Data quality, concepts and methodology: Seasonal adjustment and trend-cycle estimation
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Revisions in the raw data are required to correct known non-sampling errors. These normally include replacing imputed data with reported data, corrections to previously reported data, and estimates for new births that were not known at the time of the original estimates.
Raw data are revised, on a monthly basis, for the month immediately prior to the current reference month being published. That is, when data for December are being published for the first time, there will also be revisions, if necessary, to the raw data for November. In addition, revisions are made once a year, with the initial release of the February data, for all months in the previous year. The purpose is to correct any significant problems that have been found that apply for an extended period. The actual period of revision depends on the nature of the problem identified, but rarely exceeds three years.
Socio-economic time series such as data from the MRTS can be broken down into five main components: the trend-cycle, seasonality, the trading-day effect, the Easter holiday effect and the irregular component.
The trend represents the long-term change in the series, whereas the cycle represents a smooth, quasi-periodical movement about the trend, showing a succession of growth and decline phases (e.g., the business cycle). These two components—the trend and the cycle—are estimated together, and the trend-cycle reflects the fundamental evolution of the series. The other components reflect short-term transient movements.
The seasonal component represents sub-annual, monthly or quarterly fluctuations that recur more or less regularly from one year to the next. Seasonal variations are caused by the direct and indirect effects of the climatic seasons, institutional factors (attributable to social conventions or administrative rules; e.g., Christmas) and technological factors.
The trading day component originates from the fact that the relative importance of the days varies systematically within the week and that the number of each day of the week in a given month or a given quarter varies from year to year. This effect is present when activity varies with the day of the week. For instance, Sunday is typically less active than the other days, and the number of Sundays, Mondays, etc. in, say, July changes from year to year.
The Easter holiday effect is the variation due to the shift of part of April’s activity to March when Easter falls in March rather than April.
Lastly, the irregular component includes all other more or less erratic fluctuations not taken into account in the preceding components. It is a residual that includes errors of measurement on the variable itself as well as unusual events (e.g., strikes, drought, floods or other unexpected events causing variations in respondents’ commercial activities).
Thus, the latter four components—seasonal, irregular, trading-day and Easter holiday effect—all conceal the fundamental trend-cycle component of the series. Seasonal adjustment (correction of seasonal variation) consists in removing the seasonal, trading-day and Easter holiday effect components from the series, and it thus helps reveal the trend-cycle. While seasonal adjustment permits a better understanding of the underlying trend-cycle of a series, the seasonally adjusted series still contains an irregular component. Slight month-to-month variations in the seasonally adjusted series may be simple irregular movements. To get a better idea of the underlying trend, users should examine several months of the seasonally adjusted series.
Since April 2008, Retail Trade data are seasonally adjusted using the X-12-ARIMA 1 software. The technique that is used essentially consists of first correcting the initial series for all sorts of undesirable effects, such as the trading-day and the Easter holiday effects, by a module called regARIMA. These effects are estimated using regression models with ARIMA errors (auto-regressive integrated moving average models). The series can also be extrapolated for at least one year by using the model. Subsequently, the raw series—pre-adjusted and extrapolated if applicable— is seasonally adjusted by the X-11 method.
The X-11 method is used for analysing monthly and quarterly series. It is based on an iterative principle applied in estimating the different components, with estimation being done at each stage using adequate moving averages. 2 The moving averages used to estimate the main components—the trend and seasonality—are primarily smoothing tools designed to eliminate any undesirable component from the series. Since moving averages react poorly to the presence of atypical values, the X-11 method includes a tool for detecting and correcting atypical points. This tool is used to clean up the series during the seasonal adjustment. Outlying data points can also be detected and corrected in advance, within the regARIMA module.
To evaluate the different components of the series, taking account of the possible presence of atypical points, X-11 proceeds iteratively: estimation of components, search for unwanted effects in the irregular component, estimation of components on a corrected series, search for unwanted effects in the irregular component, etc.
Lastly, the annual totals of the seasonally adjusted series are forced to the annual totals of the original series.
Retail trade forms a system of 33 series: the Canada grand total, the 19 trade group totals, and the 13 provincial/territorial totals. For non-seasonally adjusted series, the summing of the 19 trade group totals produces the grand total (Canada) for each month and is equal to the sum of the 13 provincial/territorial totals.
Unfortunately, seasonal adjustment removes the sub-annual additivity of a system of series; small discrepancies, which generally vary between -1% and 1%, are observed between the sum of the seasonally adjusted trade groups and the sum of the seasonally adjusted provinces and territories. To restore additivity, a reconciliation process is applied to the seasonally adjusted retail trade series. The reconciliation process operates as follows:
(1) The seasonally adjusted grand total for Canada is obtained “indirectly” by summing up the trade group totals, which have previously been seasonally adjusted separately. And (2) the seasonally adjusted provincial and territorial totals are then reconciled so that their sum is equal to the seasonally adjusted grand total for Canada, obtained previously.
The procedure is such that a) the system’s seasonally adjusted components are modified as little as possible in percentage, b) the seasonally adjusted components add up to the grand total for each month, and c) the seasonally adjusted monthly values add up to the yearly totals for the non-adjusted series.
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