# Trend-cycle estimates: Frequently asked questions

By Susie Fortier, Steve Matthews and Guy Gellatly, Statistics Canada

Statistics Canada releases  graphical information on trend-cycle movements for several monthly economic indicators. Estimates of the trend-cycle are presented along with the seasonally adjusted data in selected charts in The Daily. The inclusion of trend-cycle information is intended to support the analysis and interpretation of the seasonally adjusted data.

This reference document provides information on trend-cycle data. It outlines basic concepts and definitions and discusses selected issues related to the use and interpretation of trend-cycle estimates. The document includes a specific example using data on monthly retail sales. Detailed information on the computation of the trend-cycle is also provided.

## 1. What is the trend-cycle of a time series?

Trend-cycle data represent a smoothed version of a seasonally adjusted time series. They provide information on longer-term movements, including changes in direction underlying the series.

The trend-cycle is the combination of two distinct components:

• The trend provides information on longer-term movements in the seasonally adjusted data series over several years.
• The cycle is a sequence of smoother fluctuations around the longer-term trend in part characterized by alternating periods of expansion and contraction.

Changes in trend-cycle data reflect the influence of factors that condition long-run 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 trend-cycle?

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 trend-cycle 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 trend-cycle 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 trend-cycle, 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 period-to-period movements in a seasonally adjusted time series.

By removing this irregular component from seasonally adjusted data, the trend-cycle data can yield a better picture of longer-term movements in the time series. In this sense, the trend-cycle can be interpreted as a smoothed version of the seasonally adjusted series.

## 3. What can we learn from trend-cycles?

Trend-cycle data provide information on longer-term 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 trend-cycle estimates. The trend-cycle 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 trend-cycle line.  The trend-cycle 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 mid-2014, before sales trended downward in late 2014. Trend-cycle data for early 2015 indicated a return to growth. Estimates for this most recent period are based on a preliminary estimation of the trend-cycle and should be interpreted with caution as they are subject to revision as noted above.

### Figure 1 — Retail sales

Trend-cycle data are particularly useful when the irregular component makes large contributions to the month-to-month movements in a seasonally adjusted time series. In these cases, graphical information on the trend-cycle helps to interpret the movements in the seasonally adjusted series.

## 4. Why are trend-cycle data revised?

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

## 5. Why is the trend-cycle line dotted for the most recent reference months?

The trend-cycle 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 trend-cycle 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 trend-cycle be interpreted as a means of forecasting data for future reference periods?

The trend-cycle 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 trend-cycle is subject to revision when additional reference periods are added to the series, the shape of the trend-cycle in the most recent reference periods should be viewed as a preliminary estimate.

## 7. What methods can be used to estimate the trend-cycle series?

There is no unique method that is recommended to estimate the trend-cycle 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 trend-cycle 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 trend-cycle can also be interpreted as a smoothed version of the seasonally adjusted series, a straightforward way of estimating the trend-cycle is by averaging the last three or six months of the data.   While this may yield additional insight into the long-term movement in the series, some measure of caution is warranted as this approach does not take the place of more formal trend-cycle 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 trend-cycle series?

Statistics Canada uses a weighted moving average of the data to compute the trend-cycle. 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 trend-cycle estimates will be revised.

This specific weighted moving average method was selected after an empirical analysis of different alternatives. The estimate of the trend-cycle 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 trend-cycle plotted on employment for men and women separately will sum up to the plotted trend-cycle 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 trend-cycle method work in a more technical sense?

The trend-cycle 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 trend-cycle 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 13-month symmetric moving average are not all available. The trend-cycle 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 trend-cycle estimate $T{C}_{t}$ for time $t=1,...T,$ can be derived as

$T C t = ∑ m=−6 6 w t+m S A t , t=1,...,T.$

where $S{A}_{t}$is the seasonally adjusted estimate at time $t=1,...T,$ ${w}_{t+m}$ and  are the weights of the moving average. Weights are shown in Table 1 for the various values of t.

Weights used in the moving average Footnote 1 ${w}_{t-6}$ ${w}_{t-5}$ ${w}_{t-4}$ ${w}_{t-3}$ ${w}_{t-2}$ ${w}_{t-1}$ ${w}_{t}$ ${w}_{t+1}$ ${w}_{t+2}$ ${w}_{t+3}$ ${w}_{t+4}$ ${w}_{t+5}$ Footnote 1 Rounded Dagum and Luati weights are shown for illustrative purpose. Final full-precision weights used in production are available upon request. Return to footnote 1 referrer 0 0 0 0 0 0 0.342 0.386 0.200 0.075 0.039 -0.009 -0.032 0 0 0 0 0 0.264 0.331 0.196 0.114 0.089 0.034 -0.006 -0.021 0 0 0 0 0.173 0.255 0.182 0.148 0.141 0.090 0.037 -0.003 -0.024 0 0 0 0.085 0.168 0.155 0.166 0.180 0.146 0.093 0.035 -0.005 -0.023 0 0 0.017 0.085 0.116 0.160 0.195 0.186 0.148 0.088 0.033 -0.004 -0.023 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 -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 -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 -0.023 -0.004 0.033 0.088 0.148 0.186 0.195 0.160 0.116 0.085 0.017 0 0 -0.023 -0.005 0.035 0.093 0.146 0.180 0.166 0.155 0.168 0.085 0 0 0 -0.024 -0.003 0.037 0.090 0.141 0.148 0.182 0.255 0.173 0 0 0 0 -0.021 -0.006 0.034 0.089 0.114 0.196 0.331 0.264 0 0 0 0 0 -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 trend-cycle estimates from the seventh month (July 2001) to the seventh from last month (September 2014) will be calculated using the symmetric weights. The trend-cycle estimate for September 2014 is computed as

$-0.027×S{A}_{Mar2014}-0.007×S{A}_{Apr2014}+0.031×S{A}_{May2014}+0.067×S{A}_{Jun2014}+0.136×S{A}_{Jul2014}+0.188×S{A}_{Aug2014}+0.224×S{A}_{Sep2014}$
$+0.188×S{A}_{Oct2014}+0.136×S{A}_{Nov2014}+0.067×S{A}_{Dec2014}+0.031×S{A}_{Jan2015}-0.007×S{A}_{Feb2015}-0.027×S{A}_{Mar2015}$.

Using the weights shown in the last line of the table, the trend-cycle estimate for the most current data point March 2015 is computed as

$-0.032×S{A}_{Sep2014}-0.009×S{A}_{Oct2014}+0.039×S{A}_{Nov2014}+0.075×S{A}_{Dec2014}+0.200×S{A}_{Jan2015}+0.386×S{A}_{Feb2015}+0.342×S{A}_{Mar2015}$.