ARIMA Procedure
The SCAN Method
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 for useful descriptions of the algorithm, see Box, Jenkins, and Reinsel (1994); Choi (1992).
Given a stationary or nonstationary time series 
with mean corrected form 
with a true autoregressive order of 
and with a true moving-average order of 
, 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 
and for moving-average test order 
, perform the following steps:
-
Let
. Compute the following 
matrix, 
where t ranges from
to n. Find the smallest eigenvalue,
, of 
and its corresponding normalized eigenvector, 
. The squared canonical correlation estimate is . Using the
as AR(m) coefficients, obtain the residuals for 
to n, by following the formula: 
. -
From the sample autocorrelations of the residuals,
, approximate the standard error of the squared canonical correlation estimate by 
where
.
The test statistic to be used as an identification criterion is
which is asymptotically 
if 
and 
or if 
and 
. For 
and 
, there is more than one theoretical zero canonical correlation between 
and 
. Since the 
are the smallest canonical correlations for each 
, the percentiles of 
are less than those of a ; therefore, Tsay and Tiao (1985) state that it is safe to assume a . For 
and 
, no conclusions about the distribution of are made.
A SCAN table is then constructed using to determine which of the are significantly different from zero (see Table 7). The ARMA orders are tentatively identified by finding a (maximal) rectangular pattern in which the are insignificant for all test orders 
and 
. There might be more than one pair of values (
) 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 8 depicts the theoretical pattern associated with an ARMA(2,2) series.
Table 7: SCAN Table
| MA | ||||||
| AR | 0 | 1 | 2 | 3 |  |
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| 0 |  |
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| 1 |  |
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| 2 |  |
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| 3 |  |
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Table 8: Theoretical SCAN Table for an ARMA(2,2) Series
| 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 | ||||||||