key: cord-1054176-6yv2a3wz
authors: Bógalo, Juan; Llada, Martín; Poncela, Pilar; Senra, Eva
title: Seasonality in COVID-19 times
date: 2021-12-15
journal: Econ Lett
DOI: 10.1016/j.econlet.2021.110206
sha: b9b030ea6f8fdf389a933d99c8b66ecbe90d1637
doc_id: 1054176
cord_uid: 6yv2a3wz

COVID-19 hit the economy in an unprecedented way, changing the data generating process of many series. We compare different seasonal adjustment methods through simulations, introducing outliers in the trend and seasonality to reproduce the heterogeneity in the series during COVID-19.

COVID-19 has destroyed the dynamics of many economic time series. In particular, trend and seasonality have been greatly hit by the effects of the lockdown. In this context, seasonal adjustment, a frequently used tool to monitor the state of the economy in real time, has become a difficult task generating great 5 distress about the reliability of the alternative estimations. As a consequence, the different statistical offices reacted quickly and provided some guidelines to adapt seasonal adjustment methodologies to cope with this new and unexpected situation; see, for instance, the Australian Bureau of Statistics (Australian Bureau of Statistics, 2020), the European Statistical Office (European Commis- both, the raw data as well as the data clean of seasonality. Therefore, the application of alternative seasonal adjustment methods on the simulated time series enables us to approximate the effect that COVID-19 has had on the deseasonalized data as well as the relative reliability between different procedures.

The benchmark seasonal adjustment strategy considered is the well-known X-13ARIMA-SEATS of the Census Bureau (https://www.census.gov/data/ software/x13as.html). We have also included as an alternative the newly introduced non-parametric Circulant Singular Spectrum Analysis (CiSSA), see Bógalo et al. (2021) . The comparison exercise also considers the projection of 30 last year's seasonal factors.

The remainder of the paper consists of four sections. In section 2, we revise the different methodologies for seasonal adjustment. In section 3, we introduce our simulation strategy. In section 4, we present the results. Finally, in section 5, we draw our conclusions. 35 

The most common methods for seasonal adjustment used nowadays are included in X-13ARIMA-SEATS, developed by the US Bureau of Census. This comprises enhanced versions of both, the non-parametric X-11 (Shiskin et al., 1967 ) and the ARIMA model based TRAMO-SEATS (Maravall, 1993; Gómez 40 & Maravall, 1996) . Given, for simplicity, a zero mean time series x t , t = 1, ..., T , X-11, first, estimates an initial trend using a moving average. After removing the trend from the time series, it estimates the seasonal component by also In what follows, we briefly describe the CiSSA algorithm. In the first step, we define a window length L, and transform the original vector of data x t , t = 1, ...T into a related trajectory matrix X of size L × N where N = T − L + 1, given 55 by:

In the second step, we find the eigenstructure of a matrix of second moments related to X to obtain the so called elementary matrices of rank 1. The trajectory matrix X can be recovered as the sum of the elementary matrices which are associated to different frequencies. In particular, CiSSA builds a circulant 60 matrix related to the second moments of the time series, S C given by: The eigenvalues and eigenvectors of S C are given by (see Lancaster, 1969) :

and u k = L −1/2 (u k,1, ..., u k,L ) H respectively, for k = 1, ..., L, where H indicates the conjugate transpose and u k,j = exp −i2π(j − 1) k−1 L .

The diagonalization of S C allows us to write X as sum of elementary matrices X k of rank 1 as:

In a third step, we go back from the matrices to the vectors of the time series and transform the elementary matrices X k into elementary signals of the same length as the original series for each frequency w k , k = 1, ..., L by

, where x i,j are the elements of the elementary matrix X k .

In the final fourth step, we group the extracted elementary signals according to the frequency they represent. Notice that the k-th eigenvalue in (1) is an estimate of the spectral density at w k = k−1 L , k = 1, ..., L and, therefore, the kth eigenvalue and corresponding eigenvector are associated with this frequency.

We simulate time series, modify their trend and seasonal dynamics to approximate the impact of COVID-19 and state the basis for understanding and comparing the effect that the pandemic might have had on real economic time series data. We take as the series free from COVID-19 effects, the addition of the simulated trend, cycle, seasonal, and irregular components. This can be 80 considered as the data that we should have obtained, had the pandemic not taken place, i.e. a counterfactual time series. Notice that the sum of all the 4 J o u r n a l P r e -p r o o f Journal Pre-proof components except the seasonal one constitutes the "true" time series, free of seasonality, denoted as x 0 SA,t . The knowledge of the original components enables us to assess the reliability 85 of the different methods (denoted by the subindex m) by comparing precisely the "true" seasonally adjusted time series (x 0 SA,t ) with the deseasonalized series resulting from each estimation method (x 0 m,t ). In the free-COVID-19 BSM model the data are generated as the sum of trend, cycle, seasonality and irregular components as follows 1 (we denote this 90 counterfactual time series by x 0 t ):

where µ t is the changing level or trend component, c t is the cycle, s t is the seasonal component and e t is the irregular component. The "true" deseasonalized time series will be given by x 0 SA,t = x 0 t − s t . Regarding the data generating process for x 0 t given by (2), we assume an 95 integrated random walk for the trend, see, for instance, Young (1984) , given by

with η t ∼ N (0, σ 2 η ). The cyclical and seasonal components are specified according to Durbin & Koopman (2012) . The cycle is given by the first component of the bivariate VAR (1): 1] . The seasonal component is given by

with w j = j s , j = 1, ..., [s/2] and s the seasonal period, where [·] denotes the integer part and a j,t and b j,t are two independent random walks with noise variances equal to σ 2 j . Finally, the irregular component is white noise with the variance σ 2 e . All the components are independent of each other. We set ρ c = 1, so the trend, cycle and seasonal components all have unit roots.

To approximate the COVID-19 effects on the simulated time series we address both, the impact on the trend and the seasonal components. Regarding the trend, we considered a shock at time T COV ID−19 of magnitude δ T rend drawn from a uniform distribution in [a, b], 0 < a < b < 1 where b and a would represent the minimum and maximum impact on the level of the series, respectively.

This heterogeneity in the value of δ T rend will capture the alternative effects of COVID-19 in different economic time series.

To represent the dynamics of the shock we include the possibility of transitory effects by means of including a Transitory Change intervention.

In this sense, the simulated series is of the form:

with 

To understand the possible effect of the COVID-19 pandemic on seasonal The Basic Structural Model is generated by considering equations (2 to 6), s = 12 (monthly time series) and the cyclical period equal to 1 wc = 48 months.

The noise variances of the different components are given by σ 2 η = 0.0006 2 , σ 2 j = 0.004 2 , σ 2 ε = 0.008 2 and σ 2 e = 0.06 2 . We considered three different sample sizes, T = 97, 193 and 243.

We generated R = 1000 replications from the base model and on each replication we also generated the different contamination schemes. Then, for each replication, we assessed the accuracy of the alternative procedures by means of the root mean square error (RMSE) between the generated time series free of seasonality and the estimated seasonally adjusted time series:

Afterwards, we computed the average of the RMSEs across replications.

To assess the relative performance of the different parametric and nonparametric seasonal adjustment strategies, the first set of simulations were run under the non-contaminated original Basic Structural Model. All the models considered were applied in an automated way with no outlier treatment. Regarding CiSSA, we needed to choose the window length L, such 140 that L < T /2, because the trajectory matrices with window length L and N = T − L + 1 are transposed.

We chose several values of L, multiples of s = 12, according to the different sample sizes T analyzed. Table 1 shows the average RMSE for the 1,000

replications for the different seasonal adjustment strategies and sample sizes.

145 Table 1 suggests that in normal times, there are no noteworthy differences among the alternative procedures. Table 1 

In this section we show the impact of the COVID-19 pandemic whose effect was simulated with changes in trend and seasonality according to (7) for the last 12 observations. The new parameters were δ T rend and w in (7). δ T rend is related to the COVID-19 effect of the trend which is drawn from a uniform [0.2, 0.8]. Comparing Table 2 with Table 1 we can see that the accuracy diminishes The magnitude of the worsening ranges between 1.1 and 2.5 times the RMSEs presented in Table 1 for estimations made with the whole sample, but between Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Jan-09 Jan-10 Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16 Jan-17 Jan-18 Jan-19 Jan-20 Jan-21 

Seasonal adjustment has become a very difficult task since the COVID-19

pandemic. Several proposals and recommendations have appeared to cope with this problem, among them, the use of outlier techniques and the projection of 

Circulant singular spectrum analysis: A new automated procedure for signal extraction

Time Series Analysis by State Space Methods

Guidance on treatment of covid-19-crisis effects on data

Programs TRAMO and SEATS: Instructions for the User

Theory of Matrices

Stochastic linear trends: Models and estimators

The x-11 variant of the census method ii seasonal adjustment program

Covid-19's effect on the advance monthly retail trade survey

Recursive Estimation and Time-Series Analysis

Alternative seasonal adjustment methods perform similarly in normal times

Outliers in trend and seasonality diminish the accuracy of the seasonally adjusted estimates

Projecting the previous year seasonal factors in the presence of outliers is the worst option

CiSSA performs better than X-13ARIMA-SEATS and SEATS with outlier correction