1. Stochastic Trend With Seasonality 1.Seasonal Difference 2.Multiplicative Seasonal ARMA

2. Stochastic Seasonality – No Trend (1-L s )Y t = Y t – Y (t-s) = u t Y t = Y (t-s) + u t u t is ARMA (p, q) (P, Q) s

3. Stochastic Trend and Seasonality (1-L) (1-L s )Y t = u t u t is ARMA (p, q) (P, Q) s

4. Interpretation of (1-L)(1-L s ) (1-L)(1-L s ) Y t = (1-L-L s +L (s+1) ) Y t = Y t – Y (t-1) - Y (t-s) + Y (t-s-1) = Y t – Y (t-1) – (Y (t-s) - Y (t-s-1) ) Y t = Y (t-1) + (Y (t-s) - Y (t-s-1) ) + u t

5. Multiplicative Seasonal ARMA Seasonal MA(1)(s) = Airline Model u t = (1+  L) (1+  L s )  t =  t +   t   +   t  s   +   t-s-1   Seasonal AR(1)(s) (1-  L) (1-  L s ) u t =  t u t =  u (t-1) +  u (t-s) -  u (t-s-1) +  t

6. Eviews Commands Removing both trend and seasonality: d(series_name, 1, s) s=4 for quarterly data Seasonal ARMA terms: sar(s), sma(s)