
1. What is a time series?
A time series is a sequence of observations collected at regular time intervals. These data provide information on a welldefined statistical concept for a specific reference period, and are presented at different points in time. Most economic data disseminated by Statistics Canada are presented as a time series. Examples include the monthly data on consumer prices, retail sales, employment and gross domestic product. These data correspond to monthly reference periods that are available for a long sequence of months, to facilitate comparisons over time.

2. What is a seasonally adjusted time series?
Monthly or quarterly time series data are sometimes influenced by seasonal and calendar effects. These effects can bring about changes in the data that normally occur at the same time, and in about the same magnitude, every year. For example, monthly retail sales have historically been at their highest level for the year in December as a result of holiday shopping, and then declined to lower levels in January. This occurs year after year and affects the extent to which information on trends in retail industries can be informed by comparing raw sales data for these two months. A seasonally adjusted time series is a monthly or quarterly time series that has been modified to eliminate the effect of seasonal and calendar influences. The seasonally adjusted data allow for more meaningful comparisons of economic conditions from period to period. A raw time series is the equivalent series before seasonal adjustment and is sometimes referred to as the original or unadjusted time series.

3. Why is seasonal adjustment needed?
Many users of economic and social statistics rely on time series data to understand changes in socioeconomic phenomena over time. Important statistical properties of a time series include its direction and turning points, as well as its relationship to other socioeconomic indicators. A seasonal pattern in a series can obscure these important features by making periodtoperiod movements in the data more difficult to interpret. Many users of time series data do not consider movements in the data that relate to seasonal and other calendar effects to be analytically meaningful. These seasonal and calendar effects can obscure "true" underlying movements in the data series related to the business cycle, or to nonseasonal events, such as strikes or unanticipated disruptions in production. Consequently, seasonal adjustment techniques that remove the effect of seasonal and calendar influences from the original data can sharpen the extent to which a time series can be used to evaluate meaningful changes in economic conditions over time.

4. Is seasonal adjustment always required?
Seasonal adjustment may not always be appropriate or required. It is not necessary to seasonally adjust a series that does not exhibit an identifiable seasonal pattern or other calendarbased influences. It is also not always advisable to use seasonally adjusted data when the raw estimate represents the true statistic of interest. For example, decision makers who rely on the Consumer Price Index (CPI) for indexation purposes are advised to use unadjusted data—as these reflect the actual price movements observed from periodtoperiod. However, data users who are more interested in analyzing underlying price trends in the economy are encouraged to use seasonally adjusted indexes.
Similarly, analysts who are interested in calculating the raw growth in the number of young adults working from April 2012 to May 2012 should examine the raw estimates for these two months, and calculate the difference. This monthtomonth change in raw employment might not yield much useful information about changes in the labour market conditions facing young adults if seasonal or calendar effects have a significant influence on employment levels in either or both months. However, the raw data show the extent to which actual employment for this group grew, or contracted, from April to May—which may be useful information for other purposes.

5. How common is seasonal adjustment at Statistics Canada?
Statistics Canada seasonally adjusts almost all of its major subannual economic indicators, including quarterly and monthly estimates of gross domestic product, and monthly employment estimates from the Labour Force Survey. Although the vast majority of the agency's releases highlight seasonally adjusted data, both the seasonally adjusted series and unadjusted series are often made available.

6. How are seasonally adjusted data estimated?
Seasonally adjusted data are estimated by breaking down time series data into various components. Using wellestablished statistical techniques, this process involves decomposing a time series into four separate components: (1) the trendcycle, (2) seasonal effects, (3) other calendar effects such as trading days and moving holidays, and (4) the irregular component. The seasonally adjusted series is the original time series with the estimated seasonal and calendar effects removed, or equivalently, the estimated combination of the trendcycle and the irregular components.

7. What are the time series components?
A time series can be split into four separate time series components: (1) the trendcycle, (2) seasonal effects, (3) other calendar effects such as trading days and moving holidays, and (4) the irregular component. Here is an overview of each:
The trendcycle: This represents the smoothed version of the time series and indicates its general pattern or direction. The trendcycle can be interpreted as the longterm movement in the time series, the result of different factors (or determinants) that condition longrun changes in the data over time. As its name suggests, the trendcycle also reflects periodic expansions and contractions in economic activity, such as those associated with the business cycle.
Seasonal effects: These represent regular movements or patterns in time series data that occur in the same month or quarter every year. On the basis of past movements of the time series, these regular patterns repeat themselves from year to year. These seasonal patterns are fairly stable in terms of timing, direction and magnitude. Often these seasonal effects relate to wellestablished calendarbased variations in economic activity, such as the increase in retail sales in the lead up to Christmas, or increases in construction employment in the spring. Seasonal effects identify these regularly occurring patterns in the data.
Other calendar effects such as trading days and moving holidays: Aside from seasonal effects, other systematic calendarbased effects can influence the level of economic activity in a specific period. The most important of these are the tradingday effects. These effects can be present when the level of economic activity varies depending on the day of the week. For example, retail sales are usually higher on Saturdays than on any other day of the week. Consequently, a fiveSaturday month is more likely to result in higher retail sales than a month with only four Saturdays. Another common example of a calendar effect is the date of Easter, which can be expected to increase retail sales in March or April depending on the month in which it occurs. This particular calendar effect is referred to as a moving holiday effect.
The irregular component: This component includes unanticipated movements in the data that (1) are not part of the trendcycle, and (2) are not related to current seasonal factors or calendar effects. The irregular component could relate to unanticipated economic events or shocks (for example, strikes, disruptions, unseasonable weather, etc.), or can simply arise from noise in the measurement of the unadjusted data (due to sampling and nonsampling errors).

8. Which components are included and excluded in a seasonally adjusted series?
Seasonal effects and other calendar effects such as trading days and moving holidays are excluded from seasonally adjusted series. Consequently, the seasonally adjusted series is the combination of the trendcycle and the irregular component. The contribution of the irregular component is worth emphasizing, because seasonally adjusted data are sometimes misinterpreted as providing users with "pure" information on the trendcycle.

9. Why are raw and seasonally adjusted data revised over time?
The raw data can be revised to take into consideration additional data that were reported late, to correct data that were initially misreported, or for various other reasons. In such cases, the seasonally adjusted data that are based on unadjusted data also need to be revised.
Hindsight is very important for time series analysis. Even when the raw series has not been revised, it is often useful to revise the seasonally adjusted data. To estimate the seasonal effects at any given point in time, statisticians use information from previous, current and future observations. Information about future observations is not available in real time, so seasonal adjustment is conducted using previous and current values, along with projected values. These projections are based on a statistical model that uses past information. As new data becomes available, the various time series components can be estimated more accurately. This results in revised, more accurate estimates of the seasonally adjusted data.
Periodically, the methods used to estimate time series components for specific data series are also reviewed. Each statistical program at Statistics Canada has its own revision strategy, and schedules are routinely made available to data users in advance of these revisions.

10. Do yearoveryear comparisons of raw data work as well as more formal seasonal adjustment techniques?
Comparing raw data for the same period in each year provides information on longterm trends and economic cycles, but these comparisons do not necessarily remove all the seasonal patterns from the data. Certain holidays, like Easter, do not fall on the same date or even in the same month from year to year. If the timing of these holidays influences the variable being measured, such as monthly retail sales, raw yearoveryear comparisons can be misleading. For example, in 2013, Easter was on March 31st, whereas in 2012, it was on April 8th. Thus, it may be misleading to conclude that the change in sales from March 2012 to March 2013 reflects underlying trends in retail industries, as differences in sales may have been influenced by the timing of the Easter holiday.
Similarly, yearoveryear comparisons of raw data ignore the trading day effect, which occurs in many series, and can affect the validity of yearoveryear comparisons. For example, many businesses generate less output on Saturday and Sunday than during weekdays. In 2011, October began on a Saturday, and included 5 full weekends and 21 weekdays. In 2012, October began on a Monday, and included 4 full weekends and 23 weekdays. A simple yearoveryear comparison between these two months will not account for these differences, and could affect the analysis of changes in economic output over time.
Even when no other calendar effects are present in the data, comparing the same periods in each year can still be problematic. In general, it can be shown that this type of comparison lacks timeliness for the identification of turning points (the point at which a decreasing series, for example, begins to increase).
Comparing a current value with only one past value (the value of the series 12 months before the current reference month) can also be misleading if that particular value is unusual. For example, comparing economic data for British Columbia for February 2011 to data for February 2010 (the month in which the province hosted the Winter Olympics) may not yield useful information about changes in trends. To partially mitigate this effect, data for the current month (February 2011) can be compared with an average of the data for previous Februarys (for example, the past five years). A similar technique can be applied to examine monthtomonth movements. For example, the December to January movement of this year could be compared with a historical average of December to January movements for the last five years. Although this method may yield some additional insight, some measure of caution is warranted as it does not take the place of more formal seasonal adjustment techniques.

References
Ladiray, D. and Quenneville B. (2001) Seasonal Adjustment with the X11 Method, SpringerVerlag, Lecture Notes in Statistics, vol 158.
Statistics Canada (2009) Seasonal adjustment and trendcycle estimation, Statistics Canada Quality Guideline, 5th edition, Catalogue no. 12539X
Wyman, D. (2010), Seasonal adjustment and identifying economic trends, Statistics Canada. Canadian Economic Observer, March 2010, Catalogue no. 11010X
Readers are also invited to consult the various papers
Section 1: Concepts and definitions
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