Economic analysis using higher-frequency time series: challenges for seasonal adjustment

The COVID-19 pandemic has increased the need for timely and granular information to assess the state of the economy in real time. Weekly and daily indices have been constructed using higher-frequency data to address this need. Yet the seasonal and calendar adjustment of the underlying time series is challenging. Here, we analyse the features and idiosyncracies of such time series relevant in the context of seasonal adjustment. Drawing on a set of time series for Germany—namely hourly electricity consumption, the daily truck toll mileage, and weekly Google Trends data—used in many countries to assess economic development during the pandemic, we discuss obstacles, difficulties, and adjustment options. Furthermore, we develop a taxonomy of the central features of seasonal higher-frequency time series.

The WEI has sparked the development of a series of similar weekly indicators, especially for European countries (e.g. Delle Monache et al. 2020;Eraslan and Götz 2020;Fenz and Stix 2021;Wegmüller et al. 2021). Lourenço and Rua (2021) develop a daily variant (DEI) for the Portuguese economy. Woloszko (2020) constructs countryspecific economic indicators for 46 countries based on Google Trends data.
During the last decade, the number of seasonal adjustment methods for higherfrequency time series has seen a considerable increase (for a discussion and further references see, De Livera et al. 2011;Ladiray et al. 2018;Ollech 2021;Proietti and Pedregal 2022;Webel 2020). Yet, in the construction of the WEI, DEI and some of the other indices, these methods are not employed. Instead, ad-hoc measures like yearon-year rates are used to handle the seasonality of the data. This does not usually take into account calendar effects or the structure of the periodic and seasonal effects such as the varying number of weeks per year.
The higher-frequency time series used in the construction of the economic indices often present methodological challenges that differ substantially from those encountered in lower-frequency data. In this paper, we therefore contribute to the literature by analysing and seasonally adjusting a set of higher-frequency time series that is typical of the kind of data relied upon by economists and business cycle analysts during the COVID-19 pandemic. As the literature on the adjustment and modelling of higherfrequency seasonal time series is growing rapidly, the lack of a common understanding of the features, that need to be taken into account when modelling and adjusting these data, becomes evident. Therefore, another key contribution of the analysis presented here is the derivation and systematization of those characteristics that are relevant in the context of seasonal adjustment.
To this end, Sect. 2 discusses the characteristic features of higher-frequency time series from a seasonal adjustment perspective. Section 3 presents the seasonal adjustment methods employed in this study, while Sect. 4 analyses and adjusts the daily truck toll mileage, the hourly electricity consumption and a weekly Google Trends series. These series are used as inputs to the Weekly Activity Index (WAI) devised by Eraslan and Götz (2020). Section 5 summarizes.

Key notions for the adjustment of higher-frequency data
Seasonal higher-frequency time series show features and idiosyncracies that are of immediate importance for any seasonal adjustment procedure. The time series examples discussed in the next sections include some or all of the characteristics summarized in Table 1. To guide the readers understanding and as a point of reference, this table is included at the beginning of this paper. Table 1 organizes the characteristics of seasonal higher-frequency time series into a taxonomy of relevant features.
The basic characteristics cover the most obvious features that concern the whole set of observations. The fact that higher-frequency time series contain many observations-especially in comparison with lower-frequency time series-is in itself trivial. Yet, it increases the computational burden and may render some methods inapplicable. Non-isochronicity and non-equidistance will usually either have to be

Periodic and calendar effects
Multiple periodic effects Daily time series usually contain day-of-the-week and day-of-the-year effects.
Multilevel periodic effects Seasonal structure may be hierarchical, e.g. a series with hour-of-the-day and day-of-the-week effects.
Uncommon periodic effects Higher-frequency series may contain other periodic effects, such as week-of-the-month effects.
Breaks in periodic effects Periodic effects may change rapidly, e.g. as a consequence of fundamental changes in the data generating process.

Cross-seasonality
The periodic and calendar effects can be interdependent.

Autocorrelational Seasonality
The seasonal impact of consecutive observations may be highly dependent.

Outliers and missing values
Missing values Due to data availability, some series contain (temporarily) missing values, often at the end of the series.
Unreliable data delivery Data producers do not necessarily have an obligation to deliver data or provide additional information.
Higher volatility The volatility of the time series usually decreases with a higher temporal aggregation.

Heteroskedasticity
The volatility may change over time and may be seasonal.
Non-traditional outlier patterns Observed outlier pattern can be different from lower frequency series, e.g. slower rate of decay in a TC outlire.
addressed by adapting the methods used or by transforming the data, e.g. by interpolation or time-warping. The periodic and calendar effects encompass concepts that extend the definition of seasonality and calendar effects in lower-frequency time series. As becomes clear from the examples of daily data, higher-frequency time series often contain multiple noticeable periodic effects that may be interdependent and autocorrelational. For calendar effects, we may consider including constellations that are not recommended to adjust for in lower-frequency time series, such as bridge days. This may be sensible as the higher number of observations allow us to observe and estimate such effects more directly. For weekly series, the impact of bridge days may even be inseparable from the moving holidays, if they always fall in the same week.
Higher-frequency time series are generally more volatile than traditional businesscycle indicators. This is mostly a result of the fact that irregular influences tend to partially offset each other over a longer time span and are therefore less pronounced in monthly and quarterly data. Additionally, many of the higher-frequency time series encountered are merely a by-product and are not compiled for economic analysis or official statistics and therefore do not adhere to the same data standards. Accordingly, the methods applied to higher-frequency time series usually need to be robust against outliers and irregular observations.

Seasonal adjustment methods
The focus of this paper lies on the qualitative assessment of higher-frequency time series and to this end, we employ methods for the seasonal adjustment of such series to guide our understanding. Although we discuss the seasonal adjustment results and highlight potential challenges, we do not advocate for any particular method. Instead, the findings shall carve out features of higher-frequency time series that need to be addressed regardless of the method applied.
Let {Y t } denote a series of length T with cycle length τ . For monthly series, τ equals the number of observations per year, i.e. 12. For a daily series with only a weekday pattern, the recurring pattern has a cycle length of 7. The basic time series decomposition is then given by which includes the trend-cycle (T t ), seasonal (S t ), calendar (C t ) and irregular component (I t ). It can easily be adapted to capture a multiplicative relationship between the components by log-transformation of the original time series. 1 For multiple periodic effects with cycle lengths τ 1 , τ 2 , ..., the basic time series model can be generalized to Remark 1 In Eq. 2, the periodic effects are subsumed as seasonal effects to allow a parsimonious notation. Here, effects related to calendar constellation are not considered to be periodic effects, even though the calendar constellation recurs with a cycle length of 400 years in the Gregorian calendar. However, this recurrence is not exploited in the modelling of the time series and effects arising from these calendar constellations will not usually recur in similar intensity every 400 years.
As we will see, for some series, different seasonal effects are interdependent. Likewise, calender effects and seasonal effects can be related. If all such interactions are relevant, Eq. 2 needs to be augmented so that In line with the methods used for official seasonal adjustment-namely X-13 and Tramo-Seats-the adjustment of the higher-frequency time series presented here combines a seasonal adjustment routine with a RegARIMA-based pre-adjustment. The latter is a regression model (Reg) with autoregressive integrated moving average (ARIMA) errors and is used here for the estimation and elimination of calendar and some of the interaction effects. The RegARIMA model is given by where φ(B) and θ(B) are AR and MA polynomials of order p and q, number of differences d, and capitals indicating seasonal terms while B is the backshift operator, i.e. B(y t ) = y t−1 . The parameter β i captures the impact of the ith regressor X it on the time series Y t and ε t is the error term. The ARIMA part of Eq. 4 can be abbreviated by ( p d q)(P D Q) τ . Extensions of this model to multiple seasonalities are known and available (e.g. Svetunkov 2017), but at the time of writing, these are rarely used in seasonal and calendar adjustment.

Seasonal adjustment of daily time series
The iterative daily seasonal adjustment (DSA) procedure described by Ollech (2018;2021) combines the aforementioned RegARIMA model with STL (Seasonal and Trend decomposition using Loess, Cleveland et al. 1990). DSA can integrate other seasonal adjustment methods as well, if they are flexible enough to handle daily data. In this regard,  discuss the flexibilization of X-11 to handle higher-frequency time series. STL decomposes a time series into T t , S t and I t using a series of Loess regressions and moving averages to separate out the trend and a periodic pattern from the series. Loess regressions are locally weighted linear or polynomial regressions (Cleveland and Devlin 1988). Each observation is regressed on a pre-defined set containing the γ τ closest observations. These observations are weighted, where the weight depends negatively on the distance between observations as follows: With a reference observation-i.e. the observation to be regressed on its neighbouring observations-at time t, the weight of the observation at t i is given by In the robust version of STL, after the decomposition of Y t into a trend, seasonal and irregular component (inner loop) an extreme value downweighing is added (outer loop). Observations with extreme values get a weight ω t in the local regressions in the next iterations given by The inner loop iterates through the following steps: STL only extracts one periodic pattern at a time. Therefore, DSA combines multiple runs of STL with RegARIMA:

Preliminary periodwise smoothing: Each periodwise subseries of TA
• Step I: Adjust intra-weekly seasonality with STL.
• Step II: Calendar-and outlier adjustment with RegARIMA. • Step III: Adjust intra-monthly seasonality with STL. • Step IV: Adjust intra-annual seasonality with STL.
In the intermediate steps, the remaining periodic effects are included in the trendcycle component, which by default in STL captures low-frequency variation that is lower than the considered seasonal frequency. 2 The resulting final seasonally adjusted series will be adjusted for all seasonal and moving holiday effects considered. The order of the steps in DSA follows the maxim to start with the periodic pattern with the shortest cycle length, i.e. the intra-weekly seasonality. Unlike the methods used for seasonal adjustment of lower-frequency time series, the RegARIMA part is not the first step, as it would necessitate to estimate a RegARIMA model with multiple seasonal parts. Future versions of DSA may include such an extension.
For all computations and analyses, we use R 4.0.2 (R Core Team 2021). The seasonal adjustment of daily time series is computed using the {dsa} package in version 1.1.15.

Seasonal adjustment of hourly time series
Hourly series are challenging due to the large number of observations included. For computational reasons, it will often not be possible to estimate a RegARIMA model for such series, if many regressors need to be included. Here we adapt the DSA procedure to the case of hourly data as follows: • Step I: Adjust hour-of-the-week effects with STL.
• Step II: Calendar-and outlier adjustment with RegARIMA based on daily observations. • Step III: Adjust hour-of-the-month effects with STL.
• Step IV: Adjust hour-of-the-year effects with STL.
As with DSA, single steps of the routine used to seasonally adjust hourly data may be omitted, e.g. only a few series will exhibit an hour-of-the-month effects. Also, the first step may be changed to model hour-of-the-day effects instead or in addition to day-of-the-week effects (see Remark 9).

Remark 2
The transformation from hourly to daily observations in step II only serves to reduce the computational burden. For some series, it may be feasible and preferable to directly estimate the calendar effects in the hourly series-provided an hourly RegARIMA model can be computed. If a distinct moving holiday impact for each hour of a holiday is assumed, the number of regressors may be extremely large. This is aggravated if cross-seasonal effects need to be modelled, e.g. if interactions between fixed holidays, the weekday and potentially the hour are prevalent.

Seasonal adjustment of weekly time series
On average, a year contains a non-integer number of weeks, namely 52.18. Ladiray et al. (2018) describe how, in these cases, autoregressive fractionally integrated moving average (ARFIMA) processes can be exploited that adapt the seasonal differencing part of Eq. 4 to incorporate fractionally integrated processes. The authors develop a fractional variant of the well-known airline model, i.e. the seasonal ARIMA model of order (0, 1, 1)(0, 1, 1) τ . For a non-integer seasonal period of τ = τ + α, with α ∈ [0, 1], the fractional differencing operator∇ τ can be approximated by a first-order Taylor series expansion so that The fractional airline model can be used for linearization of a time series analogously to Tramo. 3 We combine this pre-processing with a SEATS-type time series decomposition of the fractional airline model to seasonally adjust weekly time series with τ = 52.18.

Empirical illustrations
We will present a small set of higher-frequency time series that have been of high importance for economic analysis during the COVID-19 pandemic. We will discuss key features of this series and show how these might be accounted for when conducting calendar and seasonal adjustment.

Truck toll mileage index
The Federal Office for Freight Transport in Germany is responsible for a distancebased toll on trucks, which was implemented in January 2005. The truck toll mileage index has been developed together with the Federal Statistical Office. It is based on raw mileage and free of structural breaks resulting from changes in the vehicles that have to pay the toll. The data are available as a monthly and a daily index (Deutsche Bundesbank, 2020). The daily time series analysed here is available from 1 January 2005 up to 12 September 2021, and thus contains many observations.
As can be seen from Fig. 1, the series is characterized by multiple periodic effects, namely a strong weekday pattern, with a trough on Sundays and an annual pattern. These effects are interdependent: the cross-seasonality presents itself as a weekday pattern that changes throughout the year. In part, this is due to governing laws and regulations: with only a few exceptions, trucks are not permitted to drive on motorways on Sundays or on public holidays. During July and August, this is extended to Saturdays as well. We further observe different patterns around Christmas related to different changes in consumption behaviour.

Remark 3
Other series contain less typical or even uncommon periodic effects. Ollech (2021) finds a monthly recurring pattern in currency in circulation in Germany.
The series is further marked by a weakly positively sloping trend that is halted temporarily in early 2020, as a consequence of the COVID-19 pandemic. After the COVID-19-induced slowdown, we observe breaks in periodic effects, in particular the weekday pattern. At least temporarily, the difference in the truck mileage between workdays and weekend days is less pronounced. The annual seasonal pattern is nonisochronous, i.e. the number of observations per cycle is not the same for all cycles as years contain either 365 or 366 observations. Remark 4 Non-isochronicity can impact a time series in a number of ways. We may observe that the seasonal pattern in longer cycles is just a stretched-out version of the pattern in shorter cycles. This might be the case, if the seasonal pattern is a smooth pattern in the sense that the seasonal impact of neighbouring values are highly correlated. In series with a more fluctuating pattern, the seasonal impact of additional observations in a given cycle, such as February 29 in a leap year, may be less related to adjacent observations. This may imply that the estimation of the seasonal impact of these additional observations in longer cycles cannot be inferred from observations in the shorter cycle.
For higher-frequency time series, it is often possible to observe the impact of holidays more directly, and thus identify series-specific calendar effects. Figure 2 shows the truck toll mileage index on All Saints' Day (November 1) as well as the three days leading up to and the three days after the holiday. All Saints' Day is a public holiday in 5 out of 16 German federal states. In these states, all of which are located either in the south or west of the country, trucks are prohibited from driving on motorways on this day. The impact of All Saints' Day is cross-seasonal: the magnitude of the decrease depends on the weekday. 4 As restrictions already apply to the driving of trucks on Sundays, All Saints' Day does not have an additional impact on the truck toll mileage index if it falls on a Sunday. By contrast, there is a considerable reduction if it falls on any given weekday.
If All Saints' Day falls on a Tuesday (Thursday), the neighbouring Monday (Friday), i.e. any bridge days, also appear to have a reduced number of trucks on motorways. In contrast to lower-frequency time series, these bridge day effects can be seen and modelled more directly.
The COVID-19 pandemic, which hit Germany in March 2020, has a twofold impact on the time series. First, it leads to an abrupt decline in the number of kilometres driven by trucks. Second, it impacts the observed weekday pattern owing to changes in the restrictions regarding on which days trucks are allowed to drive on motorways and the consumption pattern. To obtain a calendar and seasonally adjusted series, DSA was used with γ 7 := 7 and γ 365 := 11 using an additive decomposition. For the seasonal adjustment of the monthly index, we found a multiplicative decomposition to be more appropriate. Yet due to the higher volatility of the daily index-which is typical for higher-frequency time series-the seasonal factors from a multiplicative model would inflate strong outliers in many cases, leading to extreme spikes in the seasonally adjusted series. The series does not contain day-of-the-month effects 5 ; accordingly, step III of the DSA procedure is omitted. The choice of a very short filter to estimate the day-of-theweek effect, i.e. γ 7 := 7, allows the day-of-the-week-effect to change during the year, thus making it possible to capture this particular cross-seasonal effect. As discussed above, this is important, because we already know that the weekday pattern is different in different parts of the year, due to the aforementioned driving restrictions. The dayof-the-year effect is captured using γ 365 := 11. This is appropriate because it allows the seasonal effect to change slowly over time, but does not overreact to single years. An important tool to identify an appropriate value for γ τ are visualizations of the seasonal-irregular component (Cleveland and Terpenning 1982), so-called SI-ratios. These are frequently used to visualize the volatility of the combined seasonal-irregular series and a plausible trajectory of the seasonal component. To estimate the impact of moving holiday effects and the interaction between weekdays and fixed holidays, a RegARIMA model is used. As described by Ollech (2021), this model combines a non-seasonal ARIMA model with trigonometric terms that capture deterministic seasonality. For the truck toll mileage index, 30 cosine and sine terms are used. Ollech (2021) states that multiples of 12 capture intra-monthly pattern, yet here, the high number of trigonometric terms used instead reflects the complexity of the seasonal pattern and does not indicate a day-of-the-month effect.

Remark 5
To obtain the unadjusted monthly truck toll mileage index, the daily raw truck toll mileage is summed up for each month and transformed into an index. If a temporal aggregation is likewise performed on the seasonally adjusted truck toll mileage index via summation, the resulting time series will contain a length-of-themonth effect-i.e. months with more days will have a higher value-and thus be seasonal. Using the average monthly index instead is a simple remedy, but reduces the comparability of the directly adjusted monthly and daily time series.
The calendar and seasonally adjusted truck toll mileage index is especially volatile around holidays (see Fig. 3). This heteroskedasticity is due to estimation uncertainty and the fact that truck drivers are restricted by laws with regard to the number of hours that they are allowed to drive per week and per day. The latter determines the optimal logistics and thus the interaction between holidays, consumer demands and kilometres driven by trucks on a given day resulting in the observed local volatility increases. Based on time series only adjusted for intra-weekly seasonal effects. A RegARIMA(2,1,1) model with 2 × 30 trigonometric terms has been estimated 1 NH includes the following holidays with fixed dates: Epiphany, Labour Day, Assumption Day, German Unity, Reformation Day and All Saints' Day. The weights of the regional holidays are given by: Epiphany 0.2, Assumption Day 0.1, Reformation Day (after 2017) 0.2 and All Saints' Day 0.6 Note: Ascension and Labour Day 2008 both fell on 1 May. Because the effect is not additive, the effect has been assigned to Labour Day only, i.e. the regressor for Ascension is 0 on that day. Table 2 shows the estimated impact of moving holidays and cross-seasonal effects. All moving holidays that are public holidays at the national or federal state level are included in the regression. This is extended to the surrounding days, as a transition phase can usually be observed, so that both the day before and after a given public holiday show a decrease. The only exception is the Sunday just before Pentecost Monday, which is slightly positive compared to a typical Sunday in spring. Additionally, we include the main event days of the carnival season for which a significant impact can be detected.
The impact of fixed holidays is usually captured by STL in the last step of DSA. Yet, as discussed above, the interdependency between that impact and the weekday cannot be captured by STL. Here, we model it using interaction dummies in the RegARIMA model. The impact of each weekday is combined across all national and federal statelevel holidays to increase the parsimony of the estimated model. Days that are public holidays only regionally are weighted accordingly (for details on the weights see Table  2 and Deutsche Bundesbank 2020). The remaining spikes around Christmas are not seasonal, as they do not fall on the same day of each year. Yet, with more observations it may be possible to further improve the estimation of cross-seasonal effects for that time period.
The changes in the weekday pattern after the start of the COVID-19 pandemic in Germany discussed above recur weekly and may thus be considered to be part of the S (7) -component. A different view is that as these changes are only temporary and a result of the irregular nature of the crisis they should be captured in the irregular component and thus be visible in the calendar and seasonally adjusted series. 6 For the period from 23 March to 30 August 2020, i.e. from the beginning of the lockdown until after the summer holidays, we chose the latter approach. To seasonally adjust this period, forecasted calendar and seasonal factors are used, which are obtained by restricting the estimation span from the beginning of the time series to 22 March 2020. For the period after 30 August, we again use all available data in a controlled current adjustment scheme. This means that the seasonal and calendar components are reestimated monthly, but we control weekly, whether a re-estimation is necessary. This is the case when the forecasted seasonal or calendar component no longer adequately capture the respective effect, e.g. because the seasonal pattern has evolved differently than predicted.

Remark 6
Generally, at the beginning of a crisis, it may be a good policy not to re-estimate the seasonal and calendar components immediately, in order to avoid including transitory and irregular influences into the seasonal or calendar components. Once the crisis has stabilized, a controlled current adjustment scheme as discussed above may be implemented.

Electricity consumption
The German electricity consumption is compiled by the German Federal Network Agency using data from the network providers starting in January 2015. The series analysed here ends on 30 April 2021. Energy consumption is related to industrial production and GDP. Electricity consumption is therefore of particular interest to business-cycle analysis (Arora and Shi 2016;Do et al. 2016). For the most recent observations, the series is subject to unreliable data delivery as some network providers do not always provide data immediately.
Remark 7 More broadly, unreliable data delivery stems from the issue that data providers are often not legally or contractually obligated to provide data and do not need to adhere to any standards regarding data quality or deadlines. Furthermore, if the data are merely a by-product, it may be difficult to analyse the quality of the data.

Fig. 4 Hourly electricity consumption in Germany
In the case of electricity consumption, this leads to temporarily missing values that might require interpolation, either before or as part of the seasonal and calendar adjustment.
Remark 8 Some higher-frequency time series do have structurally missing values, e.g. time series that only contain observations on working days. If this is the case, the data may be non-equidistant, i.e. the distance in time between observations is not the same for all neighbouring observations. In other words, the distance between a Monday and a Tuesday is less than between a Friday and a Monday.
Electricity consumption is available on a 15-minute basis. Ollech (2021) discusses how the daily electricity consumption can be seasonally adjusted. Here, we will analyse the hourly electricity consumption. Clearly, for business cycle analysis, it may be advantageous to focus on a lower-frequency aggregation level, such as a weekly series, as the volatility of the time series tends to decrease with aggregation level. However, discussing an hourly series will inform our understanding of typical characteristics of higher-frequency time series and thus is relevant for the modelling of daily and weekly series.
As can be seen from Fig. 4, the electricity consumption is characterized by an hourof-the-week pattern and-especially if we disregard Christmas and fixed holidays-an almost sinusoidal annual seasonality. The latter is termed autocorrelational seasonality, as the seasonal impact of each day of the year strongly correlates with the seasonal influences visible in the neighbouring observations. A dependence structure such as this is usually not exploited fully in the seasonal adjustment of lower-frequency time series. Yet, for higher-frequency time series it may improve the estimation of the seasonal effects given the high volatility of the time series and the corresponding estimation uncertainty.
Remark 9 Some series show multilevel periodic effects. Instead of an hour-of-theweek effect, an hourly series may contain an hour-of-the-day effect and an influence of the day of the week, if the pattern of the intraday-movements remains the same throughout the week and only the level of the series varies from weekday to weekday. By contrast, the magnitude of the intra-day-movements of the electricity consumption changes throughout the week, particularly when contrasting working days and weekends.
The difficulty of estimating all relevant effects is aggravated, as the daily electricity consumption is a relatively short series. 7 As a consequence, we cannot observe all possible calendar and cross-seasonal constellations. To illustrate this point, let us assume that we want to model the interaction between the weekdays and Labour Day. With just over five years of observations, not all possible interactions occurred in the time span considered, and in any case, only very few observations per constellation are included.
At times, higher-frequency time series show rather uncommon calendar effects. The daily electricity consumption is impacted by Daylight Saving Time (DST)-even in the hourly series.
The start of the COVID-19 pandemic in Germany again evokes a temporarily declining trend and some slight and gradual changes in the weekly pattern, reflecting the reduction in output in the production sector and possibly changes in the share of people working remotely (for a cross-country comparison of the impact of the pandemic on electricity consumption, see, López Prol and O 2020). For the seasonal adjustment of lower-frequency time series, the pandemic is treated using series of level shifts (LS), additive outliers (AO) and, less frequently, temporary change outliers (TC) with a pre-specified decay rate (Eurostat 2020). The downturn observed for electricity consumption is gradual enough so that one or multiple level shifts are not necessary to model the crisis based on daily data. In general, we observe non-traditional outlier pattern more frequently for higher-frequency time series than for monthly or quarterly series. This is neatly illustrated by the finding that the decay rate of 0.7 for TC, the default for monthly series, is often unsuitable for daily or weekly series.
To estimate the hour-of-the-week effect-which encompasses 24 · 7 = 168 hours a week-STL is employed with γ 168 := 7, i.e. a very short filter that is tailored towards an effect that changes throughout the year. More precisely, the difference between the daily minimum and daily maximum consumption is smaller on the weekend than on working days. This difference is especially small in winter compared to the rest of the year. 8 After the hour-of-the-week effect has been estimated, the adjusted series aggregated to daily observations serves as input to DSA. We omit the estimation of the day-ofweek and day-of-the-month effect, as these effects are not (or no longer) present in this partially adjusted series. Table 3 shows the estimated moving holiday and cross-seasonal effects for this time series. The cross-seasonal effects included are again interactions between weekday and fixed holidays. As mentioned, due to the length of the series, not all possible calendar interactions can be observed and overall, only very few observations per constellation are available.
As discussed above, a noteworthy calendar effect is Daylight Saving Time. DST has an obvious-albeit small-effect on the time series. Here, the moving holiday and cross-seasonal effects will be estimated in the series aggregated to a daily series by averaging the hourly observations. Alternatively, the hourly series could be temporally aggregated to a daily series by summing up the hourly values. Generally, if a multiplicative time series model is used, the difference between the estimated effects using daily averages comparing to those obtained using daily sums is often negligible for calendar constellations that impact the whole day. For example, electricity consumption is estimated to be 23.3 percent lower on Pentecost based on daily averages (see Table 3). If we used daily sums instead, the estimated impact was 23.2 percent.
If we consider matters that affect only single hours of a day, such as DST, the differences can be considerable. Transforming the hourly to a daily series by taking the mean will reduce the estimated impact of DST on the series, because DST will mostly be reduced to capturing the configuration of the hours in a day and the effect on the electricity consumers. If we used hourly sums instead, the estimated impact of DST would additionally include a length-of-the-day effect.
After the calendar and cross-seasonal effects have been estimated, they are broken down into hours, assuming that all hours of the day are influenced in the same way. The hourly series is then adjusted using these hourly factors.
Finally, the hour-of-the-year effect is estimated using STL in DSA with γ 24·365 := 13. The final adjusted series can be seen in Fig. 5.

Google Trends: Unemployment
Weekly Google Trends data have to be downloaded in chunks of five years. Each of these chunks is a random sample of all search queries and therefore subject to Table 3 Estimated moving holiday and cross-seasonal effects for German electricity consumption, in percent Based on time series adjusted only for hour-of-the week effects and aggregated to a daily series using daily means. A RegARIMA(5,1,1) model with 2 × 24 trigonometric terms has been estimated The definition of the week does not adhere to the ISO 8601 standard. Instead it is defined to start on Sunday and end on Saturday. Such date and time conventions are of relevance for regressor construction. For example, Easter Sunday and Easter Monday fall in the same week, and their effect cannot be disentangled. In turn, we only need one regressor that captures the joint effect of Easter Sunday and Easter Monday.
For lower-frequency time series, Christmas and New Year are seasonal effects as they fall in the same period each year. Depending on the modelling strategy, especially if the data are modified so that every year has 365 observations, this holds for daily data too. For weekly time series, as a consequence of their non-isochronicity and date conventions, Christmas' Day can fall in the 51st or 52nd week of the year, while New Year's Day falls in week 52, 53 or 1 (ISO 8601 standard) or week 0 or 1 (US convention).
The results of the RegARIMA model estimation are included in Table 4. As discussed, the construction of the data implies that some holidays always fall in the As can be seen from Fig. 6, the series is very volatile, especially at the beginning of the series. It may be difficult to assess the seasonality of the time series visually, but seasonality tests 10 and the coefficients of the ARIMA model indicate seasonality. To obtain seasonal factors, we rely on the default settings of the fractional airline decomposition.
As the original data are revised considerably every week, the revision policy of this weekly series differs from the daily series discussed above. For the Google Trends series, we follow a partial concurrent adjustment scheme, i.e. the calendar and seasonal components are re-estimated every week, but the order of the RegARIMA model is fixed and is only re-identified annually. The Appendix includes a discussion of graphical tools that can be used for weekly time series if a controlled current adjustment scheme is used.

Summary
The contribution of this paper is twofold. First, we performed an illustrative analysis of a small set of higher-frequency time series. We discussed how these data differ from lower-frequency time series and how this is relevant for seasonal adjustment in general and in light of the COVID-19 pandemic. Second, we developed a taxonomy of the central features of seasonal higher-frequency time series. This list of features can contribute to the assessment of the seasonal adjustment of higher-frequency time series and might serve as a building block in the development of quality diagnostics. Further research may evaluate different procedures that allow the seasonal adjustment of higher-frequency time series with respect to their ability to handle all of these features.
Funding Open Access funding enabled and organized by Projekt DEAL.

Declarations
Compliance with Ethical Standards This research received no external funding. Further, the author is not aware of any potential conflicts of interest.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

A Appendix
For a controlled current adjustment revision scheme, the results from the calendar and outlier adjustment are investigated and, perhaps most importantly, the previously forecasted seasonal components (Ŝ F ) are compared to the new seasonal factors (Ŝ N ) obtained after re-estimation. As reference points for the comparison, the raw seasonal factors (Ŝ raw ), i.e. the combined seasonal-irregular component, are included in the analysis. The comparison is either based on tables or graphics, with the latter often being referred to as the SI ratios. The idea is to gauge whetherŜ N better captures the systematic development ofŜ raw compared toŜ F . If this is the case to a relevant extent, the previous components are discarded and the re-estimated results are used. OtherwiseŜ F can still be used and the previously published seasonally adjusted values are not revised.
For integer-type periodic influences the construction of the SI ratios is straightforward. Each subplot shows theŜ N ,Ŝ raw andŜ F for a given period, e.g. a subplot for each month of the year. For series with a non-integer number of observations per year, the configuration is more intricate. For weekly series, the estimated seasonal component of a given observation depends both on observations (multiples of) 52 weeks ago as well as 53 weeks ago and likewise on future observations. Therefore,Ŝ raw should include all weeks that contribute to the estimation of the seasonal component. For the most recent observations, this is approximately the sequence of weeks with a step size of 52 and the respective previous week. This is shown by way of example in Fig. 8. For any period of the seasonal component, the exact representation of the relevant raw seasonal factors is given by the sequence (...,Ŝ raw t+ 2τ ,Ŝ raw t+ 2τ ,Ŝ raw t+ τ ,Ŝ raw t+ τ ,Ŝ raw t ,Ŝ raw t− τ ,Ŝ raw t− τ ,Ŝ raw t+ 2τ ,Ŝ raw t+ 2τ , ...).