Moving Average Processes

In the Moving Average process, the data points are all dependent on each other. This is unlike the Autoregressive Time Series or The Markov Property.

We define MA(1) as the 0-mean moving average process of order 1

$$Z_n={\theta }_1{ϵ}_{n-1}+{ϵ}_n$$

• We say $ϵ$ - s are independent and normally distributed with mean of 0 and variance ${\sigma }^2$
• ${\theta }_1$ is a constant

Any ${ϵ}_k$ will be independent of some $Z_j$ for $j<k$

We define MA(1) with $\mu$-mean moving average process of order 1

$$Y_n=Z_n+\mu$$

Since $ϵ$ s have mean of 0, $\mathbb{E}[Z_n]=0$.

This is because $ϵ$ s are the only random variable

We can show that $Z_n$ depends on all previous $Z$s. Note that this is a contradiction of the Markov Property.

We can show that for an MA(1) Process, the first lag autocovariance is $\text{Cov}(Z_n{Z}_{n-1)}=\theta {\sigma }^2$. For terms after this, we can show that the autocovariance is 0.

The $kth$ autocorrelation is defined as the $kth$ auto covariance divided by the variance of the process.

Simulating 1000 observations from MA(1) with ${\theta }_1=0.9$

ma1 <- arima.sim(1000, model=list(ma=c(.9)))
acf(ma1)

Given a time series plot and asked to determine whether the plot s AR(1) or MA(1) we can determine based on the sample acf function. If the acf function cuts off after lag 1 we have MA(1). If they decay exponentially then we have an AR(1).