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The ARIMA Procedure |
The Smallest CANonical (SCAN) correlation method can tentatively identify the orders of a stationary or nonstationary ARMA process. Tsay and Tiao (1985) proposed the technique, and Box et al (1994) and Choi (1990) provide useful descriptions of the algorithm.
Given a stationary or nonstationary time series
with mean corrected form
,
with a true autoregressive order of p+d,
and with a true moving-average order of q,
you can use the SCAN method to analyze eigenvalues
of the correlation matrix of the ARMA process.
The following paragraphs provide a brief description of the algorithm.
For autoregressive test order m = pmin, ... , pmax and for moving-average test order j = qmin, ... , qmax, perform the following steps.
The test statistic to be used as an identification criterion is
which is asymptotically
if m = p+d and
or
if
and j = q.
For m > p and j < q , there is more than
one theoretical zero canonical correlation between Ym,t
and Ym,t-j-1.
Since the
are the smallest canonical
correlations for each (m,j), the percentiles of c(m,j)
are less than those of a
;
therefore, Tsay and Tiao (1985) state that it is safe to assume a
.
For m < p and j < q, no conclusions about
the distribution of c(m,j) are made.
A SCAN table is then constructed using c(m,j) to
determine which of the
are significantly different from zero (see Table 7.6).
The ARMA orders are tentatively identified by finding a (maximal)
rectangular pattern in which the
are insignificant
for all test orders
and
.
There may be more than one pair of values (p+d, q)
that permit such a rectangular pattern.
In this case, parsimony and the number of insignificant items in the
rectangular pattern should help determine the model order.
Table 7.7 depicts the theoretical pattern associated with an
ARMA(2,2) series.
MA | ||||||
AR | 0 | 1 | 2 | 3 | · | · |
0 | c(0,0) | c(0,1) | c(0,2) | c(0,3) | · | · |
1 | c(1,0) | c(1,1) | c(1,2) | c(1,3) | · | · |
2 | c(2,0) | c(2,1) | c(2,2) | c(2,3) | · | · |
3 | c(3,0) | c(3,1) | c(3,2) | c(3,3) | · | · |
· | · | · | · | · | · | · |
· | · | · | · | · | · | · |
MA | ||||||||
AR | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
0 | * | X | X | X | X | X | X | X |
1 | * | X | X | X | X | X | X | X |
2 | * | X | 0 | 0 | 0 | 0 | 0 | 0 |
3 | * | X | 0 | 0 | 0 | 0 | 0 | 0 |
4 | * | X | 0 | 0 | 0 | 0 | 0 | 0 |
X = significant terms | ||||||||
0 = insignificant terms | ||||||||
* = no pattern |
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