Capturing black Friday effect in seasonal adjustment with an application to the Swedish retail sales

Abstract

Black Friday’s rapid growth in Sweden since its introduction in 2013 poses challenges for the seasonal adjustment of Swedish retail sales index. Long-term seasonal effect patterns fail to reflect the short-term presence of Black Friday, which could cause misleading seasonally adjusted results for retail sales and household consumption, in particular for November and December months. We demonstrate this problem with the increasing seasonal-irregular ratios from seasonal adjustment. We propose to use explanatory variables for capturing and estimating Black Friday effect and three explanatory variables are suggested, with gradually increasing magnitude for November months. Seasonal adjustments using those explanatory variables included in the RegARIMA model showed significantly improved results. The explanatory variable in an exponential form has extra appealing properties for our purposes.

Keywords

Black Friday retail sales index seasonal adjustment X-12-ARIMA

1. Introduction

Black Friday is originated in Philadelphia, United States, in the 1950s and takes place the Friday after Thanksgiving every year. In the 1980s, Black Friday spread from Philadelphia to the rest of the United States. In 2010, Amazon introduced Black Friday to the United Kingdom, bringing the shopping phenomenon to Europe.¹ Over the course of 15 years, Black Friday has become a global shopping phenomenon. What started as a single-day shopping event has grown significantly from its initial format. In recent years, add-ons such as Black Week and Cyber Monday further amplify their effect on retail sales. In Sweden, the shopping holiday has, since its introduction in about 2013,² become a well-established event on the Swedish consumer calendar for both retailers and consumers.

Previous studies in the field of Black Friday have focused primarily on exploring shopping rituals, consumer behavior, and related topics to help retailers in their marketing and understanding of consumers’ perspectives. Several studies have also examined consumer attitudes, shopping venue preferences, and Black Friday convenience factors in physical and online stores.¹ The main purpose of our work is nevertheless to investigate Black Friday effect on retail sales indices and how to capture and estimate the effect during seasonal adjustments.

Retail sales index (RSI) measures turnovers from retail enterprises. RSI, such as the one released by Statistics Sweden³ since 1963, is a building block for Household expenditure statistics and an important indicator used by the government and the central bank (Riksbanken) to monitor business cycles.⁴ This index is of huge interest for the retail industry in Sweden too.

In order to mitigate the seasonal fluctuations and facilitate the short-term comparison, National Statistical Institutes (NSIs) usually conduct seasonal adjustments of RSI.⁵ Although we can use methods built in the mainstream seasonal adjustment approaches, such as X-12-ARIMA⁶ or TRAMO/SEATS,⁷ to capture the more established seasonal effects, the emergence of new and rapidly evolving patterns poses a challenge in accurately seasonally adjusting such data. The new introduction of sales spikes in November, brought by Black Friday shopping, for example, makes it difficult to interpret seasonally adjusted results of RSIs over time, in particular for November and December months. Misleading seasonally adjusted result may lead to wrong decisions for decision-makers or retailers.⁴ The Office for National Statistics (ONS) in the United Kingdom (UK) faces similar challenges especially in the face of post-Brexit uncertainty and rising prices⁸ after the COVID-19 pandemic. To be able to handle Black Friday effect adequately in their RSI, ONS collects additional feedback from retailers on their Black Friday promotions to understand variations in activity across different types of stores, and analyzes each sub-sector individually.⁹

The approach used by ONS is solid. However, it is tedious and time demanding since it requires significantly more data collection to accommodate differences in consumer behavior and retail patterns. In this article, we offer the first systematic investigation of the Black Friday effect within the context of seasonal adjustment by investigating the seasonal-irregular components (SI ratios, cf.⁶) of Swedish retail sales, in particular SI ratios for November months. To address the instability caused by this rapidly emerging phenomenon, we propose three purpose-built explanatory variables to capture its evolving magnitude. The result of seasonal adjustment of the Swedish RSI shows that these explanatory variables, included in the RegARIMA model of X-12-ARIMA,⁶ can successfully capture the Black Friday effects. The practical usefulness of our approach is demonstrated through applications to both the Swedish and the UK retail sales data, showing that the proposed variables can be seamlessly integrated into standard RegARIMA-based seasonal adjustment workflows, which make them extra appealing for NSIs.

The remainder of this paper is organized as follows. In Section 2, we introduce the Swedish retails sales index and the impact of Black Friday. The main results including the proposal of three explanatory variables and the evaluation are given in Section 3, while the final section, Section 4, concludes the article.

2. The Black Friday effect and SI ratios

2.1. The Swedish retail sales

Our interest to Black Friday effect in seasonal adjustment arises when we study the Swedish retail sales,³ using data from Statistics Sweden.¹⁰ The Swedish RSI includes indices for 40 different retail sales sectors, ranging from food stores and gas stations to opticians and watchmakers. The data we studied are available monthly from January 1991 until January 2025 with 397 observations. Indices for retail sales in supermarkets, other food stores, and Systembolaget (stated-owned alcohol retailer) are not included in our analysis since these retailers usually do not offer promotions during Black Friday in Sweden. All other remaining sectors are included in the analysis, disregarding if they were expected to be influenced by Black Friday promotions or not. The list of sectors included in our study is available in Table 3. Without loss of generality our description below will mainly focus on the Swedish General RSI (i.e., General retail trade except fuel). The Swedish RSI data were extracted from Statistics Sweden on 2025-11-25, using the latest available vintage including all revisions available at that date.

2.2. The SI ratios

In this study, we utilize the SI ratios from X-12-ARIMA⁶ to investigate the Black Friday effects (nevertheless, it does not limit our proposed approach to X-12-ARIMA method). We use JDemetra+,¹¹ the free seasonal adjustment program recommended by Eurostat,¹² for seasonal adjustments in our study. SI ratios, the observed values after removing the long-term trend-cycle, are particularly interesting to observe how seasonality evolves along with the emergence of Black Friday phenomenon. To study the seasonality, it is seemingly sufficient to investigate the seasonal component alone. However, without a proper handling of Black Friday effects in the seasonal adjustment, some part of the effect might be (wrongly) decomposed into the irregular component. Therefore, our focus will be on the SI ratios which includes irregular components too.

In order to explain easier the consequences of possibly wrong SI ratios and to be self-contained, we introduce here shortly the decomposition of the X-11 algorithm, while the preprocessing step in seasonal adjustment will be discussed briefly in Section 3.1. Other details of X-11 algorithm are referred to Findley et al.⁶ and the monograph.¹³

In Stage 1 of the X11 decomposition, the initial trend, SI ratio, seasonal factor, and seasonally adjusted values are estimated from the pre-processed data, denoted by $Z_{t}$ . The trend component is obtained by applying a “centered 12-term” (1) moving average.⁶

\begin{aligned} T_{t}^{(1)} & = \frac{1}{24} Z_{t - 6} + \frac{1}{12} Z_{t - 5} + \dots + \frac{1}{12} Z_{t} \\ + \dots + \frac{1}{12} Z_{t + 5} + \frac{1}{24} Z_{t + 6} . \end{aligned}

(1)

Then, the initial SI ratios are obtained as in Eq.(2)

\begin{aligned} S I_{t}^{(1)} = \frac{Z_{t}}{T_{t}^{(1)}} . \end{aligned}

(2)

The initial preliminary seasonal factor is estimated via ”3 x 3” seasonal moving average (3)

\begin{aligned} {\hat{S}}_{t}^{(1)} & = \frac{1}{9} S I_{t - 24}^{(1)} + \frac{2}{9} S I_{t - 12}^{(1)} + \frac{3}{9} S I_{t}^{(1)} \\ + \frac{2}{9} S I_{t + 12}^{(1)} + \frac{1}{9} S I_{t + 24}^{(1)} . \end{aligned}

(3)

The initial seasonal factor and seasonally adjusted estimate, $A_{t}$ , are obtained after standardizing.

\begin{aligned} S_{t}^{(1)} & = \frac{{\hat{S}}_{t}^{(1)}}{\frac{1}{24} {\hat{S}}_{t - 6}^{(1)} + \frac{1}{12} {\hat{S}}_{t - 5}^{(1)} + \dots + \frac{1}{12} {\hat{S}}_{t + 5}^{(1)} + \frac{1}{24} {\hat{S}}_{t 6}^{(1)}}, \end{aligned}

(4)

\begin{aligned} A_{t}^{(1)} & = \frac{Z_{t}}{S_{t}^{(1)}} . \end{aligned}

(5)

In Stage 2, the symmetric Henderson filters¹⁴ are introduced during the estimation of trend and removing the seasonality. Kenny & Durbin discovered in 1982 that the surrogate Henderson weights achieve the same results as if the time series were extended using linear extrapolation.¹⁴ The $(2 H + 1)$ term presented below is the Henderson coefficient and is denoted as $h_{j}^{2 H + 1}$ where $H$ denotes half of the length of the moving average window. The goal of the Henderson coefficients is to minimize the mean square revision between the initial and the final trend estimates. The subscript $j$ varies from $- H$ to $H$ , which indicates the distance each coefficient has to the middle observation.¹⁴ Using the Henderson filters, the intermediate trend is obtained in Eq. (6) and SI ratio in Eq. (7).

\begin{aligned} T_{t}^{(2)} & = \sum_{j = - H}^{H} h_{j}^{(2 H + 1)} A_{t + j}^{(1)}, \end{aligned}

(6)

\begin{aligned} S I_{t}^{(2)} & = \frac{Z_{t}}{T_{t}^{(2)}} . \end{aligned}

(7)

Then, a preliminary seasonal factor via a ”3 x 5” seasonal moving average is calculated as

\begin{aligned} {\hat{S}}_{t}^{(2)} & = \frac{1}{15} S I_{t - 36}^{(2)} + \frac{2}{15} S I_{t - 24}^{(2)} + \dots + \frac{3}{15} S I_{t}^{(2)} \\ + \dots + \frac{2}{15} S I_{t + 24}^{(2)} + \frac{1}{15} S I_{t + 36}^{(2)}, \end{aligned}

(8)

with an updated seasonal factor as

\begin{aligned} S_{t}^{(2)} = \frac{{\hat{S}}_{t}^{(2)}}{\frac{1}{24} {\hat{S}}_{t - 6}^{(2)} + \frac{1}{12} {\hat{S}}_{t - 5}^{(2)} + \dots + \frac{1}{12} {\hat{S}}_{t + 5}^{(2)} + \frac{1}{24} {\hat{S}}_{t 6}^{(2)}} . \end{aligned}

(9)

And then lastly in Stage 2, the seasonally adjusted results are derived as follows.

\begin{aligned} A_{t}^{(2)} = \frac{Z_{t}}{S_{t}^{(2)}} . \end{aligned}

(10)

The final (Stage 3) trend, irregular component, and estimated decomposition of observed time series are given in Eq.(11) – (13).

\begin{aligned} T_{t}^{(3)} & = \sum_{j = - H}^{H} h_{j} (2 H + 1) A_{t + j}^{(2)}, \end{aligned}

(11)

\begin{aligned} I_{t}^{(3)} & = \frac{A_{t}^{(2)}}{T_{t}^{(3)}}, \end{aligned}

(12)

\begin{aligned} Z_{t} & = T_{t}^{(3)} S_{t}^{(2)} I_{t}^{(3)} . \end{aligned}

(13)

To visualize the Black Friday effect in the Swedish RSI, the SI ratios from X-12-ARIMA, Eq. (7), for November months 1991-2024 are presented in Figure 1.

Figure 1.

The SI ratios for November month for the Swedish General RSI.

Figure 1 shows that while the SI ratios are rather stable between 1991 and 2012, there is a clear increase beginning around year 2013, the first year Black Friday was introduced to Sweden. SI ratios for other months (not reported), have been stable (except for December) for the whole time period, indicating a stable seasonality for those months. While for December months, SI ratios show a longer decreasing trend, which might indicate both the shift of Christmas shopping to Black Friday and Black Week and a relative decrease of Christmas shopping compared with other months in Sweden. The investigation of Christmas shopping is outside the scope of this study.

The abnormal increase in November SI ratios reflects an evolving seasonal pattern rather than irregular fluctuations. As will be shown later, this evolving seasonality can be captured through deterministic regressors in the RegARIMA pre-processing step. Once the estimated Black Friday effect is removed from the linearised series prior to X-11 decomposition, the resulting SI ratios become substantially more stable over time, as the decomposition is then performed on a series where the evolving seasonal component has already been accounted for.

The increase of SI ratios in November months is constant, except for the COVID-19 pandemic year 2020. As can be seen from Equations (3), (4), (8) and (9), in order to estimate seasonal factor $S_{t}^{(2)}$ at the end of time series, forecasts of future SI ratios such as $S I_{t + 12}^{(2)}$ , $S I_{t + 24}^{(2)}$ , and $S I_{t + 36}^{(2)}$ , are needed, which are obtained in practice by forecasting observed time series $Y_{t}$ . However, it is difficult, if not impossible, to capture Black Friday effects sufficiently by just ordinary ARIMA models without a dedicated treatment of the effects.

We emphasise that this limitation does not concern the ARIMA model itself. In RegARIMA-based seasonal adjustment, ARIMA provides the stochastic structure for residuals and the forecasts required at the series endpoints, while deterministic effects such as calendar components, outliers, and the proposed Black Friday regressor are handled through the regression part of the model. The difficulty therefore arises from the absence of an explicit regressor describing the evolving Black Friday effect, not from any insufficiency of ARIMA modelling.

Figure 1 illustrates the limitation of relying solely on ARIMA forecasts without dedicated regressors. It can be seen from Eq.(8) and (9) that constantly increasing SI ratios at the end of time series without accurate forecasts will lead to wrong estimate of seasonal factors $S_{t}^{(2)}$ . Consequently, the revision will be larger for the final seasonal adjusted results compared with preliminary releases. It is our purpose to mitigate this problem by including some dedicated explanatory variables.

In operational X-11, SI ratios are not obtained in a single pass of the moving-average formulas above. The algorithm proceeds iteratively through the B–C–D stages, where preliminary SI ratios are refined after the detection and down-weighting of extreme-value observations. The SI ratios reported in this paper correspond to the final iteration applied to the linearised series obtained after RegARIMA pre-processing. For a full description of the iterative routine and the treatment of extreme values, see for example.¹³

3. Explanatory Variables for Black Friday effect

Before we introduce our explanatory variables, it is helpful to explain briefly our modeling choice, and the framework of RegARIMA model in seasonal adjustment.

3.1. The RegARIMA model

Our modeling choice is guided by the operational setting of official statistics. The aim of the paper is not to develop a new seasonal adjustment method, but to provide a simple and transparent way to capture the Black Friday effect as a deterministic regressor within an established seasonal adjustment workflow. In RegARIMA-based frameworks, deterministic effects such as calendar components and outliers are estimated in a pre-processing step and removed, either temporarily or permanently, before the decomposition of the linearised series by the X-11 algorithm. This is precisely the mechanism through which NSIs typically incorporate special events and evolving calendar-related patterns in routine production, with effects allocated to the appropriate components of the decomposition.

Alternative approaches are available. State-space and structural time series models can represent evolving seasonality and interventions in a unified latent-component framework, and Bayesian variants can further quantify parameter uncertainty. However, these approaches require specifying and validating a full component model and associated estimation procedure, which may reduce portability across production environments and complicate routine implementation. Since our main contribution concerns the specification and interpretation of Black Friday regressors, rather than the construction of a new decomposition model, we focus on RegARIMA pre-adjustment, which is supported by mainstream seasonal adjustment tools and can be adopted with minimal changes to existing production pipelines.

ARIMA models have been used for modelling seasonal time series and seasonal adjustment for a long time; see e.g.¹⁵ Maravall¹⁶ and Gomez et al.⁷ developed a regression with ARIMA errors (RegARIMA) in the seasonal adjustment program TRAMO/SEATS. RegARIMA was later integrated into the X-12-ARIMA method.⁶

For the observed time series $Y_{t}$ , a RegARIMA model is a regression (14) with deterministic regressors $X_{i t}$ and the error term $Z_{t}$ following an ARIMA model (15),

\begin{aligned} Y_{t} & = \sum_{i = 1}^{r} β_{i} X_{i t} + Z_{t}, \end{aligned}

(14)

\begin{aligned} ϕ_{p} (B) Φ_{P} (B^{s}) (1 - B)^{d} (1 - B^{s})^{D} Z_{t} & = θ_{q} (B) Θ_{Q} (B^{s}) a_{t}, \end{aligned}

(15)

where

(B)

indicates the back-shift operator,

B (Z_{t}) = Z_{t - 1}

, and

s

are the seasonal periods, for monthly data as in our case,

s = 12

Common regressors of $X_{i t}$ include trend constants, calendar adjustment variables such as those for trading day and moving holidays, and variables accounting for outliers. Parameters of RegARIMA models can be estimated by maximum likelihood estimation or Iterative Generalized Least Squares (IGLS) algorithm.¹⁷ During this procedure, a diagnostics check is made which includes examination of residuals as well as identification of outliers.⁶

Accordingly, the seasonal adjustment in this paper follows the standard two-step structure. First, the RegARIMA pre-processing step estimates and removes temporarily deterministic effects, including calendar components, outliers, and the proposed Black Friday regressor, producing the linearised series. Second, the X-11 algorithm decomposes this linearised series through the iterative procedure described in Section 2.2. In praxis,¹³ the deterministic effects will be allocated to different final components depending on their nature. For example, the calendar effects are usually assigned to the final seasonal component and will be removed from the final seasonally adjusted result. Level shift (LS) outliers are allocated to the Trend-cycle component, while Temporary Change (TC) or Additive Outlier (AO) effects to the Irregular Component (cf.⁶ or¹³).

The RegARIMA model plays an important role across seasonal adjustment frameworks. In both TRAMO/SEATS and X-11 based workflows, RegARIMA is used to produce the linearised series and the short-term forecasts (cf. Eq.(8) and (9)), which support the moving-average decomposition in X-11 algorithm.⁶ In TRAMO/SEATS, the ARIMA specification plays a broader role, as it is also used to derive the model-based decomposition filters.⁷ Our approach is independent of this distinction, since the proposed Black Friday variable enters the deterministic regression component and can therefore be applied in both frameworks without modification.

3.2. Explanatory variables for Black Friday effect

Our main idea in this study is to construct an explanatory variable to capture Black Friday effect. The coefficient of such a variable should be significant when fitting to the data, and the effect should be plausible and interpretable. A starting point is that an application of the explanatory variable (or variables) should mitigate the abnormal SI ratios for November (Figure 1). What is more, it will be beneficial if the value of the variable can be interpreted as the degree of Black Friday effect. Here, the ”degree of Black Friday effect” is understood as a normalized measure of how established the Black Friday phenomenon is relative to its long-run saturation level. Values close to zero correspond to an emerging effect, while values close to one indicate a mature and fully established Black Friday impact on retail sales.

There are different ways to construct such explanatory variables. One consideration is whether we should barely model the effect of Black Friday or include the whole Black Week. In the latter case, a seemingly natural choice is to split up the effect into November and December in proportion of numbers of days in that week that are falling into November respective December, analogous to how the Easter effect is handled in seasonal adjustment.⁶ However, such a variable is insignificant according to our preliminary test on the Swedish RSI. One possible explanation is that shopping on Black Friday alone dominates the consumption of whole Black Week, a shopping behavior functioning different as Easter. In addition, it is difficult to disentangle Black Week effect eventually assigned to December from the traditional Christmas shopping effect. Therefore, in our study, we will focus on creating explanatory variables with non-zero values only for November months, while values for the other months are zero.

3.2.1. Outliers for Black Friday effect?

Imposing seasonal outliers (SO) to November months is a natural option to attempt to capture Black Friday effect. Seasonal outlier for some particular time point $t_{0}$ (November 2013 in our case), denoted by $S O_{t}^{t_{0}}$ , takes value $0$ for $t < t_{0}$ , $1$ for $t \geq t_{0}$ and $t$ at the same season (November in our case) as $t_{0}$ , and $- 1 / (s - 1)$ for other time $t$ , where $s$ is the frequency of the time series.¹¹ However, such a SO is insignificant for all years of the Swedish General RSI, an expected result with respect to the increasing SI ratios for these years.

As a comparison, it is of interest to experiment adding additive outliers to each and every November month from year 2013 until 2024 for the same time series. The result is reported in Table 1. As we can see, eight of the twelve additive outliers have p-values under $0.05$ except for years 2013-2015 and 2019. Furthermore, the coefficients of those outliers generally increase from year 2013 to 2022 (except for year 2019). The coefficients don’t continue increasing in 2023 and 2024, probably affected by the decreased retail sales in Sweden (see Figure 2) these two years followed by the high inflation.

Figure 2.

SA results of the Swedish General RSI without (”Original”) respectively with (”Exp”) Var3 in the RegARIMA pre-processing.

Table 1.

Estimates of the additive outliers imposed in all November months between 2013 and 2024 for the Swedish general RSI.

Year	Coefficient	T-Stat	P[T]>t
2013	0.0170	0.94	0.3490
2014	0.0187	0.94	0.3477
2015	0.0365	1.71	0.0883
2016	0.0497	2.18	0.0301
2017	0.0629	2.60	0.0097
2018	0.0746	2.92	0.0037
2019	0.0502	1.87	0.0623
2020	0.0847	3.02	0.0027
2021	0.1077	3.70	0.0003
2022	0.1091	3.61	0.0003
2023	0.1031	3.30	0.0011
2024	0.0902	2.75	0.0062

Table 2.

Values of the three proposed explanatory variables for November month.

Year	Var1	Var2	Var3
2013	0.0	0.0803	0.0952
2014	0.1	0.1605	0.1813
2015	0.2	0.2408	0.2592
2016	0.3	0.3211	0.3297
2017	0.4	0.4014	0.3935
2018	0.5	0.4816	0.4512
2019	0.6	0.5619	0.5034
2020	0.7	0.6422	0.5507
2021	0.8	0.7224	0.5934
2022	0.9	0.8027	0.6321
2023	1.0	0.8830	0.6671
2024	1.0	0.9632	0.7534

Table 3.

Regression coefficients and p-values (in parentheses) for the Swedish retail sectors. P-values under $0.05$ in bold.

Sector	Var1	Var2	Var3
General Retail trade (except fuel)	0.0962 (0.0009)	0.1084 (0.0009)	0.1393 (0.0008)
Non-specialised stores	0.0303 (0.0813)	0.0409 (0.0357)	0.0545 (0.0271)
Retail sale with mostly durables	0.1736 (0.0001)	0.1985 (0.0001)	0.2431 (0.0002)
Other non-specialised stores	0.0809 (0.3817)	0.1048 (0.3176)	0.1523 (0.2459)
Specalized stores except state liquor stores	−0.0013 (0.9789)	−0.0083 (0.8803)	−0.0236 (0.7355)
Specialised stores for information and communication equipment	0.498 (0.0000)	0.5284 (0.0000)	0.6062 (0.0000)
Specialised stores for other household equipment	0.0448 (0.397)	0.0659 (0.2723)	0.0827 (0.2779)
Specialised stores for interior decoration	0.061 (0.3983)	0.0861 (0.2938)	0.1308 (0.2087)
Specialised stores for hardware, plumbing and building materials, paints and glass	0.0365 (0.5403)	0.0561 (0.4071)	0.0772 (0.3709)
Specialised stores for home furniture; specialised stores for office furniture	−0.0159 (0.8032)	−0.0124 (0.8641)	−0.0236 (0.7984)
Specialised stores for cultural and recreation goods	0.1974 (0.0073)	0.232 (0.0053)	0.279 (0.0083)
Bookshops; specialised stores for newspapers and stationery	0.0657 (0.5418)	0.0591 (0.6287)	0.0442 (0.7757)
Sports shops	0.2166 (0.0032)	0.2578 (0.0019)	0.32 (0.0025)
Sports shops; bicycle shops	0.2658 (0.0000)	0.3079 (0.0000)	0.3806 (0.0001)
Toyshops	-0.0205 (0.8272)	0.0032 (0.9762)	0.0607 (0.6531)
Specialised stores for other goods	0.0953 (0.0598)	0.1145 (0.0469)	0.1536 (0.0359)
Retail sale of clothes and footwear	17.0845 (0.0343)	18.6893 (0.0417)	23.6986 (0.0422)
Clothing stores	0.152 (0.0362)	0.1587 (0.0547)	0.1903 (0.071)
Shoe and leather goods shops	0.2169 (0.0158)	0.2457 (0.0157)	0.3145 (0.0154)
Dispensing chemists	2.3959 (0.2342)	2.7411 (0.2295)	3.8358 (0.1865)
Retail sale of medical, cosmetic and toilet articles	0.1974 (0.0048)	0.224 (0.0065)	0.2929 (0.0047)
Flower shops and florist’s shops	0.0064 (0.9522)	0.0165 (0.8917)	0.0468 (0.7618)
Specialised stores for watches and jewellery	0.0538 (0.5178)	0.0779 (0.4093)	0.1169 (0.3282)
Specialised stores for watches and clocks	0.1488 (0.1381)	0.1686 (0.1398)	0.2718 (0.0588)
Specialised stores for jewellery	0.0086 (0.9235)	0.0288 (0.7768)	0.0374 (0.7725)
Opticians	0.0004 (0.9955)	0.0225 (0.7617)	0.0219 (0.8178)
Specialised stores for other new goods n.e.c.	0.1284 (0.3305)	0.1633 (0.2764)	0.1795 (0.3484)
Stalls and market stands; other retail sale not in stores, stalls or markets	0.1754 (0.3341)	0.2046 (0.3210)	0.2425 (0.3534)
Retail sale via mail order houses or via internet	0.232 (0.0366)	0.2281 (0.0703)	0.2819 (0.0747)

Table 4.

BIC values of the three explanatory variables for the Swedish retail sectors.

Sector	Var1	Var2	Var3
General Retail trade (except fuel)	−7.8711	−7.8749	−7.8748
Non-specialised stores	−8.3024	−8.3045	−8.3067
Retail sale with mostly durables	0.3970	0.3810	0.3806
Other non-specialised stores	−8.4463	−8.4468	−8.4471
Specalized stores except state liquor stores	−7.0383	−7.0666	−7.0378
Specialised stores for information and communication equipment	−5.7089	−5.7086	−5.7090
Specialised stores for other household equipment	−6.2387	−6.2388	−6.2391
Specialised stores for interior decoration	−5.4524	−5.4303	−5.4236
Specialised stores for hardware, plumbing and building materials, paints and glass	−6.5173	−6.5192	−6.5189
Specialised stores for home furniture; specialised stores for office furniture	−5.9580	−5.9592	−5.9605
Specialised stores for cultural and recreation goods	−5.8395	−5.8404	−5.8406
Bookshops; specialised stores for newspapers and stationery	−5.8823	−5.8822	−5.8824
Sports shops	−6.0258	−6.0276	−6.0256
Sports shops; bicycle shops	−5.4019	−5.4016	−5.4011
Toyshops	−5.7583	−5.7615	−5.7607
Specialised stores for other goods	−5.5250	−5.5274	−5.5244
Retail sale of clothes and footwear	−5.4637	−5.4636	−5.4640
Clothing stores	−6.7983	−6.7996	−6.8006
Shoe and leather goods shops	3.3645	3.3986	3.3987
Dispensing chemists	−5.9434	−5.9417	−5.9407
Retail sale of medical, cosmetic and toilet articles	−4.9143	−4.9144	−4.9144
Flower shops and florist’s shops	1.6182	1.6178	1.6170
Specialised stores for watches and jewellery	−5.3594	−5.3707	−5.3721
Specialised stores for watches and clocks	−5.1234	−5.1234	−5.1237
Specialised stores for jewellery	−5.8681	−5.8690	−5.8698
Opticians	−5.2636	−5.2632	−5.2660
Specialised stores for other new goods n.e.c.	−5.4640	−5.4644	−5.4644
Stalls and market stands; other retail sale not in stores, stalls or markets	−5.7843	−5.7848	−5.7848
Retail sale via mail order houses or via internet	−4.6662	−4.6667	−4.6658

The AO experiment indicates a gradually increasing pattern rather than a one-time break, which supports modelling the effect with a single smooth regressor instead of a set of year-specific outliers. Combined with the insignificance of the SO specification, this suggests that the Black Friday effect is better represented as a deterministic evolving seasonal component.

3.2.2. Explanatory variables

As can be seen from Figure 1, the increase of the SI ratios is a fairly steady (except for year 2020). The comparison with AO and SO alternatives suggests that a single evolving regressor be sufficient and preferable to a set of ad-hoc outliers. Therefore, a baseline is to construct a linear variable, starting from 2013. Several step sizes are tested. The explanatory variable with a step size 0.1 (denoted as Var1 in Table 2) works fairly well in our case. Another choice to decide the step size in a linear explanatory variable is to fit a (first order) linear regression to the observed SI ratios between 2013 and 2024 and use the slope as the step size, which leads to variable Var2 in Table 2.

These two variables are very simple and easy to use. But the linearity assumption is rather arbitrary; nothing guarantees an even and equal increase of the Black Friday effect during the entire period. Another problem with linear variables is that it cannot really ensemble ”degrees of” Black Friday effect, because it reaches 1 sooner or later, which would indicate some sort of ”full effect” of a theoretical maximum Black Friday effect. The tentative Black Friday effect reaches 1 in year 2023 in Var1 and 2025 in Var2. We will then lose this desirable interpretation for time periods after. In our experiment, we tentatively truncate Var1 to 1 for year 2024 (and later).

To circumvent these problems, we propose thus an exponential explanatory variable defined as

\begin{aligned} x_{t} = 1 - e^{- b t} . \end{aligned}

(16)

In Eq. (16), $b > 0$ is a hyperparameter and ${t}$ are years with $t = 1$ for 2013. For $t > 0$ , $x_{t}$ is an increasing function of $t$ , $x_{t} < 1$ , and $lim_{t \to \infty} x_{t} = 1$ . The hyperparameter $b$ controls the speed at which the Black Friday effect approaches its long-run level, with larger values implying a faster maturation of the effect and smaller values corresponding to a more gradual adoption process. The hyperparameter $b$ can take different values depending on data property. For our data $b = 0.1$ is chosen to fit the SI ratios (up to a constant factor). In practical applications by NSIs, the parameter $b$ can be selected using simple empirical criteria, such as a regression over November SI ratios or AO coefficients in Table 1, performing sensitivity analysis over a small grid of plausible values, or assessing the stability of regression diagnostics. Since the regressor enters the RegARIMA model linearly, its scale does not affect statistical inference, making this choice operationally straightforward. In 2013, Black Friday effect is about 10 percent (0.0952) of the full effect (see Var3 in Table 2), while in 2024, it reaches about $75 %$ , which seems to be reasonable.

There are possibly other explanatory variables that fit our purpose too, but we leave the investigation for future research.

3.3. The Results and Evaluation

The use of a single Black Friday regressor in the RegARIMA equation (14) is motivated by the AO analysis in Section 3.2.1, which reveals a coherent upward trajectory of November effects rather than isolated one-time shocks. This pattern can be parsimoniously captured by one deterministic variable, whose estimated effect is allocated to the seasonal component and removed from the final seasonally adjusted series in accordance with standard seasonal adjustment practice. We included the explanatory variables in turn into the RegARIMA model (Eq. (14)) and fit the models to the Swedish RSI data, not only to the General retail trade, but also other sectors of RSI.

3.3.1. Significance of the explanatory variables

Table 3 presents the p-values of the explanatory variables for different sectors in the Swedish retail trade. The p-values are significant (less than 0.05) for the broader sectors such as the General RSI and Retail sale with mostly durables. For other sectors, the results are generally aligned to what we would expect from the Black Friday promotion practices. The explanatory variables are statistically significant across most sectors for communication equipment, clothes, and cosmetic, while less significant for sectors for hardware, office furniture, or jewelry. The p-values for a couple of sectors, interior decoration and toyshop, are surprisingly insignificant, contrary to our expectation. However, we should bear in mind that our intuitive expectation are originated mainly from the Christmas shopping. It is difficult to verify or disprove the Black Friday effect without the data and analysis of promotions and shopping behavior during Black Fridays.

Among the three explanatory variables, the results are quite similar, while one or two more sectors have significant p-values with the exponential variable (Var3) than with the linear variables (Var1 resp. Var2).

3.3.2. BIC

We computed the Bayesian Information Criterion (BIC) to compare the three explanatory variables. Once again, the results in Table 4) are quite similar. It is not unexpected since we have only 12 (meaningful) observations for the variables, and their values are very similar in the beginning years.

3.3.3. SI ratios with the explanatory variables

The SI ratios for November months, obtained from seasonal adjustment of the General Retail Trade in Sweden with the explanatory variables, are shown in Figure 3.

Figure 3.

The new SI ratios for November when seasonally adjusted with the Black Friday explanatory variables for the Swedish General RSI.

The SI ratios between 2013 and 2024 shift clearly to a more horizontal line. Except for a remaining dip in 2020, the SI ratios are quite stable from 1998 onward. This indicates that the strong seasonality (above 1) observed in Figure 1 was mainly driven by the Black Friday effect. After removing this effect, the SI ratios behave normally and the seasonality in November months for the General Retail Trade is slightly below 1. Judged from the change in SI ratios, the proposed explanatory variables successfully achieve their intended purpose. We emphasise that the stability of SI ratios is used here only as a diagnostic device to illustrate the evolving seasonal pattern, and that the quality of the seasonally adjusted series is usually assessed by for example residual seasonality diagnostics rather than by visual stability of SI ratios alone.

3.3.4. SA results with the explanatory variables

In order to illustrate the impact of the explanatory variables, we compared the original seasonally adjusted (SA) result (with same calendar variables used by Statistics Sweden), with the SA result additionally with the exponential explanatory variable included in RegARIMA (Eq. (14)). The results are shown in Figure 2. In seasonal adjustments, the praxis (cf.⁵) is that the estimated effects of regression variables in RegARIMA are removed from the final SA results, if they are calendar or seasonal effects. In our application, we treat Black Friday effects in a similar way as calendar effects and remove them from the SA results. In practice, it is acceptable since the Black Friday phenomenon is relatively new and not so established as for instance Easter.

As we can see from Figure 2, with the explanatory variable included in the RegARIMA, the SA results can be rather different, and capturing and removing the Black Friday effects affects would impact the seasonality of all months (cf. Table 5). For instance, the month-to-month growth rate for November 2020 was changed from -0.58% (without Var3) to -2.14% (with Var3), while for December 2020, the growth rate was changed from -0.35% to 4.32%.

Table 5.
Monthly growth rate (%) for the Swedish general RSI before and after applying var3 (oct., nov. and dec., 2013–2024).

Time Growth before (%) Growth after (%)

2013-10-01 −1.25 −3.19

2013-11-01 1.23 2.16

2013-12-01 0.89 1.30

2014-10-01 0.12 −1.28

2014-11-01 −0.09 1.72

2014-12-01 0.35 0.63

2015-10-01 0.34 0.10

2015-11-01 −0.05 0.03

2015-12-01 −0.45 0.42

2016-10-01 2.01 0.35

2016-11-01 −0.17 −0.11

2016-12-01 0.07 −0.26

2017-10-01 1.33 2.06

2017-11-01 −0.38 −0.30

2017-12-01 0.90 0.29

2018-10-01 −1.05 1.36

2018-11-01 1.37 −0.50

2018-12-01 0.15 1.11

2019-10-01 0.07 −1.05

2019-11-01 −2.05 1.25

2019-12-01 4.14 0.38

2020-10-01 0.51 0.08

2020-11-01 −0.58 −2.14

2020-12-01 −0.35 4.32

2021-10-01 −0.36 0.53

2021-11-01 0.49 −0.67

2021-12-01 −1.54 −0.21

2022-10-01 −2.31 −0.39

2022-11-01 0.75 0.46

2022-12-01 −1.12 −1.50

2023-10-01 1.12 −2.37

2023-11-01 −0.75 0.74

2023-12-01 0.15 −1.12

2024-10-01 2.25 0.97

2024-11-01 0.65 −0.78

2024-12-01 −0.49 0.17

Time	Growth before (%)	Growth after (%)
2013-10-01	−1.25	−3.19
2013-11-01	1.23	2.16
2013-12-01	0.89	1.30
2014-10-01	0.12	−1.28
2014-11-01	−0.09	1.72
2014-12-01	0.35	0.63
2015-10-01	0.34	0.10
2015-11-01	−0.05	0.03
2015-12-01	−0.45	0.42
2016-10-01	2.01	0.35
2016-11-01	−0.17	−0.11
2016-12-01	0.07	−0.26
2017-10-01	1.33	2.06
2017-11-01	−0.38	−0.30
2017-12-01	0.90	0.29
2018-10-01	−1.05	1.36
2018-11-01	1.37	−0.50
2018-12-01	0.15	1.11
2019-10-01	0.07	−1.05
2019-11-01	−2.05	1.25
2019-12-01	4.14	0.38
2020-10-01	0.51	0.08
2020-11-01	−0.58	−2.14
2020-12-01	−0.35	4.32
2021-10-01	−0.36	0.53
2021-11-01	0.49	−0.67
2021-12-01	−1.54	−0.21
2022-10-01	−2.31	−0.39
2022-11-01	0.75	0.46
2022-12-01	−1.12	−1.50
2023-10-01	1.12	−2.37
2023-11-01	−0.75	0.74
2023-12-01	0.15	−1.12
2024-10-01	2.25	0.97
2024-11-01	0.65	−0.78
2024-12-01	−0.49	0.17

Table 6.

Coefficients and p-values (in parentheses) of the explanatory variables including resp. excluding 2020 and 2021 years’ data.

Series	Var1	Var2	Var3
Swedish RSI incl. COVID years	0.0962 (0.0009)	0.1084 (0.0009)	0.1393 (0.0008)
Swedish RSI excl. COVID years	0.0935 (0.0009)	0.1062 (0.0009)	0.1360 (0.0008)
UK RSI incl. COVID years	0.0370 (0.0152)	0.0381 (0.0267)	0.0918 (0.0001)
UK RSI excl. COVID years	0.0320 (0.0143)	0.0354 (0.0153)	0.0503 (0.0060)

3.4. An Application to UK data

To demonstrate the generalisability of our proposed explanatory variables, we apply the three variables to the General RSI (also known as All retailing excluding automotive fuel) from ONS, UK. The UK RSI follows a similar definition as the Swedish RSI. But the reporting times are different. In Sweden, the data are collected for the whole months, while in the UK, the retailers report the sales in predefined 4-weeks periods. The calendar variables used here consist of Easter and default working days.¹¹ The SI ratios for November, prior to including the explanatory variables, are shown in Figure 4. The UK data were downloaded from the Office for National Statistics (ONS) on 2026-02-03, using the latest revised series available at the time of analysis.

Figure 4.

The SI ratios for November for the UK General RSI without Black Friday explanatory variables.

The increase of SI ratios for the UK General RSI is not as stable as that of Sweden, but a clearly visible increase starts from about 2014, except for years 2019, 2020, and 2024. For both 2019 and 2024, the Black Friday was on the 29 November and was excluded in the report periods of retail trade to ONS (cf.⁹ and¹⁸). Note that Black Friday was introduced to the UK earlier, but established around 2013.⁹ It is of interest to point out that retail trade series in the UK have a stronger seasonal factor in November than Sweden.

Because of this problem with reporting periods, we may need to adjust our approach for creating the explanatory variables to the UK data. However, to test the feasibility of the approach, we include the explanatory variables without any modification in the RegARIMA model and seasonally adjust the UK General RSI with X-12-ARIMA method. The new SI ratios from the seasonal adjustment are shown in Figure 5.

Figure 5.

The new SI ratios for November when seasonally adjusted with the Black Friday explanatory variables for the UK General RSI.

Seen from this figure, it is clear that the increase of SI ratios from 2014 onward has been smoothed out and the seasonal effects are in level with previous years, indicating that our approach successfully achieved its main purpose. In addition, our approach can be adjusted to further improve the result. For instance, one could set up an optimization framework and choose another value of the hyperparameter $b$ for the UK data. For the problem with Black Fridays excluded in report periods, one possible solution is to replace the values of the explanatory variables, for example, by zeros, for such years.

The UK application serves to illustrate the external validity of the proposed approach beyond the Swedish setting. While both Sweden and the UK experienced a rapid diffusion of Black Friday promotions in the early 2010s, there are notable differences in timing, retail structure, and statistical reporting practices. In Sweden, Black Friday became firmly established around 2013, whereas in the UK adoption occurred slightly earlier and with greater heterogeneity across retailers. Moreover, the UK retail sector exhibits a higher degree of concentration and a stronger role of online sales, and reporting periods may exclude Black Friday in some years when it falls late in November. Despite these institutional differences, the proposed explanatory variables perform similarly in stabilizing November SI ratios, suggesting that the approach is robust to cross-country variation and can be applied flexibly within standard RegARIMA-based seasonal adjustment frameworks.

3.5. Some remarks on the COVID-19 effects

One possible disturbance to the results is the COVID-19 pandemic in 2020 and 2021. The COVID-19 period is characterised by exceptionally large and abrupt shocks to retail activity. In our analysis, pandemic-related disturbances are primarily handled within the RegARIMA pre-processing stage through automatic outlier detection, allowing for AO, temporary changes (TC), and level shift (LS) outliers. For the Swedish General RSI, no outlier was identified associated with the COVID-19 period (cf. Figure 2). It is plausible since Sweden has not been locked down during the COVID-19 period. For other Swedish retail sectors, AO or LS outliers were found, mainly concentrated on March or April 2020. By contrast, AO or TC outlier was identified in March, April, and June 2020, as well as January and April 2021 in the UK General RSI, with different signs, reflecting the measures taken to fight the COVID-19 pandemic in the UK and the disturbance brought to the retail sales.

It is possible that the COVID-19 pandemic altered consumer behaviour and temporarily interacted with Black Friday promotions, for instance through the shift toward an even higher degree of online retail in 2020–2021. In our study, pandemic disturbances are treated either as outliers or irregular effects, while the Black Friday effect is modelled as a deterministic seasonal component. The eventual interaction between COVID-19 and Black Friday effects is difficult to quantify theoretically using aggregate monthly retail data in just two years.

As an empirical illustration to how the results of our approach would change if COVID-19 had never happened, we repeated the seasonal adjustment with similar settings, but removed all the observations from 2020 and 2021 and treated them as missing values in JDemetra+ for both the Swedish and the UK General RSI. The overall pattern of November SI ratios (not reported) remain qualitatively unchanged, with a coherent upward trajectory before adding our explanatory variables and a smooth trajectory after. The coefficients of Var1, var2, and Var3 decreased slightly when excluding data from 2020 and 2021, but remain very significant, as shown in Table 6. For example, the coefficient of Var3 was changed from $0.1393$ to $0.136$ for the Swedish General RSI, and from $0.0918$ to $0.0503$ for the UK General RSI. The result suggests that the COVID-19 pandemic might have slightly accelerated the Black Friday adaptation, seen from the slightly larger coefficients with the real data in 2020 and 2021. The result also show that our proposed approach remains robust and valid in either cases, in particular for the Swedish data, where the modification of the coefficients is quite small which can be found in Table 6.

4. Conclusion and discussion

Black Friday, the once single-day shopping event originated from the United States, has grown significantly from its initial format and gained popularity in many European countries such as Sweden and UK in recent years. Black Friday’s rapid growth not only affects the retail sales, but also poses challenges for the seasonal adjustment of the retail sales indexes.

In this article, we investigated the Black Friday effect in a comprehensive way by inspecting the seasonal-irregular components (SI ratios). The figure of SI ratios showed a clearly abnormal increase from around year 2013, in particular for the Swedish RSI, plausibly caused by the occurrence and growth of the Black Friday effect. Using X-11 algorithm, we explained how the estimation of seasonal factor for November would be misleading if we failed capturing the Black Friday effect and disregarding the constant increase of SI ratios.

We proposed to use explanatory variables to catch the Black Friday effect. Utilizing once again the SI ratios as an approximation to this effect and the experiments with seasonal outlier and additive outliers, three explanatory variables were created for this purpose, with gradually increased magnitude for November and zeros for other months from 2013 and onward. Including the explanatory variables into the RegARIMA model and seasonally adjusting the Swedish RSI, we showed that the proposed explanatory variables could successfully mitigate the problem of increasing SI ratios after 2013, and the seasonal factors for November between 2013 and 2024 became in level of previous years. All explanatory variables are significant for most of sectors in retail sales.

To our best knowledge, there is no study of Black Friday effect in the literature, not even a clear definition of it, which makes it difficult to handle the Black Friday effect in the practice of seasonal adjustment. In this study, we utilize the SI ratios as a proxy for Black Friday effect and propose to use single evolving explanatory variable to capture the effect. While two linear variants (Var1 and Var2) are simple and easy to implement, the exponential variable, $x_{t} = 1 - e^{- b t}$ , is extra appealing since its value can be interpreted as ”degree” of Black Friday effect, with $x_{t} < 1$ for $t > 0$ , and $lim_{t \to \infty} x_{t} = 1$ . Despite simplicity of the framework, the approach has been proved useful both for the Swedish and the UK retail sales data. What is more, as our applications showed, it needs not much extra efforts to integrate the explanatory variables to the RegARIMA and the ordinary seasonal adjustment workflow.

Admittedly, many potential improvements can be made to our approach. We have suggested linear and exponential forms for explanatory variables that work for our case, but by no means exclude other types of functions. The explanatory variables are imposed only to November with zeros for the other months in our suggestion and we have not found other combinations that perform satisfactorily. It would be interesting to investigate other combinations of values of explanatory variables, not least some zero-mean explanatory variables. Empirical study has been conducted to choose the step sizes for the linear variables, as well as parameter $b$ for the exponential variable, such that the explanatory variables fit the SI ratios for the time periods after the introduction of Black Friday. During the empirical study, We made use of the fact that since the variables will be integrated into a RegARIMA model, they are equivalent up to a constant factor. A likely more legitimate approach is to develop an optimization framework with respect to some predefined target, and then find the parameters from the optimization. We leave all such improvements for future work and hope that our study could inspire more research in this area.

In addition to improving the seasonal adjustment of Swedish and UK retail sales, our approach has broader practical implications. The proposed explanatory variables are fully portable and can be applied by any national statistical institute working with monthly retail indices. Because they operate through standard regression components in the same RegARIMA framework, they are compatible with all major seasonal adjustment frameworks, including X-11, X-12-ARIMA, X-13 and TRAMO/SEATS. Importantly, the method requires no additional data collection, which makes it particularly attractive for official statistics where methodological stability, transparency and operational efficiency are essential.

Footnotes

Acknowledgments

We are very grateful to the Editor and two anonymous referees for their careful review of our manuscript and for their constructive and insightful comments, which has lead to significant improvement of the manuscript.

ORCID iDs

Ingrid Heed

Yingfu Xie

Yukai Yang

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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