Computation Details for Sliding Spans Analysis

The X11 Procedure

Computation Details for Sliding Spans Analysis

Length and Number of Spans

The algorithm for determining the length and number of spans for a given series was developed at the U.S. Bureau of the Census, Statistical Research Division. A summary of this algorithm is as follows.

First, an initial length based on MACURVE specification is determined, then the maximum number of spans possible using this length is determined. If this maximum number exceed four, set the number of spans to four. If this maximum number is one or zero, there is not enough observations to perform the sliding spans analysis. In this case a note is written to the log and the sliding spans analysis is skipped for this variable.

If the maximum number of spans is two or three, the actual number of spans used is set equal to this maximum. Finally, the length is adjusted so that the spans begin in January (or the first quarter) of the beginning year of the span.

The remaining part of this section gives the computation formulas for the maximum percent difference (MPD) calculations along with the threshold regions.

Seasonal Factors (Table D10): S_t(k)

For the additive model, the MPD is defined as

$max_{k{\epsilon}N_{t}}S_{t}(k) - min_{k{\epsilon}N_{t}}S_{t}(k)$

For the multiplicative model, the MPD is

$MPD_{t} = \frac{max_{k{\epsilon}N_{t}}S_{t}(k) - min_{k{\epsilon}N_{t}}S_{t}(k)}{min_{k{\epsilon}N_{t}}S_{t}(k) }$

A series for which less than 15% of the MPD values of D10 exceed 3.0% is stable; between 15% and 25% is marginally stable; and greater than 25% unstable. Span reports S 2.A - S 2.C give the various breakdowns for the number of times the MPD exceeded these levels.

Trading Day Factor (Table C18): TD_t(k)

For the additive model, the MPD is defined as

$max_{k{\epsilon}N_{t}}TD_{t}(k) - min_{k{\epsilon}N_{t}}TD_{t}(k)$

For the multiplicative model, the MPD is

$MPD_{t} = \frac{max_{k{\epsilon}N_{t}}TD_{t}(k) - min_{k{\epsilon}N_{t}}TD_{t}(k)}{min_{k{\epsilon}N_{t}}TD_{t}(k) }$

The Census Bureau currently gives no recommendation concerning MPD thresholds for the Trading Day factors. Span reports S 3.A - S 3.C give the various breakdowns for MPD thresholds. When TDREGR=NONE is specified, no trading day computations are done, hence this table is skipped.

Seasonally Adjust Data (Table D11): SA_t(k)

For the additive model, the MPD is defined as

$max_{k{\epsilon}N_{t}}SA_{t}(k) - min_{k{\epsilon}N_{t}}SA_{t}(k)$

For the multiplicative model, the MPD is

$MPD_{t} = \frac{max_{k{\epsilon}N_{t}}SA_{t}(k) - min_{k{\epsilon}N_{t}}SA_{t}(k)}{min_{k{\epsilon}N_{t}}SA_{t}(k) }$

A series for which less than 15% of the MPD values of D11 exceed 3.0% is stable; between 15% and 25% is marginally stable; and greater than 25% unstable. Span reports S 4.A - S 4.C give the various breakdowns for the number of times the MPD exceeded these levels.

Month-to-month changes in the Seasonally Adjust Data: MM_t(k)

Some additional notation is needed for the month-to-month and year-to-year differences. Define N1_t by

N1_t = {k: span k contains month t and t-1 }

For the additive model the month-to-month change for span k is defined by

MM_t(k) = SA_t-SA_t-1

while for the multiplicative model

MM_t(k) = [(SA_t-SA_t-1)/(SA_t-1 )],

Since this quantity is already in percentage form, the MPD for both the additive and multiplicative model is defined by

$MPD_{t} = max_{k{\epsilon}N1_{t}}MM_{t}(k) - min_{k{\epsilon}N1_{t}}MM_{t}(k)$

The current recommendation of the Census Bureau is that if 35% or more of the MPD values of the month-to-month differences of D11 exceed 3.0% then the series is usually not stable. 40% exceeding this level clearly marks an unstable series. Span reports S 5.A.1 - S 5.C give the various breakdowns for number of times the MPD exceeds these levels.

Year-to-year changes in the Seasonally Adjust Data: YY_t(k)

First define N12_t by

N12_t = {k: span k contains month t and t-12 }

(appropriate changes in notation for a quarterly series are obvious.)

For the additive model the month-to-month change for span k is defined by

YY_t(k) = SA_t-SA_t-12

while for the multiplicative model

YY_t(k) = [(SA_t-SA_t-12)/(SA_t-12 )],

Since this quantity is already in percentage form, the MPD for both the additive and multiplicative model is defined by

$MPD_{t} = max_{k{\epsilon}N1_{t}}YY_{t}(k) - min_{k{\epsilon}N1_{t}}YY_{t}(k)$

The current recommendation of the Census Bureau is that if 10% or more of the MPD values of the month-to-month differences of D11 exceed 3.0% then the series is usually not stable. Span reports S 6.A - S 6.C give the various breakdowns for the number of times the MPD exceeds these levels.

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