# Applying spline-based phase analysis to macroeconomic dynamics

• Gadasina Lyudmila and Vyunenko Lyudmila
From the journal Dependence Modeling

## Abstract

The article uses spline-based phase analysis to study the dynamics of a time series of low-frequency data on the values of a certain economic indicator. The approach includes two stages. At the first stage, the original series is approximated by a smooth twice-differentiable function. Natural cubic splines are used as an approximating function y . Such splines have the smallest curvature over the observation interval compared to other possible functions that satisfy the choice criterion. At the second stage, a phase trajectory is constructed in ( t , y , y ) -space, corresponding to the original time series, and a phase shadow as a projection of the phase trajectory onto the ( y , y ) -plane. The approach is applied to the values of GDP indicators for the G7 countries. The interrelation between phase shadow loops and cycles of economic indicators evolution is shown. The study also discusses the features, limitations and prospects for the use of spline-based phase analysis.

MSC 2010: 65D15; 65D07; 37C50

## 1 Introduction

Traditionally, regression and autoregressive models are used to analyze the dynamics of macroeconomic indicators. Such approaches require a priori definition of the model specification and rather strict assumptions about the data properties for example stationarity. Another approach is to consider the time series of macroeconomic indicators as a function of the dynamic system states, it’s evolution being described by deterministic or stochastic differential equations [5,6,9,10,11].

The values of macroeconomic indicators are aggregated and averaged over relatively long periods. This allows us to consider them as deterministic ones and provides the basis for applying the second approach. In this case, the main task is to reconstruct the dynamic system that generated the time series. According to Takens’ theorem [8], a description of the phase space of a dynamical system can be obtained if, instead of the real variables of the system, we take finite-dimensional delay vectors composed of the values series at successive moments of time.

This study uses the approach called spline-based phase analysis to investigate a time series of low-frequency data. It is applicable to the analysis of the macroeconomic indicators dynamics, such as GDP, consumption, foreign trade balance, etc. Spline-based phase analysis can adequately assess the features of the process, which remain unnoticed when using simple regression models.

The approach includes two stages. At the first one, we replace the discrete time series Y i of economic indicators by a smooth function g ( t ) that best approximates the original discrete time series according to a certain criterion described later. The second stage consists in analyzing the joint behavior of the constructed function and it’s first derivative g ( t ) .

The criterion for choosing an approximating function is as follows. It should be continuous, adaptive and provide a minimum error in the description of the data under study. The specified requirements are simultaneously satisfied by interpolation cubic spline functions.

In a number of studies [1,2, 4,7], spline interpolation is effectively used to analyze low-frequency data. For example, Chakroun and Abid state that “the cubic spline method is a tractable and reasonably correct estimation method that we recommend in any market with infrequent trading” [2]. Roul and Prasad Goura [7] demonstrate the effectiveness of using splines for the Asian option pricing problem. The authors recommend using the cubic spline method to analyze stock quotes in any market with infrequent trading.

## 2 Data description and methodology

We consider historical annual data on GDP for G7 countries in 2000–2019. Data source is World Bank Group – International Development, Poverty, & Sustainability [12].

The collected datasets have the following properties: the data are low-frequency and highly aggregated. This means that it is free of noise and can be considered “as is.”

## 4 Discussion

Empirical analysis has shown that phase shadows are an appropriate tool for studying the dynamics of macroeconomic indicators. Herewith, interpolating by natural cubic splines is well suited for smoothing low-frequency aggregated data.

When applying the approach outlined in the article, it is necessary to take into account the peculiarities of calculating the analyzed indicators. It is important to consider the data in constant prices, and at the same time, when comparing phase shadows for different subjects, it is necessary to normalize the indicators.

The results clearly illustrate the events in the global economy. Applying the proposed method for a set of indicators study allows identifying crises and examining their phases, both a posteriori, and forward by making forecasts based on spline extrapolation of data. The outlined approach has a good development prospect for frequent data as well.

1. Funding information: This research was funded by the project Russian Foundation for Basic Research (RFBR). Project number: 20-010-00960.

2. Conflict of interest: The authors state no conflict of interest.

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Revised: 2022-04-15
Accepted: 2022-04-22
Published Online: 2022-06-15