Question

time series

3. Specify and interpret the model in terms of a time- series data Xt for the following multiplicative models ARIMA: (1, 0, 0)(1, 0, 0)4 ARIMA (0, 1, 0)(1, 1, 0)6 ARIMA (0, 1, 1)(0, 0, 1)12

Answer 1

It is Seasonal ARIMA Model

The seasonal ARIMA model incorporates both non-seasonal and seasonal factors in a multiplicative model. One shorthand notation for the model is

                        ARIMA (p,q,d)*(P,D,Q)S

with p = non-seasonal AR order,

      d = non-seasonal differencing,

      q = non-seasonal MA order,

      P = seasonal AR order,

      D = seasonal differencing,

      Q = seasonal MA order,

and S = time span of repeating seasonal pattern.

i) ARIMA(1,0,0)*(1,0,0)4

This model have repeating seasonal pattern of time span equals to 4 ( S=4 )

Here p=1 and P=1

Hence 1 non-seasonal AR order, and 1 seasonal AR order,

Here d= 0 and D = 0   hence no differencing

Also q=0 and Q = 0 implies no MA terms involved .

Thus ARIMA(1,0,0)*(1,0,0)4 model includes a non-seasonal AR(1) term, a seasonal AR(1) term, no differencing, no MA terms and the seasonal period is S = 4

The non-seasonal AR(1) polynomial is $\small \phi$ (B)= 1 - $\small \phi$ 1 B

The seasonal AR(1) polynomial is $\small \phi$ (BS) = $\small \phi$ (B4) = 1 - $\small \phi$ 1 B4

The model is (1 - $\small \phi$ 1 B)*(1 - $\small \phi$ 1 B4)(Xt - $\small \mu$ ) = Zt      where Zt ~ WN(0,$\small \sigma ^{2}$)

ii) ARIMA(0,1,0)*(1,1,0)6

Here p = 0 , d = 1 , q = 0

        P = 1 , D = 1 , Q = 0

    and S = 6

Thus ARIMA(0,1,0)*(1,1,0)6

model dose not include includes a non-seasonal AR(1) term and non-seasonal MA(1) term , and d=1 non-seasonal differencing,

It include a seasonal AR(1) term, seasonal differencing, but no seasonal MA order terms

and the seasonal period is S = 6

If trend is present in the data (d=1), we may also need non-seasonal differencing. Often (not always) a first difference (non-seasonal) will “detrend” the data. That is, we use (1-B)Xt = Xt - Xt-1 in the presence of trend.

iii) ARIMA(0,1,1)*(0,0,1)12

Here p = 0 , d = 1 , q = 1

        P = 0 , D = 0 , Q = 1

    and S = 12

Thus ARIMA(0,1,1)*(0,0,1)12

model dose not include includes a non-seasonal AR(1) term and non-seasonal differencing, but it includes non-seasonal MA(1) term

It dose not include a seasonal AR(1) term, seasonal differencing, but it includes seasonal MA(1) terms

and the seasonal period is S = 12

In presence of trend , Using (1-B)Xt = Xt - Xt-1 in the presence of trend. will “detrend” the data.

Here ,

The non-seasonal MA(1) polynomial is $\small \theta$ (B)= 1 - $\small \theta$ 1 B

The seasonal MA(1) polynomial is $\small \theta$ (BS) = $\small \theta$ (B12) = 1 + $\small \theta$ 1 B12