- Lee, K. (2018). 6. Benchmarking and Reconciliation. Quarterly National Accounts Manual (2017 Edition).
Shiskin, J. (1967). The X-11 variant of the census method II seasonal adjustment program (No. 15). US Department of Commerce, Bureau of the Census.

Findley, D. F., Monsell, B. C., Bell, W. R., Otto, M. C., & Chen, B. C. (1998). New capabilities and methods of the X-12-ARIMA seasonal-adjustment program. Journal of Business & Economic Statistics, 16(2), 127-152.

# Quarterly estimation methods

## Direct sources

7.29 The preferred method of compiling quarterly national accounts estimates is to use a high-quality data source which provides data for the aggregate being measured according to the conceptual basis required for the national accounts. In such cases both the quarterly and annual estimates may be compiled from the same source, the annual estimates being obtained simply as the sum of the quarterly estimates.

## Indirect sources

7.30 Annual national accounts estimates are considered to be superior to quarterly estimates. In the case of the income, expenditure and production components of GDP, the annual estimates are balanced in S-U tables, unlike their quarterly counterparts. Therefore, it is desirable to ensure the quarterly estimates are temporally consistent with their annual counterparts. This is achieved by using mathematical procedures to “benchmark” the quarterly estimates to the annual estimates.

7.31 Three commonly used statistical benchmarking procedures are:

- pro rata adjustment;
- Denton difference method; and
- Denton proportional method.

## Pro rata adjustment

7.32 In many cases, the quarterly data sources used to compile the national accounts are less reliable, less detailed and/or less appropriate than those used for compiling the annual national accounts benchmarks for particular aggregates. Consequently, indicator series are used to allocate (on a pro rata basis) annual estimates for such aggregates to the quarters of each financial year, and to extrapolate forward for the quarters of the latest incomplete year.

7.33 This benchmarking method simply consists of multiplying the quarterly preliminary estimates in a year by the ratio of the annual national accounts variable to the sum of the preliminary estimates of the four quarters.

7.34 While this method preserves the quarterly growth rates within the year, it changes the growth rate between the last quarter of one year and the first quarter of the next. The extent of the change to this growth rate is determined by how much the annual benchmark-to-preliminary estimate ratio has changed between the two years; that is, the ratio of the annual benchmark to the sum of the preliminary estimates for the corresponding four quarters. If the ratio were to change from 1.02 in year \(t\) to 1.00 in year \(t+1\), for example, the growth rate of the preliminary estimates from the fourth quarter of year \(t\) to the first quarter of year \(t+1\) would be reduced by two percentage points after benchmarking.

7.35 A particular problem that arises when using the indicators (pro rata) method is that the September quarter estimates can be adversely affected by what is known as the 'step problem'. A significant step problem will arise if the relationship between the annualised indicator series and the annual benchmark estimates varies significantly between any two consecutive financial years. In effect, the difference in the annual relationship between the benchmark and the indicator series is largely reflected in the September quarter.

7.36 This problem is reduced by using the 'benchmark' procedure. Given the obvious advantage of using the 'benchmark' procedure, the pro rata method is generally only used in a limited number of cases where the step problem is not significant.

## Denton difference method

7.37 The benchmarked estimates are obtained by allocating the discrepancy between the sum of four preliminary quarters and the corresponding annual national accounts estimate to the four quarters in each year, by minimizing a quadratic loss function over the whole, or overlapping lengthy spans, of the time series. Different versions of the quadratic loss function (expressed as a weight matrix) may be chosen.

7.38 The loss function is commonly defined as the sum of squares of either the first or second order differences of each preliminary quarterly estimate and the benchmarked quarterly estimate. In the first difference case, the benchmarked values are those that minimize the following

\(\large \min \sum\limits_{t = 1}^n {{{\left( {\left( {{b_t} - {p_t}} \right) - ({b_{t - 1}} - {p_{t - 1}})} \right)}^2}}\), subject to satisfying the annual constraints,

where there are \(n\) quarterly observations; \(b_t\) is the benchmarked quarterly estimate at time \(t\); and \(p_t\) is the preliminary quarterly estimate at time \(t\).

## Denton proportional method

7.39 A combination of the pro rata adjustment and the Denton difference method consists of minimizing the sum of squares of the first differences of the quotient of the benchmarked quarterly estimate and the preliminary quarterly estimate; that is:

\(\large \min {\sum\limits_{t = 2}^n {\left( {\frac{{{b_t}}}{{{p_t}}} - \frac{{{b_{t - 1}}}}{{{p_{t - 1}}}}} \right)} ^2}\), subject to satisfying the annual constraints.

7.40 This method can only be performed when the values of \(b\) and \(p\) are strictly positive.

## Characteristics of the two Denton methods

7.41 The Denton difference method minimises the differences of the absolute adjustments of two neighbouring quarters, whilst the Denton proportional method minimises the differences of proportional adjustments of two neighbouring quarters. Therefore, the Denton difference method results in a smooth additive distribution of the differences between the annualised indicator and the benchmark series, and the Denton proportional method results in a smooth multiplicative distribution of these differences. As a result, the Denton difference method tends to produce a smoother series, but the Denton proportional method changes the quarterly growth rates of the of the preliminary estimates least.

7.42 A characteristic of the quarterly national accounts series is that their seasonality and irregularity are generally more multiplicative than additive in nature, and better seasonal adjustments are generally obtained using a multiplicative rather than an additive model.

7.43 The Denton difference method can be applied to data that change sign, whilst the proportional method should only be applied to data that are strictly positive.

7.44 The methods described above are applicable to flow data, but there are other versions suitable for stock data and averages. For further details, refer to Chapter 6 of the IMF's Quarterly National Accounts Manual.³⁹

7.45 The ASNA uses the Denton proportional method for all flow series that are strictly positive. The Denton difference method is used when this is not the case, such as changes in inventories.

## Trend interpolation

7.46 Where there are no quarterly direct data sources or indicator series available it is necessary to generate a quarterly time series by adopting the most appropriate allocation procedure. One possible method would be to divide the annual estimate by four, but this would result in steps each September quarter, and no change in the other three quarters. The method used in the ASNA is to apply a linear interpolation method to calculate quarterly time series from annual series. The procedure involves forecasting annual estimates for two extra years, using a weighted average of the movements in year \(t-1\) and year \(t\). Such forecasts are used in preference to the standard projection produced by the interpolation procedure, if information is available to provide a superior forecast for the annual estimates for those two years.

7.47 A mathematical representation of the trend interpolation procedure is given below (see Table 7.1). This method is particularly appropriate for series such as consumption of fixed capital, where only annual estimates are available, and where it is reasonable to expect that movements in the quarterly series will be relatively smooth.

7.48 This type of interpolation procedure is designed to calculate quarterly series from annual series by linear trend interpolation; the annual series are projected backwards by one period, and forwards by two periods using a weighted average of the rate of increase prior to calculation of the quarterly values (the forward projection gives quarterly estimates for the current year).

Let \(\large {Y_1},{Y_2},\:\:.....,{Y_n}\) represent the annual series. Then the extrapolated annual series will be: | |

\(\large {Y_0},{Y_1},{Y_2},.....,{Y_n},{Y_{n + 1}},{Y_{n + 2}}\) | |

where \(\large{Y_1},{Y_2},{Y_3}\) are all positive | |

\(\large {Y_0} = {Y_1}\left( {0.4\frac{{2 + {Y_2}}}{{2 + {Y_3}}} + 0.6\frac{{2 + {Y_1}}}{{2 + {Y_2}}}} \right)\) | |

otherwise if \(\large {Y_1},{Y_2},{Y_3}\) are all negative, then | |

\(\large {Y_0} = {Y_1} - 0.6({Y_2} - {Y_1}) - 0.4({Y_3} - {Y_2})\) | |

And if \(\large {Y_n},{Y_{n - 1}},{Y_{n - 2}}\) are all positive | |

\(\large R = 0.4\frac{{2 + {Y_{n - 1}}}}{{2 + {Y_{n - 2}}}} + 0.6\frac{{2 + {Y_n}}}{{2 + {Y_{n - 1}}}}\) | |

\(\large {Y_{n + 1}} = R{Y_n}\) | |

\(\large {Y_{n + 2}} = R{Y_{n + 1}}\) | |

where \(\large R\) is the weighted projection factor used in order to move forward two periods when the annual series are all positive. | |

Otherwise, | |

\(\large X = 0.4({Y_{n - 1}} - {Y_{n - 2}}) + 0.6({Y_n} - {Y_{n - 1}})\) | |

\(\large {Y_{n + 1}} = X + {Y_n}\) | |

\(\large {Y_{n + 2}} = X + {Y_{n + 1}}\) | |

where \(\large X\) is the weighted projection factor used in order to move forward two periods when the annual series contain negative values. | |

The interpolation procedure which gives the required quarterly series is defined below. | |

For any year \(\large t\) , where \(\large t=1\) to \(\large n+1\) (same as above), the four quarterly observations are: | |

\(\large {q_t},1 = \frac{1}{4}(\frac{1}{4}{Y_{t - 1}} + \frac{7}{8}{Y_t} - \frac{1}{8}{Y_{t + 1}})\) | |

\(\large {q_t},2 = \frac{1}{4}(\frac{9}{8}{Y_t} + \frac{1}{8}{Y_{t + 1}})\) | |

\(\large {q_t},3 = \frac{1}{4}( - \frac{1}{8}{Y_{t - 1}} + \frac{9}{8}{Y_t})\) | |

\(\large {q_t},4 = \frac{1}{4}( - \frac{1}{8}{Y_{t - 1}} + \frac{7}{8}{Y_t} + \frac{1}{4}{Y_{t + 1}})\) |

## Seasonal adjustment and trend estimates

7.49 Quarterly time series such as those in national accounts publications are affected by three influences – calendar (mostly seasonal), trend and irregular influences – and the original series can conceptually be split into activity due to each of these components. For example, the activity in a particular December quarter can be conceptually split into:

- systematic calendar and/or seasonal related activity (e.g. Christmas related activity; October long weekend activity, etc.);
- trend activity, that is, the underlying level of the series; and
- irregular activity (e.g. impact of a short-term stimulus package, short-term non-systematic and unpredictable fluctuations).

7.50 When interpreting a quarterly series, it is helpful to assess combinations of the three components, as they each highlight different attributes of the data. In particular, the original, seasonally adjusted and trend series are seen as valuable tools for interpreting time series data. The original series contains all three components and shows 'what actually happened' (according to our survey data). The seasonally adjusted series has the seasonal component removed, leaving the trend and irregular. It shows what happened once the systematic activity that happens the same way every year has been removed, revealing more information about the underlying direction of the series, and/or the impact of irregular influences that may have been overshadowed by seasonal influences in the original series. Finally, the trend series contains only the trend component, and reflects the underlying level or long-term behaviour of the series.

7.51 The seasonal adjustment process splits the original series into estimates of the three components. It first estimates and removes the seasonal and calendar-related influences, creating the seasonally adjusted series. A further statistical process — Henderson smoothing — removes the irregular influence to reveal an estimate of the trend. The estimate of the irregular influences is the difference between the seasonally adjusted and the trend. This section summarises the methods used by the ABS to decompose quarterly national accounts series into their three components and generate the published seasonally adjusted and trend series.

## The seasonal adjustment process

7.52 Seasonal effects usually reflect the influence of the seasons themselves, either directly or through production series related to them (such as costs for generating farm production), or social conventions (such as the incidence of holidays) or administrative practices (such as the timing of tax payments). Other types of calendar variation occur as a result of influences such as the number and composition of days in the calendar period (trading day); accounting or recording practices adopted by businesses; the effect of regular paydays on activity levels; or the incidence of movable holidays (such as Easter).

7.53 Statistical techniques can be used to evaluate the effects of normal seasonal and other calendar influences operating on a series. If detectable seasonal or calendar variation is observed, the estimated effects may then be removed from the series to produce a seasonally adjusted series. Although calendar variation may be present in a series, factors applied in a particular period may vary significantly from year to year due to the variability in the number and composition of days in that particular period. This is especially evident in series affected by, say, the payment of salaries or pensions on a fortnightly basis. Seasonal or calendar variation can also move gradually over time in reaction to changing influences, and this is allowed for in the estimation of the seasonal factors.

7.54 Not all statistical series are significantly affected by seasonal or calendar influences which are regular enough to be described as 'reliable', so seasonal or calendar influences cannot always be removed from them. In such cases, the original series may be regarded as also being the seasonally adjusted series. Some examples in the quarterly national accounts are the rent component of farm costs, and the series related to the consumption of fixed capital.

## The method of seasonal adjustment

7.55 The ABS software for seasonal adjustment is the SEASABS (SEASonal analysis, ABS standards) package, a knowledge-based seasonal analysis and adjustment tool. The seasonal adjustment algorithm used by SEASABS is based on the X-11 Variant seasonal adjustment software from the U.S. Census Bureau.⁴⁰

7.56 The X-11 technique uses a filter-based approach to decompose the series to be analysed into estimated trend, seasonal and irregular components. The irregular component reflects the influence of unusual or transitory effects, for example, the effect of a major industrial dispute or of unseasonal weather conditions. It also reflects sampling and non-sampling errors which may be present in the original series, and other short-term fluctuations in the series that are neither systematic nor predictable.

7.57 The X-11 program includes a statistical procedure for automatically identifying and modifying unusually large or small values included in the original series, for the purposes of improving the estimate of the seasonal component only. Occasionally, modification of extreme values is undertaken directly prior to seasonal adjustment, in order to better stabilise the estimation of the seasonal component and minimise the extent to which both the estimated seasonal and trend components are affected by irregular influences.

7.58 Adjustments are also made prior to seasonal analysis to deal with abrupt discontinuities in the seasonal pattern or the trend where sufficient observations and/or supplementary information are available to estimate the magnitude of the effects. These 'break factors' have been employed retrospectively in the analysis of a number of national accounts series, and some series contain more than one such break. It is impossible, in most cases, to recognise and assess changes in seasonality or trend at the time they occur, and, until enough subsequent data are available to indicate otherwise, they may initially remain undetected, or be considered irregular effects.

7.59 Although based on the X-11 software, SEASABS also includes components of the U.S. Census Bureau X-12 ARIMA software package.⁴¹ For the national accounts, regression-ARIMA modelling techniques from X-12 ARIMA are used to compare actual original values to expected original values to detect possible extreme values and sudden discontinuities in the trend, and to assist with the estimation of prior adjustment factors to account for them. Additional information (such as unit record data) may also be used in the estimation of appropriate prior adjustment factors.

7.60 The seasonal adjustment process alone cannot indicate whether an unexpected movement appearing in current end seasonally adjusted figures denotes a variation in trend, or an unusual (irregular) effect, or whether it is due to an abrupt change in seasonality. However, the addition of subsequent data points to the series end and/or supplementary information about the reasons underlying series behaviour can assist in the identification and treatment of seasonal or trend discontinuities as soon as possible after they occur.

7.61 After extreme values and sudden discontinuities in a series have been accounted for, calendar and seasonal effects, where measurable, are estimated by X-11 using mainly filtering techniques, and occasionally regression procedures. The estimated seasonal and calendar influences, together with certain (but not all) prior adjustment factors, form the combined adjustment factors by which the original series is seasonally adjusted. It should be noted that only the estimates of seasonal and/or other types of calendar variation are removed from the original series to form the seasonally adjusted series, which contains the trend and irregular components. Since the irregular influences remain, an unexpectedly large movement in the seasonally adjusted series does not necessarily indicate a change in the underlying trend of the series.

## Multiplicative, additive or pseudo-additive adjustments

7.62 The SEASABS program allows for the original series to be decomposed into trend, seasonal and irregular components by using a multiplicative, additive, or pseudo-additive model. The choice of which of these models to use depends on whether it is more appropriate to consider the amplitudes of the trend, seasonal and irregular components to be proportional to or largely independent of each other. Specifically, the multiplicative model treats all three components as dependent on each other, the additive model treats them independently, and the pseudo-additive model treats the seasonal and irregular components as independent of each other but dependent upon the level of the trend.

7.63 Although most series in the national accounts are adjusted multiplicatively there are some exceptions. Series which include both positive and negative values cannot be directly adjusted using a multiplicative model. An additive or pseudo-additive model must be used if such series cannot be disaggregated into components having wholly positive (or negative) values. Several series relating to gross farm product (i.e. outputs and inputs) are affected by such extreme seasonal variations that the pseudo-additive model provides the best seasonally adjusted results. Other time series (especially inventories) are best adjusted using the additive model.

## Direct or indirect seasonal adjustments for aggregate series

7.64 It is possible to seasonally adjust an aggregate series either directly or by seasonally adjusting a number of its components and adding the results. The latter (aggregative) method has been employed for most of the major aggregates in the national accounts. Besides retaining, as far as possible, the essential accounting relationships, the aggregative approach is needed because many of the aggregates include components having different seasonal and trend characteristics, and sometimes require different methods of adjustment. Details of the methods of adjustment used for each of the quarterly national accounts aggregates are available on request.

## Concurrent adjustment

7.65 The national accounts use a concurrent adjustment methodology, under which the calendar and seasonal effects are re-estimated each quarter using all available data, including that for the most recent period. This allows for the most accurate estimate possible of the seasonal component of the series, as:

- using the data from the most recent periods allows better estimates of the calendar effects at the current end, especially when the calendar effects for a period move over time;
- it automatically takes into account revisions to the original data, resulting in appropriate revisions to the seasonally adjusted and trend data; and
- the adjustment method can be more responsive to changes in the seasonal and trend components, and identify them soon after occurrence; under concurrent adjustment; for example, turning points in the trend series are usually identified within three periods of them occurring.

7.66 The improvements to the estimation of the seasonal component result in improved estimates of the seasonally adjusted and trend series, especially at the current end, and smaller revisions in subsequent periods. Note that this method results in reduced revisions compared to the previously utilised adjustment methodology, forward factor adjustment, under which a year’s worth of seasonal factors was extrapolated at the time of the annual reanalysis, and then revised a year later.

7.67 The use of concurrent methodology minimises the risk of incorrect seasonal forward factors being used in the adjustment process and an inappropriate seasonally adjusted series being published.

7.68 In the March quarter 2020 issue of Australian National Accounts: National Income, Expenditure and Product, the method used to produce seasonally adjusted estimates was temporarily changed from concurrent methodology to the forward factors method for series with significant and prolonged impacts from COVID-19. Series will return to concurrent seasonal adjustment, when economic conditions are assessed to have returned to pre COVID-19 patterns.

7.69 Temporary switch to forward factors will be used in the future where economic shocks significantly disrupt regular seasonal patterns, as occurred with the COVID-19 pandemic.

## The annual seasonal reanalysis cycle and revisions

7.70 The characteristics of National Accounts time series are reviewed annually. During this reanalysis, the method and quality of the seasonal adjustment process are scrutinised for each series, for the purpose of identifying any changes required to improve the adjustment, and, subsequently, the seasonally adjusted and trend estimates. Such improvements could include:

- changes to decomposition models, filters, etc.;
- insertion of new prior adjustments (e.g. corrections for unusually large or small values, or adjustments for abrupt changes in the seasonal pattern or trend level); and/or
- improvements to existing prior adjustments (e.g. updating corrections in response to new supplementary information).

7.71 Significant revisions can occur as a result of the annual reanalysis, with the more recent periods likely to be most affected. However, the impact of such revisions has generally been reduced since the introduction of concurrent seasonal adjustment.

## Interpreting seasonally adjusted series

7.72 The following points need to be taken into account when using seasonally adjusted statistics:

- seasonal adjustment is a means of removing the estimated effects of seasonal and other types of calendar variation from statistical series, so that the effects of other influences on the series may be more clearly recognised;
- seasonal adjustment does not remove the effect of irregular influences from the statistics, so an unexpected movement in a seasonally adjusted series should not necessarily be regarded as a change in trend; and
- seasonally adjusted statistics will be revised following revisions to the original data and as additional original data points are included each quarter.

## The trend estimates

7.73 A statistical technique is used to dampen the irregular element in cases where the removal of only the seasonal element from an original series (resulting in the seasonally-adjusted series) may not be sufficient to allow identification of changes in its trend. This technique is known as smoothing, and the resultant smoothed series is known as trend series.

7.74 Smoothing to derive trend estimates is achieved by applying moving averages to seasonally adjusted series. A number of different types of moving averages may be used; for quarterly series, a seven-term Henderson moving average is applied. The use of Henderson moving averages leads to smoother data series relative to series that have been seasonally adjusted only. This average is symmetric, but asymmetric forms of the average are applied as the end of a time series is approached. The application of asymmetric weights is guided by an end-weight parameter, which is based on the calculation of a noise-to-signal ratio; that is, the average movement in the irregular component divided by the average movement in the trend component, known as the I/C ratio). While enabling trend estimates for recent periods to be produced, asymmetric weights result in revisions to the estimates when subsequent observations are available.

7.75 Revisions to the trend series may arise from:

- the availability of subsequent data;
- revisions to the underlying data;
- identification of and adjustment for extreme values, seasonal breaks and/or trend breaks;
- re-estimation of seasonal factors; and
- changes to the end weight parameter.

7.76 For more information about ABS procedures for deriving trend estimates and an analysis of the advantage of using them over alternative techniques for monitoring trends, see Information Paper: A Guide to Interpreting Time Series – Monitoring Trends, 2003.

## Further reading

For further information on time series analysis in the ABS, please refer to Time Series Analysis Frequently Asked Questions.