Mixed seasonal models

ARIMA Models in R

David Stoffer

Professor of Statistics at the University of Pittsburgh

Mixed Seasonal Model

  • Mixed model: SARIMA$(p, d, q) \times (P, D, Q)_s$ model

  • Consider a SARIMA$(0, 0, 1) \times (1, 0, 0)_{12}$ model

$$X_t = \Phi X_{t-12} + W_t + \theta W_{t-1}$$

  • SAR(1): Value this month is related to last year's value $X_{t-12}$

  • MA(1): This month's value related to last month's shock $W_{t-1}$

ARIMA Models in R

ACF and PACF of SARIMA(0,0,1) x (1,0,0) s=12

  • The ACF and PACF for this mixed model:

$$X_t = .8 X_{t-12} + W_t -.5 W_{t-1}$$

ch4_2.010.png

ARIMA Models in R

ACF and PACF of SARIMA(0,0,1) x (1,0,0) s=12

  • The ACF and PACF for this mixed model:

$$X_t = .8 X_{t-12} + W_t -.5 W_{t-1}$$

ch4_2.011.png

ARIMA Models in R

ACF and PACF of SARIMA(0,0,1) x (1,0,0) s=12

  • The ACF and PACF for this mixed model:

$$X_t = .8 X_{t-12} + W_t -.5 W_{t-1}$$

ch4_2.013.png

ARIMA Models in R

Seasonal Persistence

ch4_2.015.png

ARIMA Models in R

Seasonal Persistence

ch4_2.016.png

ARIMA Models in R

Seasonal Persistence

ch4_2.017.png

ARIMA Models in R

Air Passengers

  • Monthly totals of international airline passengers, 1949-1960

ch4_2.020.png

ARIMA Models in R

Air Passengers: ACF and PACF of ddlx

ch4_2.022.png

ARIMA Models in R

Air Passengers: ACF and PACF of ddlx

ch4_2.023.png

  • Seasonal: ACF cutting off at lag 1s (s = 12); PACF tailing off at lags 1s, 2s, 3s…
ARIMA Models in R

Air Passengers: ACF and PACF of ddlx

ch4_2.024.png

  • Seasonal: ACF cutting off at lag 1s (s = 12); PACF tailing off at lags 1s, 2s, 3s…
ARIMA Models in R

Air Passengers: ACF and PACF of ddlx

ch4_2.025.png

  • Seasonal: ACF cutting off at lag 1s (s = 12); PACF tailing off at lags 1s, 2s, 3s…

  • Non-Seasonal: ACF and PACF both tailing off

ARIMA Models in R

Air Passengers

airpass_fit1 <- sarima(log(AirPassengers), p = 1, 
                                           d = 1, q = 1, P = 0, 
                                           D = 1, Q = 1, S = 12)
airpass_fit1$ttable

     Estimate     SE t.value p.value
ar1    0.1960 0.2475  0.7921  0.4296
ma1   -0.5784 0.2132 -2.7127  0.0075
sma1  -0.5643 0.0747 -7.5544  0.0000

airpass_fit2 <- sarima(log(AirPassengers), 0, 1, 1, 0, 1, 1, 12)
airpass_fit2$ttable

     Estimate     SE t.value p.value
ma1   -0.4018 0.0896 -4.4825       0
sma1  -0.5569 0.0731 -7.6190       0
ARIMA Models in R

Air Passengers

ch4_2.032.png

ARIMA Models in R

Let's practice!

ARIMA Models in R

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