Time Series: AR, MA, ARMA

Time Series Analysis From Plots:

From this youtube post. Also, here is a more extensive document with simulations found online.

Examples:

Moving Average MA(1):

On this plot the ACF is significant only once (in reality the first entry in the ACF is always significant, since there is no lag in the first entry - it’s the correlation with itself), while the PACF is geometric. Hence it is an MA(1) process.

The negative values in the plot respond to a process of the form \(\large\color{blue} {y_t} = k - \color{red}{\theta}\,\epsilon_{t-1}+\epsilon_t.\)

Here is a simulation of an MA(1) process with \(\theta = - 0.7:\)

set.seed(2017)
MA = arima.sim(model=list(ma = - 0.7), n = 100)
par(mfrow = c(1,2)); acf(MA); pacf(MA)

par(mfrow=c(1,1))
library(ggplot2)
library(fpp2) 
ggtsdisplay(MA)

Moving Average MA(2):

In this example the ACF is significant in the first and second lags, while the PACF follows a geometric decay. It is again a MA process, but this time an MA(2) of the form: \(\large\color{blue} {y_t} = k+ \color{red}{\theta_1}\,\epsilon_{t-1}-\color{red}{\theta_2}\,\epsilon_{t-2}+\epsilon_t.\)

Here’s an R simulation with \(\theta_1 = 0.9\) and \(\theta_2= -0.2:\)

set.seed(2017)
MA2 = arima.sim(list(ma= c(0.9, - 0.2)), n = 100)
par(mfrow = c(1,2));acf(MA2);pacf(MA2)

par(mfrow = c(1,1))
ggtsdisplay(MA2)

Autoregressive AR(1):

Here the ACF decays geometrically, and the PACF shows only one significant lag. This is a AR(1) process of the form: \(\large \color{blue}{y_t} = c + \color{red}{\rho}\, \color{blue}{y_{t-1}}+\epsilon_t.\)

Here is the simulation in R with \(\rho = 0.9:\)

set.seed(2017)
AR = arima.sim(model=list(ar = .9), n = 100)
par(mfrow = c(1,2));acf(AR);pacf(AR)

ggtsdisplay(AR)

Autocorrelation AR(1) with faster decay:

This is again an AR(1) process, but with a faster decay, \(\rho=0.5:\)

set.seed(2017)
AR = arima.sim(model=list(ar = .5), n = 100)
par(mfrow = c(1,2));acf(AR);pacf(AR)

ggtsdisplay(AR)

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