The ARIMA Procedure

Initial Values

The syntax for giving initial values to transfer function parameters in the INITVAL= option parallels the syntax of the INPUT= option. For each transfer function in the INPUT= option, the INITVAL= option should give an initialization specification followed by the input series name. The initialization specification for each transfer function has the form

\[ C ~ \$ ~ (V_{1,1},V_{1,2}, {\ldots }) (V_{2,1}, {\ldots }) {\ldots } / (V_{\mi{i},1}, {\ldots }) {\ldots } \]

where C is the lag 0 term in the first numerator factor of the transfer function (or the overall scale factor if the ALTPARM option is specified) and ${V_{i,j}}$ is the coefficient of the ${L_{i,j}}$ element in the transfer function.

To illustrate, suppose you want to fit the model

\[ Y_{t}={\mu }+\frac{({\omega }_{0}- {\omega }_{1}{B}-{\omega }_{2}{B}^{2})}{(1-{\delta }_{1}{B}-{\delta }_{2}{B}^{2}-{\delta }_{3}{B}^{3})}X_{t-3} +\frac{1}{(1-{\phi }_{1}{B}-{\phi }_{2}{B}^{3})}a_{t} \]

and start the estimation process with the initial values ${\mu }$=10, ${\omega }_{0}$=1, ${\omega }_{1}$=0.5, ${\omega }_{2}$=0.03, ${\delta }_{1}$=0.8,
${\delta }_{2}$=–0.1, ${\delta }_{3}$=0.002, ${\phi }_{1}$=0.1, ${\phi }_{2}$=0.01. (These are arbitrary values for illustration only.) You would use the following statements: 

   identify var=y crosscorr=x;
   estimate p=(1,3) input=(3$(1,2)/(1,2,3)x)
            mu=10 ar=.1 .01
            initval=(1$(.5,.03)/(.8,-.1,.002)x);

Note that the lags specified for a particular factor are sorted, so initial values should be given in sorted order. For example, if the P= option had been entered as P=(3,1) instead of P=(1,3), the model would be the same and so would the AR= option. Sorting is done within all factors, including transfer function factors, so initial values should always be given in order of increasing lags.

Here is another illustration, showing initialization for a factored model with multiple inputs. The model is

\begin{eqnarray*} Y_{t} = {\mu } & +& \frac{{\omega }_{1,0}}{(1-{\delta }_{1,1}{B})}W_{t} + ({\omega }_{2,0}-{\omega }_{2,1}{B})X_{t-3} \nonumber \\ & +& \frac{1}{(1-{\phi }_{1}{B})(1-{\phi }_{2}{B}^{6}-{\phi }_{3}{B}^{12})}a_{t} \nonumber \end{eqnarray*}

and the initial values are ${\mu }$=10, ${{\omega }_{1,0}}$=5, ${{\delta }_{1,1}}$=0.8, ${{\omega }_{2,0}}$=1, ${{\omega }_{2,1}}$=0.5, ${{\phi }_{1}}$=0.1, ${{\phi }_{2}}$=0.05, and ${{\phi }_{3}}$=0.01. You would use the following statements: 

   identify var=y crosscorr=(w x);
   estimate p=(1)(6,12) input=(/(1)w, 3$(1)x)
            mu=10 ar=.1 .05 .01
            initval=(5$/(.8)w 1$(.5)x);