Stochastic%20Trend%20With%20Seasonality

1
Stochastic Trend With Seasonality

1. Seasonal Difference
2. Multiplicative Seasonal ARMA

2
Stochastic Seasonality No Trend

• (1-Ls)Yt Yt Y(t-s) ut
• Yt Y(t-s) ut
• ut is ARMA (p, q) (P, Q)s

3
Stochastic Trend and Seasonality

• (1-L) (1-Ls)Yt ut
• ut is ARMA (p, q) (P, Q)s

4
Interpretation of (1-L)(1-Ls)

• (1-L)(1-Ls) Yt (1-L-LsL(s1)) Yt
  Yt Y(t-1) - Y(t-s) Y(t-s-1) Yt
  Y(t-1) (Y(t-s) - Y(t-s-1))
• Yt Y(t-1) (Y(t-s) - Y(t-s-1) ) ut

5
Multiplicative Seasonal ARMA

• Seasonal MA(1)(s) Airline Model
• ut (1q L) (1Q Ls) et et q e(t-1)
  Q e(t-s) q Q e(t-s-1)
• Seasonal AR(1)(s) (1- f L) (1- F Ls) ut et
  ut fu(t-1) Fu(t-s) - f F
  u(t-s-1) et

6
Eviews Commands

• Removing both trend and seasonality
• d(series_name, 1, s) s4 for quarterly
  data
• Seasonal ARMA terms sar(s), sma(s)