# BT Question Set P1-T2-20-22-2: AR versus MA process

## P1.T2.20.22. Stationary Time Series: autoregressive (AR) and moving average (MA) processes

### Learning objectives

• Define and describe the properties of autoregressive (AR) processes.
• Define and describe the properties of moving average (MA) processes.
• Explain how a lag operator works.
library(tidyverse)
## -- Attaching packages --------------------------------------------------------------------------------- tidyverse 1.3.0 --
## v ggplot2 3.3.2     v purrr   0.3.4
## v tibble  3.0.3     v dplyr   1.0.2
## v tidyr   1.1.2     v stringr 1.4.0
## v readr   1.3.1     v forcats 0.5.0
## -- Conflicts ------------------------------------------------------------------------------------ tidyverse_conflicts() --
set.seed(25)

MA_mean = 2.0
MA_weight = 0.5
# Generate an MA(1) with mean, μ = 2.0 and weight parameter 𝜃= 0.5
MA <- arima.sim(model=list(order = c(0,0,1), ma = MA_weight), n = 200, mean = MA_mean)

AR_intercept = 3.0
AR_param = 0.6
# Generate an AR(1) with intercept δ = 3.0 and AR parameter ϕ = 0.6
AR <- arima.sim(model=list(order=c(1,0,0), ar = AR_param), n = 200, mean = AR_intercept)

color_AR = "#266935"
color_MA = "darkblue"

time_ma_ar <- bind_cols(MA, AR) %>% rowid_to_column() %>%
rename(y_MA = ...1, y_AR = ...2)
## New names:
## * NA -> ...1
## * NA -> ...2
time_ma_ar %>% ggplot(aes(x = rowid)) +
theme_bw() +
theme(
axis.title.y = element_blank(),
axis.title.x = element_blank(),
axis.text = element_text(size = 14, face = "bold"),
legend.position = c(0.8, 0.86)
) +
ggtitle("AR(1) series in GREEN. MA(1) series in BLUE.") +
geom_line(aes(y = y_AR), color = color_AR, size = 1) +
geom_line(aes(y = y_MA), color = color_MA, size = 1) +
scale_y_continuous(breaks = c(0, 2.5, 5.0, 7.5, 10, 12.5))

# scale_color_manual(name = "Simulations with arima.sim()", labels=c("MA(1)", "AR(1)"))

# MA mean is intercept
lr_mean_AR <- AR_intercept/(1 - AR_param)
variance_MA <- (1 + MA_weight^2)*1
variance_AR <- 1/(1 - AR_param^2)

lr_mean_AR
## [1] 7.5
variance_MA
## [1] 1.25
variance_AR
## [1] 1.5625
##### David Harper
###### Founder & CEO of Bionic Turtle

I teach financial risk and enjoy learning data science