^{1}

^{1}

^{2}

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 aver age (LS-ARFIMA) and level shift generalized autoregressive conditional heterosk edasticity (LS-GARCH). Test statistics that are useful to examine if mean and volatility level shifts are present in an autoregressive fractionally in tegrated moving average - generalized autoregressive conditional heteroskedas ticity (ARFIMA-GARCH) model are derived. Quasi maximum likelihood esti mation of the model is also considered.

When dealing with empirical time series from diverse fields of application, we are confronted with the phenomenon of long memory or long range dependence. A popular way to analyze a long memory time series is to use autoregressive fractionally integrated moving average (ARFIMA) processes introduced by [

Thus, 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, [_{10} concentration using a seasonal ARFIMA model with GARCH errors. Tong [

The model to be developed combines ideas from different strands of the statistical, financial and econometric literature. Autoregressive Moving Average (ARMA) models are extensively discussed in [

This article introduces detection of a mean and volatility level shifts innovation in an ARFIMA-GARCH model. The works of [

The first concern of this paper is how one would formally address modeling mean and volatility level shifts in an ARFIMA-GARCH. The second concern is derivation of test statistics that are useful to examine presence of level shifts in mean and volatility for an ARFIMA-GARCH model. The layout of the paper is organised as follows. Section 2 reviews some theoretical results of ARFIMA and GARCH. In Section 3, we introduce the class of LS-ARFIMA-LS-GARCH models. Section 4 deals with parameter estimation in LS-ARFIMA and LS-GARCH models. Section 5 is dedicated to the proposed procedure of level shift detection in ARFIMA-GARCH models. In Section 6, we perform some simulation study of the mean and volatility level shift detection procedure. The last section concludes with the main findings and limitations. Common acronyms used in this paper are given in

Acronym | Explanation |
---|---|

ARMA | autoregressive moving average |

ARFIMA | autoregressive fractionally integrated moving average |

GARCH | generalized autoregressive conditional heteroskedasticity |

LS-ARFIMA | level shift-autoregressive fractionally integrated moving average |

LS-GARCH | level shift-generalized autoregressive conditional heteroskedasticity |

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

The study of time series turned attention to incorporate long memory or long-range dependence characteristics. The ARFIMA(p, d, q) process, first introduced by [

Montanari et al. [

ϕ ( B ) ( 1 − B ) d ( y t − μ 0 ) = θ ( B ) ε t , for t ∈ ℤ , (1)

where μ 0 is the mean of the process, { 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 , ϕ ( ⋅ ) and θ ( ⋅ ) are the polynomials of degrees p and q, respectively, defined by

ϕ ( B ) = 1 + ∑ i = 1 p ( − ϕ i ) B i and θ ( B ) = 1 + ∑ j = 1 q ( θ j ) B j (2)

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

The difference operator ( 1 − B ) d is defined by means of the binomial expansion ( 1 − B ) d and can be expressed as:

( 1 − B ) d = ∑ i = 0 ∞ ( d i ) ( − B ) i = ∑ i = 0 ∞ ( i + d − 1 ) ! i ! ( d − 1 ) ! B i . (3)

The ARFIMA model is said to be stationary when − 0.5 < d < 0.5 , where the effect of shocks to ε t decays at a gradual rate to zero. The model becomes nonstationary when d ≥ 0.5 and stationary but non invertible when d ≤ − 0.5 , 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 d = 0 , where the effect of shocks decays geometrically; 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 GARCH(r, s) 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 (4)

where z t is normal distributed with mean 0 and variance 1. Bollerslev [

h t = ω 0 + ∑ i = 1 r α i ε t − i 2 + ∑ i = 1 s β i h t − i (5)

where ω 0 > 0 , α 1 , ⋯ , α r , β 1 , ⋯ , β s ≥ 0 , r and s are positive integer. Yang and Wang [

Let the ARFIMA(p, d, q)-GARCH(r, s) model be the discrete time series model of { y t } given by the following equation:

y t = μ 0 + ϕ − 1 ( B ) ( 1 − B ) − d θ ( B ) 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 s β i h t − i (6)

The following theorem shows some properties of ARFIMA(p, d, q)-GARCH(r, s) models.

Let { y t } be generated by model (6). Suppose that all roots of ϕ ( B ) and

θ ( B ) lie outside the unit circle and ∑ i = 1 r α i + ∑ j = 1 s β j < 1 .

1) If d < 1 2 , then { y t } is second-order stationary and has the following representation:

y t = μ 0 + ϕ − 1 ( B ) θ ( B ) ∑ i = 0 ∞ ( i + d − 1 ) ! i ! ( d − 1 ) ! ε t − i a . s . (7)

Hence { y t } is strictly stationary and ergodic.

2) If d > − 1 2 , then { y t } is invertible, that is, ε t can be written as

ε t = ϕ ( B ) θ − 1 ( B ) ∑ i = 0 ∞ ( i − d − 1 ) ! i ! ( − d − 1 ) ! ( y t − i − μ 0 ) a . s . (8)

For proof of Theorem (2.3) see [

By rearranging the conditional variance Equation (5) 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 (9)

where γ = α 1 + β 1 and η t = z t 2 − 1 . Ishida and Engle [

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 (10)

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 .

Ishida and Engle [

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 (11)

where φ = 1 − γ determines the speed at which the conditional variance reverts to its long run mean τ = 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 (12)

Belkhouja and Mootamri [

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 is written as

ϕ ( B ) ( 1 − B ) d ( y t − μ 0 ) = θ ( B ) ε t , for t = 1, ⋯ , n (13)

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 = i , i = 2 , ⋯ , n of the data, we write the sum of an unobserved ARFIMA process and the term for the mean level shift which we will denote as LS-ARFIMA(p, d, q)

y t = μ 0 + ϕ − 1 ( B ) ( 1 − B ) − d θ ( B ) ε t + μ 1 ( 1 − B ) − 1 I t (14)

where I t is an indicator variable taking values 1 for t = i , and 0 otherwise. The parameter μ 1 indicates the size of the mean level shift at time t = i . The mean level shift is an abrupt but permanent shift by μ 1 in the series caused by an intervention.

The extension of (14) to k level shifts is straightforward. We define μ j as the j^{th} shift in level, compared to the previous level, where j = 1, ⋯ , k . When we allow k level changes at pre-specified time t = j , we can extend (14) to

y t = μ 0 + ϕ − 1 ( B ) ( 1 − B ) − d θ ( B ) ε t + ∑ j = 1 k μ j ( 1 − B ) − 1 I t (15)

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

As indicated earlier, [

h t = ω 0 + ∑ i = 1 r α i ε t − i 2 + ∑ i = 1 s β i h t − i (16)

To allow for a volatility level shift, denoted α 1 , after time t = i , i = 2 , ⋯ , n 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(r, s).

h t = ω 0 + ∑ i = 1 r α i ε t − i 2 + ∑ i = 1 s β i h t − i + ω 1 ( 1 − B ) − 1 I t (17)

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

The extension of (17) to k volatility level shifts is straightforward. We define ω j as the j^{th} 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 = j , we can extend (17) to

h t = ω 0 + ∑ i = 1 r α i ε t − i 2 + ∑ i = 1 s β i h t − i + ∑ j = 1 k ω j ( 1 − B ) − 1 I t j (18)

The component ∑ j = 1 k ω j ( 1 − B ) − 1 I t 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 j .

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-LS-GARCH

y t = μ 0 + ϕ − 1 ( B ) ( 1 − B ) − d θ ( B ) ε t + ∑ j = 1 k μ j ( 1 − B ) − 1 I 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 s β i h t − i + ∑ j = 1 k ω j ( 1 − B ) − 1 I t (19)

The LS-ARFIMA-LS-GARCH series is shown in

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:

ϕ ( B ) ( 1 − B ) d y t = μ 0 + θ ( B ) ε t + μ 1 ( 1 − B ) − 1 I t . (20)

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

The partial derivatives evaluated under

Once the LS-ARFIMA model is estimated and the residuals

Therefore the null hypothesis of the unconditional variance constancy becomes:

The partial derivatives evaluated under

Under the null hypothesis, the “hats” indicates the maximum likelihood estimator and

The mean level shift detection test was previously derived by [

The hypothesis to be tested is

which is based on

Extension of [

where

Model (26) can be rewritten as:

This implies transforming the series by differencing once. Thus if

The distribution of the statistics is discussed in great detail in [

1) Normal Distribution:

2) Gamma Distribution:

The maximum domain of attraction of the Gumbel is shown to some extent in [

Let

Assume that the stationary component of the model

Let also the test statistics be given by

Then under

where D signifies convergence in distribution. Here,

Thus a test of hypothesis can be conducted by comparing the test statistic

The second step is a natural extension of mean level shift detection in ARFIMA-GARCH model to volatility level shift detection in ARFMA-GARCH model. After estimating the LS-ARFIMA model and the residuals

The hypothesis tested is

which is based on

The derivation is based on the statistics

where

and

Model (37) can be rewritten as

Thus if

Thus from Equation (12),

Similarly just like the mean level shift test statistic, 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:

2) Gamma Distribution:

The maximum domain of attraction of the Gumbel is shown to some extent in [

Let

For any realization

Then under

where D signifies convergence in distribution. Thus a test of hypothesis can be conducted by comparing the test statistic

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

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

b) Compute the residuals, the impact

where

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

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

b) Compute the residuals, the impact

where

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

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.

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

The critical values for the 10%, 5% and 1% level of significance are presented in

n | ||||||
---|---|---|---|---|---|---|

100 | 10% | 3.2696 | 3.1051 | 2.9734 | 2.9750 | 2.9805 |

5% | 4.0390 | 3.8715 | 3.7017 | 3.7497 | 3.7682 | |

1% | 5.8189 | 5.7323 | 5.5161 | 5.4073 | 5.2964 | |

500 | 10% | 3.5775 | 3.3900 | 3.3009 | 3.2663 | 3.2311 |

5% | 4.4122 | 4.1790 | 4.0486 | 3.9502 | 3.9976 | |

1% | 6.3285 | 6.0363 | 5.8687 | 5.6095 | 5.6986 | |

1,000 | 10% | 3.7314 | 3.5081 | 3.4206 | 3.3245 | 3.3185 |

5% | 4.5490 | 4.3108 | 4.2207 | 4.1038 | 4.0990 | |

1% | 6.3031 | 6.1391 | 5.7147 | 5.8760 | 5.7313 | |

5,000 | 10% | 3.9483 | 3.7847 | 3.6439 | 3.6160 | 3.5036 |

5% | 4.7155 | 4.5783 | 4.4396 | 4.3262 | 4.2897 | |

1% | 6.6114 | 6.3562 | 6.2494 | 5.9959 | 5.9833 | |

10,000 | 10% | 4.0949 | 3.8871 | 3.7579 | 3.6174 | 3.6262 |

5% | 4.8897 | 4.6576 | 4.5000 | 4.3393 | 4.3337 | |

1% | 6.6178 | 6.4573 | 6.2892 | 6.0755 | 5.9562 | |

20,000 | 10% | 4.1804 | 3.9571 | 3.8567 | 3.7411 | 3.7332 |

5% | 4.9806 | 4.7187 | 4.6338 | 4.5600 | 4.5402 | |

1% | 6.7149 | 6.4485 | 6.4418 | 6.2476 | 6.3332 | |

50,000 | 10% | 4.3269 | 4.1342 | 3.9570 | 3.8526 | 3.8739 |

5% | 5.1190 | 4.9377 | 4.6882 | 4.6208 | 4.6640 | |

1% | 6.7830 | 6.6396 | 6.2536 | 6.4134 | 6.3391 |

n | ||||||
---|---|---|---|---|---|---|

100 | 10% | 3.2696 | 3.2865 | 3.2884 | 3.2626 | 3.2436 |

5% | 4.0390 | 4.0229 | 4.0952 | 4.0574 | 4.0556 | |

1% | 5.8189 | 5.9099 | 5.8328 | 6.0107 | 5.8000 | |

500 | 10% | 3.5775 | 3.5126 | 3.5874 | 3.6187 | 3.5995 |

5% | 4.4122 | 4.3355 | 4.4011 | 4.4700 | 4.4322 | |

1% | 6.3285 | 6.1218 | 6.1148 | 6.2525 | 6.1946 | |

1,000 | 10% | 3.7314 | 3.7101 | 3.7431 | 3.7431 | 3.6815 |

5% | 4.5490 | 4.5165 | 4.5359 | 4.5329 | 4.4728 | |

1% | 6.3031 | 6.4385 | 6.3633 | 6.2887 | 6.2068 | |

5,000 | 10% | 3.9483 | 3.9699 | 3.9467 | 3.9659 | 3.9751 |

5% | 4.7155 | 4.8739 | 4.7202 | 4.7767 | 4.7581 | |

1% | 6.6114 | 6.6898 | 6.5888 | 6.5661 | 6.6130 |

10,000 | 10% | 4.0949 | 4.0612 | 4.0680 | 4.0834 | 4.0707 |
---|---|---|---|---|---|---|

5% | 4.8897 | 4.8223 | 4.8846 | 4.8707 | 4.8286 | |

1% | 6.6178 | 6.6805 | 6.7916 | 6.7117 | 6.5539 | |

20,000 | 10% | 4.1804 | 4.2230 | 4.1278 | 4.2206 | 4.2452 |

5% | 4.9806 | 5.0617 | 4.8998 | 4.9738 | 5.0493 | |

1% | 6.7149 | 6.7940 | 6.7940 | 6.7136 | 6.8014 | |

50,000 | 10% | 4.3269 | 4.3737 | 4.3578 | 4.3045 | 4.3737 |

5% | 5.1190 | 5.2264 | 5.1389 | 5.0943 | 5.2264 | |

1% | 6.7830 | 7.0419 | 6.8585 | 6.9424 | 7.0419 |

Before conducting the test it should be clear that the position of the mean level shift impact i.e. point

For illustration purposes, mean level shift of sizes

The probability of correctly detecting a mean level shift is the power of the test.

n = 100 | ||||||||

Freq | 191 | 492 | 1 890 | 5 137 | 8 357 | 9 739 | 9 978 | 10 000 |

Prob | 0.0678 | 0.2590 | 0.4669 | 0.9813 | 1.0000 | 1.0000 | 1.0000 | 1.000 |

1.1100 | 1.7991 | 2.3724 | 6.0735 | 13.0371 | 17.4898 | 10.4253 | 30.8862 | |

n = 1,000 | ||||||||

Freq | 251 | 516 | 1 678 | 4 727 | 8 266 | 9 790 | 9 995 | 10 000 |

Prob | 0.0897 | 0.2736 | 0.7684 | 0.9517 | 0.9981 | 1.0000 | 1.0000 | 1.0000 |

1.2197 | 1.8407 | 3.4341 | 5.1058 | 8.4128 | 24.3315 | 15.2128 | 29.5363 | |

n = 10,000 | ||||||||

Freq | 349 | 774 | 2 435 | 6 319 | 9 541 | 9 994 | 10 000 | 10 000 |

Prob | 0.2792 | 0.4755 | 0.9868 | 0.8990 | 0.9988 | 1.0000 | 1.0000 | 1.0000 |

1.8563 | 2.3964 | 6.4260 | 4.3404 | 8.8156 | 12.5498 | 16.5698 | 26.7614 |

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

The critical values for the 10%, 5% and 1% level of significance are presented in

n | ||||||
---|---|---|---|---|---|---|

100 | 10% | 3.6582 | 3.7604 | 3.5579 | 3.6815 | 3.6603 |

5% | 5.8953 | 6.1064 | 5.8003 | 5.7904 | 5.8381 | |

1% | 12.6589 | 12.5134 | 13.7865 | 12.9361 | 13.3118 | |

500 | 10% | 7.8762 | 8.1762 | 7.9626 | 8.2954 | 8.0637 |

5% | 11.6656 | 11.7909 | 12.1844 | 12.0447 | 11.7591 | |

1% | 23.8624 | 25.2815 | 25.8811 | 24.0450 | 24.5703 | |

1,000 | 10% | 11.0332 | 10.9712 | 11.2518 | 11.4327 | 11.2209 |

5% | 15.7878 | 16.1100 | 16.2858 | 16.1257 | 16.1505 | |

1% | 30.3888 | 32.6443 | 33.3903 | 31.3055 | 32.9398 | |

5,000 | 10% | 21.6356 | 22.1738 | 22.2093 | 22.1011 | 21.7998 |

5% | 30.1309 | 29.7765 | 29.7067 | 29.1700 | 29.8191 | |

1% | 57.5182 | 56.5253 | 51.5002 | 56.0019 | 54.7722 | |

10,000 | 10% | 28.9310 | 28.6193 | 28.9583 | 29.2728 | 28.3075 |

5% | 38.5022 | 37.2838 | 37.8952 | 38.4157 | 37.2962 | |

1% | 64.9144 | 68.7209 | 68.0323 | 68.8051 | 68.2281 | |

20,000 | 10% | 37.9433 | 37.2058 | 37.0357 | 37.8052 | 37.0132 |

5% | 48.9679 | 48.6394 | 48.5380 | 49.4719 | 49.1459 | |

1% | 84.9527 | 82.4076 | 80.5673 | 86.2474 | 83.3416 | |

50,000 | 10% | 51.7280 | 52.6568 | 51.5697 | 50.8732 | 51.9408 |

5% | 67.6419 | 68.0185 | 67.0430 | 66.3581 | 69.0306 | |

1% | 114.7755 | 116.1880 | 119.2706 | 115.2127 | 119.8323 | |

100,000 | 10% | 65.8681 | 65.5515 | 65.9760 | 65.1466 | 66.6319 |

5% | 83.9845 | 84.1396 | 86.4056 | 82.9548 | 86.1579 | |

1% | 148.4967 | 142.1744 | 143.9268 | 141.1822 | 147.3558 |

n | ||||||
---|---|---|---|---|---|---|

100 | 10% | 3.6582 | 3.7769 | 3.6860 | 3.8035 | 3.7577 |

5% | 5.8953 | 6.1048 | 6.1547 | 6.1189 | 6.1132 | |

1% | 12.6589 | 13.4891 | 13.9148 | 13.3196 | 13.3694 | |

500 | 10% | 7.8762 | 8.0239 | 8.2833 | 8.1952 | 8.2330 |

5% | 11.6656 | 12.1334 | 12.0662 | 11.7553 | 12.0480 | |

1% | 23.8624 | 24.8997 | 24.0587 | 24.9745 | 22.7606 | |

1,000 | 10% | 11.0332 | 11.1450 | 11.3055 | 11.1477 | 10.7323 |

5% | 15.7878 | 15.6398 | 15.9193 | 15.6644 | 15.5776 | |

1% | 30.3888 | 31.8784 | 30.2050 | 15.6644 | 31.9759 |

5,000 | 10% | 21.6356 | 21.6703 | 22.2743 | 21.6456 | 21.8549 |
---|---|---|---|---|---|---|

5% | 30.1309 | 28.7835 | 29.8158 | 28.6908 | 29.4122 | |

1% | 57.5182 | 55.4737 | 54.7683 | 50.9280 | 56.5341 | |

10,000 | 10% | 28.9310 | 28.9176 | 29.1411 | 28.8159 | 28.9125 |

5% | 38.5022 | 38.2453 | 38.9251 | 38.4272 | 38.3793 | |

1% | 64.9144 | 70.1031 | 69.5237 | 70.9082 | 70.2986 | |

20,000 | 10% | 37.9433 | 37.6648 | 37.6448 | 37.2528 | 37.8225 |

5% | 48.9679 | 48.9763 | 50.1476 | 48.8909 | 49.0337 | |

1% | 84.9527 | 89.0566 | 88.8139 | 86.6354 | 90.3910 | |

50,000 | 10% | 51.7280 | 51.7868 | 52.7127 | 51.6432 | 51.6855 |

5% | 67.6419 | 67.3734 | 69.3968 | 66.3510 | 67.0463 | |

1% | 114.7755 | 121.3809 | 124.0272 | 120.0173 | 115.1223 |

Before conducting the volatility level shift test it should be clear that the position of the volatility level shift impact i.e. point

For illustration purposes, volatility level shift of sizes

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.

3) Parameter estimation of LS-ARFIMA-LS-GARCH models was derived using quasi-maximum likehood estimation.

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.

The authors declare no conflicts of interest regarding the publication of this paper.