1. The simple moving average model
Models with autocorrelation can be constructed from white noise.
2. The simple moving average model
A weighted sum of current and previous noise is called a simple moving average process.
There are many moving average or MA processes. You will focus on the simplest, first-order case, in which Today's observation is regressed on Yesterday's noise.
If epsilon is mean zero White Noise, then Y is an MA process if it follows the displayed equation.
There are three parameters: Mu is the mean, Theta is the slope parameter, and sigma-squared is the white noise variance.
3. MA processes - I
If the slope parameter theta is zero, Y is simply a white noise process.
4. MA processes - II
If the slope parameter theta is NOT zero, Y depends on both the current and previous noise, and the process Y is autocorrelated.
Large values of the slope theta lead to greater autocorrelation. And negative values of the slope theta result in oscillatory time series.
5. MA examples
Now lets look at four examples of MA time series. In the top left you see an MA series with theta equal to 0-point-5, and to its right, one with theta equal to 0-point-9. In both cases the series show persistence in level, meaning each observation is close to its neighbors. This persistence is more pronounced on the right for the larger value of theta.
In the bottom left you see an MA series with theta equal to -0-point-5. MA series with negative MA coefficients tend to exhibit an alternating or oscillatory patterns, as you see here.
Finally, in the bottom right you see an MA series with theta equal to 0. Here, there is no autocorrelation. Recall, in this case, the process is white noise.
6. Autocorrelations
The MA process has autocorrelation as determined by the slope parameter theta, however, it only has dependence for one period. The autocorrelation is zero at lags 2 and higher.
If theta is positive, the lag 1 autocorrelation will be positive, if theta is negative, the lag 1 autocorrelation will be negative.
7. Let's practice!
Great! Now you will practice simulating simple MA processes and estimating their autocorrelation functions.