1. Describe Model
In a Moving Average, or MA model,
2. Mathematical Description of MA(1) Model
today's value equals a mean plus noise, plus a fraction theta of yesterday's noise. Since there is only one lagged error on the right hand side, this is called an MA model of order 1, or simply an MA(1) model.
If the MA parameter, theta, is zero, then the process is white noise. MA models are stationary for all values of theta.
3. Interpretation of MA(1) Parameter
Suppose R t is a time series of stock returns. If theta is negative, then a positive shock last period, represented by epsilon t-1, would have caused last period's return to be positive, but this period's return is more likely to be negative. A shock two periods ago would have no effect on today's return - only the shock now and last period. Also, note that the lag-1 autocorrelation turns out not to be theta, but theta over 1 plus theta squared.
4. Comparison of MA(1) Autocorrelation Functions
Here are four autocorrelation functions for different MA parameters. In each case, there is zero autocorrelation for an MA(1) beyond lag-1. When theta is positive, the lag-1 autocorrelation is positive and when theta is negative, the lag-1 autocorrelation is negative.
5. Example of MA(1) Process: Intraday Stock Returns
Higher frequency stock returns are a nice example of an MA(1) process. Here is an intraday plot for the stock price of Sprint Corporation for one day. The frequency of the data is one minute. Stocks trade at discrete one-cent increments rather than at continuous prices, and you can see that the stock can bounce back and forth over a one-cent range for long periods of time. This is sometimes referred to as the "bid/ask bounce".
6. Autocorrelation Function of Intraday Stock Returns
The bid/ask bounce induces a negative lag-1 autocorrelation, but no autocorrelation beyond lag-1. You can see this with the autocorrelation function plot of intraday returns. The lag-1 autocorrelation is significantly negative, and the other autocorrelations are not significantly different from zero.
7. Higher Order MA Models
So far, we've been only looking at MA(1) models. The model can be extended to include more lagged errors and more theta parameters. Here we show an MA(1), an MA(2), and an MA(3) model.
8. Simulating an MA Process
Just like in the last chapter with AR models, you may want to simulate a pure MA process. You can use the same statsmodels module, ArmaProcess. This time, for an MA(1), the AR order is just an array containing 1. The MA order is an array containing 1 and the MA(1) parameter theta. Unlike with the AR simulation, you don't need to reverse the sign of theta.
As before, you create an instance of the class ArmaProcess. To simulate data, use the method generate_sample, with the number of simulated samples as an argument.
9. Let's practice!
Now let's try some examples.