### General Note

Dear Researchers

I have been preparing an application for my unit root tests; therefore, you can easily manage to employ these tests through this application. Briefly you have to follow the below instructions to employ tests: NON-STAT (Nonlinear Statistics or Nonlinear Econometrics) You will reach all the programs from this link: https://econometricsletters.com/non-stat/

1. Do not care about the notification first of all, which are written in red. They are not serious problems,
2. Prepare an Excel file with xlsx with one column,
3. Use Browser to upload the file you have prepared,
4. Then all the estimations will be done automatically.
5. If you want to take the values you can easily copy and paste them to word file. For the graphics, left click the mouse and carry to a word file.
6. For the data at the end of the output window you will find the original data and the estimated nonlinear trends. If you want you can draw your figures b using other applications such as Excel. In that case, first of all copy and paste the data in to a notepad then copy and paste to Excel. If you directly paste it to Excel you will see the transpose of the data.

#### Summary of OSH test with selected lag BIC Criteria/data2[, 2] is OSH test Result

You will find the Unit Root test result for the OSH test. This first output window is giving the result by using BIC criteria the first parameters t-value is the test value of OSH test with selected lag. You can use the critical value Table to come to conclusion about the test.

#### OSH test Automatic:(lag_1)/ Estimated Nonlinear Parameters of Model C: , gamma, threshold and Beta Parameters: Linear and Non-linear intercept and trend

In this window, you will see the OHS test with lag one in the first column for further check of the test. Gamma is giving the ESTR trend transition speed. Threshold is the half way of the transition function as it is explained in the article. The Beta parameters are intercept, intercept*ESTR function, Trend and ESTR function (x) Trend.

#### Exponential Smooth Transition Trend Model_C and Original Series

The red line is nonlinear ESTR trend and black one is the original series.

#### Observations of Original Series and Exponential Smooth Transition trend

The first one y11 is the original series and the STR2 is the ESTR trend data. You can use them as it is explained above.

Now the NON-STAT application covers the below tests and applications:

1. FFFFF test Omay (2015): Omay, T. (2015) Fractional Frequency Flexible Fourier Form to approximate smooth breaks in unit root testing, Economics Letters, 134(C), p. 123-126. (Model Intercept and Trend)

2. LNV-KSS Omay and Yildirim (2014): Omay, T. Yildirim, D, (2014) Nonlinearity and Smooth Breaks in Unit Root Testing Econometrics Letters, 1(1), p. 2-9. ( Model C)

3. OSH ESTR-KSS and AESTR (2020) : Omay, T., , Shahbaz M, Hasanov M (2020) PPP hypothesis and temporary structural breaks with nonlinear adjustment Applied Economics, (Forthcoming). (ESTR-KSS Model C)

4. Easy-Finance: This application uses the Yahoo Finance data stream to show you all the financial data at first and at the end of the Graphics you can find a one day head forecast of relevant variable that you write the code or symbol from Yahoo finance. The base code or symbol is EURUSD=x. If you change it to USDTRY=x you will see the Turkish US exchange rate and so on. If you write Bond then you will see the ten year treasury bond of US. If you write gold you will see gold prices. If you write for example SPY you will see a stock price from New-york stock exchange.

5. Corona Contamination Speed and One step ahead forecast app: In this application again you will Browse the data as Excel xlsx which is the daily infected cases from a country which you are interested in. Then program automatically is giving the transition speed and the inflection point for you as gamma and threshold respectively. And in the final window you will see a day ahead forecasts. I hope you will enjoy with the program or application. I will update it in the coming days. The server which I am using asks for a membership for deploying more applications, hence, first of all I have to fulfil their requirements in this week then I will first deploy the Model A and B of OSH KSS test then I will deploy OHS AESTR unit root test Model A, B and C. Most probably, I will also include a Help menu as I have given for FFFFF test. Moreover, the journal which the NON-STAT appeared is founded at 2014 by me and my colloquies and we will appreciate any contribution to the Journal (Econometrics Letters) for this year. It is indexed in Econ-Lit and we are putting the issues in to Ideas REPEC.

Sincerely, Tolga Omay (Phd) Chief Editor of Econometrics Letters, Board of Trustee in Turkish Economic Association, Affiliated to researcher to METU, Department of Economics, Atilim University, Turkey.

### 1. Introduction

In the recent literature, multiple smooth breaks have been modeled by Flexible Fourier Transforms by Becker et al. (2006), Enders and Lee (2012a,b), and Rodrigues and Taylor (2012). The advantages of the Fourier approach include being able to capture the behavior of a deterministic function of unknown form even if the function itself is not periodic, working better than dummy variable methods irrespective of whether the breaks are instantaneous or smooth, and avoiding the problems of selecting the dates, number and form of breaks (Becker et al., 2006, Enders and Lee, 2012a,b; Rodrigues and Taylor, 2012). All these papers pointed out that the single frequency component of the Fourier Transforms should be used for structural break determination; otherwise, the over-filtration problem arises. However, none of them seems toconsider the Fractional Frequency version of their test in their studies.

The Fractional Frequency Flexible Fourier Form (FFFFF) is used in Becker et al. (2004) for the structural break test, namely Trig-test. They attempt to prove that their methodology is better than the conventionally used break tests. Moreover, in their study Becker et al. (2004) show that the best fitting frequency for the US inflation rate between the period 1947:1 to 2011:11 is f * = 1.178, which is fractionally determined when testing for structural breaks. Therefore, by combining the methodologies of Becker et al. (2004) and Enders and Lee (2012b, Henceforth, EL test), this study aims to improve the unit root testing with Fourier Transforms.

### 2. Unit Root Test with FFFFF

The following Fourier de-trending first take place in the FFFFF-ADF testing procedure:

$y_{t}=\beta+\beta_{2}\sin (2\pi k^f t/T)+\beta_{3}\cos (2\pi k t/T)+\epsilon_{t}$

In the second step classical ADF test without any deterministic component employed.

$\Delta\epsilon_{t}=\alpha_{1}\epsilon_{t-1}+\nu_{t}$
Finally, Omay(2015) has used the Davies (1987) methodlogy in order to select the appropirate frequency selection:

De- Trended by FFFFF and IFFFF

Critical Values for FFFFF

See Omay (2015) for details

Citation of the Omay(2015) in Google Scholar

### 1. Introduction

In this study we have developed a unit root test by combining Kapetanois et al (2003) (henceforth KSS) and Leybourne, Newbold and Vougas (1998) (henceforth LNV). KSS (2003) employ exponential smooth transition autoregressive (ESTAR) models to propose tests of the null hypothesis of a unit root that allow under the alternative hypothesis for stationary nonlinear adjustment towards a fixed mean. Thus, we extent the KSS tests to the case of a nonlinear attractor . Section 2 of this paper develops the proposed test statistics and represents their critical values. Section 3 provides the small sample performance of proposed test in comparison with the power of the ADF, LNV, Sollis, KSS and EG tests. Section 4 presents the application of our aforementioned tests to PPP hypothesis.

### 2. Unit Root Test with LNV KSS Omay Yildirim (2014)

The following Fourier de-trending first take place in the LNV-KSS testing procedure:

In the second step classical KSS test without any deterministic component employed. LNV-KSS Equation in the second step

De- Trended by LSTR Model A