Autoregressive Fractionally Integrated Moving Average-Generalized Autoregressive Conditional Heteroskedasticity Model with Level Shift Intervention

In this paper, we introduce the class of autoregressive fractionally integrated moving average-generalized autoregressive conditional heteroskedasticity (ARFIMA-GARCH) models with level shift type intervention that are capable of capturing three key features of time series: long range dependence, volatility and level shift. The main concern is on detection of mean and volatility level shift in a fractionally integrated time series with volatility. We will denote such a time series as level shift autoregressive fractionally integrated moving average (LS-ARFIMA) and level shift generalized autoregressive conditional heteroskedasticity (LS-GARCH). Test statistics that are useful to examine if mean and volatility level shifts are present in an autoregressive fractionally integrated moving average-generalized autoregressive conditional heteroskedasticity (ARFIMA-GARCH) model are derived. Quasi maximum likelihood estimation of the model is also considered.


Some Theoretical Results
This section presents some theoretical literature on ARFIMA models and GARCH models. An overview of ARFIMA-GARCH models is also presented.

The ARFIMA Model
The study of time series turned attention to incorporate long memory or longrange dependence characteristics. The ARFIMA(p, d, q) process, first introduced by [1] and [2], present this property when the differencing parameter d is in the interval (0, 0.5). This feature is reflected by the hyperbolic decay of its autocorrelation function or by the unboundedness of its spectral density function, while in the ARMA model, dependency between observations decays at a geometric rate.
Montanari et al. [19] introduced a special form of the generalized ARFIMA model and also considered by [20]. This formulation is able to reproduce shortand long-memory periodicity in the autocorrelation function of the process. Using the [11] notation, let { } and can be expressed as: The ARFIMA model is said to be stationary when 0.5 0.5 d − < < , where the effect of shocks to t ε decays at a gradual rate to zero. The model becomes nonstationary when 0.5 d ≥ and stationary but non invertible when 0.5 d ≤ − , which means the time series is impossible to model for any AR process. With regard to the modeling of data dependencies, the ARFIMA model represents a short memory if 0 d = , where the effect of shocks decays geometrically; and a unit root process is shown when 1 d = . Furthermore, the model has a positive dependence among distance observations or the so called long memory process if 0 0.5 d < < ; and it also has an anti-persistent property or has an intermediate

The GARCH(r, s) Model
The GARCH(r, s) model can be obtained from Equation (1) where t z is normal distributed with mean 0 and variance 1. Bollerslev [4] introduced the GARCH(r, s) model which defines the conditional variance equation as follows: Note that the GARCH model defined by (5) can be replaced by other conditional heteroscedastic models.

The General ARFIMA(p, d, q)-GARCH(r, s) Model
Let the ARFIMA(p, d, q)-GARCH(r, s) model be the discrete time series model of { } t y given by the following equation: The following theorem shows some properties of ARFIMA(p, d, q)-GARCH(r, s) models.
Let { } t y be generated by model (6). Suppose that all roots of ( ) For proof of Theorem (2.3) see [22].

Variance of Variance in the Standard GARCH(1, 1) Model
By rearranging the conditional variance Equation (5) for a GARCH(1, 1) we obtain: where z κ denotes the conditional kurtosis of t z , which we assume to be finite constant. If the distribution of t z is standard normal, then 1 2 z κ − = .
Ishida and Engle [23] further rearranged the terms in Equation (9), the conditional variance equation becomes: where 1 ϕ γ = − determines the speed at which the conditional variance reverts to its long run mean Belkhouja and Mootamri [24] performed a long memory and structural change in the G7 inflation dynamics. The following section presents a natural extension of ARFIMA-GARCH models to the case with level shift.

ARFIMA-GARCH Models with Level Shift
This section presents a natural extension of the ARFIMA-GARCH models to a case with level shift. We start with a shift in the mean, then a shift in volatility and finally shift in both mean and volatility.

The ARFIMA(p, d, q) Model with Level Shift
The ARFIMA(p, d, q) model is written as where t I is an indicator variable taking values 1 for t i = , and 0 otherwise. The The extension of (14) to k level shifts is straightforward. We define j µ as the j th shift in level, compared to the previous level, where 1, , j k =  . When we allow k level changes at pre-specified time t j = , we can extend (14) to The component ( )

The GARCH(r, s) Model with Level Shift
As indicated earlier, [4] introduced the GARCH(r, s) model which defines the conditional variance equation as follows: ( ) where t I is an indicator variable taking values 1 for t i = , and 0 otherwise. The The extension of (17) to k volatility level shifts is straightforward. We define j ω as the j th shift in volatility level, compared to the previous level, where 1, , j k =  . When we allow k volatility level changes at pre-specified time t j = , we can extend (17) to ( ) The component ( )

The General ARFIMA(p, d, q)-GARCH(r, s) Model with Level Shift
Extension of the ARFIMA(p, d, q)-GARCH(r, s) model to the case with level shift is given by the following equation which we will denote as LS-ARFIMA- The LS-ARFIMA-LS-GARCH series is shown in Figure 1.

Estimation of LS-ARFIMA Model Parameters
The first step of estimation consists in estimating the ARFIMA(p, d, q) assuming that the conditional variance is constant over time. By rearranging Equation (14) for one mean level shift we have: Therefore the null hypothesis of unconditional mean constancy becomes: be the approximate likelihood estimator (MLE) 1 ψ of 1 ψ that maximizes the conditional log-likelihood: The partial derivatives evaluated under 0 H are given by:

Estimation of LS-GARCH Parameters
Once the LS-ARFIMA model is estimated and the residuals t ε are obtained, we test the alternative of LS-GARCH specification with one volatility level shift against the null hypothesis of GARCH model. Let us rearrange model (17) with one volatility level shift: Therefore the null hypothesis of the unconditional variance constancy becomes: be the vector of the LS-GARCH model parameters and the quasi-likelihood function is given by: The partial derivatives evaluated under 0 H are given by: Under the null hypothesis, the "hats" indicates the maximum likelihood estimator and 0 t h denotes the conditional variance estimated at time t.

Mean Level Shift Detection in ARFIMA-GARCH
The mean level shift detection test was previously derived by [25] for ARFIMA(p, d, q) models assuming conditional variance is constant over time. In order to derive the test statistic, let us rewrite model (15), with only one mean level change: The hypothesis to be tested is which is based on 1 2 , , , n y y y  a realization of time series { } t y satisfying ARFIMA-GARCH model with mean level shift.
Extension of [26] test statistics can be written as: where ( ) ( ) Model (26) can be rewritten as: This implies transforming the series by differencing once. Thus if 1 0 µ = , The distribution of the statistics is discussed in great detail in [25] The maximum domain of attraction of the Gumbel is shown to some extent in [27] and in greater detail in [28].
Let also the test statistics be given by

Volatility Level Shift Detection in ARFIMA-GARCH Model
The second step is a natural extension of mean level shift detection in ARFIMA- Thus The maximum domain of attraction of the Gumbel is shown to some extent in [27] and in greater detail in [28].

Mean and Volatility Level Shift Detection in ARFIMA-GARCH
Summary of the detection procedure is presented below: 1) Plot the data to get a picture of the type of series and possible level shift in the data.
2) Assume that the underlying ARFIMA-GARCH series { } t y contains no level shift and use maximum likelihood procedure to estimate its parameters.
3) The first test is performed to check the mean level shift which can be conducted as follows: a) State the hypothesis being tested, which is

Simulation Study of the Level Shift Detection Procedure
To appreciate the procedure we derived a simulation study consisting of simulation of critical values for mean and volatility level shift, simulating different sizes of mean and volatility level shift impact, performing detection test and conducting the power of the mean level shift detection procedure.

Critical Values for Mean Level Shift Detection Test
Simulation of the critical values was done using R software. An assumption that there are mean level shifts was made, then simulations conducted. This is based on an estimate of the statistic n C as shown in Equation (34) with norming constants given in Equation (31).
The critical values for the 10%, 5% and 1% level of significance are presented in   Figure 2 shows the graph of critical values for detecting mean level shift using 5% level of significance. It can be depicted from the graph that the critical values depend on the fractional differencing parameter d and sample size. As the sample size increases the critical value appears to be converging. The same scenario is also the case for 1% and 10% level of significance.

Mean Level Shift Detection Test
Before conducting the test it should be clear that the position of the mean level shift impact i.e. point t i = is not known. The level shift impact 1 µ is tested for significance using the hypotheses

Power of the Mean Level Shift Detection Test
The probability of correctly detecting a mean level shift is the power of the test. Table 4 shows the frequency (denoted Freq) with which the location of a mean level shift is correctly detected, the probability (denoted Prob) of correctly de-    Table 4 depicts the probability of correctly detecting a mean level shift is high as long as the mean level shift ˆn C is significantly different from the 95% Gumbel critical value of 5.1348 but it is low as long as the resulting level shift is low.
The frequencies of the detection of mean level shift approaches 10 000 as the size of mean level shift increases.
DOI: 10.4236/ojs.2020.102023 Figure 3 is a graph showing the power of the detection test of mean level shift using 95% Gumbel critical value of 5.1348. This is the general behaviour for 90% and 99% Gumbel critical value.

Critical Values for Volatility Level Shift Detection Test
As with critical values for the mean level shift, similar simulation of the critical values for the volatility level shift was done using R programs. An assumption that there are volatility level shifts was made, then simulations conducted. This is based on an estimate of the statistic n C as shown in Equation (45) with norming constants given in Equation (43).
The critical values for the 10%, 5% and 1% level of significance are presented in the critical values are the same, they only increase with the sample size n as depicted in Table 6. Samples of sizes 100, 500, 1,000, 5,000, 10,000, 20,000 and 50,000 were used. It can be noted that, for example, at 5% level of significance with 0.0 d = the critical value ranges from 5.8953 for a sample of size 100 to 67.6419 for a sample of size 50,000. We can conclude without loss of generality that the simulated critical values in Table 5 can be observed to be diverging critical values as the sample size increases. Figure 4 shows the graph of critical values for detecting volatility level shift using 5% level of significance. Unlike the mean level shift, it can be depicted from the graph that the critical values do not depend on the fractional differencing parameter d. But as the sample size increases the critical value appears to be diverging. The same scenario is also the case for 1% and 10% level of significance.

Conclusions
In this study, we derive and extend level shift detection test to the case of ARFIMA-GARCH models, the resulting models were denoted as LS-ARFIMA-LS-GARCH models. The derivation was in both the mean and volatility, such that a natural extension to LS-ARFIMA-LS-GARCH models was established. Then parameter estimation of LS-ARFIMA-LS-GARCH models was derived.
Step by step detection procedure for level shift was also suggested and presented. Finally a simulation study of the critical values was performed using sample sizes of up-to 50 000 for mean level shift detection test and up to 100 000 for volatility level shift detection test. Some concluding remarks can be summarized as follows: 1) A natural extension of level shift models in ARFIMA-GARCH models (denoted LS-ARFIMA-LS-GARCH models) was established.
2) Level shift detection tests for both the mean and volatility in models with ARFIMA-GARCH using step by step procedure were established.
4) The simulation study shows that critical values of the mean level shift detection test converges to Gumbel whereas the critical values of volatility level shift detection test diverge.
5) Power of the test was also conducted and results for mean level shift shows that the probability of correctly detecting a mean level shift is high as long as the mean level shift impact is significantly different from the 95% Gumbel critical values of 5.1348.
6) It was observed that critical values of volatility level shift detection procedure fail to converge to a Gumbel distribution. Further derivation and establishment of the normalizing constants of the test statistics and distribution which converges is still work in progress.