Time series analysis has turned attention to the studies with long memory or long-range 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. [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 { x t } t ∈ ℤ be a stochastic process, then { x t } t ∈ ℤ is a zero mean ARFIMA ( p , d , q ) process given by the expression

ϕ ( B ) ( 1 − B ) d x t = θ ( B ) ε t ,     for   t ∈ ℤ (1)

where { ε t } t ∈ ℤ is a white noise process with zero mean and variance σ ε 2 = E ( ε t 2 ) , B is the backward-shift operator, that is, B k x t = x t − k and ( 1 − B ) d is the non seasonal difference, ϕ ( ⋅ ) and θ ( ⋅ ) are the non-seasonal polynomials of degrees p and q, respectively, defined by:

ϕ ( B ) = ∑ i = 0 p ( − ϕ i ) B i ,     θ ( B ) = ∑ j = 0 q ( − θ j ) B j , (2)

where ϕ i , 1 ≤ i ≤ p , and θ j , 1 ≤ j ≤ q are constants and ϕ 0 = − 1 = θ 0 .

The difference operator ( 1 − B ) d is defined by means of the binomial expansion:

( 1 − B ) d = ∑ j = 0 ∞ ( d j ) ( − B ) j , (3)

where;

( d j ) = Γ ( 1 + d ) Γ ( 1 + j ) Γ ( 1 + d − j ) and Γ ( ⋅ ) 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 ≥ 0.5 and stationary but non invertible when d ≤ − 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 .

The seasonal autoregressive fractionally integrated moving average process, denoted hereafter by SARFIMA ( p , d , q ) × ( P , D , Q ) s , is an extension of the long range dependence in the mean ARFIMA ( p , d , q ) process, proposed by [1] and [2]. The SARFIMA ( p , d , q ) × ( P , D , Q ) 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 { x t } t ∈ ℤ be a stochastic process, then { x t } t ∈ ℤ is a zero mean SARFIMA ( p , d , q ) × ( P , D , Q ) s process given by the expression

ϕ ( B ) Φ ( B s ) ( 1 − B ) d ( 1 − B s ) D x t = θ ( B ) Θ ( B s ) ε t ,     for   t ∈ ℤ (4)

where s ∈ ℕ is the seasonal period, B is the backward-shift operator, that is, B s k x t = x t − s k , ( 1 − B s ) D is the seasonal difference operator, Φ ( ⋅ ) and Θ ( ⋅ ) are the polynomials of degrees P and Q, respectively, defined by:

Φ ( B s ) = ∑ i = 0 P ( − Φ i ) B s i ,     Θ ( B s ) = ∑ j = 0 Q ( − Θ j ) B s j (5)

where Φ i , 1 ≤ i ≤ P and Θ j , 1 ≤ j ≤ Q are constants and Φ 0 = − 1 = Θ 0 .

The seasonal difference operator ( 1 − B s ) D , with seasonality s ∈ ℕ , for all D > − 1 , is defined by means of the binomial expansion;

( 1 − B s ) D = ∑ j = 0 ∞ ( D j ) ( − B s ) j , (6)

where;

( D j ) = Γ ( 1 + D ) Γ ( 1 + j ) Γ ( 1 + D − j ) (7)

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

ϕ ( B ) Φ ( B s ) ∇ d x t = θ ( B ) Θ ( B s ) ε t ,     for   t ∈ ℤ (8)

In Equation (8), the operator ∇ d is defined by

∇ d = ( 1 − B ) d ( 1 − B s ) D (9)

where d = ( d , D ) ∈ 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:

( 1 − B k ) l = ∑ j = 0 ∞ ( l j ) ( − B k ) j , k = 1 , s     and     l = d , D (10)

where;

( l j ) = Γ ( 1 + l ) Γ ( 1 + j ) Γ ( 1 + l − j ) (11)

Suppose in Equation (8), ϕ ( B ) Φ ( B s ) = 0 and θ ( B ) Θ ( B s ) = 0 have no common zeros. Let also | d + D | < 0.5 and | D | < 0.5 , the process { x t } t ∈ ℤ has spectral density function is given by;

f x ( λ ) = σ ε 2 2 π | θ ( e − i λ ) | 2 | Θ ( e − i λ s ) | 2 | ϕ ( e − i λ ) | 2 | Φ ( e − i λ s ) | 2 [ 2 sin ( λ 2 ) ] − 2 d [ 2 sin ( λ s 2 ) ] − 2 D , (12)

where 0 < λ ≤ π .

Note that according to [15]:

1) The processs { x t } t ∈ ℤ is stationary if d + D < 0.5 , D < 0.5 and ϕ ( B ) Φ ( B s ) ≠ 0 , for | B | ≤ 1 .

2) The stationary process { x t } t ∈ ℤ has a long memory property if 0 < d + D < 0.5 , 0 < D < 0.5 and ϕ ( B ) Φ ( B s ) ≠ 0 , for | B | ≤ 1 .

3) The stationary process { x t } t ∈ ℤ has an intermediate property if − 0.5 < d + D < 0 , − 0.5 < D < 0 and ϕ ( B ) Φ ( B s ) ≠ 0 , for | B | ≤ 1 .

For convenience, we introduce the notation ℤ ≥ = { k ∈ ℤ | k ≥ 0 } , ℤ ≤ = { k ∈ ℤ | k ≤ 0 } and A = { 1, ⋯ , s − 1 } ⊂ ℕ .

Theorem 2.1. Let { x t } t ∈ ℤ be the SARFIMA ( 0, d ,0 ) ( 0, D ,0 ) s process given by expression:

( 1 − B ) d ( 1 − B s ) D x t = ε t (13)

with zero mean, s ∈ ℕ as the seasonal period. Then the process { x t } t ∈ ℤ has autocovariance function of order h, h ∈ ℤ ≥ , given by

γ x ( h ) = { σ ε 2 ∑ ν ∈ ℤ ≥     γ z ( s ν ) γ y ( h − s ν ) , if   h = s l , l ∈ ℤ ≥ ; 0, if   h = s l + ζ , ζ ∈ A , (14)

where

1) The process { y t } t ∈ ℤ is an ARFIMA ( 0, d ,0 ) with autocovariance function of order h, h ∈ ℤ ≥ , given by

γ y ( h ) = ( − 1 ) h Γ ( 1 − 2 d ) Γ ( h − d + 1 ) Γ ( 1 − h − d ) (15)

2) The process { z t } t ∈ ℤ is an SARFIMA ( 0, D ,0 ) s with autocovariance function of order ν , ν ∈ ℤ ≥ , given by

γ z ( s ν + ξ ) = { ( − 1 ) ν Γ ( 1 − 2 D ) Γ ( ν − D + 1 ) Γ ( 1 − ν − D ) = γ x ( ν ) , if   ξ = 0 0 , if   ξ ∈ A (16)

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

Theorem 2.2. Let { x t } t ∈ ℤ be a causal and invertible SARFIMA ( p , d , q ) ( P , D , Q ) s process given by the expression:

ϕ ( B ) Φ ( B s ) ( 1 − B ) d ( 1 − B s ) D x t = θ ( B ) Θ ( B s ) ε t     for   t ∈ ℤ (17)

with zero mean, s ∈ ℕ as the seasonal period. Suppose ϕ ( B ) Φ ( B s ) = 0 and θ ( B ) Θ ( B s ) = 0 have no common zeros. For | d + D | < 0.5 , | D | < 0.5 , then the process { x t } t ∈ ℤ has autocovariance function of order h, h ∈ ℤ ≥ , given by

γ x ( h ) = { σ ε 2 ∑ ν ∈ ℤ ≥ γ z ( s ν ) γ y ( h − s ν ) , if   h = s l , l ∈ ℤ ≥ ; 0 , if   h = s l + ζ , ζ ∈ A , (18)

where

1) the process { y t } t ∈ ℤ is an ARFIMA ( p , d , q ) with autocovariance function given by γ y ( ⋅ ) .

2) the process { z t } t ∈ ℤ is an SARFIMA ( P , D , Q ) s with autocovariance function given by γ z ( ⋅ ) .

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

The GARCH ( m , r ) model can be obtained from Equation (1) by letting E [ ε t | F t − 1 ] = 0 and the conditional variance, E [ ε t 2 | F t − 1 ] = h t where F t − 1 is the σ field generated by the past information { ε t − 1 , ε t − 2 , ⋯ } . Let also ε t | F t − 1 ~ N ( 0, h t ) and

ε t = z t h t (19)

where z t is normal distributed with mean 0 and variance 1. [4] introduced the GARCH ( m , r ) model which defines the conditional variance equation as follows;

h t = ω 0 + ∑ i = 1 r     α i ε t − i 2 + ∑ i = 1 m     β i h t − i (20)

where ω 0 > 0 , α 1 , ⋯ , α r , β 1 , ⋯ , β m ≥ 0 , r and m are positive integer. Note that the GARCH model defined by (20) can be replaced by other conditional heteroscedastic models.

Let the SARFIMA ( p , d , q ) ( P , D , Q ) s -GARCH ( m , r ) model be the discrete time series model of { y t } given by the following equation.

y t = μ 0 + ϕ − 1 ( B ) Φ − 1 ( B s ) ( 1 − B ) − d ( 1 − B s ) − D θ ( B ) Θ ( B s ) ε t

ε t = z t h t , ε t | F t − 1 ~ N ( 0, h t )

h t = ω 0 + ∑ i = 1 r     α i ε t − i 2 + ∑ i = 1 m     β i h t − i (21)

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

h t = ω 0 + ( α 1 + β 1 ) h t − 1 + α 1 ( ε t − 1 2 − h t − 1 ) = ω 0 + γ h t − 1 + α 1 h t − 1 η t − 1 (22)

where γ = α 1 + β 1 and η t = z t 2 − 1 . [16] have shown that the variance of variance is given by:

V a r ( h t ) = α 1 2 h t − 1 E [ η t − 1 ] = ( κ z − 1 ) α 1 2 h t − 1 2 (23)

where κ 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 κ z − 1 = 2 .

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

h t − h t − 1 = φ ( τ − h t − 1 ) + α 1 h t − 1 η t − 1 = ω 0 + γ h t − 1 + α 1 h t − 1 η t − 1 (24)

where φ = 1 − γ determines the speed at which the conditional variance reverts to its long run mean τ , that is, E ( τ ) = ω 0 ( 1 − γ ) − 1 and its corresponding variance becomes:

V a r ( h t − h t − 1 ) = ( κ z − 1 ) α 1 2 h t − 1 (25)

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 SARIMA ( p , d , q ) ( P , D , Q ) s model:

ϕ ( B ) Φ ( B s ) ( 1 − B ) d ( 1 − B s ) D ( y t − μ 0 ) = θ ( B ) Θ ( B s ) ε t ,     for   t = 1, ⋯ , n (26)

and let x t be the contaminated series containing k outliers represented by:

x t = y t + ∑ j = 1 k     μ j G j ( B ) I t τ j (27)

where μ j is the initial impact of the outlier at time t = τ j ; I t τ j is an indicator variable such that it is 1 for t = τ j and 0 otherwise; and G j ( B ) determines the dynamics of the intervention occurring at time t = τ j according to the following schemes:

1) AO: G j ( B ) = 1 .

2) LS: G j ( B ) = ( 1 − B ) − 1 .

3) TC: G j ( B ) = ( 1 − δ B ) − 1 ,   0 < δ < 1 .

4) IO: G j ( B ) = ϕ − 1 ( B ) Φ − 1 ( B s ) ( 1 − B ) − d ( 1 − B s ) − D θ ( B ) Θ ( B s ) .

[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 SARIMA ( p , d , q ) ( P , D , Q ) s at some point t = τ 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 + μ G ( B s ) I t τ     where     G ( B s ) = ( 1 − B s ) − 1 (28)

However G ( B s ) = ( 1 − B s ) − 1 causes an impact on trend which can only be removed by defining dynamic weights as

G ( B s ) = 1 1 − B s − 1 s ( 1 − B ) (29)

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

G ( B s ) = s s − 1 [ ( 1 − B s ) − 1 − 1 s ( 1 − B ) − 1 ] = s s − 1 [ ( 1 + B s + B 2 s + B 3 s + ⋯ ) − 1 s ( 1 + B + B 2 + B 3 + ⋯ ) ] = [ 1 − s s − 1 B + s s − 1 B s ] ( 1 − B s ) − 1 ( 1 − B ) − 1 (30)

Thus model (28) becomes

x t = y t + μ G ( B s ) I t τ = y t + μ [ 1 − s s − 1 B + s s − 1 B s ] ( 1 − B s ) − 1 ( 1 − B ) − 1 I t τ (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 μ . The outlier is introduced in the model by generating a variable:

G ( B s ) I t τ = { 0 for     t < τ 1 at     t = τ + s j − 1 s − 1 at     t = τ + s j + 1 , τ + s j + 2 , ⋯ , τ + s j + s − 1         j = 1 , 2 , ⋯ (32)

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

The SARFIMA ( p , d , q ) ( P , D , Q ) s model is written as

ϕ ( B ) Φ ( B s ) ( 1 − B ) d ( 1 − B s ) D ( y t − μ 0 ) = θ ( B ) Θ ( B s ) ε t ,   for     t = 1 , ⋯ , n (33)

where y t is the time series at time t, μ 0 is the unconditional mean of the process. We assume the noise process ε t to be Gaussian, with expectation zero and variance σ ε 2 .

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 LS-SARFIMA ( p , d , q ) ( P , D , Q ) s

x t = y t + μ 11 ( 1 − B ) − 1 I t τ (34)

= y t + μ 11 G 11 ( B ) I t τ (35)

where x t denotes the observed contaminated series; y t follows the SARFIMA process; I t τ is an indicator variable taking values 1 for t = τ , and 0 otherwise. The parameter μ 11 indicates the size of the mean level shift at time t = τ ; G 11 ( B ) determines the dynamics of the intervention occurring at time t = τ . The mean level shift is an abrupt but permanent shift by μ 11 in the series caused by an intervention.

The extension of (34) to k level shifts is straightforward. We define μ 1 j as the τ j t h shift in level, compared to the previous level, where j = 1, ⋯ , k . When we allow k level changes at pre-specified time t = τ j , we can extend Equation (34) and Equation (35) to

x t = y t + ∑ j = 1 k     μ 1 j ( 1 − B ) − 1 I t τ j (36)

= y t + ∑ j = 1 k     μ 1 j G 1 j ( B ) I t τ j (37)

The component ∑ j = 1 k     μ 1 j ( 1 − B ) − 1 I t τ j allows the intercept of the SARFIMA model to fluctuate over time between μ 0 and μ 0 + ∑ j = 1 k     μ 1 j .

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 = τ would affect the series y t and is represented by model (38):

x t = μ 21 G 21 ( B s ) I t τ + y t       where     G 21 ( B s ) = ( 1 − B s ) − 1 (38)

Assuming the observed series contains k seasonal level shifts at time t = τ j for j = 1 , ⋯ , k , their combined effect can be expressed in general form as:

x t = ∑ j = 1 k     μ 2 j G 2 j ( B s ) I t τ j + y t       where     G 2 j ( B s ) = ( 1 − B s ) − 1 (39)

The component ∑ j = 1 k     μ 2 j ( 1 − B s ) − 1 I t τ j allows the intercept of the SARFIMA model to fluctuate over time between μ 0 and μ 0 + ∑ j = 1 k     μ 2 j .

As indicated earlier, [4] introduced the GARCH(m, r) model which defines the conditional variance equation as follows;

h t = ω 0 + ∑ i = 1 r     α i ε t − i 2 + ∑ i = 1 m     β i h t − i (40)

To allow for a volatility level shift, denoted ω 11 , after time t = τ 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 = ω 0 + ∑ i = 1 r     α i ε t − i 2 + ∑ i = 1 m     β i h t − i + ω 11 ( 1 − B ) − 1 I t τ (41)

= ω 0 + ∑ i = 1 r     α i ε t − i 2 + ∑ i = 1 m     β i h t − i + ω 11 G 11 ( B ) I t τ (42)

where I t τ is an indicator variable taking values 1 for t = τ , and 0 otherwise. The parameter ω 11 indicates the size of the volatility level shift at time t = τ .

The extension of Equation (41) and Equation (42) to k volatility level shifts is straightforward. We define ω 1 j as the τ j t h shift in volatility level, compared to the previous level, where j = 1 , ⋯ , k . When we allow k volatility level changes at pre-specified time t = τ j , we can extend (41) and (42) to

h t = ω 0 + ∑ i = 1 r     α i ε t − i 2 + ∑ i = 1 m     β i h t − i + ∑ j = 1 k     ω 1 j ( 1 − B ) − 1 I t τ j (43)

= ω 0 + ∑ i = 1 r     α i ε t − i 2 + ∑ i = 1 m     β i h t − i + ∑ j = 1 k     ω 1 j G 1 j ( B ) I t τ j (44)

The component ∑ j = 1 k     ω 1 j ( 1 − B ) − 1 I t τ j governs the level shift movement of GARCH model intercept, that is baseline volatility, over time between ω 0 and ω 0 + ∑ j = 1 k     ω 1 j .

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 = τ would affect the series h t and is represented by model (45) and (46):

h t = ω 0 + ∑ i = 1 r     α i ε t − i 2 + ∑ i = 1 m     β i h t − i + ω 21 ( 1 − B s ) − 1 I t τ (45)

= ω 0 + ∑ i = 1 r     α i ε t − i 2 + ∑ i = 1 m     β i h t − i + ω 21 G 21 ( B s ) I t τ (46)

Assuming the observed series contains k seasonal level shifts at time t = τ j for j = 1 , ⋯ , k , their combined effect can be expressed in general form as:

h t = ω 0 + ∑ i = 1 r     α i ε t − i 2 + ∑ i = 1 m     β i h t − i + ∑ j = 1 k     ω 2 j ( 1 − B s ) − 1 I t τ j (47)

= ω 0 + ∑ i = 1 r     α i ε t − i 2 + ∑ i = 1 m     β i h t − i + ∑ j = 1 k     ω 2 j G 2 j ( B s ) I t τ j (48)

The component ∑ j = 1 k     ω 2 j ( 1 − B s ) − 1 I t τ j allows the intercept of the GARCH model to fluctuate over time between ω 0 and ω 0 + ∑ j = 1 k     ω 2 j .

Extension of y t the SARFIMA ( p , d , q ) ( P , D , Q ) s -GARCH ( m , r ) 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 + ∑ j = 1 k     μ 1 j ( 1 − B ) − 1 I t τ j

ε t = z t h t , ε t | F t − 1 ~ N ( 0 , h t )

h t = ω 0 + ∑ i = 1 r     α i ε t − i 2 + ∑ i = 1 m     β i h t − i + ∑ j = 1 k     ω 1 j ( 1 − B ) − 1 I t τ j (49)

Similarly, extension of y t the SARFIMA ( p , d , q ) ( P , D , Q ) s -GARCH ( m , r ) 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 + ∑ j = 1 k     μ 2 j ( 1 − B s ) − 1 I t τ j

ε t = z t h t , ε t | F t − 1 ~ N ( 0, h t )

h t = ω 0 + ∑ i = 1 r     α i ε t − i 2 + ∑ i = 1 m     β i h t − i + ∑ j = 1 k     ω 2 j ( 1 − B s ) − 1 I t τ j (50)

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

ϕ ( B ) Φ ( B s ) ( 1 − B ) d ( 1 − B s ) D y t = μ 0 + θ ( B ) Θ ( B s ) ε t + μ 21 ( 1 − B s ) − 1 I t . (51)

Therefore the null hypothesis of unconditional mean constancy becomes: H 0 : μ 21 = 0 . Let ψ 1 = ( d , D , μ 0 , μ 21 , ϕ ′ , Φ ′ , θ ′ , Θ ′ , σ ε 2 ) ′ be the approximate likelihood estimator (MLE) ψ ^ 1 of ψ 1 that maximizes the conditional log-likelihood:

l t ( ψ 1 ) = − 1 2 ln 2 π − 1 2 ln σ ε 2 − 1 2 ε t 2 σ ε 2 . (52)

The partial derivatives evaluated under H 0 are given by:

∂ l t ∂ ψ 1 | H 0 = − ε ^ t σ ε 2 ∂ ε ^ t ∂ ψ 1 | H 0 (53)

· ∂ ε t ∂ d | H 0 = ( 1 − B ) d ( 1 − B s ) D ϕ ^ ( B ) Φ ^ ( B s ) ∑ j t − 1 y t − j j + ∑ j = 1 Q     Θ ^ j ∂ ε ^ t − j s ∂ d                         + ∑ j = 0 Q   ∑ i = 0 q     Θ ^ j θ ^ i ∂ ε ^ t − i − j s ∂ d

· ∂ ε t ∂ D | H 0 = ( 1 − B ) d ( 1 − B s ) D ϕ ^ ( B ) Φ ^ ( B s ) ∑ j t − 1 y t − j s j s + ∑ j = 1 Q     Θ ^ j ∂ ε ^ t − j s ∂ D                         + ∑ j = 0 Q   ∑ i = 0 q     Θ ^ j θ ^ i ∂ ε ^ t − i − j s ∂ D

· ∂ ε t ∂ μ 0 | H 0 = − 1 + ∑ j = 1 Q     Θ ^ j ∂ ε ^ t − j s ∂ μ 0 + ∑ j = 0 Q   ∑ i = 0 q     Θ ^ j θ ^ i ∂ ε ^ t − i − j s ∂ μ 0

· ∂ ε t ∂ μ 21 | H 0 = − ( 1 − B s ) − 1 I t + ∑ j = 1 Q     Θ ^ j ∂ ε ^ t − j s ∂ μ 21 + ∑ j = 0 Q   ∑ i = 0 q     Θ ^ j θ ^ i ∂ ε ^ t − i − j s ∂ μ 21

· ∂ ε t ∂ ϕ | H 0 = Φ ^ ( B s ) ( 1 − B ) d ( 1 − B s ) D ( y t − 1 , ⋯ , y t − p ) + ∑ j = 1 Q     Θ ^ j ∂ ε ^ t − j s ∂ ϕ                         + ∑ j = 1 Q   ∑ i = 0 q     Θ ^ j θ ^ i ∂ ε ^ t − i − j s ∂ ϕ

· ∂ ε t ∂ Φ | H 0 = ϕ ^ ( B ) ( 1 − B ) d ( 1 − B s ) D ( y t − s , ⋯ , y t − p s ) + ∑ j = 1 Q     Θ ^ j ∂ ε ^ t − j s ∂ Φ                         + ∑ j = 0 Q   ∑ i = 0 q     Θ ^ j θ ^ i ∂ ε ^ t − i − j s ∂ Φ

· ∂ ε t ∂ θ | H 0 = ( ∑ j = 0 Q     Θ ^ j ε t − 1 − j s , ⋯ , ∑ j = 0 Q     Θ ^ j ε t − q − j s ) + ∑ j = 1 Q     Θ ^ j ∂ ε ^ t − j s ∂ θ                           + ∑ j = 1 Q   ∑ i = 0 q     Θ ^ j θ ^ i ∂ ε ^ t − i − j s ∂ θ

· ∂ ε t ∂ Θ | H 0 = ( ε t − s , ε t − 2 s , ⋯ , ε t − Q s ) + ( ∑ i = 0 q     θ ^ i ε t − i − s , ⋯ , ∑ i = 0 q     θ ^ i ε t − i − Q s )                         + ∑ j = 1 Q     Θ ^ j ∂ ε ^ t − j s ∂ Θ + ∑ j = 0 Q   ∑ i = 0 q     Θ ^ j θ ^ i ∂ ε ^ t − i − j s ∂ Θ

· ∂ ε t ∂ σ ε 2 | H 0 = − 1 2 σ ε 2 + ε ^ t 2 2 σ ε 4

Under H 0 , the LM-type statistics is asymptotically distributed as χ 2 with one degree of freedom:

L M 1 = 1 2 ∑ t = 1 n ∂ l t ∂ ψ 1 | H 0 ( ∑ t = 1 n [ ∂ ε t ∂ ψ 1 | H 0 ] [ ∂ ε t ∂ ψ 1 | H 0 ] ′ ) − 1 ∑ t = 1 n ∂ l t ∂ ψ 1 | H 0 (54)

Once the SLS-SARFIMA model is estimated and the residuals ε 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 = ω 0 + α ( B ) ε t + β ( B ) h t + ω 21 ( 1 − B s ) − 1 I t (55)

Therefore the null hypothesis of the unconditional variance constancy becomes: H 0 : ω 21 = 0 . Let ψ 2 = ( ω 0 , ω 21 , α ′ , β ′ ) ′ be the vector of the SLS-GARCH model parameters and the quasi-likelihood function is given by:

l t ( ψ 2 ) = − 1 2 l n 2 π − 1 2 l n h t − 1 2 ε t 2 h t . (56)

The partial derivatives evaluated under H 0 are given by:

∂ l t ∂ ψ 2 | H 0 = 1 2 [ ε ^ t 2 h ^ 0 t − 1 ] ∂ ln h t ∂ ψ 2 | H 0 (57)

· ∂ ln h t ∂ ω 0 | H 0 = ( h ^ 0 t ) − 1 [ 1 + ∑ j = 1 m β ^ j ∂ h ^ t − j ∂ ω 0 ]

· ∂ ln h t ∂ ω 21 | H 0 = ( h ^ 0 t ) − 1 [ ( 1 − B s ) − 1 I t + ∑ j = 1 m β ^ j ∂ h ^ t − j ∂ ω 21 ]

· ∂ ln h t ∂ α | H 0 = ( h ^ 0 t ) − 1 [ ( ε t − 1 2 , ⋯ , ε t − r ) ′ + ∑ j = 1 m β ^ j ∂ h ^ t − j ∂ α ]

· ∂ ln h t ∂ β | H 0 = ( h ^ 0 t ) − 1 [ ( h t − 1 , ⋯ , h t − m ) ′ + ∑ j = 1 m β ^ j ∂ h ^ t − j ∂ β ]

Under the null hypothesis, the “hats” indicates the maximum likelihood estimator and h ^ 0 t denotes the conditional variance estimated at time t. Under H 0 , the LM-type statistics is asymptotically distributed as χ 2 with one degree of freedom:

L M 2 = 1 2 ∑ t = 1 n ∂ l t ∂ ψ 2 | H 0 ( ∑ t = 1 n [ ∂ ln h t ∂ ψ 2 | H 0 ] [ ∂ ln h t ∂ ψ 2 | H 0 ] ′ ) − 1 ∑ t = 1 n ∂ l t ∂ ψ 2 | H 0 (58)

Let ψ 1 = ( d , D , μ 0 , ϕ ′ , Φ ′ , θ ′ , Θ ′ , σ ε 2 ) ′ be the vector of parameters in model (59)

ϕ ( B ) Φ ( B s ) ( 1 − B ) d ( 1 − B s ) D y t = μ 0 + θ ( B ) Θ ( B s ) ε t , t = 1 , ⋯ , n (59)

and let us suppose, for the moment, that it is known. Further, suppose that the observed series is subject to the influence of a perturbation at time t = τ such that:

x t = μ 21 G ( B s ) I t τ + y t (60)

where we assume that model (59) is stationary. Model (60) can be rewritten as a linear regression model presented in model (61) and model (62)

x t = μ 21 Z t * ( τ ) + y t (61)

where Z t * ( τ ) = G ( B s ) I t τ is an n × 1 . Writing model (61) in matrix form:

x = μ 21 Z * + y (62)

where x = ( x 1 , ⋯ , x n ) ′ ; y = ( y 1 , ⋯ , y n ) ′ and Z * = ( Z 1 * ( τ ) , ⋯ , Z n * ( τ ) ) ′ .

According to [21], model (62) is a regression model with autocorrelation residuals and therefore the problem of estimating μ 21 can be solved by Generalized Least Squares (GLS).

Let V a r ( x ) = σ ε 2 Σ with Σ a n × n matrix which depends on ψ 1 and which is assumed to be positive definite, and let Σ = L ′ L be the Cholesky decomposition of Σ with L lower triangular. Pre-multiplying Equation (62) by L − 1 , and setting e * = L − 1 x , Z = L − 1 Z * and e = L − 1 y , we obtain the Ordinary Least Square (OLS) model

e * = μ 21 Z + e (63)

where V a r ( e ) = σ ε 2 I n . The OLS estimator of μ 21 and its variance are obtained from Equation (63) as

μ ^ 21 = ( Z ′ Z ) − 1 Z ′ e *       and     V a r ( μ ^ 21 ) = ( Z ′ Z ) − 1 σ ε 2 (64)

As argued in [24], to move from the GLS model in (62) to the OLS model in (63), there is no need to evaluate the matrix Σ , since the application of the Kalman filter on the observed series x yields the vector of standardized residuals e * = L − 1 x . Similarly, the application of the same filter on vector Z * provides the vector Z = L − 1 Z * from which Equation (64) can be computed.

To test the null hypothesis that the observation at time t = τ is not an intervention, one can use the standardized statistic.

λ = μ ^ 21 V a r ( μ ^ 21 ) (65)

which, for known ψ 0 , follows a standard normal distribution. By setting appropriate starting conditions (see, for example, ( [24] [25], or [26] ), the previous scheme extends in a straightforward manner to non-stationary series, for which y t follows model (59).

The mean level shift detection test was previously derived by [27] for ARIMA models and extended by [20] for the fractionally integrated ARFIMA ( p , d , q ) models assuming conditional variance is constant over time. [28] extended the level shift detection test of the mean for a realization of time series { x t } satisfying LS-SARFIMA-LS-GARCH model. For our purpose a natural extension of the seasonal level shift detection test of the mean for a realization of time series { x t } satisfying SLS-SARFIMA and SLS-GARCH model was derived. In order to derive the test statistic, let us rewrite model (38), with only one mean seasonal level shift:

x t = y t + μ 21 ( 1 − B s ) − 1 I t τ (66)

The hypothesis to be tested is

H 0 : μ 21 = 0     against       H 1 : μ 21 ≠ 0 (67)

which is based on x 1 , x 2 , ⋯ , x n a realization of time series { x t } satisfying SARFIMA-GARCH model with mean seasonal level shift.

Model (66) can be rewritten as

( 1 − B s ) x t = ( 1 − B s ) y t + μ 21 I t τ

This implies transforming the series by differencing of ( 1 − B s ) . Thus if μ 21 = 0 , ( 1 − B s ) x t = ( 1 − B s ) y t . The intervention parameter μ 21 can be estimated using various methods like the maximum likelihood estimation and least square estimation. The least square estimate of μ 21 if the mean intervention is at time t = τ is given by

μ ^ 21 = ∑ t = 2 n ( 1 − B s ) x t I t τ ∑ t = 2 n I t 2 = ( 1 − B s ) x τ = ( 1 − B s ) y τ , τ = s + 1 , s + 2 , ⋯ , n (68)

Extension of [27] test statistics can be written as:

T n = max { | T n ( t ) | } = max { | μ ^ 21 ( t ) | V a r [ μ ^ 21 ( t ) ] } , t = s + 1 , s + 2 , ⋯ , n (69)

where μ ^ 21 ( t ) = x t − x t − s is the estimated intervention or impact at time t = τ and V a r [ μ ^ 21 ( t ) ] is an estimate of the variance of μ ^ 21 ( t ) .

The distribution of the statistics is based on the fact that it is originally Normally distributed and then transformed to the Gamma distribution both of which belong to the Domain of Attraction of the Gumbel distribution with normalizing constants:

1) Normal Distribution:

d n = 2 ln ( n ) − ln ( ln ( n ) ) + ln ( 4 π ) 2 2 ln ( n )   and   c n = 1 / 2 ln ( n ) (70)

2) Gamma Distribution:

d n = 2 ln ( n ) − ln ( ln ( n ) ) − 2 ln Γ ( π )   and   c n = 2 (71)

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

S n = T n 2 − d n c n (72)

Then under H 0 : μ 21 = 0 , the statistics S n satisfies

S n → D F ( y ) = exp ( − e − ( y − λ ) / δ ) , as   n → ∞ (73)

where D signifies convergence in distribution. Here, λ ∈ R is location parameter and δ is scale parameter. The location parameter is also the mode of the distribution. Inverse of the F ( y ) in Equation (73), is given by:

y = λ − δ ln ( − ln ( F ) ) (74)

Thus a test of hypothesis can be conducted by comparing the test statistic S n in Equation (72) with an appropriate critical value. The largest T n 2 ( t ) statistic is considered an intervention at the α significance if the S n value exceeds the critical value.

The second step is a natural extension of mean seasonal level shift detection in SARFIMA-GARCH model to volatility seasonal level shift detection in SARFMA-GARCH model. After estimating the SLS-SARFIMA model and the residuals ε t are obtained, we test, the alternative hypothesis of SLS-GARCH volatility level shift against the null hypothesis of GARCH model. Let us rewrite model (47) with one volatility seasonal level shift:

h t = g t + ω 21 ( 1 − B s ) − 1 I t τ where     g t = ω 0 + ∑ i = 1 r     α i ε t − i 2 + ∑ i = 1 s     β i h t − i (75)

The hypothesis tested is

H 0 : ω 21 = 0       against     H 1 : ω 21 ≠ 0 (76)

which is based on h 1 , h 2 , ⋯ , h n a realization of time series { h t } from a GARCH model with seasonal level shift.

The derivation is based on the statistics

T n = max { | T n ( 1 ) | , ⋯ , | T n ( n ) | } = max { | ω ^ 21 ( 1 ) | V a r [ ω ^ 21 ( 1 ) ] , ⋯ , | ω ^ 21 ( n ) | V a r [ ω ^ 21 ( n ) ] } (77)

where ω ^ 21 ( t ) = h t − h t − s is the estimated intervention or impact at time t = τ and V a r [ ω ^ 21 ( t ) ] is an estimate of the variance of ω ^ 21 ( t ) .

Model (75) can be rewritten as

( 1 − B s ) h t = ( 1 − B s ) g t + ω 21 I t τ (78)

Thus if ω 21 = 0 , ( 1 − B s ) h t = ( 1 − B s ) g t . The intervention parameter ω 21 can be estimated using various methods like the maximum likelihood estimation and least square estimation. The least square estimate of ω 21 if the volatility intervention is at time t = τ is

ω ^ 21 = ∑ t = 2 n ( 1 − B s ) h t I t τ ∑ t = 2 n ( I t τ ) 2 = ( 1 − B s ) h τ = ( 1 − B s ) g τ , τ = s + 1 , s + 2 , ⋯ , n (79)

The distribution of the statistics is based on the fact that it is originally Normally distributed and then transformed to the Gamma distribution both of which belong to the Domain of Attraction of the Gumbel distribution with normalizing constants:

1) Normal Distribution:

d n = 2 ln ( n ) − ln ( ln ( n ) ) + ln ( 4 π ) 2 2 ln ( n )   and   c n = 1 / 2 ln ( n ) (80)

2) Gamma Distribution:

d n = 2 ln ( n ) − ln ( ln ( n ) ) − 2 ln Γ ( π )   and   c n = 2 (81)

The maximum domain of attraction of the Gumbel is shown to some extent in [29] and in greater detail in [30].

Let { h t } be a time series satisfying the volatility seasonal level shift model

( 1 − B s ) h t = ( 1 − B s ) g t + ω 21 I t τ (82)

For any realization h 1 , h 2 , ⋯ , h n of this time series, let the test statistics be given by:

S n = T n 2 − d n c n (83)

Then under H 0 : ω 21 = 0 , the statistics S n satisfies

S n → D F ( y ) = e x p ( − e − ( y − λ ) / δ ) , as   n → ∞ (84)

where D signifies convergence in distribution. Thus a test of hypothesis can be conducted by comparing the test statistic S n Equation (83) with an appropriate critical value. The largest T n 2 ( i ) statistic is considered as volatility intervention at the α level of significance if the test statistic S n value exceeds the critical value.

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 { y t } 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:

a) State the hypothesis being tested, which is

H 0 : μ 21 = 0       against       H 1 : μ 21 ≠ 0 (85)

b) Compute the residuals, the impact μ 21 and the test statistics like the popular [27] ’s likelihood ratio test statistics given by

T n = max { | T n ( t ) | } = max { | μ ^ 21 ( t ) | V a r [ μ ^ 21 ( t ) ] } , t = s + 1 , s + 2 , ⋯ , n

Then compute the statistics:

S n = T n 2 − d n c n

c) Determine the critical values to use in the test.

d) Determine whether observations are seasonal level shifts and remove each from the series by subtracting the value of the impact μ 2 i , i = 1 , ⋯ , k then apply the SARFIMA-GARCH modeling procedure to obtain the adequate model.

4) The second test is performed to check the volatility level shift which can be conducted as follows:

a) State the hypothesis being tested, which is

H 0 : ω 21 = 0       against       H 1 : ω 21 ≠ 0 (86)

b) Compute the residuals, the impact ω 21 and the test statistics like the popular [27] ’s likelihood ratio test statistics given by

T n = max { | T n ( 1 ) | , ⋯ , | T n ( n ) | } = max { | ω ^ 21 ( 1 ) | V a r [ ω ^ 21 ( 1 ) ] , ⋯ , | ω ^ 21 ( n ) | V a r [ | ω ^ 21 ( n ) | ] } (87)

Then compute the statistics:

S n = T n 2 − d n c n

c) Determine the critical values to use in the test.

d) Determine whether observations are level shifts and remove each from the series by subtracting the value of the impact ω 2 i , [ i = 1 , ⋯ , k ] then apply the SARFIMA-GARCH modeling procedure to obtain the adequate model.