SETAR (Self-exciting Threshold Autoregressive) Non-linear Currency Modelling in EUR/USD, EUR/TRY and USD/TRY Parities

In economies that are open to foreign markets the numerical value of the currencies as a macroeconomic variable is of great importance especially when the mutual dependency among the economies is concerned. When it is considered in terms of political economy, the targeted level of the currencies have vital importance especially in economies that have the characteristics of export-driven growth and in economies that struggle not to disrupt the picture in macroeconomic design. When it is considered that each time series has a structure that is sensitive to its own internal dynamics (sometimes these dynamics are expressed as the time series components), these dynamics provide us with coordinates for estimations and may eliminate the compulsory dependency on the outsourced variables at a serious level. This is exactly what has been done in this study. First of all, the non-linear time series analyses are examined in terms of linearity tests, and the linearity tests are applied for all parties and for different time periods. Then, the SETAR Modelling, which is the title of the study, has been applied in order to explain the non-linear pattern in detail. The SETAR Modelling process and other definitions statistical analyses of this model have been applied in relevant parities for separate time periods. The SETAR model, which is one of the TAR Group modeling, shows a better performance than many other linear and non-linear modeling. In this study, the secondary purpose is to express that the SETAR model performance is superior to the other models by considering the observation values of the parities.


Introduction and the Literature
The TAR (Threshold Autoregressive) Model Family, which is important in non-linear time series modelling because of being more practical, keeps its popularity in academic literature. The basic logic of TAR Modelling is allowing different regimes that allow autoregressive analysis of different levels. TAR Modelling was first handled by Tong (1978) in [21] in the literature. Then, parallel studies were conducted by Tong and Lim (1980) in [19]. Again, [23] presented the SETAR (Self Excited Autoregressive Models), which has a popular usage area, to the literature, and with the help of this a light was cast on many non-linear analyses. For example, the SETAR Modelling was used by Tong and Yeung (1991) 1 in [20], who examined the data on the assets market prices; by Tahir İsmail and Zaidi (2006) in [12], who examined the currency incomes by using the currencies of Malaysia, Singapore and Thailand; by Krager and Kugler (1993) in [11], who studied currencies, by Chappell, Mistry, Padmore, Ellis (1996) in [7], who examined the real currencies; by Feng and Liu (2002) [8] in a study conducted on the GDP of Canada covering the period between 1965 and 2000 period; and by P. Clements and Smith (1997) [14] in a study that was based on currencies, GNP and on the estimation performance among the other variables. Chan and Tsay (1998) in [6], Potter (1995) in [16], Tong (1990) in [23], Tiao and Tsay (1994) in [17], Hansen (1997) in [9] also have studies conducted on these group modellings. The definition, estimation and interpretation in TAR models are simpler when compared with the other non-linear models. Bratcikoviene (2012) in [4] conducted a study on Real Harmonized Consumer Price Index in Lithuania, and used the adapted SETAR models for January/1996 -December/2009 period. He reported that the properties or the characteristics of the adapted SETAR of time series with complex nature were observed. Aydın and Güneri (2015) in [1] conducted a comparative study for Export Volume Index and Domestic Producer Price Index Series in Turkey, and 34 SETAR (Self-exciting Threshold Autoregressive) Non-linear Currency Modelling in EUR/USD, EUR/TRY and USD/TRY Parities evaluated the performances of Hybrid AR, SETAR and ARM models. As a result of the evaluations, they reported that the best modeling for the time series in question was the AAR-SETAR modeling. Boero and Lampis (2016) in [3] conducted a study for 4 major European countries and applied SETAR modeling by using Industrial Production Index (IPI) values, and investigated the belief that claimed that the dynamic determination process increased the prediction performance (Terӓsvirta et al. (2005)). However, the literature is not rich in terms of the properties of the sampling on this model group and in terms of test statistics (Hansen, 1997).

TAR Group Models
It is possible to show a TAR Model with k regime as follows; (Montgomery and others (1998) Here, According to the different forms of the t d x − threshold variable, the TAR group models also show differences. In this sense, the SETAR Modelling will be examined in the next title.

Self-excited Autoregressive Models (SETAR)
Although there are few origin studies in the field of threshold autoregressive models, several studies attract attention. [2] and [21] conducted studies and expressed the following procedure about the SETAR models and parameter estimations, which we will explain.

Hansen's Approach
The Structure of the Model These are the models that are valid in case the threshold variable in the TAR modelling is determined as the delayed value of the relevant time series.
A Double-Regime SETAR Model may be shown in a simple way as follows. (Yılancı, (2007), in [26] ) Or it may be shown as follows based on the notation given in the study of Hansen (1997) in [9]; Here, again p denotes the autoregressive level; γ denotes the threshold parameter or the threshold value. The threshold variable is expressed as t d When expressed in a different way, the following model may be considered; (Hansen (1997) Here, if the relevant model is described by separating it into terms, the ω parameter will be as follows; ( ) The expression ( ) t X γ , on the other hand, will be expressed as follows in Hansen (1997) As a consequence, the relevant model will be shown as follows, as it is expressed in the study of Hansen (1997) in [9], When the (5) model is considered, the method for estimating the ω parameter is the Successive Least Squares Method because the model parameters are non-linear (Hansen, 1997) The Estimation of the Parameters The following formulas are given by Hansen (1997) in [9] for the estimation of the ω parameter and for the inclusion.
The inclusion variance that is dealt with in the Successive Least Squares Method is as follows; (Hansen, (1997 In the successive least squares estimation of the threshold parameter the minimization of equality (11) is the basic principle. The situation has been expressed as follows, again in Hansen (1997); The search in (12) is important for the selection of the threshold parameter. The possible threshold parameter in the time series being more in number makes the error variance to be obtained become various. For this reason, it would be proper to select the threshold parameter ( ) γ that will minimize the inclusion variance (12) as shown. Right at this point, the process will follow an algorithmic system for selecting the threshold variable t d Y − and the threshold parameter ( ) γ , which will minimize the error variance. 3

Tong's Approach
The Structure of the Model 3 When t d Y − variable is considered, it is important to select the delay length. Hansen (1997) expressed in his study that the d delay length might not be known, and for this reason, the iterations would be followed as if the d numerical value were known. Tong (1983) in [22] took the above model as the basis and recommended the 3 steps explained as follows.

The Estimation of the Parameters
Step 1 It is assumed that the d and γ values are known as the first step. Based on these assumptions, the observation values are separated into small sub-groups, and the AIC data criterion for each sub-group is calculated with the This is shown as follows; [ ] In this situation, firstly the i p value of each regime is Step 2 In the second step, the d value is kept constant (it is assumed that it corresponds to a certain value; in other words, it is known), this time, the threshold parameters that will minimize the AIC data criterion value are tested. In other words, the γ value that minimizes the ( , ) AIC d γ  value is selected from among the other threshold parameters. This situation is shown in Tong (1983) in [22] as follows.
Step 3 In the first two steps, the i p and γ values are determined.
In the remaining third step, the d value will be determined. The d value that minimizes the ( ) NAIC d value will be found from the d choice in k number. 5 After the 3 steps that are evaluated by using the data criterion, the model will be estimated by depending on the abovementioned parameters. Tsay (1989) presented the successive regression and local estimation as a different method to the SETAR Modelling. Tsay explained this estimation procedure both in his own study in 1989 and in the study about the assets volatility he conducted with [2]. The modelling steps in the study of Tsay (1989) were presented as follows; 4 Although Tong suggested the maximum level in the regimes as ( (1 / 2)) L n α α = <

Tsay's Approach
, the selection of this value is optional. 5 Here, since the values of the variable d will influence the number of the observations (n) in different regimes, the NAIC value will be used instead of the usual AIC . The NAIC here is shown as  Tsay (1989) in [25]. The linearity tests mentioned here are the linearity tests belonging to Petrucelli and Davies (1986) in [15], Keenan (1985) in [10], and Tsay (1986) 6 Tsay suggested the partial autocorrelation coefficients function ( PACF ) instead of the AIC data criterion in determining the autoregressive level because the process included a non-linear dynamism. 7 Tsay expressed the selection of the delay parameter with Here, 1, 2,....., The formula which is the subject matter of F test is In Tsay (1989) in [25], the inclusion formula that is the subject matter of the least square is shown as; selected as the threshold parameter.)

Determining the autoregressive level of each regime with linear auto regression techniques.
In summary, Tsay (1989) conducted a study and determined firstly the AR level in modeling process, and then applied threshold linearity test for predefined p and probable d value. Through this Linearity Test, he decided whether the process was linear or not, and selected the delay value (d) that gave the highest F test value for the relevant p value, which was selected as the threshold value. After the ( t d Y − ) value of the threshold variable was determined, the threshold value was determined by making use of the distribution graphics. The distribution graphics residual estimations mentioned here are the t values of the standardized residual estimations and consecutive estimations of the model coefficients. After this stage, the autoregressive level of each regime was determined with the linear modeling approaches.

Application on Currencies
In our study, the parities that are expressed as major parties, which are many in number among the currency parities, and the EUR/USD parity, which is active in operations in spot currency market, and the USD/TRY, EUR/TRY parities, which are convertible into TL will be taken as the basis. The parities will be examined for descriptive statistics in 3-period intervals as D1 (1-day), W1 (1-week), MN (1-month); and for linearity tests, the examination will be made as D1 (1-day), W1 (1-week), MN (1-month). Since the time intervals of the observation values were taken from the history section of the Forex platform that was provided to the users by intermediary institution acting in Forex market, and since the maximum observation number was limited with the data given here, they were expressed as given in the table. In addition, since investors are inclined to take long-term position, daily (D1), weekly (W1) and monthly (MN) data were taken. The parities are given as follows with the relevant intervals and the start and end dates. If we express the justification of using the EUR/USD, USD/TRY and EUR/TRY parities in this paper; firstly, the EUR/USD parity was used because it is the most frequently processed parity in Forex Market. The USD/TRY and EUR/TRY parities, on the other hand, were used because of the volatility experienced in the Turkish Lira in recent years. Especially the exchange policy and monetary policy of the economy administrator in Turkey require more sensitive investigation in terms of mathematical modeling, and therefore SETAR, which is a more sensitive modeling tool, was used. The descriptive statistics of linear time series, modellings and diagnostic methods will be applied separately to the observation values mentioned in this study.
It is important that before the SETAR modelling is applied to the datasets, the issue of whether they are linear or not must be tested. For this purpose, firstly, the linearity tests will be dealt with, and then the SETAR modellings will be given together with parameter estimations.

Linearity Tests
Although the linearity tests for non-linear time series are various, they show parallelism with the method used in relevant studies. The most popular and qualified linearity tests will be explained in this study; and eventually, the Likelihood Ratio Linearity Test, which tests the threshold linearity hypotheses, will especially be dealt with.

Keenan's One-Degree Test for Nonlinearity
It denotes the linearity test against Second-Level Volterra Expansion. In this context, the Keenan Test examines and tests the Quadratic Nonlinearity Hypothesis and provides information on threshold nonlinearity (Keenan, 1985 in [10]). This situation refers to the F test in the following model: , In case the 3 rd Term of the model is zero, it gives proof on linearity. In this situation, the null hypothesis will express the situation in which the 3 rd Term in Model (16) is equal to zero.
In [10]'s test, first of all a regression is formed between t  (17) and then the F test statistics will be expressed as follows in Keenan (1985): Here, HKT denotes the sum of the error squares expressed in the first model. (

Tsay's Test for Nonlinearity
Tsay's (Tsay, 1986) linearity test is based on recursive auto regression and destructive term estimators, and firstly, the recursive auto regressions are established starting from b. observation value in return for the p and the relevant d values with AR level, and then the model is established between the t e  values and 1 2 (1, , ,..., ) 9 Then the following test is obtained among the inclusions of the model formed 9 Here, b is expressed as ( /10) b n p = + .
Here, it is expressed that In the F test of Tsay, the null hypothesis claiming that the relevant time series is ( ) AR p .
The recursive regression mentioned here is the modeling approach that was focused on the change in the model parameters. The sharp changes in the numerical values of the parameters in the model that occur as a result of each additional iteration, degenerations or refractions are considered in recursive regression.

Likelihood Ratio Test for Threshold Nonlinearity
Keenan's and Tsay's quadratic nonlinear tests above are insufficient in dealing with the threshold non-linearity hypothesis. Therefore, Tong (1986) in [18] developed a different approach to test the opposite hypotheses.  The null hypothesis: "Model is an autoregressive process. In other words ( ) AR p ", and  The opposite hypothesis "Model is the TAR model with constant variance at p level with two regimes".
In their studies, Tong (1986) dealt with the probability ratio that focused the numerical approach.
When the excessiveness of the observation value of the dataset in question is considered, it is inevitable that the threshold autoregressive processes and the iterative mathematical processes that must be performed about the modelling of these processes are done in a package program that includes statistical notations. Especially the "R" statistical package program has been used because of its having open-source systematic and its being used in a great extent that cannot be underestimated in the field of statistics in the literature. This program provides vast opportunities in terms of data and graphics performance. It was necessary to develop the commands or the codes that were dealt with in time series to be able to form the threshold autoregressive model with R, which is the subject matter of this study. These codes include both the time series analysis commands and the more up-to-date structures in the developing academic literature.
The linearity tests that will be dealt with in the further parts of the study were found by finding the model hyper parameters; and the descriptive statistics part and the SETAR modelling results were found with R Project 3.1.2 Version.

Applying the Linearity Tests to the Relevant Parameters
When the significance values in Table 2 are examined, it is observed that the null hypothesis in the 3 parities in the Keenan's (1985) test and Tsay's (1986) test are not denied. The null hypothesis was denied in the three parities that were examined in the Likelihood Ratio test except for the EURUSD D1, EURTRY D1 and EURTRY MN parities. Therefore, the expression that may be uttered about linearity is as follows; "Some of the parities in different time periods show linear characteristics, while some others show non-linear pattern. The linearity tests of Keenan (1985) and Tsay (1986) give similar results in relevant parities. It is necessary to point out that, here, while the null hypothesis is tested against a non-alternative hypothesis in the linearity tests of Keenan and Tsay, the null hypothesis is tested against a specific alternative hypothesis in the Likelihood Ratio test".

Applying the SETAR Modelling to Parities
In this part of the study, the SETAR modelling in Tong's approach will be applied separately in the relevant time period for each parity, and the findings will be discussed in the light of descriptive statistical notations. For this purpose, firstly, the hyper parameters of the SETAR modelling (SETAR hyper parameters) of the relevant parities will be given with the R Project (version 3.1.2) printout.     The descriptive properties of the SETAR model of the relevant parities are given in D1 time periods in the following tables. The "x$ff" expression in the graphics is written for ease of encoding, and denotes the relevant time series. Graphic 7 expresses the red pattern which is over the threshold value and the black patterns which is over the threshold value. When Graphic 8 is examined the first graphic threshold value indicates the observation values. Again, when the second graphic is examined in Graphic 8, it is observed that the position of the threshold value may be determined provided that 15% of the observation values from both ends are kept hidden. The delay (d) of the threshold value of the model and the threshold value are defined in Graphic 9-10-11 (within relevant time range) where the pooled-AIC value is at the minimum level. Graphic 15 expresses the red pattern which is over the threshold value and the black patterns which is over the threshold value. When Graphic 16 is examined the first graphic threshold value indicates the observation values. Again, when the second graphic is examined in Graphic 16, it is observed that the position of the threshold value may be determined provided that 15% of the observation values from both ends are kept hidden.   The SETAR Model, which is one of the TAR Group modeling, shows a better performance than many other linear and non-linear modeling. In this study, the purpose is to express that the SETAR model performance is superior to the other models by considering the observation values of the parities. In this context, the AIC (Akaike Information Criteria) values of the SETAR Model and the other models were compared, and it was revealed that the SETAR model is more influential in terms of the relevant parities.