General Concepts

Lag Operator

$$ LX_t = X_{t-1}\\ L^{k}X_t = X_{t-k} $$

Stationary

Constant statistical properties:

1. Constant mean over time
2. Constant variance over time
3. No seasonality

How to check:

• Global vs local test
• Augmented Dicky-Fuller Test

How to make a time series stationary?

Take difference of value at time t with value at time t-1

Auto Regressive Model - AR(p) model

Represents the linear regression of the current value of the time series on its past values.

$\\epsilon_t$ is white noise

$$ X_t = \sum_{i=1}^p \phi_i X_{t-i} + \epsilon_t\\ X_t = \sum_{i=1}^p \phi_i L^i X_t + \epsilon_t\\ $$

Moving Average Model - MA(q) model

Represents linear regression of the current value of the time series on the past prediction errors

$$ X_t = \mu+\epsilon_t+\sum_{i=1}^q \theta_i \epsilon_{t-i}\\ X_t = \mu+\epsilon_t+\sum_{i=1}^q \theta_i L^i \epsilon_t\\ $$

ARMA(p,q) Model

$$ X_t = \epsilon_t + \sum_{i=1}^{p}\phi_i X_{t-i} + \sum_{i=1}^{q}\theta_i \epsilon_{t-i} $$

With lag operator: