate topic (Quant Analysis, Chapter 10, Stationary Time Series).