Modeling Seasonal Fractionally Integrated Autoregressive Moving Average-Generalized Autoregressive Conditional Heteroscedasticity Model with Seasonal Level Shift Intervention

This paper introduces the class of seasonal fractionally integrated autoregressive moving average-generalized conditional heteroskedastisticty (SARFIMAGARCH) models, with level shift type intervention that are capable of capturing simultaneously four key features of time series: seasonality, long range dependence, volatility and level shift. The main focus is on modeling seasonal level shift (SLS) in fractionally integrated and volatile processes. A natural extension of the seasonal level shift detection test of the mean for a realization of time series satisfying SLS-SARFIMA and SLS-GARCH models was derived. Test statistics that are useful to examine if seasonal level shift in an SARFIMA-GARCH model is statistically plausible were established. Estimation of SLS-SARFIMA and SLS-GARCH parameters was also considered.


Introduction
The phenomenon of long memory or long range dependence in time series processes has been of interest in time series research. A popular way to analyze a long memory time series is to use seasonal autoregressive fractionally integrated moving average (SARFIMA) processes introduced by [1] and [2]. The works of [1] and [2] assume that the conditional variance of the time series is constant over time. However, non constant variance in non-linear time series is a chal-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. [12] introduced a special form of the generalized ARFIMA model considered by [13]. This formulation is able to reproduce short-and long memory periodicity in the autocorrelation function of the process. Using the [14] notation, the general form of the ARFIMA model is defined as follows: The difference operator ( )  A special form of the generalized ARFIMA model was considered by [13].
This formulation is able to reproduce short and long memory periodicity in the autocorrelation function of the process. Using the [14] where s ∈  is the seasonal period, B is the backward-shift operator, that is, are the polynomials of degrees P and Q, respectively, defined by: are constants and 0 The seasonal difference operator ( ) , with seasonality s ∈  , for all 1 D > − , is defined by means of the binomial expansion; A compact form of Equation (1) and Equation (4) is given by: In Equation (8), the operator ∇ d is defined by  is the memory vector parameter, d and D are the fractionally parameters at non seasonal and seasonal frequencies, respectively. The fractional filters are: Suppose in Equation (8) where 0 λ < ≤ π .
Note that according to [15]: 2) The stationary process { } t t x ∈ has a long memory property if 3) The stationary process { } t t x ∈ has an intermediate property if For convenience, we introduce the notation with zero mean, s ∈  as the seasonal period. Then the process { } For proof of Theorem (2.1) see [15] where t z is normal distributed with mean 0 and variance 1.
, r and m are positive integer. Note that the GARCH model defined by (20) can be replaced by other conditional heteroscedastic models.

The General SARFIMA-GARCH Model
Let the

Variance of Variance in the Standard GARCH(m, r) Model
By rearranging the conditional variance Equation (20) for a GARCH(1,1) we obtain: [16] have shown that the variance of variance is given by: 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 − = .
[16] further rearranged the terms in Equation (22), the conditional variance equation becomes: where 1 ϕ γ = − determines the speed at which the conditional variance reverts to its long run mean τ , that is, ( ) ( ) and its corresponding variance becomes:

Intervention in ARIMA Models
Traditional time series analysis has considered four types of interventions, see for instance, [17] [18] [19] and [20]. The four types of interventions are: 1) Additive Outlier (AO): represents an isolated spike.
3) Transitory or Temporary Change (TC): represents a spike that takes a few periods to disappear. According to [21], the effect of an AO, a LS, or a TC on an observed series is independent of the ARIMA model whereas the effect of an IO on an observed series consist of an initial shock that propagates in the subsequent observations with the weights of the ARIMA model.
and let t x be the contaminated series containing k outliers represented by: where j µ is the initial impact of the outlier at time 2) LS: 3) TC: sons. The basic model with SLS suggested by [21] is given by The dynamic impact was normalized as suggested by [22]: Thus model (28) becomes However in this research, for a SLS intervention at period t τ = , we shall stick to the simple specification model (28).

SARFIMA-GARCH Models with Level Shift
This section presents a natural extension of the SARFIMA-GARCH models to a case with level shift. We start with a standard shift in the mean, then a level shift in seasonality. We will also consider level shift in the volatility with its corresponding shift in seasonality.

The SARFIMA Model with Level Shift
The ( )( ) SARFIMA , , , , p d q P D Q s model is written as where t y is the time series at time t, 0 µ is the unconditional mean of the process. We assume the noise process t ε to be Gaussian, with expectation zero and variance 2 ε σ . To allow for a mean level shift, after time t τ = of the data, we write the sum of an unobserved SARFIMA process and the term for the mean level shift which we will denote as The mean level shift is an abrupt but permanent shift by 11 µ in the series caused by an intervention.
The extension of (34) to k level shifts is straightforward. We define 1 j µ as

The SARFIMA Model with Seasonal Level Shift
Seasonal Level Shift in SARFIMA models denoted as SLS-SARFIMA type intervention display seasonal features in its pattern. Many economic series display breaks and anomalies within the previous restrictions. To illustrate the argument, the SLS is presented in Figure 1. This paper covers the methodologically and computationally less complex approach by extending intervention detection procedures to cover shifts in the seasonal component in fractionally integrated SARFIMA models. The Seasonal Level Shift (SLS) intervention has an effect on the trend given by the step function of Figure 2(a), and an effect on the seasonal component shown in Figure  2

The GARCH(m, r) Model with Level Shift
As indicated earlier, [4] introduced the GARCH(m, r) model which defines the conditional variance equation as follows; ( ) ( ) The component ( )

The General SARFIMA-GARCH Model with Level Shift
Extension of t y the

Estimation of SLS-SARFIMA Model Parameters
The The partial derivatives evaluated under 0 H are given by:

Estimation of SLS-GARCH Parameters
Once the SLS-SARFIMA model is estimated and the residuals t ε are obtained, we test the alternative of SLS-GARCH specification with one volatility level shift against the null hypothesis of GARCH model. Let us rearrange model (41) with one volatility level shift: Therefore the null hypothesis of the unconditional variance constancy becomes: Under the null hypothesis, the "hats" indicates the maximum likelihood estimator and 0 t h denotes the conditional variance estimated at time t. Under 0 H , the LM-type statistics is asymptotically distributed as 2 χ with one degree of freedom:

Level Shift Detection and Estimation in SARFIMA-GARCH
In this section we discuss how the iterative detective procedure described in [23] and [24] can be extended to allow for both detection and estimation of SLS in SARFIMA-GARCH intervention to be denoted SLS-SARFIMA and SLS-GARCH.

Let
is an 1 n × . Writing model (61) in matrix form:  . According to [21], model (62)  nd Var As argued in [24], to move from the GLS model in (62)   The maximum domain of attraction of the Gumbel is shown to some extent in [29] and in greater detail in [30].
Let the test statistics be given by where D signifies convergence in distribution. Here, R λ ∈ is location parameter and δ is scale parameter. The location parameter is also the mode of the distribution. Inverse of the ( )

Volatility Seasonal Level Shift Detection in SARFIMA-GARCH Model
For any realization 1

Mean and Volatility Level Shift Detection in SARFIMA-GARCH
Summary of the detection procedure is presented below: 1) Plot the data to get a picture of the type of series and possible seasonal level shift in the data.
2) Assume that the underlying SARFIMA-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 seasonal level shift which can be conducted as follows:

Conclusions
This paper focused on the theoretical derivation of the class of seasonal fractionally integrated autoregressive moving average-conditional heteroskedastisticty (SARFIMA-GARCH) models, with level shift type intervention. The following derivations were established: 1) A natural extension of the seasonal level shift detection test of the mean for a time series satisfies SLS-SARFIMA.
2) A natural extension of the seasonal level shift detection test of the volatility for a time series satisfies SLS-GARCH.
3) Test statistics that are useful to examine if seasonal level shift in an SARFIMA-GARCH model was established. 4) Estimation of SLS-SARFIMA and SLS-GARCH parameters was derived using quasi maximum likelihood estimation.
To appreciate the procedure, we derived a simulation study consisting of simulation of critical values for mean and volatility seasonal level shift, simulating different sizes of mean and volatility seasonal level shift impact, performing detection test and conducting the power of the mean level shift detection procedure which are considered in a separate paper by the same authors.