ABSTRACT

This paper introduces the class of seasonal fractionally integrated autoregressive moving average-generalized conditional heteroskedastisticty (SARFIMA- GARCH) 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.

Keywords:

Seasonality, Fractional Integration, Long-Memory, Level Shift, SLS-SARFIMA, SLS-GARCH, Volatility

1. 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 challenging modelling exercise, considered among other things by [3]. In particular, the stylized fact that the volatility of financial time series is non constant has been long recognized in literature, see for example [4] [5] and [6].

The methodology for modelling time series with long memory behavior has been extended to long memory time series with time varying conditional variance, see for instance, [7] who developed the ARFIMA model with generalized autoregressive conditional heteroskedasticity (GARCH) type innovations, and [8] examine the daily average PM₁₀ concentration using a seasonal ARFIMA model with GARCH errors. Tong [9] analyzed the nonlinear time series using GARCH models and [10] used GARCH models for testing market efficiency. These models do not capture level shifts both in mean and variance, in this paper we introduce a new class of SARFIMA-GARCH models with seasonal level shift intervention in mean and volatility. This approach allows us to model mean and volatility seasonal level shifts in an SARFIMA-GARCH model which is often observed in financial or economics time series.

This article introduces detection of a mean and volatility level shifts innovation in an ARFIMA-GARCH model. The works of [11] first applied ARFIMA-GARCH models to price indices then [7] derived conditions for asymptotic normality of the approximate (Gaussian) maximum likelihood (ML) estimator in the ARFIMA-GARCH model. This paper also extends parameter estimation for an SARFIMA-GARCH model to case with seasonal level shift which we will denote Seasonal Level Shift SARFIMA (SLS-ARFIMA) and Seasonal Level Shift GARCH (SLS-GARCH) using quasi-maximum likelihood estimation.

The first concern of this paper is how one would formally address modeling mean and volatility seasonal level shifts in an SARFIMA-GARCH. The second concern is derivation of test statistics that are useful to examine presence of seasonal level shifts in mean and volatility for an SARFIMA-GARCH model. The layout of the paper is organised as follows. Section 2 reviews some theoretical results of SARFIMA-GARCH and intervention. In Section 3, we introduce the class of SLS-ARFIMA-SLS-GARCH models. Section 4 deals with parameter estimation in SLS-ARFIMA and SLS-GARCH models. Section 5 is dedicated to the proposed procedure of level shift detection in SARFIMA-GARCH models. The last section concludes with the main findings and limitations. Common acronyms used in this paper are given in Table 1.

2. SARFIMA-GARCH Models

This section presents review of the ARFIMA models, SARFIMA models, GARCH models and the SARFIMA-GARCH models. The variance of GARCH model and intervention in ARFIMA models is also presented.

2.1. ARFIMA Process

Time series analysis has turned attention to the studies with long memory or long-range dependence characteristics. The $\text{ARFIMA}\left(p\mathrm{,}d\mathrm{,}q\right)$ 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. [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:

Let ${\left\{x_t\right\}}_{t\in \mathbb{Z}}$ be a stochastic process, then ${\left\{x_t\right\}}_{t\in \mathbb{Z}}$ is a zero mean $\text{ARFIMA}\left(p\mathrm{,}d\mathrm{,}q\right)$ process given by the expression

$\varphi \left(B\right){\left(1-B\right)}^d{x}_t=\theta \left(B\right){\epsilon }_t\mathrm{,}\text{for}t\in \mathbb{Z}$ (1)

where ${\left\{{\epsilon }_t\right\}}_{t\in \mathbb{Z}}$ is a white noise process with zero mean and variance ${\sigma }_{\epsilon }^2=\mathbb{E}\left({\epsilon }_t^2\right)$, B is the backward-shift operator, that is, $B^k{x}_t=x_{t-k}$ and ${\left(1-B\right)}^d$ is the non seasonal difference, $\varphi (\cdot )$ and $\theta (\cdot )$ are the non-seasonal polynomials of degrees p and q, respectively, defined by:

$\varphi \left(B\right)={\displaystyle \underset{i=0}{\overset{p}{\sum }}}\left(-{\varphi }_i\right)B^i,\theta \left(B\right)={\displaystyle \underset{j=0}{\overset{q}{\sum }}}\left(-{\theta }_j\right)B^j,$ (2)

where ${\varphi }_i\mathrm{,\; <mi>\; </mi>1}\le i\le p$, and ${\theta }_j\mathrm{,\; <mi>\; </mi>1}\le j\le q$ are constants and ${\varphi }_0=-1={\theta }_0$.

The difference operator ${\left(1-B\right)}^d$ is defined by means of the binomial expansion:

${\left(1-B\right)}^d={\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}\left(\begin{array}{c}d\\ j\end{array}\right){\left(-B\right)}^j,$ (3)

where;

$\left(\begin{array}{c}d\\ j\end{array}\right)=\frac{\Gamma \left(1+d\right)}{\Gamma \left(1+j\right)\Gamma \left(1+d-j\right)}$ and $\Gamma (\cdot )$ is the well known gamma function.

The ARFIMA model is said to be stationary when $-0.5<d<0.5$. The model becomes nonstationary when $d\ge 0.5$ and stationary but non invertible when $d\le -0.5$. The ARFIMA model represents a short memory if $d=0$ and a unit root process is shown when $d=1$. Furthermore, the model has a positive dependence among distance observations or the so called long memory process if $0<d<0.5$ ; and it also has an anti-persistent property or has an intermediate memory if $-0.5<d<0$.

2.2. $SARFIMA\left(p,d,q\right)\times \left(P,D,Q\right)s$ Process

The seasonal autoregressive fractionally integrated moving average process, denoted hereafter by $\text{SARFIMA}\left(p\mathrm{,}d\mathrm{,}q\right)\times \left(P\mathrm{,}D\mathrm{,}Q\right)s$, is an extension of the long range dependence in the mean $\text{ARFIMA}\left(p\mathrm{,}d\mathrm{,}q\right)$ process, proposed by [1] and [2]. The $\text{SARFIMA}\left(p\mathrm{,}d\mathrm{,}q\right)\times \left(P\mathrm{,}D\mathrm{,}Q\right)s$ process describes time series with long memory or long range dependence or persistent periodical behavior at finite number of spectrum frequencies.

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] notation, the general form of the SARFIMA model is defined below:

Let ${\left\{x_t\right\}}_{t\in \mathbb{Z}}$ be a stochastic process, then ${\left\{x_t\right\}}_{t\in \mathbb{Z}}$ is a zero mean $\text{SARFIMA}\left(p\mathrm{,}d\mathrm{,}q\right)\times \left(P\mathrm{,}D\mathrm{,}Q\right)s$ process given by the expression

$\varphi \left(B\right)\Phi \left(B^s\right){\left(1-B\right)}^d{\left(1-B^s\right)}^D{x}_t=\theta \left(B\right)\Theta \left(B^s\right){\epsilon }_t,\text{for}t\in \mathbb{Z}$ (4)

where $s\in \mathbb{N}$ is the seasonal period, B is the backward-shift operator, that is, $B^{sk}{x}_t=x_{t-sk}$,${\left(1-B^s\right)}^D$ is the seasonal difference operator, $\Phi (\cdot )$ and $\Theta (\cdot )$ are the polynomials of degrees P and Q, respectively, defined by:

$\Phi \left(B^s\right)={\displaystyle \underset{i=0}{\overset{P}{\sum }}}\left(-{\Phi }_i\right)B^{si},\Theta \left(B^s\right)={\displaystyle \underset{j=0}{\overset{Q}{\sum }}}\left(-{\Theta }_j\right)B^{sj}$ (5)

where ${\Phi }_i\mathrm{,\; <mi>\; </mi>1}\le i\le P$ and ${\Theta }_j\mathrm{,\; <mi>\; </mi>1}\le j\le Q$ are constants and ${\Phi }_0=-1={\Theta }_0$.

The seasonal difference operator ${\left(1-B^s\right)}^D$, with seasonality $s\in \mathbb{N}$, for all $D>-1$, is defined by means of the binomial expansion;

${\left(1-B^s\right)}^D={\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}\left(\begin{array}{c}D\\ j\end{array}\right){\left(-B^s\right)}^j,$ (6)

where;

$\left(\begin{array}{c}D\\ j\end{array}\right)=\frac{\Gamma \left(1+D\right)}{\Gamma \left(1+j\right)\Gamma \left(1+D-j\right)}$ (7)

A compact form of Equation (1) and Equation (4) is given by:

$\varphi \left(B\right)\Phi \left(B^s\right){\nabla }^d{x}_t=\theta \left(B\right)\Theta \left(B^s\right){\epsilon }_t\mathrm{,}\text{for}t\in \mathbb{Z}$ (8)

In Equation (8), the operator ${\nabla }^d$ is defined by

${\nabla }^d={\left(1-B\right)}^d{\left(1-B^s\right)}^D$ (9)

where $d=\left(d\mathrm{,}D\right)\in {ℝ}^2$ is the memory vector parameter, d and D are the fractionally parameters at non seasonal and seasonal frequencies, respectively. The fractional filters are:

${\left(1-B^k\right)}^l={\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}\left(\begin{array}{c}l\\ j\end{array}\right){\left(-B^k\right)}^j,\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}k=1,s\text{and}l=d,D$ (10)

where;

$\left(\begin{array}{c}l\\ j\end{array}\right)=\frac{\Gamma \left(1+l\right)}{\Gamma \left(1+j\right)\Gamma \left(1+l-j\right)}$ (11)

Suppose in Equation (8), $\varphi \left(B\right)\Phi \left(B^s\right)=0$ and $\theta \left(B\right)\Theta \left(B^s\right)=0$ have no common zeros. Let also $\left|d+D\right|<0.5$ and $\left|D\right|<0.5$, the process ${\left\{x_t\right\}}_{t\in \mathbb{Z}}$ has spectral density function is given by;

$f_x\left(\lambda \right)=\frac{{\sigma }_{\epsilon }^2}{2\pi }\frac{{\left|\theta \left({\text{e}}^{-i\lambda }\right)\right|}^2{\left|\Theta \left({\text{e}}^{-i\lambda s}\right)\right|}^2}{{\left|\varphi \left({\text{e}}^{-i\lambda }\right)\right|}^2{\left|\Phi \left({\text{e}}^{-i\lambda s}\right)\right|}^2}{\left[2\sin \left(\frac{\lambda }{2}\right)\right]}^{-2d}{\left[2\sin \left(\frac{\lambda s}{2}\right)\right]}^{-2D},$ (12)

where $0<\lambda \le \pi$.

Note that according to [15]:

1) The processs ${\left\{x_t\right\}}_{t\in \mathbb{Z}}$ is stationary if $d+D<0.5$,$D<0.5$ and $\varphi \left(B\right)\Phi \left(B^s\right)\ne 0$, for $\left|B\right|\le 1$.

2) The stationary process ${\left\{x_t\right\}}_{t\in \mathbb{Z}}$ has a long memory property if $0<d+D<0.5$,$0<D<0.5$ and $\varphi \left(B\right)\Phi \left(B^s\right)\ne 0$, for $\left|B\right|\le 1$.

3) The stationary process ${\left\{x_t\right\}}_{t\in \mathbb{Z}}$ has an intermediate property if $-0.5<d+D<0$,$-0.5<D<0$ and $\varphi \left(B\right)\Phi \left(B^s\right)\ne 0$, for $\left|B\right|\le 1$.

For convenience, we introduce the notation ${\mathbb{Z}}_{\ge }=\left\{k\in \mathbb{Z}\mathrm{<mo>\; |\; </mo>}k\ge 0\right\}$,${\mathbb{Z}}_{\le }=\left\{k\in \mathbb{Z}\mathrm{<mo>\; |\; </mo>}k\le 0\right\}$ and $A=\left\{\mathrm{1,}\cdots \mathrm{,}s-1\right\}\subset \mathbb{N}$.

Theorem 2.1. Let ${\left\{x_t\right\}}_{t\in \mathbb{Z}}$ be the $\text{SARFIMA}\left(\mathrm{0,}d\mathrm{,0}\right)\left(\mathrm{0,}D\mathrm{,0}\right)s$ process given by expression:

${\left(1-B\right)}^d{\left(1-B^s\right)}^D{x}_t={\epsilon }_t$ (13)

with zero mean, $s\in \mathbb{N}$ as the seasonal period. Then the process ${\left\{x_t\right\}}_{t\in \mathbb{Z}}$ has autocovariance function of order h, $h\in {\mathbb{Z}}_{\ge }$, given by

${\gamma }_x\left(h\right)=\{\begin{array}{ll}{\sigma }_{\epsilon }^2{\displaystyle \underset{\nu \in {\mathbb{Z}}_{\ge }}{\sum }}{\gamma }_z\left(s\nu \right){\gamma }_y\left(h-s\nu \right)\mathrm{,}\hfill & \text{if}h=sl\mathrm{,\; <mi>\; </mi>}l\in {\mathbb{Z}}_{\ge }\mathrm{<mo>\; ;\; </mo>}\hfill \\ \mathrm{0,}\hfill & \text{if}h=sl+\zeta \mathrm{,\; <mi>\; </mi>}\zeta \in A\mathrm{,}\hfill \end{array}$ (14)

where

1) The process ${\left\{y_t\right\}}_{t\in \mathbb{Z}}$ is an $\text{ARFIMA}\left(\mathrm{0,}d\mathrm{,0}\right)$ with autocovariance function of order h, $h\in {\mathbb{Z}}_{\ge }$, given by

${\gamma }_y\left(h\right)=\frac{{\left(-1\right)}^h\Gamma \left(1-2d\right)}{\Gamma \left(h-d+1\right)\Gamma \left(1-h-d\right)}$ (15)

2) The process ${\left\{z_t\right\}}_{t\in \mathbb{Z}}$ is an $\text{SARFIMA}{\left(\mathrm{0,}D\mathrm{,0}\right)}_s$ with autocovariance function of order $\nu$,$\nu \in {\mathbb{Z}}_{\ge }$, given by

${\gamma }_z\left(s\nu +\xi \right)=\{\begin{array}{ll}\frac{{\left(-1\right)}^{\nu }\Gamma \left(1-2D\right)}{\Gamma \left(\nu -D+1\right)\Gamma \left(1-\nu -D\right)}={\gamma }_x\left(\nu \right),\hfill & \text{if}\xi =0\hfill \\ 0,\hfill & \text{if}\xi \in A\hfill \end{array}$ (16)

For proof of Theorem (2.1) see [15] Theorem 2.1.

Theorem 2.2. Let ${\left\{x_t\right\}}_{t\in \mathbb{Z}}$ be a causal and invertible $\text{SARFIMA}\left(p\mathrm{,}d\mathrm{,}q\right)\left(P\mathrm{,}D\mathrm{,}Q\right)s$ process given by the expression:

$\varphi \left(B\right)\Phi \left(B^s\right){\left(1-B\right)}^d{\left(1-B^s\right)}^D{x}_t=\theta \left(B\right)\Theta \left(B^s\right){\epsilon }_t\text{for}t\in \mathbb{Z}$ (17)

with zero mean, $s\in \mathbb{N}$ as the seasonal period. Suppose $\varphi \left(B\right)\Phi \left(B^s\right)=0$ and $\theta \left(B\right)\Theta \left(B^s\right)=0$ have no common zeros. For $\left|d+D\right|<0.5$,$\left|D\right|<0.5$, then the process ${\left\{x_t\right\}}_{t\in \mathbb{Z}}$ has autocovariance function of order h, $h\in {\mathbb{Z}}_{\ge }$, given by

${\gamma }_x\left(h\right)=\{\begin{array}{ll}{\sigma }_{\epsilon }^2{\displaystyle \underset{\nu \in {\mathbb{Z}}_{\ge }}{\sum }}{\gamma }_z\left(s\nu \right){\gamma }_y\left(h-s\nu \right),\hfill & \text{if}h=sl,\mathrm{<mi>\; </mi>}l\in {\mathbb{Z}}_{\ge };\hfill \\ 0,\hfill & \text{if}h=sl+\zeta ,\mathrm{<mi>\; </mi>}\zeta \in A,\hfill \end{array}$ (18)

where

1) the process ${\left\{y_t\right\}}_{t\in \mathbb{Z}}$ is an $\text{ARFIMA}\left(p\mathrm{,}d\mathrm{,}q\right)$ with autocovariance function given by ${\gamma }_y(\cdot )$.

2) the process ${\left\{z_t\right\}}_{t\in \mathbb{Z}}$ is an $\text{SARFIMA}{\left(P\mathrm{,}D\mathrm{,}Q\right)}_s$ with autocovariance function given by ${\gamma }_z(\cdot )$.

For proof of Theorem (2.2) see [15] Theorem 2.2.

2.3. The GARCH(m, r) Model

The $\text{GARCH}\left(m\mathrm{,}r\right)$ model can be obtained from Equation (1) by letting $\mathbb{E}\left[{\epsilon }_t\mathrm{<mo>\; |\; </mo>}{F}_{t-1}\right]=0$ and the conditional variance, $\mathbb{E}\left[{\epsilon }_t^2\mathrm{<mo>\; |\; </mo>}{F}_{t-1}\right]=h_t$ where $F_{t-1}$ is the $\sigma$ field generated by the past information $\left\{{\epsilon }_{t-1}\mathrm{,}{\epsilon }_{t-2},\cdots \right\}$. Let also ${\epsilon }_t\mathrm{<mo>\; |\; </mo>}{F}_{t-1}~N\left(\mathrm{0,}{h}_t\right)$ and

${\epsilon }_t=z_t\sqrt{h_t}$ (19)

where $z_t$ is normal distributed with mean 0 and variance 1. [4] introduced the $\text{GARCH}\left(m\mathrm{,}r\right)$ model which defines the conditional variance equation as follows;

$h_t={\omega }_0+{\displaystyle \underset{i=1}{\overset{r}{\sum }}}{\alpha }_i{\epsilon }_{t-i}^2+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}{\beta }_i{h}_{t-i}$ (20)

where ${\omega }_0>0$,${\alpha }_1\mathrm{,}\cdots \mathrm{,}{\alpha }_r\mathrm{,}{\beta }_1\mathrm{,}\cdots \mathrm{,}{\beta }_m\ge 0$, r and m are positive integer. Note that the GARCH model defined by (20) can be replaced by other conditional heteroscedastic models.

2.4. The General SARFIMA-GARCH Model

Let the $\text{SARFIMA}\left(p\mathrm{,}d\mathrm{,}q\right)\left(P\mathrm{,}D\mathrm{,}Q\right)s\text{-GARCH}\left(m\mathrm{,}r\right)$ model be the discrete time series model of $\left\{y_t\right\}$ given by the following equation.

$y_t={\mu }_0+{\varphi }^{-1}\left(B\right){\Phi }^{-1}\left(B^s\right){\left(1-B\right)}^{-d}{\left(1-B^s\right)}^{-D}\theta \left(B\right)\Theta \left(B^s\right){\epsilon }_t$

${\epsilon }_t=z_t\sqrt{h_t}\mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}{\epsilon }_t\mathrm{<mo>\; |\; </mo>}{F}_{t-1}~N\left(\mathrm{0,}{h}_t\right)$

$h_t={\omega }_0+{\displaystyle \underset{i=1}{\overset{r}{\sum }}}{\alpha }_i{\epsilon }_{t-i}^2+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}{\beta }_i{h}_{t-i}$ (21)

2.5. Variance of Variance in the Standard GARCH(m, r) Model

By rearranging the conditional variance Equation (20) for a GARCH(1,1) we obtain:

$\begin{array}{c}{h}_t={\omega }_0+\left({\alpha }_1+{\beta }_1\right)h_{t-1}+{\alpha }_1\left({\epsilon }_{t-1}^2-h_{t-1}\right)\\ ={\omega }_0+\gamma h_{t-1}+{\alpha }_1{h}_{t-1}{\eta }_{t-1}\end{array}$ (22)

where $\gamma ={\alpha }_1+{\beta }_1$ and ${\eta }_t=z_t^2-1$. [16] have shown that the variance of variance is given by:

$Var\left(h_t\right)={\alpha }_1^2{h}_{t-1}\mathbb{E}\left[{\eta }_{t-1}\right]=\left({\kappa }_z-1\right){\alpha }_1^2{h}_{t-1}^2$ (23)

where ${\kappa }_z$ denotes the conditional kurtosis of $z_t$, which we assume to be finite constant. If the distribution of $z_t$ is standard normal, then ${\kappa }_z-1=2$.

[16] further rearranged the terms in Equation (22), the conditional variance equation becomes:

$\begin{array}{c}{h}_t-h_{t-1}=\phi \left(\tau -h_{t-1}\right)+{\alpha }_1\sqrt{h_{t-1}}{\eta }_{t-1}\\ ={\omega }_0+\gamma h_{t-1}+{\alpha }_1\sqrt{h_{t-1}}{\eta }_{t-1}\end{array}$ (24)

where $\phi =1-\gamma$ determines the speed at which the conditional variance reverts to its long run mean $\tau$, that is, $\mathbb{E}\left(\tau \right)={\omega }_0{\left(1-\gamma \right)}^{-1}$ and its corresponding variance becomes:

$Var\left(h_t-h_{t-1}\right)=\left({\kappa }_z-1\right){\alpha }_1^2{h}_{t-1}$ (25)

2.6. 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.

2) Level Shift (LS): represents a step function.

3) Transitory or Temporary Change (TC): represents a spike that takes a few periods to disappear.

4) Innovative Outlier (IO): represents effects that depend on the ARIMA model for the observed series.

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.

Let $y_t$ be a time series that can be described with the $\text{SARIMA}\left(p\mathrm{,}d\mathrm{,}q\right)\left(P\mathrm{,}D\mathrm{,}Q\right)s$ model:

$\varphi \left(B\right)\Phi \left(B^s\right){\left(1-B\right)}^d{\left(1-B^s\right)}^D\left(y_t-{\mu }_0\right)=\theta \left(B\right)\Theta \left(B^s\right){\epsilon }_t\mathrm{,}\text{for}t=\mathrm{1,}\cdots \mathrm{,}n$ (26)

and let $x_t$ be the contaminated series containing k outliers represented by:

$x_t=y_t+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{\mu }_j{G}_j\left(B\right)I_t^{{\tau }_j}$ (27)

where ${\mu }_j$ is the initial impact of the outlier at time $t={\tau }_j$ ; $I_t^{{\tau }_j}$ is an indicator variable such that it is 1 for $t={\tau }_j$ and 0 otherwise; and $G_j\left(B\right)$ determines the dynamics of the intervention occurring at time $t={\tau }_j$ according to the following schemes:

1) AO: $G_j\left(B\right)=1$.

2) LS: $G_j\left(B\right)={\left(1-B\right)}^{-1}$.

3) TC: $G_j\left(B\right)={\left(1-\delta B\right)}^{-1},0<\delta <1$.

4) IO: $G_j\left(B\right)={\varphi }^{-1}\left(B\right){\Phi }^{-1}\left(B^s\right){\left(1-B\right)}^{-d}{\left(1-B^s\right)}^{-D}\theta \left(B\right)\Theta \left(B^s\right)$.

[21] came up with a new intervention type, Seasonal Level Shift (SLS), that can describe a perturbation mostly related to the seasonal component. Seasonal level shifts are the interventions that affect only certain quarters or months of a year. The SLS is a special kind of level shift that occurs in $\text{SARIMA}\left(p\mathrm{,}d\mathrm{,}q\right)\left(P\mathrm{,}D\mathrm{,}Q\right)s$ at some point $t=\tau$ in time and reoccur regularly every year at same season say s and its effect carries up to subsequent seasons. The basic model with SLS suggested by [21] is given by

$x_t=y_t+\mu G\left(B^s\right)I_t^{\tau }\mathrm{<mi>\; </mi>}\text{where}G\left(B^s\right)={\left(1-B^s\right)}^{-1}$ (28)

However $G\left(B^s\right)={\left(1-B^s\right)}^{-1}$ causes an impact on trend which can only be removed by defining dynamic weights as

$G\left(B^s\right)=\frac{1}{1-B^s}-\frac{1}{s\left(1-B\right)}$ (29)

The dynamic impact was normalized as suggested by [22]:

$\begin{array}{c}G\left(B^s\right)=\frac{s}{s-1}\left[{\left(1-B^s\right)}^{-1}-\frac{1}{s}{\left(1-B\right)}^{-1}\right]\\ =\frac{s}{s-1}\left[\left(1+B^s+B^{2s}+B^{3s}+\cdots \right)-\frac{1}{s}\left(1+B+B^2+B^3+\cdots \right)\right]\\ =\left[1-\frac{s}{s-1}B+\frac{s}{s-1}{B}^s\right]{\left(1-B^s\right)}^{-1}{\left(1-B\right)}^{-1}\end{array}$ (30)

Thus model (28) becomes

$\begin{array}{c}{x}_t=y_t+\mu G\left(B^s\right)I_t^{\tau }\\ =y_t+\mu \left[1-\frac{s}{s-1}B+\frac{s}{s-1}{B}^s\right]{\left(1-B^s\right)}^{-1}{\left(1-B\right)}^{-1}{I}_t^{\tau }\end{array}$ (31)

Equation (31) is the observed series indicating series that the occurrence of SLS affects the time series for several seasons at same quarters or month with magnitude of outlier $\mu$. The outlier is introduced in the model by generating a variable:

$G\left(B^s\right)I_t^{\tau }=\{\begin{array}{ll}0\hfill & \text{for}t<\tau \hfill \\ 1\hfill & \text{at}t=\tau +s_j\hfill \\ \frac{-1}{s-1}\hfill & \text{at}t=\tau +s_j+1,\mathrm{<mi>\; </mi>}\tau +s_j+2,\mathrm{<mi>\; </mi>}\cdots ,\mathrm{<mi>\; </mi>}\tau +s_j+s-1\hfill \\ \hfill & j=1,2,\cdots \hfill \end{array}$ (32)

However in this research, for a SLS intervention at period $t=\tau$, we shall stick to the simple specification model (28).

3. 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.

3.1. The SARFIMA Model with Level Shift

The $\text{SARFIMA}\left(p\mathrm{,}d\mathrm{,}q\right)\left(P\mathrm{,}D\mathrm{,}Q\right)s$ model is written as

$\varphi \left(B\right)\Phi \left(B^s\right){\left(1-B\right)}^d{\left(1-B^s\right)}^D\left(y_t-{\mu }_0\right)=\theta \left(B\right)\Theta \left(B^s\right){\epsilon }_t,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\text{for}t=1,\cdots ,n$ (33)

where $y_t$ is the time series at time t, ${\mu }_0$ is the unconditional mean of the process. We assume the noise process ${\epsilon }_t$ to be Gaussian, with expectation zero and variance ${\sigma }_{\epsilon }^2$.

To allow for a mean level shift, after time $t=\tau$ 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 $\text{LS-SARFIMA}\left(p\mathrm{,}d\mathrm{,}q\right)\left(P\mathrm{,}D\mathrm{,}Q\right)s$

$x_t=y_t+{\mu }_{11}{\left(1-B\right)}^{-1}{I}_t^{\tau }$ (34)

$=y_t+{\mu }_{11}{G}_{11}\left(B\right)I_t^{\tau }$ (35)

where $x_t$ denotes the observed contaminated series; $y_t$ follows the SARFIMA process; $I_t^{\tau }$ is an indicator variable taking values 1 for $t=\tau$, and 0 otherwise. The parameter ${\mu }_{11}$ indicates the size of the mean level shift at time $t=\tau$ ; $G_{11}\left(B\right)$ determines the dynamics of the intervention occurring at time $t=\tau$. The mean level shift is an abrupt but permanent shift by ${\mu }_{11}$ in the series caused by an intervention.

The extension of (34) to k level shifts is straightforward. We define ${\mu }_{1j}$ as the ${\tau }_j^{th}$ shift in level, compared to the previous level, where $j=\mathrm{1,}\cdots \mathrm{,}k$. When we allow k level changes at pre-specified time $t={\tau }_j$, we can extend Equation (34) and Equation (35) to

$x_t=y_t+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{\mu }_{1j}{\left(1-B\right)}^{-1}{I}_t^{{\tau }_j}$ (36)

$=y_t+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{\mu }_{1j}{G}_{1j}\left(B\right)I_t^{{\tau }_j}$ (37)

The component ${\displaystyle {\sum }_{j=1}^k}{\mu }_{1j}{\left(1-B\right)}^{-1}{I}_t^{{\tau }_j}$ allows the intercept of the SARFIMA model to fluctuate over time between ${\mu }_0$ and ${\mu }_0+{\displaystyle {\sum }_{j=1}^k}{\mu }_{1j}$.

3.2. 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(b).

A Seasonal Level Shift intervention at period $t=\tau$ would affect the series $y_t$ and is represented by model (38):

$x_t={\mu }_{21}{G}_{21}\left(B^s\right)I_t^{\tau }+y_t\text{where}{G}_{21}\left(B^s\right)={\left(1-B^s\right)}^{-1}$ (38)

Assuming the observed series contains k seasonal level shifts at time $t={\tau }_j$ for $j=1,\cdots ,k$, their combined effect can be expressed in general form as:

$x_t={\displaystyle \underset{j=1}{\overset{k}{\sum }}}{\mu }_{2j}{G}_{2j}\left(B^s\right)I_t^{{\tau }_j}+y_t\text{where}{G}_{2j}\left(B^s\right)={\left(1-B^s\right)}^{-1}$ (39)

The component ${\displaystyle {\sum }_{j=1}^k}{\mu }_{2j}{\left(1-B^s\right)}^{-1}{I}_t^{{\tau }_j}$ allows the intercept of the SARFIMA model to fluctuate over time between ${\mu }_0$ and ${\mu }_0+{\displaystyle {\sum }_{j=1}^k}{\mu }_{2j}$.

3.3. 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;

$h_t={\omega }_0+{\displaystyle \underset{i=1}{\overset{r}{\sum }}}{\alpha }_i{\epsilon }_{t-i}^2+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}{\beta }_i{h}_{t-i}$ (40)

To allow for a volatility level shift, denoted ${\omega }_{11}$, after time $t=\tau$ of the data, we write $h_t$ as the sum of an unobserved GARCH process and the term of the volatility level shift which we will denote as LS-GARCH(m, r).

$h_t={\omega }_0+{\displaystyle \underset{i=1}{\overset{r}{\sum }}}{\alpha }_i{\epsilon }_{t-i}^2+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}{\beta }_i{h}_{t-i}+{\omega }_{11}{\left(1-B\right)}^{-1}{I}_t^{\tau }$ (41)

$={\omega }_0+{\displaystyle \underset{i=1}{\overset{r}{\sum }}}{\alpha }_i{\epsilon }_{t-i}^2+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}{\beta }_i{h}_{t-i}+{\omega }_{11}{G}_{11}\left(B\right)I_t^{\tau }$ (42)

where $I_t^{\tau }$ is an indicator variable taking values 1 for $t=\tau$, and 0 otherwise. The parameter ${\omega }_{11}$ indicates the size of the volatility level shift at time $t=\tau$.

The extension of Equation (41) and Equation (42) to k volatility level shifts is straightforward. We define ${\omega }_{1j}$ as the ${\tau }_j^{th}$ shift in volatility level, compared to the previous level, where $j=1,\cdots ,k$. When we allow k volatility level changes at pre-specified time $t={\tau }_j$, we can extend (41) and (42) to

$h_t={\omega }_0+{\displaystyle \underset{i=1}{\overset{r}{\sum }}}{\alpha }_i{\epsilon }_{t-i}^2+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}{\beta }_i{h}_{t-i}+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{\omega }_{1j}{\left(1-B\right)}^{-1}{I}_t^{{\tau }_j}$ (43)

$={\omega }_0+{\displaystyle \underset{i=1}{\overset{r}{\sum }}}{\alpha }_i{\epsilon }_{t-i}^2+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}{\beta }_i{h}_{t-i}+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{\omega }_{1j}{G}_{1j}\left(B\right)I_t^{{\tau }_j}$ (44)

The component ${\displaystyle {\sum }_{j=1}^k}{\omega }_{1j}{\left(1-B\right)}^{-1}{I}_t^{{\tau }_j}$ governs the level shift movement of GARCH model intercept, that is baseline volatility, over time between ${\omega }_0$ and ${\omega }_0+{\displaystyle {\sum }_{j=1}^k}{\omega }_{1j}$.

3.4. The GARCH(r, m) Model with Seasonal Level Shift

Seasonal Level Shift in volatility denoted as SLS-GARCH type intervention display seasonal features in its pattern. The Seasonal Level Shift (SLS) intervention has an effect on the trend given by the step function and an effect on the seasonal component.

A Seasonal Level Shift intervention in GARCH model at period $t=\tau$ would affect the series $h_t$ and is represented by model (45) and (46):

$h_t={\omega }_0+{\displaystyle \underset{i=1}{\overset{r}{\sum }}}{\alpha }_i{\epsilon }_{t-i}^2+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}{\beta }_i{h}_{t-i}+{\omega }_{21}{\left(1-B^s\right)}^{-1}{I}_t^{\tau }$ (45)

$={\omega }_0+{\displaystyle \underset{i=1}{\overset{r}{\sum }}}{\alpha }_i{\epsilon }_{t-i}^2+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}{\beta }_i{h}_{t-i}+{\omega }_{21}{G}_{21}\left(B^s\right)I_t^{\tau }$ (46)

Assuming the observed series contains k seasonal level shifts at time $t={\tau }_j$ for $j=1,\cdots ,k$, their combined effect can be expressed in general form as:

$h_t={\omega }_0+{\displaystyle \underset{i=1}{\overset{r}{\sum }}}{\alpha }_i{\epsilon }_{t-i}^2+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}{\beta }_i{h}_{t-i}+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{\omega }_{2j}{\left(1-B^s\right)}^{-1}{I}_t^{{\tau }_j}$ (47)

$={\omega }_0+{\displaystyle \underset{i=1}{\overset{r}{\sum }}}{\alpha }_i{\epsilon }_{t-i}^2+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}{\beta }_i{h}_{t-i}+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{\omega }_{2j}{G}_{2j}\left(B^s\right)I_t^{{\tau }_j}$ (48)

The component ${\displaystyle {\sum }_{j=1}^k}{\omega }_{2j}{\left(1-B^s\right)}^{-1}{I}_t^{{\tau }_j}$ allows the intercept of the GARCH model to fluctuate over time between ${\omega }_0$ and ${\omega }_0+{\displaystyle {\sum }_{j=1}^k}{\omega }_{2j}$.

3.5. The General SARFIMA-GARCH Model with Level Shift

Extension of $y_t$ the $\text{SARFIMA}\left(p\mathrm{,}d\mathrm{,}q\right)\left(P\mathrm{,}D\mathrm{,}Q\right)s\text{-GARCH}\left(m\mathrm{,}r\right)$ process to the case with standard level shift is given by the following equation which we will denote as LS-SARFIMA-LS-GARCH

$x_t=y_t+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{\mu }_{1j}{\left(1-B\right)}^{-1}{I}_t^{{\tau }_j}$

${\epsilon }_t=z_t\sqrt{h_t},\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}{\epsilon }_t|F_{t-1}~N\left(0,h_t\right)$

$h_t={\omega }_0+{\displaystyle \underset{i=1}{\overset{r}{\sum }}}{\alpha }_i{\epsilon }_{t-i}^2+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}{\beta }_i{h}_{t-i}+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{\omega }_{1j}{\left(1-B\right)}^{-1}{I}_t^{{\tau }_j}$ (49)

Similarly, extension of $y_t$ the $\text{SARFIMA}\left(p\mathrm{,}d\mathrm{,}q\right)\left(P\mathrm{,}D\mathrm{,}Q\right)s\text{-GARCH}\left(m\mathrm{,}r\right)$ process to the case with seasonal level shift is given by the following equation which we will denote as SLS-SARFIMA-SLS-GARCH

$x_t=y_t+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{\mu }_{2j}{\left(1-B^s\right)}^{-1}{I}_t^{{\tau }_j}$

${\epsilon }_t=z_t\sqrt{h_t}\mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}{\epsilon }_t\mathrm{<mo>\; |\; </mo>}{F}_{t-1}~N\left(\mathrm{0,}{h}_t\right)$

$h_t={\omega }_0+{\displaystyle \underset{i=1}{\overset{r}{\sum }}}{\alpha }_i{\epsilon }_{t-i}^2+{\displaystyle \underset{i=1}{\overset{m}{\sum }}}{\beta }_i{h}_{t-i}+{\displaystyle \underset{j=1}{\overset{k}{\sum }}}{\omega }_{2j}{\left(1-B^s\right)}^{-1}{I}_t^{{\tau }_j}$ (50)

4. Estimation of SLS-SARFIMA-SLS-GARCH Model Parameters

4.1. Estimation of SLS-SARFIMA Model Parameters

The first step of estimation consists in estimating the $\text{SARFIMA}\left(p\mathrm{,}d\mathrm{,}q\right)\left(P\mathrm{,}D\mathrm{,}Q\right)s$ assuming that the conditional variance is constant over time. By rearranging Equation (38) for one mean seasonal level shift we have:

$\varphi \left(B\right)\Phi \left(B^s\right){\left(1-B\right)}^d{\left(1-B^s\right)}^D{y}_t={\mu }_0+\theta \left(B\right)\Theta \left(B^s\right){\epsilon }_t+{\mu }_{21}{\left(1-B^s\right)}^{-1}{I}_t\mathrm{.}$ (51)

Therefore the null hypothesis of unconditional mean constancy becomes: $H_0:{\mu }_{21}=0$. Let ${\psi }_1={\left(d\mathrm{,}D\mathrm{,}{\mu }_0\mathrm{,}{\mu }_{21}\mathrm{,}{\varphi }^{\prime }\mathrm{,}{\Phi }^{\prime }\mathrm{,}{\theta }^{\prime }\mathrm{,}{\Theta }^{\prime }\mathrm{,}{\sigma }_{\epsilon }^2\right)}^{\prime }$ be the approximate likelihood estimator (MLE) ${\hat{\psi }}_1$ of ${\psi }_1$ that maximizes the conditional log-likelihood:

$l_t\left({\psi }_1\right)=-\frac{1}{2}\ln 2\pi -\frac{1}{2}\ln {\sigma }_{\epsilon }^2-\frac{1}{2}\frac{{\epsilon }_t^2}{{\sigma }_{\epsilon }^2}.$ (52)

The partial derivatives evaluated under $H_0$ are given by:

${\frac{\partial l_t}{\partial {\psi }_1}|}_{H_0}={-\frac{{\hat{\epsilon }}_t}{{\sigma }_{\epsilon }^2}\frac{\partial {\hat{\epsilon }}_t}{\partial {\psi }_1}|}_{H_0}$ (53)

· $\begin{array}{l}{\frac{\partial {\epsilon }_t}{\partial d}|}_{H_0}={\left(1-B\right)}^d{\left(1-B^s\right)}^D\hat{\varphi }\left(B\right)\hat{\Phi }\left(B^s\right){\displaystyle {\sum }_j^{t-1}}\frac{y_{t-j}}{j}+{\displaystyle {\sum }_{j=1}^Q}{\hat{\Theta }}_j\frac{\partial {\hat{\epsilon }}_{t-js}}{\partial d}\\ +{\displaystyle {\sum }_{j=0}^Q}{\displaystyle {\sum }_{i=0}^q}{\hat{\Theta }}_j{\hat{\theta }}_i\frac{\partial {\hat{\epsilon }}_{t-i-js}}{\partial d}\end{array}$

· $\begin{array}{l}{\frac{\partial {\epsilon }_t}{\partial D}|}_{H_0}={\left(1-B\right)}^d{\left(1-B^s\right)}^D\hat{\varphi }\left(B\right)\hat{\Phi }\left(B^s\right){\displaystyle {\sum }_j^{t-1}}\frac{y_{t-js}}{js}+{\displaystyle {\sum }_{j=1}^Q}{\hat{\Theta }}_j\frac{\partial {\hat{\epsilon }}_{t-js}}{\partial D}\\ +{\displaystyle {\sum }_{j=0}^Q}{\displaystyle {\sum }_{i=0}^q}{\hat{\Theta }}_j{\hat{\theta }}_i\frac{\partial {\hat{\epsilon }}_{t-i-js}}{\partial D}\end{array}$

· ${\frac{\partial {\epsilon }_t}{\partial {\mu }_0}|}_{H_0}=-1+{\displaystyle {\sum }_{j=1}^Q}{\hat{\Theta }}_j\frac{\partial {\hat{\epsilon }}_{t-js}}{\partial {\mu }_0}+{\displaystyle {\sum }_{j=0}^Q}{\displaystyle {\sum }_{i=0}^q}{\hat{\Theta }}_j{\hat{\theta }}_i\frac{\partial {\hat{\epsilon }}_{t-i-js}}{\partial {\mu }_0}$

· ${\frac{\partial {\epsilon }_t}{\partial {\mu }_{21}}|}_{H_0}=-{\left(1-B^s\right)}^{-1}{I}_t+{\displaystyle {\sum }_{j=1}^Q}{\hat{\Theta }}_j\frac{\partial {\hat{\epsilon }}_{t-js}}{\partial {\mu }_{21}}+{\displaystyle {\sum }_{j=0}^Q}{\displaystyle {\sum }_{i=0}^q}{\hat{\Theta }}_j{\hat{\theta }}_i\frac{\partial {\hat{\epsilon }}_{t-i-js}}{\partial {\mu }_{21}}$

· $\begin{array}{l}{\frac{\partial {\epsilon }_t}{\partial \varphi }|}_{H_0}=\hat{\Phi }\left(B^s\right){\left(1-B\right)}^d{\left(1-B^s\right)}^D\left(y_{t-1},\cdots ,y_{t-p}\right)+{\displaystyle {\sum }_{j=1}^Q}{\hat{\Theta }}_j\frac{\partial {\hat{\epsilon }}_{t-js}}{\partial \varphi }\\ +{\displaystyle {\sum }_{j=1}^Q}{\displaystyle {\sum }_{i=0}^q}{\hat{\Theta }}_j{\hat{\theta }}_i\frac{\partial {\hat{\epsilon }}_{t-i-js}}{\partial \varphi }\end{array}$

· $\begin{array}{l}{\frac{\partial {\epsilon }_t}{\partial \Phi }|}_{H_0}=\hat{\varphi }\left(B\right){\left(1-B\right)}^d{\left(1-B^s\right)}^D\left(y_{t-s},\cdots ,y_{t-ps}\right)+{\displaystyle {\sum }_{j=1}^Q}{\hat{\Theta }}_j\frac{\partial {\hat{\epsilon }}_{t-js}}{\partial \Phi }\\ +{\displaystyle {\sum }_{j=0}^Q}{\displaystyle {\sum }_{i=0}^q}{\hat{\Theta }}_j{\hat{\theta }}_i\frac{\partial {\hat{\epsilon }}_{t-i-js}}{\partial \Phi }\end{array}$

· $\begin{array}{l}{\frac{\partial {\epsilon }_t}{\partial \theta }|}_{H_0}=\left({\displaystyle {\sum }_{j=0}^Q}{\hat{\Theta }}_j{\epsilon }_{t-1-js},\cdots ,{\displaystyle {\sum }_{j=0}^Q}{\hat{\Theta }}_j{\epsilon }_{t-q-js}\right)+{\displaystyle {\sum }_{j=1}^Q}{\hat{\Theta }}_j\frac{\partial {\hat{\epsilon }}_{t-js}}{\partial \theta }\\ +{\displaystyle {\sum }_{j=1}^Q}{\displaystyle {\sum }_{i=0}^q}{\hat{\Theta }}_j{\hat{\theta }}_i\frac{\partial {\hat{\epsilon }}_{t-i-js}}{\partial \theta }\end{array}$

· $\begin{array}{l}{\frac{\partial {\epsilon }_t}{\partial \Theta }|}_{H_0}=\left({\epsilon }_{t-s},{\epsilon }_{t-2s},\cdots ,{\epsilon }_{t-Qs}\right)+\left({\displaystyle {\sum }_{i=0}^q}{\hat{\theta }}_i{\epsilon }_{t-i-s},\cdots ,{\displaystyle {\sum }_{i=0}^q}{\hat{\theta }}_i{\epsilon }_{t-i-Qs}\right)\\ +{\displaystyle {\sum }_{j=1}^Q}{\hat{\Theta }}_j\frac{\partial {\hat{\epsilon }}_{t-js}}{\partial \Theta }+{\displaystyle {\sum }_{j=0}^Q}{\displaystyle {\sum }_{i=0}^q}{\hat{\Theta }}_j{\hat{\theta }}_i\frac{\partial {\hat{\epsilon }}_{t-i-js}}{\partial \Theta }\end{array}$

· ${\frac{\partial {\epsilon }_t}{\partial {\sigma }_{\epsilon }^2}|}_{H_0}=-\frac{1}{2{\sigma }_{\epsilon }^2}+\frac{{\hat{\epsilon }}_t^2}{2{\sigma }_{\epsilon }^4}$

Under $H_0$, the LM-type statistics is asymptotically distributed as ${\chi }^2$ with one degree of freedom:

$L{M}_1={\frac{1}{2}{\displaystyle \underset{t=1}{\overset{n}{\sum }}}\frac{\partial l_t}{\partial {\psi }_1}|}_{H_0}{\left({\displaystyle \underset{t=1}{\overset{n}{\sum }}}\left[{\frac{\partial {\epsilon }_t}{\partial {\psi }_1}|}_{H_0}\right]{\left[{\frac{\partial {\epsilon }_t}{\partial {\psi }_1}|}_{H_0}\right]}^{\prime }\right)}^{-1}{{\displaystyle \underset{t=1}{\overset{n}{\sum }}}\frac{\partial l_t}{\partial {\psi }_1}|}_{H_0}$ (54)

4.2. Estimation of SLS-GARCH Parameters

Once the SLS-SARFIMA model is estimated and the residuals ${\epsilon }_t$ 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:

$h_t={\omega }_0+\alpha \left(B\right){\epsilon }_t+\beta \left(B\right)h_t+{\omega }_{21}{\left(1-B^s\right)}^{-1}{I}_t$ (55)

Therefore the null hypothesis of the unconditional variance constancy becomes: $H_0:{\omega }_{21}=0$. Let ${\psi }_2={\left({\omega }_0\mathrm{,}{\omega }_{21}\mathrm{,}{\alpha }^{\prime }\mathrm{,}{\beta }^{\prime }\right)}^{\prime }$ be the vector of the SLS-GARCH model parameters and the quasi-likelihood function is given by:

$l_t\left({\psi }_2\right)=-\frac{1}{2}ln2\pi -\frac{1}{2}ln{h}_t-\frac{1}{2}\frac{{\epsilon }_t^2}{h_t}\mathrm{.}$ (56)