### Moving Average Processes MA(q)

- Stock price depends on announcements of last two days
- Auto correlation function cuts off at q

### Auto regressive Processes AR(p)

- Below are the plots for AR(2) process
- Depending upon the value of phi1 and phi2 ACF has alternative positive and negative values

Writing AR(p) process as MA process by substituting values of X(t-1). And yes phi is constant, we don’t need phi1, phi2 anymore.

Mean, variance and auto-correlation of AR(p) process, we have assumed Z = Norm(0, sigma2)

### ACF of AR-p using Yule-Walker Equation

- It is a method of solving difference equation in recursive relation
- We first obtained auxiliary equation (also known as characteristic equation) which is polynomial and find root of that
- Using these root we get weighted geometric series and find weights using some initial condition
- We had learned in mathematics that this way of solving difference equation also related to solving differential equations
- In the course they had solved it for Fibonacci series and root had come out to be golden ratio
- For AR(p) ACF comes out to be difference equation, solving which can give us ACF for different values of lag

### Reference

https://www.coursera.org/learn/practical-time-series-analysis/home/welcome