By Susie Fortier, Steve Matthews and Guy Gellatly, Statistics Canada
Statistics Canada releases graphical information on trendcycle movements for several monthly economic indicators. Estimates of the trendcycle are presented along with the seasonally adjusted data in selected charts in The Daily. The inclusion of trendcycle information is intended to support the analysis and interpretation of the seasonally adjusted data.
This reference document provides information on trendcycle data. It outlines basic concepts and definitions and discusses selected issues related to the use and interpretation of trendcycle estimates. The document includes a specific example using data on monthly retail sales. Detailed information on the computation of the trendcycle is also provided.

1. What is the trendcycle of a time series?
Trendcycle data represent a smoothed version of a seasonally adjusted time series. They provide information on longerterm movements, including changes in direction underlying the series.
The trendcycle is the combination of two distinct components:
 The trend provides information on longerterm movements in the seasonally adjusted data series over several years.
 The cycle is a sequence of smoother fluctuations around the longerterm trend in part characterized by alternating periods of expansion and contraction.
Changes in trendcycle data reflect the influence of factors that condition longrun movements in the economic indicator over time, along with fluctuations in economic activity associated with the business cycle. These two components, the trend and the cycle, are often paired together because of the difficulty involved in estimating them individually.

2. What is the difference between a seasonally adjusted series and its trendcycle?
A seasonally adjusted data series is a series that has been modified to eliminate the effect of seasonal and calendar influences in order to facilitate comparisons of underlying conditions from period to period. Seasonally adjusted data series can also be defined as the combination of the trendcycle and the irregular component of a time series.
In much the same way as a seasonally adjusted series represents the raw series with seasonal and calendar effects removed, the trendcycle estimates represent the seasonally adjusted series with the irregular component removed. As its name suggests, the irregular component is the part of the time series that is not in line with the usual or expected pattern of the series. This irregular component is not part of the trendcycle, nor is it related to current seasonal factors or calendar effects.
The irregular component of a time series can represent unanticipated economic events or shocks (for example, strikes, disruptions, natural disasters, unseasonable weather, etc.) or can simply arise from noise in the measurement of the unadjusted data. In some cases, this irregular component can make large contributions to the periodtoperiod movements in a seasonally adjusted time series.
By removing this irregular component from seasonally adjusted data, the trendcycle data can yield a better picture of longerterm movements in the time series. In this sense, the trendcycle can be interpreted as a smoothed version of the seasonally adjusted series.

3. What can we learn from trendcycles?
Trendcycle data provide information on longerterm movements in a seasonally adjusted time series, including changes in the direction of the data. These smoothed data make it easier to identify periods of positive change (growth) or negative change (decline) in the time series, as the noise of the irregular component has been removed. This allows for a more accurate identification of turning points in the data.
For example, the accompanying graph presents data on monthly retail sales in Canada from July 2010 to July 2015. Two data lines are shown: the seasonally adjusted time series and the trendcycle estimates. The trendcycle estimates for the most recent reference months are more subject to revision than the estimates for previous periods, and are presented as a dotted line (see question 5).
While the seasonally adjusted data can be used to examine basic changes in the direction of the time series, it is easier to see the longer term movement in these data from the trendcycle line. The trendcycle estimates show that retail sales trended upward at a relatively constant rate during 2010 and 2011, and then slowed in 2012. Growth resumed from late 2012 until mid2014, before sales trended downward in late 2014. Trendcycle data for early 2015 indicated a return to growth. Estimates for this most recent period are based on a preliminary estimation of the trendcycle and should be interpreted with caution as they are subject to revision as noted above.
Figure 1 — Retail sales
Sources: CANSIM tables 0800020 extracted on October 14, 2015; and trendcycle computations.
Description for Figure 1
Table 1 — Retail sales $ billion Seasonally adjusted Trendcycle *preliminary estimate July 2010 36.295 36.51 August 2010 36.515 36.64 September 2010 36.633 36.79 October 2010 36.880 36.97 November 2010 37.568 37.15 December 2010 37.393 37.30 January 2011 37.392 37.45 February 2011 37.438 37.55 March 2011 37.617 37.64 April 2011 37.755 37.73 May 2011 37.724 37.81 June 2011 38.228 37.92 July 2011 37.926 38.03 August 2011 37.977 38.18 September 2011 38.182 38.34 October 2011 38.624 38.54 November 2011 38.780 38.74 December 2011 39.088 38.89 January 2012 39.069 38.99 February 2012 38.942 39.02 March 2012 39.179 39.00 April 2012 38.906 38.94 May 2012 38.774 38.90 June 2012 38.798 38.89 July 2012 38.901 38.91 August 2012 38.918 38.96 September 2012 39.083 39.04 October 2012 39.203 39.14 November 2012 39.314 39.22 December 2012 39.041 39.31 January 2013 39.467 39.44 February 2013 39.673 39.56 March 2013 39.731 39.72 April 2013 39.624 39.88 May 2013 40.337 40.06 June 2013 40.078 40.25 July 2013 40.428 40.41 August 2013 40.612 40.54 September 2013 40.802 40.67 October 2013 40.689 40.73 November 2013 40.929 40.80 December 2013 40.627 40.88 January 2014 40.987 41.00 February 2014 41.196 41.19 March 2014 41.196 41.41 April 2014 41.766 41.70 May 2014 41.840 41.98 June 2014 42.591 42.27 July 2014 42.585 42.48 August 2014 42.419 42.59 September 2014 42.799 42.61 October 2014 42.619 42.55 November 2014 42.886 42.43 December 2014 42.124 42.28 January 2015 41.523 42.22 February 2015 42.184 42.30 March 2015 42.585 42.45 April 2015 42.564 42.63* May 2015 42.937 42.82* June 2015 43.129 43.00* July 2015 43.345 43.16* Trendcycle data are particularly useful when the irregular component makes large contributions to the monthtomonth movements in a seasonally adjusted time series. In these cases, graphical information on the trendcycle helps to interpret the movements in the seasonally adjusted series.

4. Why are trendcycle data revised?
Existing estimates of the trendcycle are revised with each release of new seasonally adjusted data. As new seasonally adjusted data becomes available, the trendcycle data for previous months can be better estimated. If the trendcycle data were not revised along with the seasonally adjusted series, the resulting trendcycle data could contain series breaks, and would likely be inconsistent with the seasonally adjusted series in terms of levels, periodtoperiod movements, or both. It is necessary to revise the trendcycle data to maintain their analytical value.

5. Why is the trendcycle line dotted for the most recent reference months?
The trendcycle line that is published graphically is dotted in the most recent reference periods, as these periods are more likely to be subject to revisions. This is done to signal that the trendcycle data in this period is a preliminary estimate, and subject to change as new data becomes available. New data make it possible to more accurately estimate the various components that make up the time series. These revisions can change the location of economic turning points, as well as reverse movements between individual months. These types of revisions are more likely to occur in the most recent reference months.

6. Can the trendcycle be interpreted as a means of forecasting data for future reference periods?
The trendcycle should not be viewed as a way to forecast the underlying seasonally adjusted data. These estimates are based solely on the historical values of the seasonally adjusted series and do not take into account any other information that could be used to project data for future reference periods. Furthermore, since the trendcycle is subject to revision when additional reference periods are added to the series, the shape of the trendcycle in the most recent reference periods should be viewed as a preliminary estimate.

7. What methods can be used to estimate the trendcycle series?
There is no unique method that is recommended to estimate the trendcycle that underlies a time series. A variety of methods have been developed in the literature, ranging from very simple to highly complex. Some methods introduce restrictions on the shape of the trend (for example a linear trend of several years), others are based on explicit models that estimate a trendcycle component, and others, still, are based on variations of moving averages, where the mean of the data is calculated from successive sub spans or intervals of the data.
Since the trendcycle can also be interpreted as a smoothed version of the seasonally adjusted series, a straightforward way of estimating the trendcycle is by averaging the last three or six months of the data. While this may yield additional insight into the longterm movement in the series, some measure of caution is warranted as this approach does not take the place of more formal trendcycle estimation techniques. It can be shown that indicators of the economic cycle derived from this simplified method tend to shift in time and may be artificially dampened.

8. How does Statistics Canada estimate the trendcycle series?
Statistics Canada uses a weighted moving average of the data to compute the trendcycle. This method is based on the Cascade Linear Filter of Dagum and Luati (2008). This weighted average is computed using the previous six months, the current month and (for older estimates) up to six of the subsequent months in the series. In real time, for the most recent reference month in the series, only data for the six previous months and current month are used, as data for subsequent months are not yet known. As these data become available, the trendcycle estimates will be revised.
This specific weighted moving average method was selected after an empirical analysis of different alternatives. The estimate of the trendcycle obtained with the selected method exhibits good statistical properties, as it provides smooth results with limited revisions, and has a low incidence of falsely identifying turning points. As well, it is a linear process and will preserve additive relationship in the data. This implies, for example, that the trendcycle plotted on employment for men and women separately will sum up to the plotted trendcycle line for both sexes. The method is easy to replicate as the weights used in the calculation of the weighted average are available.

9. How does the trendcycle method work in a more technical sense?
The trendcycle is estimated by applying moving averages weighted according to the cascade linear filter to the seasonally adjusted series. In general, the moving average used to calculate the trendcycle for a specific reference month is a weighted average of up to 13 consecutive months, which are centered on the reference month—referred to as a symmetric moving average.
Near the beginning of a series, and the end of the series (the most recent reference month currently available), the seasonally adjusted data required to apply the 13month symmetric moving average are not all available. The trendcycle estimates for the final few months of a series are typically important in economic analysis, so these estimates are produced based on available months. For the final reference month in the series, seven months of data are used. A similar process is used to produce estimates for the initial reference periods in a series. The weights used in these situations are adjusted to counterbalance for the missing periods and are referred to as asymmetric weights.
Mathematically, the trendcycle estimate $T{C}_{t}$ for time $t=1,\mathrm{...}T,$ can be derived as
$$T{C}_{t}={\displaystyle \sum _{m=6}^{6}{w}_{t+m}S{A}_{t}},\text{\hspace{1em}}\text{\hspace{1em}}t=1,\mathrm{...},T.$$where $S{A}_{t}$is the seasonally adjusted estimate at time $t=1,\mathrm{...}T,$ ${w}_{t+m}$ and are the weights of the moving average. Weights are shown in Table 1 for the various values of t.
Table 1: Moving average weights used to extract a trendcycle estimate at time t for t=1, …T, from seasonally adjusted monthly data. Weights used in the moving average ^{Footnote 1} ${w}_{t6}$ ${w}_{t5}$ ${w}_{t4}$ ${w}_{t3}$ ${w}_{t2}$ ${w}_{t1}$ ${w}_{t}$ ${w}_{t+1}$ ${w}_{t+2}$ ${w}_{t+3}$ ${w}_{t+4}$ ${w}_{t+5}$ ${w}_{t+6}$  Footnote 1

Rounded Dagum and Luati weights are shown for illustrative purpose. Final fullprecision weights used in production are available upon request.
For t=1 0 0 0 0 0 0 0.342 0.386 0.200 0.075 0.039 0.009 0.032 For t=2 0 0 0 0 0 0.264 0.331 0.196 0.114 0.089 0.034 0.006 0.021 For t=3 0 0 0 0 0.173 0.255 0.182 0.148 0.141 0.090 0.037 0.003 0.024 For t=4 0 0 0 0.085 0.168 0.155 0.166 0.180 0.146 0.093 0.035 0.005 0.023 For t=5 0 0 0.017 0.085 0.116 0.160 0.195 0.186 0.148 0.088 0.033 0.004 0.023 For t=6 0  0.024 0.021 0.071 0.130 0.180 0.200 0.187 0.143 0.086 0.034 0.004 0.023 For t=7,…,T6
(symmetric)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 For t=T5 0.023 0.004 0.034 0.086 0.143 0.187 0.200 0.180 0.130 0.071 0.021 0.024 0 For t=T4 0.023 0.004 0.033 0.088 0.148 0.186 0.195 0.160 0.116 0.085 0.017 0 0 For t=T3 0.023 0.005 0.035 0.093 0.146 0.180 0.166 0.155 0.168 0.085 0 0 0 For t=T2 0.024 0.003 0.037 0.090 0.141 0.148 0.182 0.255 0.173 0 0 0 0 For t=T1 0.021 0.006 0.034 0.089 0.114 0.196 0.331 0.264 0 0 0 0 0 For t=T 0.032 0.009 0.039 0.075 0.200 0.386 0.342 0 0 0 0 0 0 For example, if the seasonally adjusted data for one series are available from January 2001 to March 2015, the trendcycle estimates from the seventh month (July 2001) to the seventh from last month (September 2014) will be calculated using the symmetric weights. The trendcycle estimate for September 2014 is computed as
$0.027\times S{A}_{Mar2014}0.007\times S{A}_{Apr2014}+0.031\times S{A}_{May2014}+0.067\times S{A}_{Jun2014}+0.136\times S{A}_{Jul2014}+0.188\times S{A}_{Aug2014}+0.224\times S{A}_{Sep2014}$
$+0.188\times S{A}_{Oct2014}+0.136\times S{A}_{Nov2014}+0.067\times S{A}_{Dec2014}+0.031\times S{A}_{Jan2015}0.007\times S{A}_{Feb2015}0.027\times S{A}_{Mar2015}$.Using the weights shown in the last line of the table, the trendcycle estimate for the most current data point March 2015 is computed as
$0.032\times S{A}_{Sep2014}0.009\times S{A}_{Oct2014}+0.039\times S{A}_{Nov2014}+0.075\times S{A}_{Dec2014}+0.200\times S{A}_{Jan2015}+0.386\times S{A}_{Feb2015}+0.342\times S{A}_{Mar2015}$.

10. How can I learn more about this topic?
The following references provide more information on the topic of seasonal adjustment, including trendcycle estimation.
Dagum, E. B. and Luati, A. 2008. "A Cascade Linear Filter to Reduce Revisions and False Turning Points for Real Time TrendCycle Estimation," Econometric Reviews. 28:13, 4059.
Statistics Canada. 2014. "Seasonally Adjusted Data — Frequently asked questions," Behind the data.
Statistics Canada. 2009. "Seasonal adjustment and trendcycle estimation," Statistics Canada Quality Guidelines. 5th edition. Catalogue no. 12539X.