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Fractional unitroot tests allowing for a fractional frequency flexible Fourier form trend: predictability of Covid19
Advances in Difference Equations volume 2021, Article number: 167 (2021)
Abstract
In this study we propose a fractional frequency flexible Fourier form fractionally integrated ADF unitroot test, which combines the fractional integration and nonlinear trend as a form of the Fourier function. We provide the asymptotics of the newly proposed test and investigate its smallsample properties. Moreover, we show the best estimators for both fractional frequency and fractional difference operator for our newly proposed test. Finally, an empirical study demonstrates that not considering the structural break and fractional integration simultaneously in the testing process may lead to misleading results about the stochastic behavior of the Covid19 pandemic.
Introduction
The forecasts of daily events lead many decisionmaking processes to be more manageable. In timeseries analysis, forecasts are generally made using the Box–Jenkins method. In the Box–Jenkins method, the prerequisite for making longterm forecasting or setting up an ARIMA model is the stationarity of the series under investigation. It is also essential to make longterm forecasts in the current Covid19 outbreak. These forecasts contain crucial information to eliminate the uncertainties that may arise during the process. For example, forecasting the peak number of infected cases in the long term may give valuable information about the health care system. If these numbers can be accurately predicted, then the intensive care unit bed capacities and other resources can be allocated efficiently. This vital information can also be used by the other sectors which are affected by the Covid19 outbreak. Besides, longterm forecasting can also be made for all other natural phenomena. Reliable forecasts of earthquakes, meteorology, biodiversity, and others are needed to manage disasters. The timeseries literature has described covariance stationarity as a steady state in which the mean, variance, and covariance do not change over time. The stochastic difference equation’s stationarity is determined by using the unitroot test of [1]. The test’s basic principle is to see if the firstdegree stochastic difference equation’s parameter is statistically equal to 1 or not. If it is equal to 1, the series is a unitroot process or simply not stationary.
In this study, we need to add the dynamics of this natural outbreak to the [1] (henceforth, ADF) method to examine the outbreak’s stochastic features and test its longterm predictability. If the epidemic’s data generation process is substituted correctly into the test methodology leading to a stationarity test result, then we can claim that the correct longterm forecast model is achieved. It is recognized that the number of daily cases in the outbreak models conforms to exponential function patterns. However, it is not easy to generalize the epidemic model to different functional designs, such as the second wave that may occur in later stages. This functional pattern will create a double exponential model or a more complex functional form. The complexities that arise obtained in this way can also decrease the effectiveness of the longterm forecasts. We have used the Fourier function to overcome this problem, thereby providing a remarkable convergence to any functional form whose structure is uncertain. In the literature, many researchers have employed the Fourier function to capture smooth structural breaks with integer frequency. Nevertheless, some studies have shown that this should be handled within a fractional frequency structure. In addition to the importance of using low frequency, previous studies have also emphasized the problems of using cumulative frequency in Fourier type of unitroot testing. A wellknown problem associated with the traditional unitroot tests is that the power of the test decreases if too many variables are added into the testing equation when the cumulative frequency is employed.
So how can the Fourier function capture the shortterm oscillations in the daily cases without using cumulative frequency? In the consecutive days of pandemics, different dynamics or numbers of infected patients are detected. Temporary or permanent jumps are a prevailing dynamic of daily infected cases that the firstorder difference equation cannot capture. The fractional difference equation employed recently in the literature is seen to solve such dynamics. It has been observed that the number of daily cases exhibits fractionalorder difference equation features. After detrending the daily infected cases data with the Fourier function, the remaining series exhibit the features of a fractional firstorder difference equation. Therefore, in the light of these explanations, the pretest of the longterm predictability of the number of daily Covid19 cases must have considered the fractional frequency Fourier functional form with a fractional difference equation. Let us now turn to discussing the methodology used in the paper and literature available until now.
Following the influential work of [1], testing the stationarity characteristics of variables has attracted a great deal of attention among researchers. This testing methodology can be broadly classified into three categories; linear unitroot tests, unitroot tests that permit a break in mean and/or trend (this can be termed timedependent nonlinearity, or structural break (SB)), and finally unitroot tests that permit statedependent nonlinearity. However, after recognizing the longmemory features of the stochastic processes, the fractionally integrated unitroot tests have attracted a great deal of attention in the recent literature. Therefore, in this study, we will focus on combining the unitroot tests that permit structural break and fractional integration (FI).
A typical exercise in most timeseries investigations is to check whether the drift part of a series is correctly characterized as deterministic or stochastic. Naturally, the stochastic drift is considered as a unitroot process. In contrast, the deterministic one is particularly time trends. It is generally concluded that traditional methods developed for fractionally integrated processes could drive spurious FI response if employed to short memory processes encompassing structural breaks. The reverse outcome is also entirely recognized; standard methods for identifying and measuring break dates lead to spurious structural change, generally at the midpoint of the series, in fact, there is only fractional integration in the sample (see [2, 3]). Therefore, fractional integration and structural break indicate vastly diverse long and mediumrun dynamics, making it hard to discriminate among them. As it is also recommended in [4], these two methods, FI and SB, are alternative methods for difference stationarity (DST) and trend stationarity (TST). It is well known that to avoid spurious estimates of the parameters and biases in the timeseries studies, the data must be differenced to make them stationary. Therefore, the decision of optimal differencing is vital for obtaining correct information from the data under investigation. By introducing these two alternatives, we have to choose the correct differencing among DST, TST, fractional difference stationarity (FDST), and structural break stationarity (SBST). In addition to these, it is well documented that in the DST case memory is infinite, and past shocks are perfectly remembered. In the case of TST, memory is short, and the autocorrelation function decays exponentially. [2] indicates that the FI processes or the FDST case establish an interesting alternative to this separation as they are capable of linking the gap between these two possibilities. Therefore, FDST has a long memory but not as much as the DST, which indicates that \(d = 0.1\) has a short memory with respect to \(d = 0.9\). These methods can fulfil the gap between the shortlasting and unchanging effect of shocks in the TST and DST models, respectively, by providing transitional behaviors such as long memory and nonstationary meanreversion (see [2]). So finding the exact differencing order is vital to limit information losses. As we have mentioned above, fractional integration and structural break indicate very different medium and longrun dynamics. Thus, it is hard to differentiate between them, so it is essential first to distinguish these two methodologies. To this end, we propose a procedure that combines these two methods using a simple but yet efficient way to identify the SB and FI processes correctly. Therefore, we can eliminate the problems which are explained in the above paragraph efficiently.
The unitroot tests which are permitting for a break in mean and/or trend are as follows; [5–8], and [9]. These have acknowledged alternative trend models in examining for the unitroot testing, and have concentrated on models with segmented line trends; and single or multiple breaks [10]. However, recent studies have proposed unitroot tests where the alternative hypothesis is stationarity around a smoothly changing trend. [11] (LNV, hereafter) and [12] used logistic smooth trend functions that permit a smooth break in the data’s deterministic trend. [13] specified nonlinear trend employing Chebyshev polynomials. Reference [14] employed trigonometric functions in Fourier form to define probable smooth breaks in the data. Numerous problems were encountered with these types of unitroot tests.^{Footnote 1} Nevertheless, the simplest and most accurate one has been the Fourier function, which was used by [14–16], and [17].
The second strand of literature deals with the fractionally integrated unitroot test proposed by [18] (henceforth, DGM). One stated that that both null hypotheses were rejected frequently in the previous studies, and concluded that many timeseries were not well characterized as either \(I ( 1 )\) or \(I ( 0 )\). Therefore, the group of fractionally integrated processes, represented as \(FI ( d )\), has proved to be very suitable in catching the persistence features of many longmemory processes (see [19, 20], and [21]). Reference [18] has pointed out the shortcomings of the alternative methodologies used and suggested a simple Waldtype test in the time domain with adequate power properties. As a byproduct of its application, this test delivers knowledge about the values of d under the alternative hypothesis. Therefore, this methodology is a generalization of the wellknown Dickey–Fuller (DF) test, which was originally developed for the case of \(I ( 1 )\) versus \(I ( 0 )\), to the more general case of \(FI ( d_{0} )\) versus \(FI ( d_{1} )\) with \(d_{1} < d_{0}\) and, thus, is denoted as the fractional Dickey–Fuller (FDF) test. DGM test is based on the normalizedOLS estimates, or on its tratio, of the parameter on \(\Delta ^{d_{1}}y_{t  1}\) in a regression of \(\Delta ^{d_{0}}y_{t}\) on \(\Delta ^{d_{1}}y_{t  1}\) and possibly some lags of \(\Delta ^{d_{0}}y_{t}\). Depending on the alternative hypothesis \(H_{1}:d < d_{0}\), the preestimation is needed for the order of d. DGM has shown that the choice of a \(T^{1/2}\) consistent estimator of d in its appropriate range suffices to make the FDF test possible, while preserving asymptotic normality. Reference [18] has highlighted the advantages of their testing procedure as follows. The first one is theorizing the simple DF framework to obtain simplicity for testing unit roots with a fractional difference operator. The second one is that the LM tests proposed contain a different structure than the traditional LM tests. The proposed LM test does not assume any known density for errors, which makes it more robust to fundamental ones. The third one is that in the exact case where \(d_{0} = 1\), the FDF method inherits the flexibility of the standard DF test. This provides a usual framework for testing the \(I(1)\) null hypothesis against some interesting compound alternative. According to [18], producing a fractional integration unitroot test by including a structural break does not seem feasible with other FI unitroot tests. However, the flexible FDF structure that they propose will make this study much easier and feasible. The final one is that [18] has found a good finite sample properties with respect to other competing tests.
Following [18], the third advice, we have extended this methodology to the structural break set up by using the [17] method. As we have mentioned above, the [17] procedure employs trigonometric functions in the form of Fourier form to define presumable smooth breaks in the data. Numerous difficulties are encountered with structural break type of unitroot tests. Nevertheless, the easiest and accurate one is the Fourier function used by [17] with which extended it to fractional frequency case. Therefore, [17] is another simple generalization of the ADF test like the DGM test. Combining these two simple methodologies will emerge as a more generalized and simple set up without facing any unnecessary details to test stationarity in a composite alternative hypothesis. The composite hypothesis of the series under investigation is a fractionally integrated series around a smoothly changing trend.
Other attempts have been made in the literature to combine these two methodologies (namely SB and FI) by using different techniques. References [22] and [23], following [24] and [25], derived a Lagrange multiplier test in the time domain, and [26] and [3] have considered Waldtype tests for a unitroot null hypothesis against fractional integration following [18]. The traditional unitroot tests usually reject the null hypothesis when the actual process is a series that is integrated fractionally with \(d= (0.5,1)\). We will see later that such series are not stationary. Therefore, the results of these studies become questionable. Moreover, it is well known that short memory processes with level shifts display features that lead one to conclude that long memory is present in the data generating process (e.g., [23], among many others). On the other hand, it was also recognized that longmemory processes cause the null hypothesis of no structural change to be rejected when traditional structural change tests are used (see, [2, 3, 23], among many others). To overcome these problems in the SBFI literature and to address the reasons mentioned earlier, we propose the SBFI unitroot test in the form of a fractionally integrated series around a smoothly changing fractional frequency flexible Fourier form. Therefore, we have obtained the following contributions from this newly proposed methodology:

1.
The confusion about structural break and fractional integration, which we explained above, has been resolved with the most appropriate methods.

2.
The twostep methodology allowed us to obtain the asymptotic distribution of the unitroot test easily.

3.
It has been shown that the Fourier function can represent the deterministic structure of the Covid19 outbreak. Also the best optimization algorithm that should be used with the fractional frequency Fourier function is found.

4.
For fractional integration, a new estimator has been proposed that minimizes information losses. It has also been shown that predictions can be made with the least loss of information with this new estimator.

5.
Finally, how to design the optimal forecast model for outbreaks by combining all of these methodologies has been shown.
The structure of the article is as follows. Section 2 presents fractional frequency Fourier form fractionally integrated ADF test with its asymptotic distribution and presents an extensive simulation study to show the smallsample features. Section 3 discusses the various optimization algorithms that can be used with the fractional frequency estimation along with the parametric and semi parametric estimation of the difference operator d. Section 4 applies the FFFFFFIADF test to pretest the longterm predictability of the Covid19 cases. Section 5 is devoted to concluding remarks.
The methodology for the fractional frequency flexible Fourier form fractionally integrated ADF test: FFFFFFIADF
In the introduction, we gave some basic ideas about the testing procedure. The main concern is to be simple in deriving the test, and its asymptotic. Hence, we have started with the Fourier approach in which we can detrend the series at first and assume the remaining part has fractionally integrated stationarity or nonstationarity of the series. Apart from [15, 16] and [17], this twostep approach provides a straightforward setting for obtaining the testing procedure and asymptotic distribution of the proposed test statistics. Therefore, we will start with the Fourier approach and include the fractionally integrated ADF test in the second step.
References [16] and [17] consider the following augmented Dickey–Fuller (DF) test:
where \(\varepsilon _{t}\) is a stationary error term with a variance of \(\sigma ^{2}\), and \(\varphi (t)\) denotes the deterministic intercept and trend. Reference [16] claims that it is problematic to estimate Eq. (1) directly and study the unitroot hypothesis \(\psi = 1\) without knowing the functional structure of \(\varphi (t)\). Following [14, 16, 27] and [17], we assume that \(\varphi (t)\) includes the following Fourier components:
where \(\alpha _{0}\), \(\alpha _{1}\), and \(\alpha _{2}\) are changing intercept parameters, T is the number of observations, and t gives the trend term. The term k denotes the particular frequency to be determined over a pregiven interval. The trigonometric components \(\sin ( \frac{2\pi kt}{T} )\) and \(\sin ( \frac{2\pi kt}{T} )\) are utilized to approximate smooth breaks. If \(\alpha _{1} = \alpha _{2} = 0\), then there are no smooth breaks. Through the gridsearch method, [15, 16] use \(k = k^{*}\) to minimize the residual sum of squares (SSR) in Eq. (1), where \(k^{*}\) indicates the value of k that achieves the minimum SSR. Besides, Becker et al. (2006) show that it is acceptable to set \(k=1\) or \(k=2\) to find the substantial structural changes in the data. Using a datadriven technique, [14] set the maximum number of breaks to be 5. Reference [15] further recommends the usage of low frequency to capture the smooth structural changes in the data. Reference [17] mentions the flexibility of the integer, but argues that it has many drawbacks in estimating the smooth trends (i.e., over filtration, type two error etc.). Hence, we follow [17] and use the fractional version of the test in this paper. To this end, instead of searching for a single integer frequency k in Eq. (2) we try to find the fractional frequency in Eq. (3), which is also employed in [14] and [15, 16] for integer values. The largest frequency applied is \(k_{\max } \), and \(\Delta k = 0.1\) is used in the 0.1 range and other smaller increments, and the accuracy of the fractional frequency search was increased. The optimal fractional frequency is obtained at the point where the SSR is the lowest. This optimization process is carried out by applying the algorithm described above for Eq. (1). Moreover, we can also employ this to define the fractional frequency Fourier trend by using an Ftest as proposed in [14] and [15, 16]. The model is as follows:
The null hypothesis of linear unit root is obtained when \(\delta = 0\), which is suggested by [16]. The twostep testing process is as follows.
In the first step of the two stages procedure the following regression is run:
where \(k^{fr}\) indicates the fractional Fourier frequency. The above equation assumes that \(\omega _{t}\) is a random walk process and after being demeaned or detrended it can be used in the second step as \(\bar{\omega }_{t}\),
where \(u_{t} \sim iidN ( 0,\sigma _{u}^{2} )\) and the initial condition \(\bar{\omega }_{0}\) is a constant. Notice that this technique is asymptotically the same as the one step procedure of [17].
As we have mentioned above instead of assuming the case of \(I ( 1 )\) versus \(I ( 0 )\), the more general case of \(FI ( d_{0} )\) versus \(FI ( d_{1} )\) with \(d_{1} < d_{0}\) can be used following [18]. The DGM test is based on the normalizedOLS estimates, or on its tratio, of the coefficient on \(\Delta ^{d_{1}}\bar{\omega }_{t  1}\) in a regression of \(\Delta ^{d_{0}}\bar{\omega }_{t}\) on \(\Delta ^{d_{1}}\bar{\omega }_{t  1}\) and possibly some lags of \(\Delta ^{d_{0}}\bar{\omega }_{t}\).^{Footnote 2} The definition of the \(FI(d)\) process that we will implement is that of an (asymptotically) stationary process when \(d < 0.5\), and that of a nonstationary (truncated) process when \(d > 0.5\).
For the asymptotic distribution of \(\delta = 1\), the twostep process will be used with the following demeaned and detrended series ω̄:
where \(\hat{\alpha }_{0}\), \(\hat{\alpha }_{1}\), \(\hat{\alpha }_{2}\) and λ̂ are OLS estimators for demeaned and detrended cases, respectively. Next, we build the fractional Fourier unitroot test by using the demeaned and detrended series \(\bar{\omega }_{t}\) in the second step. Although the DF test is coherent when compared to the fractional alternatives, its low power makes it an appropriate ground for studying the new test procedures. Thus, we extend the regression model in (3) and (5) to test the null hypothesis that a series is \(FI(d_{0})\) against the alternative that it is \(FI(d_{1})\). The variable ω̄ is thought to be a unitroot process under the null hypothesis, but it constitutes a fractionally integrated stationary process in the alternative. Precisely, our suggestion is built upon testing for the statistical significance of β in the following FIDF equation:
where \(\xi _{t} \sim iid ( 0,\sigma _{\xi }^{2} )\) \(I(0)\) process. Keep in mind that (6) is still an unbalanced regression where the dependent and independent variables are differenced with respect to their degrees of integration under the null and the alternative hypothesis. The \(\bar{\omega }_{t}\) series follows the following process assuming that \(u_{t} = \xi _{t}\) and \(\beta = 0\) in (6):
This implies that \(\bar{\omega }_{t}\) in (7) is \(FI(d_{0})\). When \(\beta < 0\), \(\bar{\omega }_{t}\) can be expressed as
where \(\bar{\omega }_{t}\) is a \(FI(d_{1})\) process. By using these arguments, we can write the normalizedOLS estimated coefficient or its tratio as in the standard DF testing methodology as follows:
The test and its asymptotic properties
Now we allow for \(d_{0} = 1\) and \(u_{t} = \xi _{t}\) in (7), where \(\{ \xi _{t} \} \) is a sequence of zeromean i.i.d. random variables with unknown variance \(\sigma _{\xi }^{2}\) and finite fourthorder moment. The OLS estimator \(\hat{\beta }_{ols}\) and its tratio, \(t_{FF}\), are given by their usual leastsquares formulas;
In order to obtain the asymptotic distribution of the \(t_{FF} ( i = \mu ,\tau )\) test, we need the subsequent outcomes, where we let \([ rT ]\), \(r \in [ 0,1 ]\) be an integer close to rT. During the course of the derivation → implies weak convergence as T approaches ∞.
Proposition 1
We have assumed that the remaining part or detrended series is a fractionally integrated series. Thus, we preserve the notation of [18] hereafter to derive the asymptotics of the proposed test.
Lemma 1
Let \(\{ \xi _{t} \} \) be a sequence of zeromean i.i.d. random variables with variance \(\sigma _{\xi }^{2}\) such that \(E \vert \xi _{t}^{4} \vert < \infty \) implies the following linear processes:
where \(I(\cdot )\) is an indicator function and
Then the following process verifies:
and
where \(\overset{p}{\rightarrow}\) denotes convergence in probability,
where \(\overset{p}{\rightarrow}\) denotes weak convergence.
Lemma 2
Let \(\xi _{t}\), \(z_{t}\), \(z_{t}^{*}\), and \(g_{t}^{*}\) be identified as in Lemma 1. Then the subsequent processes are martingale differences and confirm:
When we impose the Fourier form to the FI process:
where \(B(\cdot )\) denotes a standard Brownian motion, and \(B_{d}(\cdot )\), \(W_{d}(\cdot )\) or \(W_{d}(k^{fr},r)\) are standard fractional Brownian motions. Depending on Lemma 2, the subsequent two theorems derive the asymptotic distribution and prove the consistency of an appropriately standardizedOLS estimator of β̂ and its tratio, under the null hypothesis of \(I ( 1 )\).
Theorem 1
Under the null hypothesis of unit root and with \(\bar{\omega }_{t}\) as a random walk, the asymptotic distribution of \(t_{FF}\) is as follows:
where \(W_{i,  d_{1}} ( k^{fr},r )\) for \(i = \mu ,\tau \) give the demeaned and detrended standard fractional Brownian motions.
We can derive the asymptotics of the other cases; \(d_{1} = 0.5\) and \(0.5 < d_{1} < 1\) in a similar fashion with the \(0 \le d_{1} < 0.5\). As pointed out before, since we are following twostep approaches the other distributions are the same as [18]. Therefore, we concentrate on the nondegenerated distribution of case 1 and give its distribution explicitly in Theorem 1. This asymptotic distribution obtained for fractional frequency is the general form of the integer frequency case, and it can be easily converted to an integer form with the values given in [15].
Proof
The proof of Theorem 1 is given explicitly in Appendix A. □
Apparently, the asymptotic distribution of the obtained test statistics under the null depends on the fractional Fourier frequency, \(k^{fr}\), and integration order, \(d_{1}\), but it is invariant to the other parameters in the testing equation. The fractional frequency versions of the critical values are tabulated in Appendix B and for integer frequency the critical values tabulated in Tables 1–3 as follows:
Small sample properties of the fractional frequency flexible Fourier form fractionally integrated ADF test FFFFFFIADF (FFFFFFIADF)
First, we will examine the smallsample size features of the test statistics. To assess the size of the test statistics, we investigate the following data generating process (DGP):
where \(\upsilon _{t}\) is stationary with the above given distribution. The size features of the tests were simulated with 2000 replications via the sample dimension \(T = \{ 100,200,300,400,500, 600,700,800,900,1000 \} \). The results of these simulation exercises are presented below in Fig. 1 and suggest that the proposed test statistics have satisfactory size properties.
As can be seen from Fig. 1, the newly proposed test exhibits good size properties similar to the previous tests [18, 26, 28]. Considering the scale next to the figure, the minimum and maximum size values are in the range of 0.02 and 0.08, respectively. Since the size analysis is performed for the 5 percent significance level, this scale indicates that the newly proposed test approaches the correct size value with a minimal error rate. As can be seen from the color spectrum given above, instead of the extreme values of yellow and dark blue, the size results were obtained with light green and blue intensity, and the real value of size was mostly 5%. Thus, in the light of Fig. 1 we can safely conclude that the newly proposed test has strong size properties.
Therefore, we can proceed with the power analysis without any size adjustment. Now, we turn to the smallsample power properties of the proposed tests. We have done an extensive simulation study to see the proposed unitroot tests’ power surface using the model
Following [15, 16], we set \(\alpha _{1} = ( 0,3 )\) and \(\alpha _{2} = ( 0,5 )\). The results presented in Fig. 2 suggest that the proposed FFFFFFIADF test clearly exhibits a similar behavior to [18, 26, 28].
As it can be seen from Fig. 2, the power performance of the test is working well. The test’s power increases with the time dimension T and decreases as the difference operator parameter varies from 0.1 to 0.9. Especially after 0.8, the power has started to decline from 1.00 to 0.2. and the lowest as 0.0. Towards 1, the color spectrum turns to yellow and black, while power weakens with blue and dark blue tones. As Fig. 2 shows, the high power of the test is justified with the abundance of yellow or black areas. Overall, this analysis proves that the test is powerful in capturing fractional integration data dynamics with a structural break. Furthermore, with these power analysis results we can distinguish between structural break and fractional integration because the detrended series in the first stage will not give a pseudointegration order in the second stage.
The method of estimations for Fourier fractional frequency and fractional integration parameter d
Estimation of fractional frequency for Fourier function
The authors of [29] have conducted an extensive study analyzing the BFGS, BHHH, Genetic, Simplex and Grid Search (GS) algorithms in the estimation of the fractional frequency. They used the alternative hypothesis of the test [17] to evaluate the effects of using different algorithms on the parameter estimates. They have noticed that in the earlier studies, comparison of different optimization algorithm evaluation is commonly made on the critical value accuracy. Yet the Fourier unitroot test depends on the fractional frequency. Thus, the frequency is specified at first and then the critical values are acquired. Consequently, producing the critical values with a different optimization algorithm will not lead to different set of critical values. In our simulation study, the issues in [29] will be taken into consideration. In addition, since the subject to be examined should imitate the data generation process of the Covid19 pandemic, the following model will be used:
where \(T = 100\). Subsequently, investigating different schemes of experiments, the authors of [29] have decided to use the SSR of the estimation results. The authors of [29] have classified the fractional frequency values that they obtain in terms of the stages of the pandemic. According to this classification, the fractional frequency was estimated to be between 0–0.75 in the early stages of the pandemic, 0.75–1.0 near the peak day, 1–1.25 in the second stage, and 1.5 around the plateau stage. We follow their study and use Eq. (25) to obtain Fig. 3 and Table 3.
Like [29], we have found that the best estimation algorithms with nonlinear trends are simplex and genetic, which are indifferent in terms of SSR. As reported in [29], the second best approach appears to be the GS gridsearch algorithm, while the third one is the derivative free methods of BHHH and BFGS. Consequently, following our results and the ones obtained in [29], we use the simplex algorithm for the estimation of the fractional frequency.
Estimation of the fractional difference d
In this study, we have used Andrews and [30] (henceforth, AG), [24]) (henceforth, RE) and [31]) (henceforth, GPH). The authors of [31] suggest a biasreduced logperiodogram regression estimator, \(\hat{d}_{r}\), of the longmemory parameter, d, that eliminates the first and higherorder biases of the GPH estimator of [31]. The biasreduced estimator is identical to the GPH estimator except that the pseudoregression model that produces the GPH estimator contains as extra regressors the frequencies to the power 2k for \(k = 1,\ldots,r\) where r is a particular positive integer. The bias decrease is acquired by the assumptions made on the spectrum solitary in the neighborhood of the zero frequency. The authors of [30] following [24] found that the asymptotic bias, variance, and meansquared error (MSE) of \(\hat{d}_{r}\). These outcomes show that the bias of \(\hat{d}_{r}\) goes to zero at a faster rate than that of the GPH. Therefore, the most suitable estimator for our FFFFFFIADF test among these estimators is AG, which manages to catch the \(T^{1/2}\) convergence and satisfies the unbiasedness property. There are other estimators which may be used in our study such as the [32]’s simple search algorithm. This algorithm depends on the SSR minimization and considers both the structural break and the estimation of d. However, since we are using the twostep procedure which considers the structural break and integration order separately, this procedure creates problems in our study. Despite this fact, we have tried the SSR approach in obtaining an estimate for the d parameter but found poor results with respect to the other estimators.^{Footnote 3}
In the light of all these results, we propose a new estimator by using a simple search algorithm, which may be more suitable in our case and many other cases. In the interval \(d = [0,1]\) plenty of different dynamics are available including \(d = 0\), which corresponds to stationarity, \(0 < d < 0.5\), which gives difference stationarity, \(0.5 \le d < 1.0\), which refers to a nonstationary but meanreverting process, and \(d = 1\), which corresponds to a unitroot process. In our case, instead of using a priori estimate of d, we estimate it simultaneously within the unitroot testing procedure. For this purpose, we utilize both a search algorithm and a simple bootstrap algorithm as follows.

Step 1: Estimate \(y_{t} = \alpha _{0} + \varphi _{1} \sin ( \frac{2\pi {kt}}{{T}} ) + \varphi _{1}\cos ( \frac{2\pi {kt}}{{T}} ) + \bar{\omega} _{{t}}\) for the series under investigation by using the optimal \(k_{fr}^{*}\) and use the series thereby obtained in the second step estimation,

Step 2: For a predetermined value of \(d_{1}\), starting from \(d_{1} = 0.1\), estimate the FFFFFFIADF test value by running \(\Delta ^{d_{0}}\bar{\omega }_{t} = \beta \Delta ^{d_{1}}\bar{\omega }_{t t  1} + u_{t}\). Also introduce lags of the dependent variable using the AIC or SIC,

Step 3: Obtain critical values for this predetermined value of \(d_{1}\) using 2000 centered residuals from step 2 and a simple bootstrap algorithm,

Step 4: Use steps 2 and 3 to obtain the pvalues of the test statistics for the series under consideration,

Step 5: Repeat steps 2 to 4 using the interval \(d = (0,1)\) and increments \(\Delta d = 0.001\). Then obtain all available pvalues in this range,

Step 6: If collected pvalues truncate the 0.1 significance level, then the first truncation will be the estimate, \(\hat{d}_{1}\). If there is no such truncation, then select the minimum pvalue for the estimated \(\hat{d}_{1}\) parameter.
As an example, we have obtained the estimates for Germany, Italy, Russia, Spain, Turkey, and US in Fig. 4.
Therefore, using our new methodology, we can also provide the best procedure which leads to the minimum information loss.
Let us now elaborate more on the information loss. One major drawback of differencing is that it leads to information loss. In the most extreme case, by taking the firstorder difference, that is, with \(d = 1\) we lose valuable information contained in a series. Taking differences is in some ways analogous to differentiation. Before taking the firstorder derivative of the function we have information on its time path or the primitive function. By taking the firstorder derivative of the series with respect to time, we gain information about the rate of change or growth of the series (or derived function) while passing this time path. If the subject we want to examine includes the information of the time path, a firstorder derivative with respect to time will enable us to examine the series’ growth relationship.
In this sense, a researcher who wants to use the gross national product (GNP) of a country must consider its growth rate because GNP in levels is not stationary. As another example, suppose we want to forecast the temperature, but the temperature data is not stationary. In order to make longterm forecasts, the series analyzed should be stationary. Otherwise, the forecast error will grow so rapidly after the one step ahead forecast that it will not allow the longterm forecast to be possible. From a forecasting perspective, it may not be relevant for the researcher to predict the growth rate of GNP instead of its level. When we take the difference from a lower order, valuable data including the growth and time path of the series is retrieved. If the d parameter is close to 0, the series that we obtain contains more information about the time path of the series; otherwise (for \(d=1\)) it conveys the growth rate of the series. On the other hand, when the difference is taken at order \(d = 0.5\), an optimal mix of these two will be obtained. Figure 5 shows how the primitive function converges to the derived function as the order of differencing changes. It is obtained using the following data generation:
Figure 6 visualizes the isomorphism among series with difference orders as the difference operator converges to 1.
While \(d = 0.1\), the series still preserves almost all features of the original series or the time path; that is, it preserves the information about the original state of the series (primitive function) at the maximum level. However, when \(d = 0.5\), the resultant series seem to resemble the series’ rate of change, although still preserving some time path information. This information has important implications in the timeseries econometrics literature. Suppose the series is stationary in the interval \(d = 0.10.5\). In that case, we can continue our work with the level of the series, i.e., time path, and obtain unbiased estimates with respect to this level information. Moreover, the traditional distribution theory is still valid while conducting regression analysis with this group of data or differenced series. But if \(d> 0.5\), the series obtained no longer contains information about its time path, and we will have to comment on the rate of change. After this point, while the traditional asymptotic theory ceases to maintain its validity, we should also be careful about the different integration orders.^{Footnote 4}
Empirical example
In this section, the daily infected case forecasts of the Coronavirus (Covid19) pandemic, which started as of 01/01/2020 and spread worldwide, will be performed. Since the Covid19 epidemic is on the agenda, many empirical and theoretical studies were conducted on the subject. Empirical studies on the subject include in the literature [33–37], and [38]. In addition, studies close to the theoretical structure of this article are [39–47], and [48]. The Coronavirus daily infected case numbers are collected from the European Health Organization database for 204 countries. The newly proposed FFFFFFIADF type of unitroot test and the one developed in [17] were applied to the existing data of these countries. We have investigated the fit of fractional and integer Fourier functions to the daily infected case series for some selected countries by using the SSR estimates and graphed them below in Fig. 7.^{Footnote 5} The countries with a longer time span that exhibit different dynamics were selected.
As Fig. 7 shows, the FFFFF method gives better results than the IFFFF method for all selected countries using the SSR criterion. Thus, these results are not tabulated. It is clear from Fig. 7 and the SSR results that the FFFFF method captures better the dynamics of the daily infected cases due to its highly flexible structure. Therefore, the FFFFF methodology can be used in obtaining the longrun forecasts of the daily Covid19 cases. Of course, to confirm this claim, it is necessary to look at the results of both tests, namely FFFFFADF and FFFFFFIADF.
As aforementioned, longterm forecasting is only possible with stationary data. Thus, the FFFFF methods must be used when pretesting the stationarity of the daily infected cases. Moreover, since the daily cases were found stationary using both the FFFFFADF and the newly proposed FFFFFFIADF type of unitroot tests, a forecast model constructed for these daily cases must also include the FFFFF type of flexible function using fractional integration.
Since it would take a lot of space to tabulate the unitroot test results for the entire dataset of 240 countries, we preferred to visualize them using a world map in Fig. 8. Countries with nonstationary and stationary daily cases were colored with blue and red tones, respectively. According to the FFFFFADF test, the daily infected cases of 124 (out of 240) countries were found to be stationary. When we estimated the fractional frequency of the Fourier functions for these countries, 91 countries’ fractional frequencies were found in the interval 1.7 and 4.13. The high frequencies found in these countries can be attributed to the random oscillations caused by irregular testing, wrong protection measures adopted, and similar situations arising in these countries. In some countries extremely high case numbers are seen in one day, whereas the next day no tests are run, and no numbers are announced. This behavior of the health authorities leads to the irregular distribution of jump discontinuities. Despite these irregular oscillations, the fractional frequency Fourier function captures the unknown deterministic functional forms extremely well. Besides, fractional integration is also useful for capturing these random oscillations. Therefore, it is better to use the FFFFFFIADF test in countries where the unitroot null could not be rejected. For this reason, we selected the ten countries with the highest daily case numbers that were not found stationary with the FFFFFADF test.
As can be seen from Table 4, the FFFFFFIADF test results demonstrate that the daily cases of all countries, except Russia and Spain, are fractionally integrated and stationary. When it comes to Russia and Spain, their Covid19 cases are found to be fractionally integrated, meanreverting but nonstationary. The AG method produced stationary test results for Brazil, France, Germany, and the UK. The GPH method, which is the closest method to the AG method, yielded similar results except for Germany and the UK. Besides, the RP method leads to stationarity test results for Brazil, Chile, France, and the UK. On the other hand, the newly proposed method seems to be the most efficient one when compared to these other methods. It rejects the null hypothesis of unit root for Brazil, Chile, France, Germany, Italy, Turkey, the UK, and the US. The fractional integration dynamics that the FFFFFADF could not represent were caught with different methods. The oscillations that we mentioned in the introduction part was modeled correctly with FI. In this sense, it will be beneficial to use the FFFFFFI model to forecast Covid19’s longterm potentially infected number of cases. These efficient longterm forecasts will enable policy authorities to control the outbreak better. Moreover, we can also see that the method we have just proposed provided the lowest difference order estimates. This obtained lowest difference order allows us to perform the most accurate unitroot test with the lowest information loss.
Conclusion
In this study, we have proposed a fractional frequency flexible Fourier form fractionally integrated ADF test. By implementing an extensive simulation study, we have showed that the newly proposed test has good size and power properties. Moreover, we have demonstrated that the best estimators for our unitroot testing procedure are both fractional frequency and newly proposed fractional difference operator. The newly proposed fractional difference estimator has shown to be the best estimator with respect to the minimum information loss criteria. Finally, the empirical study has demonstrated that not considering the structural break and fractional integration simultaneously in the testing process may lead to misleading results about the stochastic behavior of the series under investigation. Therefore, our proposed FFFFFFIADF test will help policy authorities to control any natural disaster by providing an efficient method for pretesting the disaster’s longterm predictability.
Moreover, the fractional frequency and fractional difference estimation methodologies given in Sect. 3 shed light on the areas for future research. First of all, different functional forms could be used for the structural breaks. In this study, we showed that fractional frequency fits the structure of the Covid19 epidemic quite well. However, another functional form can be recommended for another data type. Furthermore, different methodologies may be developed for implementing fractional difference estimation. Section 3.2 tried to examine the fractional differencing meaning and suggested an estimator that minimizes the information loss. The importance of taking differencing in different orders shows that new estimators and difference operators can be developed for various purposes in future studies.
Availability of data and materials
Data sharing not applicable to this article as publicly open datasets are available from the European Health Organization.
Notes
 1.
See [10] for more details.
 2.
\(\Gamma (\cdot )\) denotes the gamma function, and \(\{\pi _{i}(d)\}\) represents the sequence of coefficients associated with the expansion of \(\Delta ^{d}\) in powers of L, which are defined as \(\pi _{i}(d) = \frac{\Gamma (i  d)}{\Gamma (  d)\Gamma (i  1)}\).
 3.
The results can be provided upon request.
 4.
Since the research of these topics is beyond the scope of the study, the subject is left at this maturity level. However, more detailed discussions on these issues are considered for future studies.
 5.
The country codes of these countries are given in the Appendix.
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Appendices
Appendix A
This appendix provides asymptotic distribution of FFFFFFIADF test statistics given in the text.
Proof of Proposition 1
The proofs of (i) and (ii) are known (see Hamilton, 1994). By using the continuous mapping theorem, we can obtain the proofs of (iii) and (iv) as follows:
□
Proof of Theorem 1
Let \(\sin (t) = \sin ( 2\pi k^{fr}t/T )\) and \(\cos (t) = \cos ( 2\pi k^{fr}t/T )\). We first examine the demeaned case with \(\beta = 0\) in Eq. (6). Let \(y_{t}^{\mu }\) denote the OLS residuals from the demeaned case in the text with \(w_{t} = ( 1,\sin (t),\cos (t) )'\)
where \(\theta = ( \alpha _{0},\alpha _{1},\alpha _{2} )^{\prime } \), θ̂ is the OLS estimator of θ. We let \(w = ( w_{1},\ldots,w_{T} )^{\prime } \), \(\bar{\omega } = ( \bar{\omega }_{1},\ldots,\bar{\omega }_{T} )^{\prime } \) and \(\mathbf{M}_{T} = \operatorname{diag} ( \sqrt{T},\sqrt{T},\sqrt{T} )\) to have
Applying some algebra to (A.1) and (A.2),
Depending on FCLT, the first term of (A.3) becomes
The second component in (A.3) becomes
where \(\Delta _{1} = s_{2}c_{2}  s_{2}c_{0}^{2}  s_{0}^{2}c_{2}  m_{0}^{2}\), \(a_{11} = s_{2}c_{2}  m_{0}^{2}\), \(a_{12} = c_{0}m_{0}  s_{0}c_{2}\), \(a_{13} = s_{0}m_{0}  s_{2}c_{0}\), \(a_{22} = c_{2}  c_{0}^{2}\), \(a_{23} = s_{0}c_{0}  m_{0}\) and \(a_{33} = s_{2}  s_{0}^{2}\).
Then
Finally, merging the outcomes in Eqs. (A.4) and (A.7), we obtain the demeaned Brownian motion by
For the detrended case similar arguments follow so we skip the algebra. Using the above given results, under the null we can obtain the demeaned Brownian’s. Now we can proceed with the fractionally integrated part in the second step. Under the null hypothesis that \(\bar{\omega }_{t}\) is a random walk and applying Lemma 1 and Lemma 2 and results in [18] and in [15], we obtain
□
Appendix B: Critical values
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Omay, T., Baleanu, D. Fractional unitroot tests allowing for a fractional frequency flexible Fourier form trend: predictability of Covid19. Adv Differ Equ 2021, 167 (2021). https://doi.org/10.1186/s13662021033179
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JEL Classification
 C01
 C02
 C14
 C15
 C22
Keywords
 Structural break
 Stochastic fractional difference equation
 Stationarity
 Covid19 forecast