Forecasting

1. Introduction to Time Series Analysis
AR, MA, ARMA, Stationary Data, & Differencing

2. Time Series Data
• Definition
Data that change over time, e.g., stock price, sales growth.
• Stock Price Analysis Example

3. AR, MA, and ARMA Model
• Autoregressive (AR) Model
• Moving Average (MA) Model
• Autoregressive Moving Average (ARMA) Model
𝑦𝑡 = β0 + β1𝑦𝑡−1 + β2𝑦𝑡−2 + β3𝑦𝑡−3 + ⋯ + β𝑝𝑦𝑡−𝑝
𝑦𝑡 = ε𝑡 + ɵ1ε𝑡−1 + ɵ2ε𝑡−2 + ⋯ + ɵ𝑞ε𝑡−𝑞
𝑦𝑡 = β1𝑦𝑡−1 + β2𝑦𝑡−2 + β3𝑦𝑡−3 + ⋯ + β𝑝𝑦𝑡−𝑝 +
ε𝑡 + ɵ1ε𝑡−1 + ɵ2ε𝑡−2 + ⋯ + ɵ𝑞ε𝑡−𝑞

4. Stationary Data Assumption
• The mean and variance of a time series are constant for
the whole series, no matter where you choose a period.

5. Differencing
• The process of subtracting one observation from another.
• Used for transforming non-stationary data into stationary
data.
• Example
X=[5, 4, 6, 7, 9, 12]
What should be the values of X after 1-lag differencing?
X* =[1, -2, -1, -2, -3]

6. 1-Lag Differencing Twice vs. 2-Lag
Differencing Once
X=[5, 4, 6, 7, 9, 12]
What should be the values of X after 1-lag twice?
1-lag differencing once:
X*=[1, -2, -1, -2, -3]
1-lag differencing again (i.e., 1-lag differencing twice):
X**=[3, -1, 1, 1]

7. 1-Lag Differencing Twice vs. 2-Lag
Differencing Once
X=[5, 4, 6, 7, 9, 12]
What should be the values of X after 2-lag once?
2-lag differencing once:
X’=[-1, -3, -3, -5]
We can also do 4-lag differencing once, 12-lag differencing
once, etc. Can you see where to use them?
Answer: Processing seasonal data!