Analysis of approaches to identification of trend in the structure of the time series

Objectives. The study set out to compare the forecasting quality of time series models that describe the trend in different ways and to form a conclusion about the applicability of each approach in describing the trend depending on the properties of the time series.

Methods. A trend can be thought of as the tendency of a given quantity to increase or decrease over the long term. There is also an approach in which a trend is viewed as some function, reflecting patterns in the behavior of the time series. In this case, we discuss the patterns that characterize the behavior of the series for the entire period under consideration, rather than short-term features. The experimental part involves STL decomposition, construction of ARIMA models (one of the stages of preparation for which includes differentiation, i.e., removal of the trend and transition to a weakly stationary series), construction of ACD models (average conditional displacement) and other approaches. Time-series models based on various trend models are compared with respect to the value of the maximum likelihood function. Many of the combinations have not been constructed before (Fourier series as a trend model, combination of ACD model for trend with seasonal models). Example forecasts of macroeconomic statistics of the Russian Federation and stock prices of Sberbank on the Moscow Exchange in the time range of 2000–2021 are presented.

Results. In the experiments, The LOESS method obtained the best results. A combination of polynomial model for trend description and ARIMA for seasonally description and combination of ACD algorithm for trend and ETS for seasonal model obtained good forecasts in case of seasonal time series, while Fourier time series as a trend model also achieved close quality of prediction.

Conclusions. Since the LOESS method for groups of seasonal and non-seasonal series gives the best results for all indicators, this method can be recommended for obtaining the most accurate results for series of different nature. Trend modeling using Fourier series decomposition leads to quite accurate results for time series of different natures. For seasonal series, one of the best results is given by the combination of modeling a trend on the basis of a polynomial and seasonality in the form of the ARIMA model.

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