Seasonal Adjustment Estimates Model-based seasonally adjusted estimates and sampling error Estimating certain CPS series with a model that filters out sampling error may reduce volatility in the time series, facilitating more meaningful trend analysis Richard Tiller he Current Population Survey (CPS) is the quite challenging. The monthly estimates pro- and source of the Nation’s official estimates of duced for the national aggregated series, such as Marisa Di Natale Ttotal employment and unemployment. The total employment and total unemployment, are CPS is a nationally representative, scientifically highly reliable relative to smaller, disaggregated selected monthly sample survey of approximately series. Many of the more detailed demographic 60,000 households. The survey yields data that series, such as employment and unemployment for are rich in demographic detail, including such blacks, are based on relatively small sample sizes, characteristics as age, sex, race, and Hispanic or so that survey error dominates movements in the Latino ethnicity. Estimates from the survey are underlying level of the series. The standard error published monthly in the BLS news release Em- for a (not seasonally adjusted) month-to-month ployment Situation and in the BLS publication change based on the CPS can be quite high for some Employment and Earnings. of these series. For example, the standard error for In order to make the time-series data collected a change in the unemployment rate of adult black from the CPS more useful to analysts and policy- males can be as large as 0.8 to 0.9 percentage point, makers, the monthly data from the survey are compared with 0.2 percentage point for the unem- adjusted for seasonal fluctuations. As is well ployment rate for all persons aged 16 years and known, the purpose of seasonally adjusting a older. As a result, drawing meaningful conclusions series is to remove seasonal fluctuations in the about trends or month-to-month changes is diffi- data so that users can more easily observe funda- cult, even after the data have been adjusted for mental changes in the level or trend of the series seasonal movements. that are associated with business cycle contrac- As an alternative to conventional seasonal ad- tions and expansions. Approximately 116 time justment, the study reported in this article applies series from the CPS are directly seasonally ad- an experimental model-based method to selected justed, and many more are indirectly seasonally CPS demographic series. The method is designed Richard Tiller is a adjusted, as sums or ratios of the original 116. to remove the effects of sampling error, as well as mathematical statistician on the There is, however, a source of spurious random those of seasonality, from the series, thereby Statistical fluctuations in the CPS data that arises because making it easier to discern underlying trends in the Methods Staff, and Marisa Di the CPS samples only a fraction—1 in 2,200, on data. Natale is an average, of the working-age population each economist formerly in the month: sampling error—the difference between Approaches to seasonal adjustment Division of Labor the survey estimates and the values that would be Force Statistics, Office of Current produced by a complete census of the population. The presence of large survey errors in the detailed Employment Simultaneously removing both seasonality in CPS series represents a major challenge to conven- Analysis, Bureau of Labor Statistics. the data and noise due to sampling error can prove tional methods of seasonal adjustment. Currently, Monthly Labor Review September 2005 27 Seasonal Adjustment Estimates the Bureau of Labor Statistics uses a seasonal adjustment and critics. The major concerns with the model-based program called X-12-ARIMA to seasonally adjust its CPS series. approach are that it may be difficult to develop good models This program is based on the empirical moving-average for some series and that the model may fail occasionally when approach to seasonal adjustment. new data become available. These concerns raise issues An alternative that is gaining increasing attention is the about the robustness of the adjustment and the associated model-based approach to seasonal adjustment. A comparison statistical measures. of the two approaches suggests that the model-based One major criticism of the moving-average approach is approach provides much-needed flexibility in controlling for that it lacks standard statistical measures. The absence of the effects of sampling error. Such flexibility is not possible standard errors for published seasonally adjusted data tends with the conventional moving-average approach. to promote the mistaken impression that the final seasonally Conventional approaches to seasonal adjustment are based adjusted values are exact rather than estimates. Moreover, on the classical decomposition of a time series, which assumes the lack of confidence intervals makes analysis of change in that the series is composed of trend (or trend-cycle), seasonal, the estimates and the location of turning points more difficult. and irregular components, in either an additive or a multiplicative Still, supporters of the method argue that it is robust and relationship. The first two components respectively account for nonparametric; thus, its lack of an explicit statistical model is the long- and short-run systematic variation in the series. The viewed as an advantage. Even so, the absence of statistical irregular component is a residual, usually assumed to be purely measures of reliability remains a major shortcoming.5 random variation with a fixed variance. The model-based approach makes (testable) assumptions Because the three components of the classical de- about the underlying probability distribution generating the composition are not directly observable, they must be esti- data. Along with estimates of the model parameters, these mated in order to perform seasonal adjustment. The moving- assumptions provide the means for constructing confidence average method uses weighted moving averages of the intervals and other statistical measures to quantify the original data over a period of many years to produce a smooth uncertainty in the estimates. Non-model-based estimates do trend and a seasonal pattern. The estimated trend and not, in general, afford a basis for producing measures of seasonal components are removed from the series, and the uncertainty in the estimates. residual is the irregular component. This approach makes no Another criticism of the moving-average approach is that attempt to define, in any formal statistical way, what is being it is not tailored to the specific properties of the series being estimated, but rather applies a series of moving averages adjusted.6 In contrast, the model-based approach develops a directly to a series. While some of the moving averages are model on the basis of goodness-of-fit diagnostics. The result- chosen to satisfy a mathematical smoothness criterion, the ing seasonal adjustment is based on the properties of the method was derived largely from empirical work with a wide series as represented by the model. In theory, under the range of series. assumptions of the model, the seasonal adjustment is By far the most successful application based on the “optimal” for the specific series. While the moving-average moving-average approach is the X-11 program,1 which has method can make no such theoretical claim, its moving gone through several major revisions. The latest, enhanced averages were originally selected because they work well for version is X-12-ARIMA.2 The original X-11 program, however, a very large number of series. This more generic approach remains at the core of X-12. continues to work well in practice and may have an advantage As an alternative to the moving-average approach, model- over the model-based approach when good models for sea- based seasonal adjustment has been gaining increased sonal adjustment cannot be developed. attention. The model-based approach specifies explicit Clearly, both approaches to seasonal adjustment have statistical models of the trend, seasonal, and irregular their merits and limitations. Indeed, there have been a number components of the classical decomposition.3 To seasonally of studies of the relative performance of the two approaches, adjust the data, weighted moving averages of the observed but no general agreement as to how to interpret the results. data actually are used in the model-based approach, but with Perhaps a more balanced approach is to treat them as comple- the important difference that they are derived directly from mentary tools for performing seasonal adjustment.7 the model. An essential characteristic of the approach is its use of standard statistical procedures to estimate the un- Dealing with “noisy” CPS data series observed components of the time series and to provide associated statistical measures such as confidence intervals The types of data series that are the focus of the study and significance tests. discussed in this article—survey series with large sampling There is a large body of literature on the comparative errors—represent a class of series that presents special properties of the two approaches.4 Each has its supporters problems for the moving-average approach to seasonal 28 Monthly Labor Review September 2005 adjustment. For these series, that approach (specifically, the time series for which a separate model can be specified, as is X-12 program) performs poorly, not because it has trouble done with the trend, seasonal, and irregular components of removing seasonality from the series, but because it cannot the series. Then there is the further advantage, in modeling adequately remove the effects of sampling error. The result is survey error, of having external information from the survey a seasonally adjusted series that often is dominated by on the standard errors and correlations. This information can sampling error, masking the underlying trend in the series. be directly used to specify the parameters of the model.