–14–
/ EUROPEAN COMMISSION / /Statistical Office of the European Communities / Directorate General Statistics
16 April 2007
TF-SAQNA-20
Task Force on Seasonal Adjustment
of Quarterly National Accounts
Second Meeting
Agenda Item 6
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Chain-Linking and Seasonal Adjustment
Comparison of Alternative Chain-Linking Methods:
Empirical Example for Austria
Draft contribution to a paper on chain-linking by Marcus Scheiblecker
Austrian Institute of Economic Research 2nd April 2007
Marcus Scheiblecker
Draft
Chain-linking, seasonal adjustment and the business cycle
Different chain-linking methods lead to different output and sometimes they can not be considered as time series in a strict sense. As most seasonal adjustment techniques are based on time series analysis, this can lead to problems in identifying outliers, working day effects and the seasonal component. As a consequence, the residual trend-cycle component can vary over different chain-linking techniques and lead to problems for economic policy concerning size and timing of measures. This paper is a case study about how the different chain-linking methods applied to the Austrian GDP affect the output of time series analysis based seasonal adjustment techniques and a following dating process of the business cycle.
In Austrian national accounts, quarterly estimates are compiled for all aggregates at current prices, at annual average prices of the current year and at annual average prices of the previous year. This allows applying all three chain-linking methods which are recommended by Eurostat for compiling volume estimates in quarterly national accounts.
By definition, quarterly estimates at current prices and at annual average prices of the current year have to sum up to published annual estimates at current prices. Those at annual average prices of the previous year have to sum up to published annuals at previous year's prices. This consistency in the time domain (vertical consistency) can be established by benchmarking techniques before chain-linking takes place.
In the annex, all three types of price bases are given for the Austrian GDP. As these quarters do not represent time series in a narrower sense, chain-linking techniques have to be applied in order to make them interpretable for economic analysis.
For doing this, in a first step growth rates have to be constructed. In the case of the over-the-year technique, the growth rate of a specific quarter is calculated by comparing its value at annual average prices of the previous year to the value at annual average prices of the current year of the respective quarter in the previous year. The annual overlap technique instead compares the values at annual average prices of the previous year to the fourth of the sum over all four quarters of the values at annual average prices of the current year of the previous year (as already mentioned, this sum equals the annuals at previous year's prices).
For the quarterly overlap method, the growth rate of the specific quarter is constructed by comparing the value at annual average prices of the previous year to the value at annual average prices of the current year of the fourth quarter in the previous year.
With growth rates constructed according to these different methods, an index is constructed by applying these rates in the same manner as they have been derived (this is the actual chain-linking process). In most cases, these index series are transformed later on to currency units by determining one year at current prices as the reference period, with all growth rates related to this.
All three approaches have their pros and cons. The annual overlap method has the advantage that all quarters measuring volumes constructed this way, sum up perfectly to their annual totals, derived independently by an analogous annual chain-linking method taking the same year as the reference. The disadvantage lies in the fact that as the average previous year forms the price basis, an abrupt change of the price basis takes place with the beginning of a new year. This can cause breaks in the first quarters of the series, which hamper the time series property of the output. These breaks are the larger, the higher the relative price changes and the relative volumes deviate from the first quarters. As for that, these breaks are difficult to adjust as their size and direction is unclear.
The quarterly overlap method instead does not show such breaks, as all quarters and with this also the first quarter, are compared to the forth quarter of the previous year. This advantage comes at cost of its vertical non-additivity which means that quarters do not sum up to annuals derived independently by an analogous annual chain-linking method taking the same year as the reference.
The over-the-year technique instead involves both shortcomings. Quarters do not sum up perfectly to annual totals and the quarterly change of the price basis allows no meaningful interpretation of the outcome in a time series sense.
The drawback of vertical non-additivity can be overcome by benchmarking the series with statistical or mathematical methods[1]. Theoretically, it would be best to aim at an output featuring the time series properties of the quarterly overlap method and the additivity of the annual overlap method. In order to maintain the break free time series property of the quarterly overlap method, a benchmarking technique should be selected which added differences show no breaks either as a necessary condition[2]. However, it has to be borne in mind that all benchmarking techniques imply some kind of deliberate change in the original data. As for that, applying seasonal adjustment or business cycle analysis to unbenchmarked data seems to be a sensible option.
In the following, all three chain-linking methods are applied to Austrian GDP data as given in the annex. In order to maintain time series properties and to show the effect of the different chain-linking methods on seasonal adjustment alone, no benchmarking is done after the chain-linking process. This is no shortcoming in the analysis, as benchmarking methods can only distort but not improve time series properties.
For seasonal adjustment the Tramo-Seats approach, recommended by Eurostat and implemented in the DEMETRA software package, has been used. For all three series, the default settings have been accepted with considering the working day effect by two regressors: the number of working days at a whole and the weekend days. The Easter effect and the leap year are considered, too, by accepting default options.
Tables 1a to 1 c in the annex give the statistical output informing about the test statistics of the respective models. GDPoty marks the output for Austrian GDP chained by the over-the-year method; GDPao names the annual overlap procedure and GDPqo the quarterly overlap technique. In all cases the famous Airline model which models the trend-cycle as well the seasonal part of the series as an ARIMA (0,1,1) process has been considered best by the program. The estimated coefficients are quite similar between the GDPao and the GDPqo models. This goes for the MA-term of the trend-cycle component as well as for seasonal parts. For the GDPoty model, however, the MA-term coefficient is much higher.
The consequences of the various models on the seasonal factors can be seen at Figure 1 in the annex (for graphical reasons only values starting from 1998 onwards are given). Whereas the seasonal factors of the annual and the quarterly overlap method are practically identical, the over-the-year method gives higher values for the first and lower values for the third quarter.
In all three cases working day effects have either been considered as insignificant or not existing. The same goes for the leap year and the Easter effect[3]. Interestingly, the automatic outlier detection procedure TRAMO found in the case of the annual overlap method an additive outlier in the first quarter of 1993. For the quarterly overlap method, for the same period a transitory component of the same size was found, which is slowly decaying over the following quarters. For the over-the-year model no outlier was found at all. A theoretical explanation can be that the over-the-year method has such poor time series properties, that a detection of outliers seems to be hampered by its unsystematic movements. For the annual overlap, the step problem in the first quarter could also result in problems concerning the task of outlier detection.
It is not the aim of this study to judge which model is best and also the view on test statistics would only be a poor guide for that. Just the differences and their consequences are in the focus of interest. Especially the consequences for guiding economic policy actions are to be explored, as institutions responsible for that are the most important user of quarterly national accounts. In order to pay attention to this fact, not only the growth rates formed by seasonally and working day adjusted series are of interest, but also business cycle turning points. Even looking at the quarter-on-quarter growth rates of the adjusted data (for detected outliers, working days and the seasonal component), as given in Figure 2, confirms the first impression, that chain-linking can be decisive in this field. Looking at Figure 2, one can observe that local extremes of growth rates can be found in different quarters according to the various chaining techniques. Admittedly, the methods which would require a following benchmarking procedure in order to meet their annual counterparts should be biased. But if a benchmarking process is selected which balances differences by values showing the least variation from quarter to quarter, this should not alter the pattern of peaks and troughs in growth rates but only their size.
In order to check the possible consequences for economic policy measures, all output series are submitted to a Hodrick-Prescott filter with a λ-weight of 1600, which is used very often in business cycle analysis based on quarterly data. This Hodrick-Prescott filter cancels out frequencies representing cycles longer than 32 quarters, which can be regarded as trend variations, without amplifying higher frequencies.
The adjusted series transformed by this filter are given in Figure 3. From visual inspection it becomes apparent, that not only growth rates can vary substantially, but also their local maxima and minima are very often not located in the same quarter. This is not only true for the periods after the beginning of 1993 due to differences in outlier recognition, but also for other periods like 1999 and 2000.
In order to underline the possible consequences of different chain-linking methods for business cycle analysis, the cross-correlations as well their equivalent in the frequency domain, the coherence, is given in Table 2. Cross-correlations suggest, that there is a high comovement between the HP1600 transformed GDPoty series and both others. The highest cross-correlations can be found at zero lags, stating that there is no systematic phase shift between them. Observed in the frequency domain, which allows considering not only integers for leads and lags, results for the annual overlap method confirm this but for the quarterly overlap method a small lead of 0.1 quarters was found.
A further method used in business cycle analysis is to compile a dating calendar for peaks and troughs in the series. These business cycle turning points mark local minima and maxima and can be regarded as ideal time points for setting economic policy actions. There exist several methods for doing this. Here the famous routine developed by the NBER economists Bry and Boschan (1971) is applied as it is implemented in the BUSY software package[4]. Table 3 shows in the first row the peaks and troughs found in the HP1600 transformed GDPoty series, which acts as the reference series. Four complete cycles were found. Both other series show one cycle less. This is due to the fact, that the undetected negative outlier in the first quarter of 1993 in the over-the-year chain-linked series was erroneously regarded as a trough, which has to be followed necessarily by a peak. Regarding the lead and lag structure, the other series turning points hint more to a leading than to a lagging property.
Summary
To summarize, the different chain-linking techniques used in quarterly national accounts lead to differences in the quality of time series properties in the data. As for that, further processing with time series methods lead to different output. In the underlying case, the outlier detection and, linked to this, the seasonal adjustment process gave different results. Whereas differences were minor between the annual and the quarterly overlap method, the over-the-year method results deviated rather substantially.
For the underlying data, cross-correlations of the cyclical components (isolated by a Hodrick-Prescott filter with a smoothing weight of 1600) are hinting to a comoving property between the observed series, or to a small lead of the quarterly overlap method. The same goes for detected business cycle turning points. Here again, outliers in the series which remain undetected or are estimated imprecisely seems to matter most but no significant shifts in turning points were found.
Consequently, this results not automatically in delayed signals for economic policy measures for some chain-linking methods, but in an imprecision concerning the informational content. However, especially at the recent time margin, where uncertainty concerning quarterly national accounts data is quite high, some chain-linking methods can amplify this uncertainty.
Annex
GDP at current prices / GDP at current year's average prices / GDP at previous year's average prices198801 / 27.425,2 / 27.627,4 / 27.184,4
198802 / 28.936,6 / 29.000,5 / 28.535,5
198803 / 30.511,4 / 30.436,9 / 29.986,9
198804 / 31.509,1 / 31.317,5 / 30.855,0
198901 / 29.559,0 / 29.642,4 / 28.870,8
198902 / 31.019,5 / 30.984,7 / 30.064,4
198903 / 32.444,1 / 32.405,7 / 31.344,8
198904 / 33.460,7 / 33.450,5 / 32.282,5
199001 / 31.760,0 / 32.005,3 / 31.153,3
199002 / 33.472,5 / 33.436,3 / 32.462,4
199003 / 34.957,0 / 34.935,1 / 33.868,7
199004 / 36.136,8 / 35.949,7 / 34.823,1
199101 / 34.089,7 / 34.375,2 / 33.135,9
199102 / 35.903,3 / 35.892,6 / 34.557,5
199103 / 37.886,3 / 37.820,1 / 36.429,6
199104 / 38.713,5 / 38.505,0 / 37.106,8
199201 / 36.584,4 / 36.875,7 / 35.601,6
199202 / 38.482,5 / 38.490,3 / 37.153,7
199203 / 39.798,3 / 39.716,2 / 38.338,9
199204 / 40.609,5 / 40.392,6 / 38.960,4
199301 / 37.370,4 / 37.673,0 / 36.633,3
199302 / 39.615,0 / 39.654,5 / 38.557,6
199303 / 41.393,0 / 41.196,7 / 40.083,4
199304 / 41.896,3 / 41.750,4 / 40.718,4
199401 / 40.002,2 / 40.391,5 / 39.273,8
199402 / 41.428,2 / 41.488,0 / 40.381,9
199403 / 43.016,3 / 43.005,8 / 41.917,8
199404 / 44.496,1 / 44.057,6 / 42.966,0
199501 / 41.490,1 / 41.608,6 / 40.811,8
199502 / 43.381,0 / 43.190,6 / 42.333,4
199503 / 44.471,2 / 44.589,3 / 43.726,9
199504 / 46.183,2 / 46.137,0 / 45.297,4
199601 / 43.046,6 / 43.226,2 / 42.774,6
199602 / 44.980,7 / 44.857,9 / 44.376,6
199603 / 46.184,9 / 46.141,6 / 45.692,7
199604 / 47.659,7 / 47.646,1 / 47.278,9
199701 / 43.570,5 / 43.678,2 / 43.606,0
199702 / 45.557,6 / 45.548,6 / 45.490,0
199703 / 47.141,3 / 47.077,7 / 47.154,6
199704 / 48.871,3 / 48.836,0 / 48.966,6
199801 / 45.836,6 / 45.670,9 / 45.352,0
199802 / 47.813,5 / 47.630,7 / 47.490,8
199803 / 48.381,0 / 48.754,8 / 48.526,3
199804 / 50.353,1 / 50.327,8 / 50.364,2
199901 / 46.758,9 / 46.723,7 / 46.654,6
199902 / 49.225,3 / 48.956,0 / 48.968,4
199903 / 51.242,3 / 51.352,7 / 50.897,0
199904 / 52.798,9 / 52.992,9 / 52.253,9
200001 / 49.493,7 / 49.745,6 / 49.282,0
200002 / 52.225,8 / 52.280,1 / 51.172,3
200003 / 53.266,8 / 53.292,1 / 52.317,0
200004 / 55.406,0 / 55.074,6 / 53.967,3
200101 / 52.038,3 / 52.122,5 / 51.422,7
200102 / 53.339,1 / 53.188,0 / 52.335,7
200103 / 54.534,3 / 54.641,5 / 53.530,8
200104 / 55.966,2 / 55.925,8 / 54.852,1
200201 / 52.684,3 / 52.885,2 / 52.473,2
200202 / 54.837,0 / 54.760,7 / 54.063,4
200203 / 55.963,0 / 56.015,6 / 54.932,7
200204 / 57.356,6 / 57.179,5 / 56.257,9
200301 / 54.151,7 / 54.452,3 / 53.563,2
200302 / 55.948,5 / 55.855,8 / 55.157,9
200303 / 57.241,2 / 57.188,9 / 56.547,1
200304 / 58.901,9 / 58.746,4 / 57.977,6
200401 / 55.316,9 / 55.883,8 / 54.749,8
200402 / 58.206,7 / 58.323,3 / 57.223,9
200403 / 60.165,1 / 60.055,6 / 59.066,7
200404 / 62.129,8 / 61.555,8 / 60.727,7
200501 / 57.257,1 / 57.433,6 / 56.310,3
200502 / 60.614,1 / 60.743,6 / 59.604,4
200503 / 62.513,3 / 62.528,7 / 61.398,3
200504 / 64.718,3 / 64.397,0 / 63.324,3
200601 / 59.452,3 / 58.996,7
200602 / 63.595,4 / 62.623,8
200603 / 65.441,6 / 64.489,3
200604 / 67.899,6 / 66.494,5
Table 1a : Test statistics of GDPoty