ON EQUAL PREDICTIVE ABILITY AND PARALLELISM OF SELF-EXCITING THRESHOLD AUTOREGRESSIVE MODEL
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ON EQUAL PREDICTIVE ABILITY AND PARALLELISM OF SELF-EXCITING THRESHOLD AUTOREGRESSIVE MODEL
ABSTRACT
Several writers have devised statistical approaches for determining model similarity. This work not only introduces the concept of
We expand the concept of equivalency to quantify the prediction power of a time series using a stationary self-exciting threshold autoregressive (SETAR) process.
A thesis and lemma were utilised to connect the predictability measure to the coefficients and sample autocorrelation of the SETAR process. Examples demonstrate how to conduct the test, assisting practitioners in making informed decisions.
Chapter One: Introduction
1.1 Introduction.
Yule’s Autoregressive models, used to examine sunspot numbers, sparked widespread interest and research into linear time series modelling in 1927. Linear time series models have been widely used in several fields due to its ability to provide accurate one-step forward predictions.
However, Sunspot numbers (covered below) demonstrate that this is not always the case. The reasons for this are discussed later on.
Nonlinear time series analysis became popular in the 1970s. The interest stemmed from the requirement to model nonlinear changes in real-world time series data.
Limit cycles, time-irreversibility, amplitude-frequency dependency, and jump phenomena are not properly described by autoregressive Integrated Moving Average (ARIMA) models, as demonstrated by the Sunspot statistics. Tong (1978) proposed a method for modelling nonlinear changes in time series data using different Autoregressive (AR) processes.
The switch between AR models is based on the delay parameter and threshold value(s), which are specific time lag values from the given time series. Tong and Lim (1980) and Tong (1983) provided a detailed description of the process.
Tsay (1989) presented a more straightforward procedure. According to Tsay (1989), Tong’s (1983) approach is insufficient for deciding if data can be explained using a threshold model.
The Threshold Autoregressive (TAR) model, which linearizes nonlinear models over state space using thresholds, has gained popularity due to its ability to accurately model nonlinear data. Other nonlinear time series models, such as the Nonlinear Autoregressive (AR) and Closedloop Threshold Autoregressive (TARSC), have also been proposed.
Priestly (1965) and Ozaki and Tong (1975) used the concept of local stationarity to analyse non-stationary time series and time-dependent systems. This is similar to our current concept of local linearity.
Nonlinear processes involve at least two regimes with varying parameters and/or process order. Tong and Lim (1980) proposed the following conditions for modelling nonlinear time series, listed in order of preference:
To choose a suitable model, it should not need too much computation and be broad enough to account for nonlinear occurrences.
The fitted model should provide one-step-ahead prediction and, if nonlinear, improve overall prediction performance.
It should also reflect the structure of the data generation mechanism based on theories outside statistics and be general enough to generalise to the multivariate case.
Predictive capacity in time series determines the extent to which past data may predict future outcomes. Predictive ability is critical in time series analysis. Applied researchers have traditionally focused on determining predictability among macroeconomic variables.
Many studies have been conducted to determine if money predicts output. Several tests have been used to address this subject, including simple linear Granger Causality (GC) tests (e.g., Stock and Watson, 1989) and non-linear predictive tests (e.g., Amato and Swanson, 2001; Stock and Watson, 1999).
Predictive ability has been studied and applied in a variety of sectors, including tourism and banking. Research on the predictive capacity of several series is limited (Otranto and Traccia, 2007).
Testing for identical forecast performance indicates equal predicting abilities. Testing equal prediction capacity is crucial in risk management. It’s possible to determine if time series with similar variables (e.g. economic, climate) recorded in different locations or using different procedures have comparable predictive powers.
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This study examines the equal prediction performance of the Self-Exciting Threshold Autoregressive (SETAR) model during parallel simulation. We employ the Wald test (Steece and Wood, 1985; Otranto and Triacca, 2007) to compare the similarity of SETAR processes.
1.2 Statement of Problem
Previous research on parallelism and equal prediction ability focuses on ARIMA and GARCH models. We examine parallelism and equal prediction abilities for the Self-Exciting Threshold Autoregressive model.
We use autocorrelation and model coefficients to link equal predicting ability to model structure. It’s important to assess whether the transformations are parallel to the original data.
Box and Jenkins (1970) developed an iterative technique for creating time series models, including model identification, estimation, and diagnostic checks. Identifying typical patterns of behaviour or structure in sample autocorrelation and partial autocorrelation is a crucial step in the model building process. This work delves into these issues.
1.3 Research Objectives
This work focuses on the forecasting ability of nonlinear time series. The study intends to achieve four objectives:
1. Develop an equal predictive ability test for nonlinear time series. 2. Determine the conditions for parallelism and equal predictive ability. 3. Validate the test with real-world data.
1.4 Significance of the Study
When testing for equal predictive abilities, the question is whether one forecast model outperforms another. To answer this question, examine the null hypothesis that both series have similar structure.
This testing problem is crucial for applied analysts as multiple ideas and specifications are typically employed before selecting a model. This test focuses on determining if various series are parallel, indicating structural similarities between them. To test for similarity in the structure of a series, we might use parallelism rather than predictive equality.
It’s crucial to determine if two or more time series are comparable in various scenarios. Predicting demand for common items in multiple marketplaces requires demonstrating that demand models are comparable across markets.
By accepting the hypothesis of parallelism between two time series, it is possible to improve model parameter estimates by pooling data sets. Using series with similar structure can also help forecast volatility and select seasonal adjustment procedures.
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1.5 Scope of the Study
We consider the Self-Exciting Threshold Autoregressive models’ parallelism and predictive performance. The R2 index, which tests predictive ability, can be defined as a function of time series model parameters and autocorrelation.
This helps describe the series’ structure. This index compares the prediction ability and parallelism of different models. We tested the hypothesis using Steece and Wood’s (1985) simple method for testing the equivalence of k time series. We then compared this to the predicting abilities of different time series.
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