Details on calculation of trend-cycle estimates at Statistics Canada

This document details the way Statistics Canada calculates trend-cycle estimates. It is intended as a technical document to support users who wish to apply trend-cycle estimation to monthly series available on CANSIM. General formulas outlining the mathematical calculations are presented below, and the derivation and application of the moving average weights are provided. For more information on the use of trend-cycle estimates in analysis, please refer to "Trend-cycle estimates: Frequently asked questions" (Statistics Canada 2015).

The trend-cycle estimation method used at Statistics Canada applies moving averages and does not remove seasonal patterns. This procedure is intended for monthly series with at least 13 data points that do not exhibit seasonal patterns (either because no seasonality exists, or because it has been removed by seasonal adjustment).

1.0 The general formula

For each month t, the following formula is applied to estimate the trend-cycle, denoted as T C t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubGaam4qa8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacaGG 6aaaaa@39E3@ :

Formula 1

T C t = j=t6 t+6 I j W j ( t ) Y j k=t6 t+6 I k W k ( t ) ,      (1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubGaam4qa8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacqGH 9aqpdaWcaaWdaeaapeWaaubmaeqal8aabaWdbiaadQgacqGH9aqpca WG0bGaeyOeI0IaaGOnaaWdaeaapeGaamiDaiabgUcaRiaaiAdaa0Wd aeaapeGaeyyeIuoaaOGaamysa8aadaWgaaWcbaWdbiaadQgaa8aabe aak8qacaWGxbWdamaaDaaaleaapeGaamOAaaWdaeaapeWaaeWaa8aa baWdbiaadshaaiaawIcacaGLPaaaaaGccaWGzbWdamaaBaaaleaape GaamOAaaWdaeqaaaGcbaWdbmaavadabeWcpaqaa8qacaWGQbGaeyyp a0JaamiDaiabgkHiTiaaiAdaa8aabaWdbiaadshacqGHRaWkcaaI2a aan8aabaWdbiabggHiLdaakiaadMeapaWaaSbaaSqaa8qacaWGQbaa paqabaGcpeGaam4va8aadaqhaaWcbaWdbiaadQgaa8aabaWdbmaabm aapaqaa8qacaWG0baacaGLOaGaayzkaaaaaaaakiaacYcacaWLjaGa aCzcaiaacIcacaaIXaGaaiykaaaa@6212@

where Y j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGzbWdamaaBaaaleaapeGaamOAaaWdaeqaaaaa@383E@ is the value of the input series for month j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGQbaaaa@3706@ , available for j=1,,T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGQbGaeyypa0JaaGymaiaacYcacqGHMacVcaGGSaGaamivaaaa @3C8E@ . If the value of the input series for month j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGQbaaaa@3706@ is available, I j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGjbWdamaaBaaaleaapeGaamOAaaWdaeqaaaaa@382E@ is an indicator equal to 1. Otherwise, it is equal to 0. Applying this formula near the beginning or end of a series leads to undefined terms (e.g. Y 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGzbWdamaaBaaaleaapeGaaGimaaWdaeqaaaaa@3809@ ), discussed in Section 1.1. The quantities shown as W j ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGxbWdamaaDaaaleaapeGaamOAaaWdaeaapeWaaeWaa8aabaWd biaadshaaiaawIcacaGLPaaaaaaaaa@3AEE@ represent the moving average weights applied to month j  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGQbGaaiiOaaaa@382A@ for the calculation of the trend-cycle for month t. This is referred to as the cascade linear filter, as derived by Dagum and Luati (2009) and presented in Table 1.

Table 1  Full-precision moving average weights for calculating the trend-cycle for month t
Months Weights
t-6 and t+6 -0.027
t-5 and t+5 -0.007
t-4 and t+4 0.031
t-3 and t+3 0.067
t-2 and t+2 0.136
t-1 and t+1 0.188
t 0.224

Applying formula (1) for each month t=1,,T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG0bGaeyypa0JaaGymaiaacYcacqGHMacVcaGGSaGaamivaaaa @3C98@ yields the trend-cycle series.

1.1 Applying the general formula

The weights presented in Table 1 were developed by combining several filters that are each optimal for specific purposes. The combined results provide robust trend-cycle estimates with good statistical properties. For more information, see Dagum and Luati (2009).

By design, aggregating the weights in Table 1 for all months from (t-6) to (t+6) gives an exact total of 1. This is necessary so that the level of the trend-cycle series is the same on average as the level of the input series. Note that when we derive the trend-cycle estimates for the first six and the last six months of the input series, formula (1) includes terms that are not defined, such as Y0. However, these terms can be assumed to be 0 since they disappear when multiplied by the corresponding indicator coefficient, I0, which equals 0 by definition. In these cases, the denominator in formula (1) represents an adjustment to the moving average weights. Referred to as the cut-and-normalize approach, this ensures that the moving average weights used to estimate the trend-cycle for each month add up to 1. In the cut-and-normalize approach, the weights of the months for which data are unavailable are redistributed proportionally to the months for which data are available. For all other months, the denominator in formula (1) is equal to 1, and the formula is reduced to a simple symmetric moving average with the weights specified in Table 1.

1.2 Alternate expression

In formula (1), a cut-and-normalize approach is used to derive the moving average weights for the first and last six months of the input series. In effect, the cut-and-normalize approach to estimating the trend-cycle employs modified moving average weights to calculate the trend-cycle of the first six months of the series and the final six months. For example, when the trend-cycle estimate for the final month of a series is calculated, the values of the six subsequent months are not yet known. The cut-and-normalize approach proportionally rescales the weights for the months that are available so that their sum is 1. An equivalent expression to formula (1) is given in formula (2), which is based on the rescaled moving average weights, W ˜ j ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaWGQbaapaqaa8qadaqadaWd aeaapeGaamiDaaGaayjkaiaawMcaaaaaaaa@3AFD@ , given in formula (3).

Formula 2

T C t = j=t6 t+6 I j W ˜ j ( t ) Y j ,      (2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubGaam4qa8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacqGH 9aqpdaqfWaqabSqaaiaadQgacqGH9aqpcaWG0bGaeyOeI0IaaGOnaa qaaiaadshacqGHRaWkcaaI2aaaneaacqGHris5aaGccaWGjbWdamaa BaaaleaapeGaamOAaaWdaeqaaOWdbiqadEfapaGbaGaadaqhaaWcba WdbiaadQgaa8aabaWdbmaabmaapaqaa8qacaWG0baacaGLOaGaayzk aaaaaOGaamywa8aadaWgaaWcbaWdbiaadQgaa8aabeaak8qacaGGSa GaaCzcaiaaxMaacaGGOaGaaGOmaiaacMcaaaa@50DE@

Formula 3

where W ˜ j ( t ) =  W j ( t ) k=t6 t+6 I k W k ( t ) .      (3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG3bGaaeiAaiaabwgacaqGYbGaaeyzaiaaysW7caaMe8UaaGjb VlqadEfapaGbaGaadaqhaaWcbaWdbiaadQgaa8aabaWdbmaabmaapa qaa8qacaWG0baacaGLOaGaayzkaaaaaOGaeyypa0JaaeiOamaalaaa paqaa8qacaWGxbWdamaaDaaaleaapeGaamOAaaWdaeaapeWaaeWaa8 aabaWdbiaadshaaiaawIcacaGLPaaaaaaak8aabaWdbmaavadabeWc paqaa8qacaWGQbGaeyypa0JaamiDaiabgkHiTiaaiAdaa8aabaWdbi aadshacqGHRaWkcaaI2aaan8aabaWdbiabggHiLdaakiaadMeapaWa aSbaaSqaa8qacaWGQbaapaqabaGcpeGaam4va8aadaqhaaWcbaWdbi aadQgaa8aabaWdbmaabmaapaqaa8qacaWG0baacaGLOaGaayzkaaaa aaaakiaac6cacaWLjaGaaCzcaiaacIcacaaIZaGaaiykaaaa@6073@

2.0 Illustration

To illustrate how to compute and apply the moving average weights, we consider a monthly series, Y t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGzbWdamaaBaaaleaapeGaamiDaaWdaeqaaaaa@3848@ , that spans from January 2010 to July 2015—a total of 67 months. The rescaled moving average weights in the formulas below have been rounded to six decimal places. This rounding will sometimes lead to an overestimation or an underestimation at the beginning and end of the series. Because of rounding errors, the trend-cycle obtained using these rounded weights will not be precise within these sections. The full-precision weights must be used to reproduce published trend-cycle estimates exactly.

Three examples below illustrate trend-cycle estimates near the beginning, middle and end of the series. These demonstrate the application of the moving averages when not all months of the 13-term moving average are available (examples 1 and 3), as well as when they are (example 2).

Example 1

For the March 2010 trend-cycle estimate (t=3 of the series), the Y j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGzbWdamaaBaaaleaapeGaamOAaaWdaeqaaaaa@383E@ values are available only for j=1,...,9, which correspond to January 2010 to September 2010. Therefore, a nine-term moving average is used.

According to formula (3), the rescaled weight applied to January 2010 for this moving average is

W ˜ 1 ( 3 ) = 0.136 ( 0.136+0.188+0.224+0.188+0.136+0.067+0.0310.0070.027 )  0.145299. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaIXaaapaqaa8qadaqadaWd aeaapeGaaG4maaGaayjkaiaawMcaaaaakiabg2da9maalaaapaqaa8 qacaaIWaGaaiOlaiaaigdacaaIZaGaaGOnaaWdaeaapeWaaeWaa8aa baWdbiaaicdacaGGUaGaaGymaiaaiodacaaI2aGaey4kaSIaaGimai aac6cacaaIXaGaaGioaiaaiIdacqGHRaWkcaaIWaGaaiOlaiaaikda caaIYaGaaGinaiabgUcaRiaaicdacaGGUaGaaGymaiaaiIdacaaI4a Gaey4kaSIaaGimaiaac6cacaaIXaGaaG4maiaaiAdacqGHRaWkcaaI WaGaaiOlaiaaicdacaaI2aGaaG4naiabgUcaRiaaicdacaGGUaGaaG imaiaaiodacaaIXaGaeyOeI0IaaGimaiaac6cacaaIWaGaaGimaiaa iEdacqGHsislcaaIWaGaaiOlaiaaicdacaaIYaGaaG4naaGaayjkai aawMcaaaaacqGHijYUcaqGGcGaaGimaiaac6cacaaIXaGaaGinaiaa iwdacaaIYaGaaGyoaiaaiMdacaGGUaaaaa@7292@ .

Approximate values of the rescaled moving average weights, W ˜ j ( 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaWGQbaapaqaa8qadaqadaWd aeaapeGaamiDaaGaayjkaiaawMcaaaaaaaa@3AFD@ , calculated for the months included in the moving average to estimate the trend-cycle for month 3, are shown in Table 2.

Table 2  Rescaled moving average weights (approximate values) for March 2010
Reference month Weight Approximate value
j=1 (Jan. 2010) W ˜ 1 ( 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaIXaaapaqaa8qadaqadaWd aeaapeGaaG4maaGaayjkaiaawMcaaaaaaaa@3A8D@ 0.145299
j=2 (Feb. 2010) W ˜ 2 ( 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaIYaaapaqaa8qadaqadaWd aeaapeGaaG4maaGaayjkaiaawMcaaaaaaaa@3A8E@ 0.200855
j=3 (Mar. 2010) W ˜ 3 ( 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaIZaaapaqaa8qadaqadaWd aeaapeGaaG4maaGaayjkaiaawMcaaaaaaaa@3A8F@ 0.239316
j=4 (Apr. 2010) W ˜ 4 ( 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaI0aaapaqaa8qadaqadaWd aeaapeGaaG4maaGaayjkaiaawMcaaaaaaaa@3A90@ 0.200855
j=5 (May 2010) W ˜ 5 ( 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaI1aaapaqaa8qadaqadaWd aeaapeGaaG4maaGaayjkaiaawMcaaaaaaaa@3A91@ 0.145299
j=6 (June 2010) W ˜ 6 ( 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaI2aaapaqaa8qadaqadaWd aeaapeGaaG4maaGaayjkaiaawMcaaaaaaaa@3A92@ 0.071581
j=7 (July 2010) W ˜ 7 ( 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaI3aaapaqaa8qadaqadaWd aeaapeGaaG4maaGaayjkaiaawMcaaaaaaaa@3A93@ 0.033120
j=8 (Aug. 2010) W ˜ 8 ( 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaI4aaapaqaa8qadaqadaWd aeaapeGaaG4maaGaayjkaiaawMcaaaaaaaa@3A94@ -0.007479
j=9 (Sept. 2010) W ˜ 9 ( 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaI5aaapaqaa8qadaqadaWd aeaapeGaaG4maaGaayjkaiaawMcaaaaaaaa@3A95@ -0.028846

Finally, applying formula (2) to the input series Y j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGzbWdamaaBaaaleaapeGaamOAaaWdaeqaaaaa@383E@ gives the following expression for T C 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubGaam4qa8aadaWgaaWcbaWdbiaaiodaa8aabeaaaaa@38CF@ , the trend-cycle estimate for March 2010:

T C 3 = Y 1 *( 0.145299 )+  Y 2 *( 0.200855 )+  Y 3 *( 0.239316 )+  Y 4 *( 0.200855 )+  Y 5 *( 0.145299 )+ Y 6 *( 0.071581 )+  Y 7 *( 0.033120 )+  Y 8 *( 0.007479 ) +  Y 9 *( 0.028846 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqaaaaaaaaaWdbe aafaqabeWabaaabaGaamivaiaadoeapaWaaSbaaSqaa8qacaaIZaaa paqabaGcpeGaeyypa0Jaamywa8aadaWgaaWcbaWdbiaaigdaa8aabe aak8qacaqGQaWaaeWaa8aabaWdbiaaicdacaGGUaGaaGymaiaaisda caaI1aGaaGOmaiaaiMdacaaI5aaacaGLOaGaayzkaaGaey4kaSIaae iOaiaadMfapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaaeOkamaa bmaapaqaa8qacaaIWaGaaiOlaiaaikdacaaIWaGaaGimaiaaiIdaca aI1aGaaGynaaGaayjkaiaawMcaaiabgUcaRiaabckacaWGzbWdamaa BaaaleaapeGaaG4maaWdaeqaaOWdbiaabQcadaqadaWdaeaapeGaaG imaiaac6cacaaIYaGaaG4maiaaiMdacaaIZaGaaGymaiaaiAdaaiaa wIcacaGLPaaacqGHRaWkcaqGGcGaamywa8aadaWgaaWcbaWdbiaais daa8aabeaak8qacaqGQaWaaeWaa8aabaWdbiaaicdacaGGUaGaaGOm aiaaicdacaaIWaGaaGioaiaaiwdacaaI1aaacaGLOaGaayzkaaGaey 4kaSIaaeiOaiaadMfapaWaaSbaaSqaa8qacaaI1aaapaqabaaak8qa baGaaeOkamaabmaapaqaa8qacaaIWaGaaiOlaiaaigdacaaI0aGaaG ynaiaaikdacaaI5aGaaGyoaaGaayjkaiaawMcaaiabgUcaRiaadMfa paWaaSbaaSqaa8qacaaI2aaapaqabaGcpeGaaeOkamaabmaapaqaa8 qacaaIWaGaaiOlaiaaicdacaaI3aGaaGymaiaaiwdacaaI4aGaaGym aaGaayjkaiaawMcaaiabgUcaRiaabckacaWGzbWdamaaBaaaleaape GaaG4naaWdaeqaaOWdbiaabQcadaqadaWdaeaapeGaaGimaiaac6ca caaIWaGaaG4maiaaiodacaaIXaGaaGOmaiaaicdaaiaawIcacaGLPa aacqGHRaWkcaqGGcGaamywa8aadaWgaaWcbaWdbiaaiIdaa8aabeaa k8qacaqGQaWaaeWaa8aabaWdbiabgkHiTiaaicdacaGGUaGaaGimai aaicdacaaI3aGaaGinaiaaiEdacaaI5aaacaGLOaGaayzkaaaabaGa ey4kaSIaaeiOaiaadMfapaWaaSbaaSqaa8qacaaI5aaapaqabaGcpe GaaeOkamaabmaapaqaa8qacqGHsislcaaIWaGaaiOlaiaaicdacaaI YaGaaGioaiaaiIdacaaI0aGaaGOnaaGaayjkaiaawMcaaaaaaaa@A74C@

Example 2

For the August 2012 trend-cycle estimate (t=32 of the series), the values of Y j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGzbWdamaaBaaaleaapeGaamOAaaWdaeqaaaaa@383E@ are available for each month from j=26,...,38, which correspond to February 2012 to February 2013. Therefore, a complete 13-term moving average is used. Because the denominator of formula (3) is exactly equal to 1 in this case, the rescaling has no effect and the weights used in the moving average are identical to the weights in Table 1.

The rescaled weight applied to August 2012 in this moving average is given by

W ˜ 32 ( 32 ) = 0.224  ( 0.0270.007+0.031+0.067+0.136+0.188+0.224+0.188+0.136+0.067+0.0310.0070.027 ) =0.224. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaIZaGaaGOmaaWdaeaapeWa aeWaa8aabaWdbiaaiodacaaIYaaacaGLOaGaayzkaaaaaOGaeyypa0 ZaaSaaa8aabaWdbiaaicdacaGGUaGaaGOmaiaaikdacaaI0aGaaeiO aaWdaeaapeWaaeWaa8aabaWdbiabgkHiTiaaicdacaGGUaGaaGimai aaikdacaaI3aGaeyOeI0IaaGimaiaac6cacaaIWaGaaGimaiaaiEda cqGHRaWkcaaIWaGaaiOlaiaaicdacaaIZaGaaGymaiabgUcaRiaaic dacaGGUaGaaGimaiaaiAdacaaI3aGaey4kaSIaaGimaiaac6cacaaI XaGaaG4maiaaiAdacqGHRaWkcaaIWaGaaiOlaiaaigdacaaI4aGaaG ioaiabgUcaRiaaicdacaGGUaGaaGOmaiaaikdacaaI0aGaey4kaSIa aGimaiaac6cacaaIXaGaaGioaiaaiIdacqGHRaWkcaaIWaGaaiOlai aaigdacaaIZaGaaGOnaiabgUcaRiaaicdacaGGUaGaaGimaiaaiAda caaI3aGaey4kaSIaaGimaiaac6cacaaIWaGaaG4maiaaigdacqGHsi slcaaIWaGaaiOlaiaaicdacaaIWaGaaG4naiabgkHiTiaaicdacaGG UaGaaGimaiaaikdacaaI3aaacaGLOaGaayzkaaaaaiabg2da9iaaic dacaGGUaGaaGOmaiaaikdacaaI0aaaaa@8372@

The rescaled moving average weights, W ˜ j ( 32 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaWGQbaapaqaa8qadaqadaWd aeaapeGaamiDaaGaayjkaiaawMcaaaaaaaa@3AFD@ , calculated for the months included in the moving average used to calculate the trend-cycle estimate for month 32, are presented in Table 3.

Table 3  Rescaled moving average weights to calculate trend-cycle for August 2012
Reference month(s) Weight Value
j=26 (Feb. 2012) and j=38 (Feb. 2013) W ˜ 26 ( 32 ) , W ˜ 38 ( 32 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaIYaGaaGOnaaWdaeaapeWa aeWaa8aabaWdbiaaiodacaaIYaaacaGLOaGaayzkaaaaaOGaaiilai qadEfapaGbaGaadaqhaaWcbaWdbiaaiodacaaI4aaapaqaa8qadaqa daWdaeaapeGaaG4maiaaikdaaiaawIcacaGLPaaaaaaaaa@42BA@ -0.027
j=27 (Mar. 2012) and j=37 (Jan. 2013) W ˜ 27 ( 32 ) , W ˜ 37 ( 32 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaIYaGaaG4naaWdaeaapeWa aeWaa8aabaWdbiaaiodacaaIYaaacaGLOaGaayzkaaaaaOGaaiilai qadEfapaGbaGaadaqhaaWcbaWdbiaaiodacaaI3aaapaqaa8qadaqa daWdaeaapeGaaG4maiaaikdaaiaawIcacaGLPaaaaaaaaa@42BA@ -0.007
j=28 (Apr. 2012) and j=36 (Dec. 2012) W ˜ 28 ( 32 ) , W ˜ 36 ( 32 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaIYaGaaGioaaWdaeaapeWa aeWaa8aabaWdbiaaiodacaaIYaaacaGLOaGaayzkaaaaaOGaaiilai qadEfapaGbaGaadaqhaaWcbaWdbiaaiodacaaI2aaapaqaa8qadaqa daWdaeaapeGaaG4maiaaikdaaiaawIcacaGLPaaaaaaaaa@42BA@ 0.031
j=29 (May 2012) and j=35 (Nov. 2012) W ˜ 29 ( 32 ) , W ˜ 35 ( 32 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaIYaGaaGyoaaWdaeaapeWa aeWaa8aabaWdbiaaiodacaaIYaaacaGLOaGaayzkaaaaaOGaaiilai qadEfapaGbaGaadaqhaaWcbaWdbiaaiodacaaI1aaapaqaa8qadaqa daWdaeaapeGaaG4maiaaikdaaiaawIcacaGLPaaaaaaaaa@42BA@ 0.067
j=30 (June 2012) and j=34 (Oct. 2012) W ˜ 30 ( 32 ) , W ˜ 34 ( 32 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaIZaGaaGimaaWdaeaapeWa aeWaa8aabaWdbiaaiodacaaIYaaacaGLOaGaayzkaaaaaOGaaiilai qadEfapaGbaGaadaqhaaWcbaWdbiaaiodacaaI0aaapaqaa8qadaqa daWdaeaapeGaaG4maiaaikdaaiaawIcacaGLPaaaaaaaaa@42B1@ 0.136
j=31 (July 2012) and j=33 (Sept. 2012) W ˜ 31 ( 32 ) , W ˜ 33 ( 32 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaIZaGaaGymaaWdaeaapeWa aeWaa8aabaWdbiaaiodacaaIYaaacaGLOaGaayzkaaaaaOGaaiilai qadEfapaGbaGaadaqhaaWcbaWdbiaaiodacaaIZaaapaqaa8qadaqa daWdaeaapeGaaG4maiaaikdaaiaawIcacaGLPaaaaaaaaa@42B1@ 0.188
j=32 (Aug. 2012) W ˜ 32 ( 32 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaIZaGaaGOmaaWdaeaapeWa aeWaa8aabaWdbiaaiodacaaIYaaacaGLOaGaayzkaaaaaaaa@3C07@ 0.224

Applying formula (2) to the input series Y j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGzbWdamaaBaaaleaapeGaamOAaaWdaeqaaaaa@383D@ produces the following expression for the August 2012 trend-cycle estimate, T C 32 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubGaam4qa8aadaWgaaWcbaWdbiaaiodaa8aabeaaaaa@38CF@ :

T C 32 = Y 26 *( 0.027 )+ Y 27 *( 0.007 )+ Y 28 *( 0.031 )+ Y 29 *( 0.067 )+ Y 30 *( 0.136 )+ Y 31 *( 0.188 )+ Y 32 *( 0.224 )+ Y 33 *( 0.188 )+  Y 34 *( 0.136 )+ Y 35 *( 0.067 )+  Y 36 *( 0.031 )+ Y 37 *( 0.007 )+ Y 38 *( 0.027 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqaaaaaaaaaWdbe aafaqabeWabaaabaGaamivaiaadoeapaWaaSbaaSqaa8qacaaIZaGa aGOmaaWdaeqaaOWdbiabg2da9iaadMfapaWaaSbaaSqaa8qacaaIYa GaaGOnaaWdaeqaaOWdbiaabQcadaqadaWdaeaapeGaeyOeI0IaaGim aiaac6cacaaIWaGaaGOmaiaaiEdaaiaawIcacaGLPaaacqGHRaWkca WGzbWdamaaBaaaleaapeGaaGOmaiaaiEdaa8aabeaak8qacaqGQaWa aeWaa8aabaWdbiabgkHiTiaaicdacaGGUaGaaGimaiaaicdacaaI3a aacaGLOaGaayzkaaGaey4kaSIaamywa8aadaWgaaWcbaWdbiaaikda caaI4aaapaqabaGcpeGaaeOkamaabmaapaqaa8qacaaIWaGaaiOlai aaicdacaaIZaGaaGymaaGaayjkaiaawMcaaiabgUcaRiaadMfapaWa aSbaaSqaa8qacaaIYaGaaGyoaaWdaeqaaOWdbiaabQcadaqadaWdae aapeGaaGimaiaac6cacaaIWaGaaGOnaiaaiEdaaiaawIcacaGLPaaa cqGHRaWkcaWGzbWdamaaBaaaleaapeGaaG4maiaaicdaa8aabeaak8 qacaqGQaWaaeWaa8aabaWdbiaaicdacaGGUaGaaGymaiaaiodacaaI 2aaacaGLOaGaayzkaaGaey4kaSIaamywa8aadaWgaaWcbaWdbiaaio dacaaIXaaapaqabaaak8qabaGaaeOkamaabmaapaqaa8qacaaIWaGa aiOlaiaaigdacaaI4aGaaGioaaGaayjkaiaawMcaaiabgUcaRiaadM fapaWaaSbaaSqaa8qacaaIZaGaaGOmaaWdaeqaaOWdbiaabQcadaqa daWdaeaapeGaaGimaiaac6cacaaIYaGaaGOmaiaaisdaaiaawIcaca GLPaaacqGHRaWkcaWGzbWdamaaBaaaleaapeGaaG4maiaaiodaa8aa beaak8qacaqGQaWaaeWaa8aabaWdbiaaicdacaGGUaGaaGymaiaaiI dacaaI4aaacaGLOaGaayzkaaGaey4kaSIaaeiOaiaadMfapaWaaSba aSqaa8qacaaIZaGaaGinaaWdaeqaaOWdbiaabQcadaqadaWdaeaape GaaGimaiaac6cacaaIXaGaaG4maiaaiAdaaiaawIcacaGLPaaacqGH RaWkcaWGzbWdamaaBaaaleaapeGaaG4maiaaiwdaa8aabeaak8qaca qGQaWaaeWaa8aabaWdbiaaicdacaGGUaGaaGimaiaaiAdacaaI3aaa caGLOaGaayzkaaGaey4kaSIaaeiOaiaadMfapaWaaSbaaSqaa8qaca aIZaGaaGOnaaWdaeqaaaGcpeqaaiaabQcadaqadaWdaeaapeGaaGim aiaac6cacaaIWaGaaG4maiaaigdaaiaawIcacaGLPaaacqGHRaWkca WGzbWdamaaBaaaleaapeGaaG4maiaaiEdaa8aabeaak8qacaqGQaWa aeWaa8aabaWdbiabgkHiTiaaicdacaGGUaGaaGimaiaaicdacaaI3a aacaGLOaGaayzkaaGaey4kaSIaamywa8aadaWgaaWcbaWdbiaaioda caaI4aaapaqabaGcpeGaaeOkamaabmaapaqaa8qacqGHsislcaaIWa GaaiOlaiaaicdacaaIYaGaaG4naaGaayjkaiaawMcaaiaac6caaaaa aa@BE13@

Example 3

For the July 2015 trend-cycle estimate (t=67 of the series), the values of Y j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGzbWdamaaBaaaleaapeGaamOAaaWdaeqaaaaa@383E@ are known only for each month from j=61,...,67, which correspond to January 2015 to July 2015. Therefore, a seven-term moving average is used.

The rescaled weight applied to July 2015 in this moving average is

W ˜ 67 ( 67 ) = 0.224  ( 0.224+0.188+0.136+0.067+0.0310.0070.027 ) 0.366013. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaI2aGaaG4naaWdaeaapeWa aeWaa8aabaWdbiaaiAdacaaI3aaacaGLOaGaayzkaaaaaOGaeyypa0 ZaaSaaa8aabaWdbiaaicdacaGGUaGaaGOmaiaaikdacaaI0aGaaeiO aaWdaeaapeWaaeWaa8aabaWdbiaaicdacaGGUaGaaGOmaiaaikdaca aI0aGaey4kaSIaaGimaiaac6cacaaIXaGaaGioaiaaiIdacqGHRaWk caaIWaGaaiOlaiaaigdacaaIZaGaaGOnaiabgUcaRiaaicdacaGGUa GaaGimaiaaiAdacaaI3aGaey4kaSIaaGimaiaac6cacaaIWaGaaG4m aiaaigdacqGHsislcaaIWaGaaiOlaiaaicdacaaIWaGaaG4naiabgk HiTiaaicdacaGGUaGaaGimaiaaikdacaaI3aaacaGLOaGaayzkaaaa aiabgIKi7kaaicdacaGGUaGaaG4maiaaiAdacaaI2aGaaGimaiaaig dacaaIZaGaaiiOaaaa@6B6E@

The rescaled moving average weights, W j ( 67 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGxbWdamaaDaaaleaapeGaamOAaaWdaeaapeWaaeWaa8aabaWd biaadshaaiaawIcacaGLPaaaaaaaaa@3AEE@ , calculated for the other months included in the moving average used to calculate the trend-cycle estimate for month 67, are presented in Table 4.

Table 4  Rescaled moving average weights (approximate values) for July 2015
Reference month Weight Approximate value
j=61 (Jan. 2015) W ˜ 61 ( 67 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaI2aGaaGymaaWdaeaapeWa aeWaa8aabaWdbiaaiAdacaaI3aaacaGLOaGaayzkaaaaaaaa@3C11@ -0.044118
j=62 (Feb. 2015) W ˜ 62 ( 67 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaI2aGaaGOmaaWdaeaapeWa aeWaa8aabaWdbiaaiAdacaaI3aaacaGLOaGaayzkaaaaaaaa@3C12@ -0.011438
j=63 (Mar. 2015) W ˜ 63 ( 67 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaI2aGaaG4maaWdaeaapeWa aeWaa8aabaWdbiaaiAdacaaI3aaacaGLOaGaayzkaaaaaaaa@3C13@ 0.050654
j=64 (Apr. 2015) W ˜ 64 ( 67 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaI2aGaaGinaaWdaeaapeWa aeWaa8aabaWdbiaaiAdacaaI3aaacaGLOaGaayzkaaaaaaaa@3C14@ 0.109477
j=65 (May 2015) W ˜ 65 ( 67 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaI2aGaaGynaaWdaeaapeWa aeWaa8aabaWdbiaaiAdacaaI3aaacaGLOaGaayzkaaaaaaaa@3C15@ 0.222222
j=66 (June 2015) W ˜ 66 ( 67 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaI2aGaaGOnaaWdaeaapeWa aeWaa8aabaWdbiaaiAdacaaI3aaacaGLOaGaayzkaaaaaaaa@3C16@ 0.307190
j=67 (July 2015) W ˜ 67 ( 67 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaaI2aGaaG4naaWdaeaapeWa aeWaa8aabaWdbiaaiAdacaaI3aaacaGLOaGaayzkaaaaaaaa@3C17@ 0.366013

Applying formula (2) to the input series Y j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGzbWdamaaBaaaleaapeGaamOAaaWdaeqaaaaa@383E@ gives the following expression for the July 2015 trend-cycle estimate, T C 67 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubGaam4qa8aadaWgaaWcbaWdbiaaiAdacaaI3aaapaqabaaa aa@3993@ :

T C 67 =  Y 61 *( 0.044118 ) +  Y 62 *( 0.011438 ) +  Y 63 *( 0.050654 ) +  Y 64 *( 0.109477 ) +  Y 65 *( 0.222222 ) +  Y 66 *( 0.307190 ) +  Y 67 *( 0.366013 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqaaaaaaaaaWdbe aafaqabeGabaaabaGaamivaiaadoeapaWaaSbaaSqaa8qacaaI2aGa aG4naaWdaeqaaOWdbiabg2da9iaacckacaWGzbWdamaaBaaaleaape GaaGOnaiaaigdaa8aabeaak8qacaqGQaWaaeWaa8aabaWdbiabgkHi TiaaicdacaGGUaGaaGimaiaaisdacaaI0aGaaGymaiaaigdacaaI4a aacaGLOaGaayzkaaGaaeiOaiabgUcaRiaabckacaWGzbWdamaaBaaa leaapeGaaGOnaiaaikdaa8aabeaak8qacaqGQaWaaeWaa8aabaWdbi abgkHiTiaaicdacaGGUaGaaGimaiaaigdacaaIXaGaaGinaiaaioda caaI4aaacaGLOaGaayzkaaGaaeiOaiabgUcaRiaabckacaWGzbWdam aaBaaaleaapeGaaGOnaiaaiodaa8aabeaak8qacaqGQaWaaeWaa8aa baWdbiaaicdacaGGUaGaaGimaiaaiwdacaaIWaGaaGOnaiaaiwdaca aI0aaacaGLOaGaayzkaaGaaeiOaiabgUcaRiaabckacaWGzbWdamaa BaaaleaapeGaaGOnaiaaisdaa8aabeaak8qacaqGQaWaaeWaa8aaba WdbiaaicdacaGGUaGaaGymaiaaicdacaaI5aGaaGinaiaaiEdacaaI 3aaacaGLOaGaayzkaaaabaGaey4kaSIaaeiOaiaadMfapaWaaSbaaS qaa8qacaaI2aGaaGynaaWdaeqaaOWdbiaabQcadaqadaWdaeaapeGa aGimaiaac6cacaaIYaGaaGOmaiaaikdacaaIYaGaaGOmaiaaikdaai aawIcacaGLPaaacaqGGcGaey4kaSIaaeiOaiaadMfapaWaaSbaaSqa a8qacaaI2aGaaGOnaaWdaeqaaOWdbiaabQcadaqadaWdaeaapeGaaG imaiaac6cacaaIZaGaaGimaiaaiEdacaaIXaGaaGyoaiaaicdaaiaa wIcacaGLPaaacaqGGcGaey4kaSIaaeiOaiaadMfapaWaaSbaaSqaa8 qacaaI2aGaaG4naaWdaeqaaOWdbiaabQcadaqadaWdaeaapeGaaGim aiaac6cacaaIZaGaaGOnaiaaiAdacaaIWaGaaGymaiaaiodaaiaawI cacaGLPaaacaGGUaaaaaaa@9D3D@

3.0 Exact weights

A summary of the rescaled moving average weights, W ˜ j ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaiaWaa0baaSqaa8qacaWGQbaapaqaa8qadaqadaWd aeaapeGaamiDaaGaayjkaiaawMcaaaaaaaa@3AFD@ , that are used in calculating the trend-cycle estimates over the entire series are given in Table 5.

Table 5  Rescaled weights for each month, derived under the cut-and-normalize approach
  j=t-6 j=t-5 j=t-4 j=t-3 j=t-2 j=t-1 j=t j=t+1 j=t+2 j=t+3 j=t+4 j=t+5 j=t+6
t=1 0 0 0 0 0 0 =(0.224) / (0.612) =(0.188) / (0.612) =(0.136) / (0.612) =(0.067) / (0.612) =(0.031) / (0.612) =(-0.007) / (0.612) =(-0.027) / (0.612)
t=2 0 0 0 0 0 =(0.188) / (0.8) =(0.224) / (0.8) =(0.188) / (0.8) =(0.136) / (0.8) =(0.067) / (0.8) =(0.031) / (0.8) =(-0.007) / (0.8) =(-0.027) / (0.8)
t=3 0 0 0 0 =(0.136) / (0.936) =(0.188) / (0.936) =(0.224) / (0.936) =(0.188) / (0.936) =(0.136) / (0.936) =(0.067) / (0.936) =(0.031) / (0.936) =(-0.007) / (0.936) =(-0.027) / (0.936)
t=4 0 0 0 =(0.067) / (1.003) =(0.136) / (1.003) =(0.188) / (1.003) =(0.224) / (1.003) =(0.188) / (1.003) =(0.136) / (1.003) =(0.067) / (1.003) =(0.031) / (1.003) =(-0.007) / (1.003) =(-0.027) / (1.003)
t=5 0 0 =(0.031) / (1.034) =(0.067) / (1.034) =(0.136) / (1.034) =(0.188) / (1.034) =(0.224) / (1.034) =(0.188) / (1.034) =(0.136) / (1.034) =(0.067) / (1.034) =(0.031) / (1.034) =(-0.007) / (1.034) =(-0.027) / (1.034)
t=6 0 =(-0.007) / (1.027) =(0.031) / (1.027) =(0.067) / (1.027) =(0.136) / (1.027) =(0.188) / (1.027) =(0.224) / (1.027) =(0.188) / (1.027) =(0.136) / (1.027) =(0.067) / (1.027) =(0.031) / (1.027) =(-0.007) / (1.027) =(-0.027) / (1.027)
t=7,…,T-6 -0.027 -0.007 0.031 0.067 0.136 0.188 0.224 0.188 0.136 0.067 0.031 -0.007 -0.027
t=T-5 =(-0.027) / (1.027) =(-0.007) / (1.027) =(0.031) / (1.027) =(0.067) / (1.027) =(0.136) / (1.027) =(0.188) / (1.027) =(0.224) / (1.027) =(0.188) / (1.027) =(0.136) / (1.027) =(0.067) / (1.027) =(0.031) / (1.027) =(-0.007) / (1.027) 0
t=T-4 =(-0.027) / (1.034) =(-0.007) / (1.034) =(0.031) / (1.034) =(0.067) / (1.034) =(0.136) / (1.034) =(0.188) / (1.034) =(0.224) / (1.034) =(0.188) / (1.034) =(0.136) / (1.034) =(0.067) / (1.034) =(0.031) / (1.034) 0 0
t=T-3 =(-0.027) / (1.003) =(-0.007) / (1.003) =(0.031) / (1.003) =(0.067) / (1.003) =(0.136) / (1.003) =(0.188) / (1.003) =(0.224) / (1.003) =(0.188) / (1.003) =(0.136) / (1.003) =(0.067) / (1.003) 0 0 0
t=T-2 =(-0.027) / (0.936) =(-0.007) / (0.936) =(0.031) / (0.936) =(0.067) / (0.936) =(0.136) / (0.936) =(0.188) / (0.936) =(0.224) / (0.936) =(0.188) / (0.936) =(0.136) / (0.936) 0 0 0 0
t=T-1 =(-0.027) / (0.8) =(-0.007) / (0.8) =(0.031) / (0.8) =(0.067) / (0.8) =(0.136) / (0.8) =(0.188) / (0.8) =(0.224) / (0.8) =(0.188) / (0.8) 0 0 0 0 0
t=T =(-0.027) / (0.612) =(-0.007) / (0.612) =(0.031) / (0.612) =(0.067) / (0.612) =(0.136) / (0.612) =(0.188) / (0.612) =(0.224) / (0.612) 0 0 0 0 0 0

References

Dagum, E.B., and Luati, A. 2009. “A cascade linear filter to reduce revisions and false turning points for real time trend-cycle estimation.” Econometric Reviews 28 (1–3): 40–59.