A Random Forest-based Approach to Combining and Ranking Seasonality Tests

Daniel Ollech; Karsten Webel

doi:10.1515/jem-2020-0020

Abstract

Virtually every seasonal adjustment software includes an ensemble of tests for assessing whether a given time series is in fact seasonal and hence a candidate for seasonal adjustment. However, such tests are certain to produce either agreeing or conflicting results, raising the questions how to identify the most accurate tests and how to aggregate the results in the latter case. We suggest a novel random forest-based approach to answer these questions. We simulate seasonal and non-seasonal ARIMA processes that are representative of the macroeconomic time series analysed regularly by the Bundesbank. Treating the time series’ seasonal status as a classification problem, we use the p-values of the seasonality tests implemented in the seasonal adjustment software JDemetra+ as predictors to train conditional random forests on the simulated data. We show that this aggregation approach avoids the size distortions of the JDemetra+ tests without sacrificing too much power compared to the most powerful test. We also find that the modified QS and Friedman tests are the most accurate ones in the considered ensemble.

Keywords: binary classification; conditional inference trees; correlated predictors; simulation study; supervised machine learning

JEL Classification: C12; C22; C14; C45; C63

Corresponding author: Daniel Ollech, Deutsche Bundesbank, Central Office, Directorate General Statistics and Research Centre, Wilhelm-Epstein-Strasse 14, 60431 Frankfurt, Germany, E-mail: daniel.ollech@bundesbank.de

Appendix

We provide basic information about the six JD+ tests against the null hypothesis (H ₀) of absence of seasonality in a weakly stationary time series {z _t} of length T with τ observations per year. Non-stationary time series are made stationary by applying appropriate orders of (non-seasonal) differencing.

Modified QS Test

The modified QS test checks {z _t} for significant positive autocorrelation at seasonal lags. Let γ ( h ) = E ( z t + h z t ) − E 2 ( z t ) and ρ(h) = γ(h)/γ(0) denote the lag-h autocovariance and autocorrelation, respectively, of {z _t}. Then, H ₀ : ρ(k) ≤ 0 for k ∈ {τ, 2τ}, and the QS-statistic is given by

Q S = T ( T + 2 ) ρ ̂ 2 ( τ ) T − τ + max { 0 , ρ ̂ ( 2 τ ) } 2 T − 2 τ × 1 ( 0 , ∞ ) ρ ̂ ( τ ) ,

where ρ ̂ ( h ) is the lag-h sample autocorrelation of {z _t}. The null distribution of QS is approximated by a χ ²-distribution with two degrees of freedom (Maravall 2011).

Friedman Test

The Friedman (FD) test checks for significant differences between the period-specific mean ranks of the values of {z _t}. Let r _ij be the within-year rank of the observation in the ith period of the jth year, so that 1 ≤ r _ij ≤ τ, and μ i = E ( r i j ) . Then, H ₀ : μ ₁ = μ ₂ = … = μ _τ, and the FD-statistic is calculated as a one-way ANOVA with repeated measures, which asymptotically follows a χ ²-distribution with τ − 1 degrees of freedom under H ₀ (Friedman 1937).

Kruskal–Wallis Test

The Kruskal–Wallis (KW) test results from replacing the within-year ranks of the FD test with within-span ranks, so that 1 ≤ r _ij ≤ T and the KW-statistic is calculated as a one-way ANOVA without repeated measures. Hence, the same asymptotic null distribution applies (Kruskal and Wallis 1952).

Periodogram Test

The periodogram (PD) test checks if a weighted sum of the spectral density f(ω) = (2π)⁻¹ ∑ _h γ(h) e^−ihω evaluated at the seasonal frequencies S ( τ ) = ω 1 ⋆ , … , ω τ / 2 ⋆ with ω j ⋆ = 2 π j / τ for j ∈ {1, …, τ/2} is significantly different from zero. Let Σ S ( τ ) = 2 f ω 1 ⋆ + ⋯ + f ω τ / 2 − 1 ⋆ + f ω τ / 2 ⋆ be this weighted sum. Then, H 0 : Σ S ( τ ) = 0 , and the PD-statistic is obtained through plugging the periodogram estimates of {z _t} into Σ S ( τ ) and proper rescaling, so that PD follows an F-distribution with τ − 1 − 1 E [ T ] and T − τ + 1 E [ T ] degrees of freedom under H ₀, where E is the set of even integers.

Seasonal Peaks Test

The seasonal peaks (SP) test checks if f(ω) displays visually significant peaks within S ( τ ) . Let n _vsp be the number of such peaks and S P = 1 { 1 , … , τ / 2 } [ n vsp ] . Then, H ₀ : SP = 0, and the SP-statistic is obtained from replacing n _vsp with a count estimator n ̂ vsp that aggregates information from the Tukey and AR(30) estimators of f(ω). Criteria for visual significance, including simulated critical values, and the aggregation rule are derived in Maravall (2011).

Seasonal Dummies Test

Dropping the stationarity assumption on {z _t}, the seasonal dummies (SD) test checks if the joint effects of the τ − 1 seasonal dummies, denoted by β ∈ R τ − 1 , are significantly different from zero in a time series regression with a constant mean and ARIMA (pdq)(000) disturbances. Thus, H ₀ : β = 0, and the SD-statistic is the usual F-statistic based on the GLS estimator of β , which follows an F-distribution with τ − 1 and T − d − p − q − τ − 1 degrees of freedom under H ₀ (Gómez and Maravall 2001).

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/jem-2020-0020).

Received: 2020-11-19

Revised: 2022-05-12

Accepted: 2022-06-01

Published Online: 2022-06-23

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