2.1 Huber SVR

Support vector regression (SVR) is a statistical learning method that originated from [41], which aims to precisely estimate the structure of a given dataset nonparametrically. Unlike many traditional parametric time series models, the strength of estimating time series model using SVR arises from its innate versatility to model dataset whose underlying structure is unknown or contains a severe degree of nonlinearity. This strength is particularly beneficial especially when estimating the structure of time series with heteroscedastic volatility, such as financial indices, as conditional volatility is unobservable in general and can be complex due to the interdependent nature of the current global economy. Moreover, utilizing SVR to estimate the structure of the time series can help to simultaneously avoid deploying a sophisticated parametric model, such as the family of exponential GARCH (EGARCH) models [42].

In this study, we utilize the Huber SVR (HSVR) when modeling nonlinear GARCH models to escalate the robustness. HSVR is advantageous over the standard SVR as it can effectively offset the effects of sporadic outliers, frequently observed in time series. Although many versions regarding HSVR exist in the literature, we utilize the ϵ-insensitive HSVR proposed by [33, 43].

Given a set of training observations {(x_i, y_i)} (i = 1, …, n, ), HSVR seeks to find a function of the form: (1) where , is a matrix whose j-th row consists of (j = 1, …, n), and are parameters to be estimated, and with K₁(x₁, x₂) = 〈ϕ₁(x₁), ϕ₁(x₂)〉 for some predefined kernel K. Here, 〈⋅, ⋅〉 and ϕ₁(x) respectively denote the inner product of the feature space and the implicit kernel operator corresponding to the kernel K₁.

At first glance, (1) seems to deviate from the function form of the standard SVR, which is given as, (2) where K and ϕ is defined analogously to K₁ and ϕ₁, respectively. However, we can see that (1) and (2) are actually identical, which implies that HSVR can be viewed as a direct extension of the standard SVR. This is because by using the definition of K₁(x, A^T), (1) can also be rewritten as where w_j is the j-th element of the vector w. Therefore, analogous to SVR, estimating (1) using HSVR can be understood as implicitly performing the following two-stage procedure: (i) first map the given observations (x₁, ⋯, x_n) onto some feature space using the prescribed mapping function ϕ₁ to produce (z₁, ⋯, z_n) ≔ (ϕ(x₁), ⋯, ϕ(x_n)), (ii) then estimate the linear function on that feature space that best describes the mapped observations z_i. For more details, we refer to [20, 28] for the standard SVR and [33] for HSVR.

To estimate the model parameters w and b in (1), we solve (3) where C₁ is a tuning parameter that balances between the model risk and the flatness of the estimated function, and L is an ϵ-insensitive Huber loss function defined as follows: (4) where x₊ ≔ max(x, 0) for , and ϵ, γ ≥ 0 are tuning parameters that determine the degree of robustness, see Fig 1 for an illustration.

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Fig 1. A plot of ϵ-insensitive Huber loss function in (4) for (ϵ, γ) = (0.1, 0.2) (left) and (ϵ, γ) = (0.5, 1.5) (right).

https://doi.org/10.1371/journal.pone.0299120.g001

As depicted in Fig 1, ϵ-insensitive Huber loss function returns zero when x ∈ [−ϵ, ϵ], increases quadratically in (−γ, −ϵ] and (ϵ, γ], then increases linearly otherwise. Moreover, in a statistical perspective, the optimization problem (3) attains the shrinkage estimator regarding w and b rather than only w, as seen in the traditional SVR models. This alteration is employed to derive a unique solution of w and b in solving (3) while simultaneously maintaining the strong convexity, refer to [44].

Despite the similarities between HSVR and SVR, the former deviates from the latter in terms of the way of solving the given optimization problem. Indeed, HSVR directly solves the primal problem by taking an iterative approach. To attain the solution, we first explicitly expand (3) by substituting (4), then reparametrize the problem with respect to z ≔ (w^T, b)^T: (5) where y = (y₁, …, y_n)^T, B = [K₁ 1_n], K₁ is an n × n matrix, whose (i, j)-th element is K_1,ij = K₁(x_i, x_j), A is a matrix of explanatory variables, whose i-th row is defined as , and is a vector of ones. As the objective function is strongly convex with respect to z, we attain the optimality by obtaining the solution of the first-order condition, that is, Therefore, (5) reduces to the problem of the following iterative method that computes the updated solution z_i+1 from the previous iteration z_i: for i = 0, 1, …. The general process of using the aforementioned HSVR for GARCH models is presented in Section 3.

2.2 Support vector data description algorithm

To formulate the control chart that integrates the one-class classification algorithm, we use the support vector data description (SVDD) model in this study, which is a complex of support vector machine (SVM) and the data description algorithm proposed by [35, 45]. SVDD is a method to find a boundary that encompasses given observations with a single hypersphere in the feature space. Because the proportion of observations inside the hypersphere can be settled in advance, we can readily employ SVDD as an anomaly detection method. For a general overview, we refer to [36].

The fundamental concept of SVDD stems from constructing a hypersphere with a minimum possible volume that includes the desired number of observations within such a boundary. Namely, given the training dataset (i = 1, …, n), SVDD solves the following optimization problem with respect to the center and the radius R ≥ 0 of the hypersphere: (6) where ξ_i ≥ 0 (i = 1, ⋯, n) denotes a slack variable that penalizes observations lying outside the hypersphere, C₂ ≥ 0 is a tuning parameter which controls the balance between the misclassification errors and the volume of the hypersphere, and ϕ₂(x) is an implicit kernel operator that is defined analogously to the kernel function K₁ in Section 2.1, namely, In practice, the following Gaussian kernel function is widely selected: (7) where κ > 0 is a tuning parameter that determines the complexity of the decision boundary. In practice, the decision boundary becomes more nonlinear and sophisticated when κ is small.

The general procedure of solving (6) is similar to that of the classical SVM. To elaborate, we first construct the unconstrained problem as below: (8) where α_i ≥ 0 and γ_i ≥ 0 (i = 1, …, n) are Lagrange multipliers. Then, by using the Karush-Khun-Tucker conditions, we obtain the following equivalent dual problem with respect to α_i: (9) Notice that γ_i vanishes due to the relationship α_i = C₂ − γ_i, (i = 1, …, n), which is obtainable using the first-order conditions. Using the optimal α_i of the problem (9), the radius R² can be calculated by the formula: where x_sv denotes one of the observations that is precisely located on the boundary of the constructed hypersphere. Similarly, we measure the kernel distance between a new observation and the center a by computing (10) and z is classified as an anomaly when D² is larger than R².

A primary reason why SVDD can be deployed in constructing a control chart is the usage of kernels, which enables practitioners to design a tailored control chart suitable to their needs. Indeed, the control limit of the chart can be determined by the tuning parameter C₂ in (6), as it can be exploited to precisely specify the rate of the misclassified datasets. To illustrate, by adjusting C₂, we can reset the upper bound ρ > 0 of the misclassification error rate through the following relationship: (11) and it is known that ρ matches the misclassification rate when the number of observations are sufficiently large and the distribution of the dataset is continuous [45]. Control charts using SVDD can be remarkably favorable when the underlying structure of the dataset is sophisticated, as SVDD seeks the boundary from a higher-dimensional feature space that can be further adjusted using κ in (7). Therefore, tuning C₂ is analogous to adjusting the control limit of control charts, see Sections 3 and 4 for more details.

3.1 HSVR-GARCH model

We extend the GARCH model to capture the nonlinear structure of the conditional volatility specified as follows: (12) where f is an unknown nonlinear function with the form in (1), which would be obtained by solving (3), is the conditional volatility at time t = 1, …, n, and ϵ_t are independent and identically distributed (iid) errors with E(ϵ₁) = 0 and Var(ϵ₁) = 1. Observe that, as is a (conditional) variance, both and σ_t must be nonnegative by definition.

Model (12) includes a broad class of stationary GARCH time series models. For a class of parametric GARCH models, [46] studied the estimation of their parameters and presented a method of conducting the change point test as an application. Their research later offered a theoretical foundation that based the OCC control charts of [47].

Although it seems plausible to directly deploy HSVR to model obtain the function f in (12), it is actually impractical because HSVR does not automatically guarantee the conditional variance to be nonnegative. To rectify the problem, we modify the above model by taking the logarithm to ensure the nonnegativity of as follows: (13)

As is not observable in most practical circumstances when estimating g in (13), we substitute in (12) with (14) for t = 1, …, n, and s ≥ 1 denotes a degree of smoothness. When t < s, (14) is defined as . is one of the most widely accepted proxy of and can be readily deployed when predicting the conditional volatility [48] and constructing monitoring schemes [28]. In this study, we set s = 20, as it renders the to be stabilized around the level of the unconditional variance, thereby enhancing the quality of residuals utilized in the formulation of control charts.

Another repercussion due to the unknown is that an adequate source, purposed to compare with for computing the empirical loss, is absent when aiming to determine the optimal set of tuning parameters, as is always estimated to be zero for pure GARCH-type models. This definitely prohibits practitioners from using standard loss functions, such as the mean absolute error (MAE), when performing the tuning parameter optimization. We overcome this problem by introducing a likelihood-based loss ℓ(g) for a function g with a regularization term regarding w. Specifically, by temporarily assuming that the error term ϵ_t is normally distributed, we consider the negative log-likelihood of g in (13) as follows: with . Then, by appending the regularization term regarding w to prevent overfitting, we obtain the following loss function [49]: (15) where δ ≥ 0 is a tuning parameter that balances between the model’s flatness and the goodness of fit. This approach is conceptually similar to that of the quasi maximum-likelihood estimator (QMLE), which is the standard method of estimation in parametric GARCH models [13].

The general procedure to fitting a HSVR-GARCH model is based on the sequential k-fold cross-validation method for time series. To elaborate, we follow the steps below:

1. Step 1. Partition the time series (x_t, y_t) into chunks of k ≥ 2, and denote the time series of the m-th chunk as (m = 1, ⋯, k). Moreover, let n_m be the number of observations in the chunk m.
2. Step 2. Use the first m − 1 chunks to obtain the estiamted function and the estimated conditional volatility in (13) as .
3. Step 3. Using the m-th chunk, compute the loss in (15) for the m-th chunk and denote the result as , namely,
4. Step 4. Repeat Steps 2 and 3 for m = 2, ⋯, k.
5. Step 5. The loss function that determines the optimal tuning parameter is finally defined as and the best set of tuning parameters is then set to those that minimize .

This process of obtaining via HSVR-GARCH models is encapsulated in Algorithm 1 below.

The set of tuning parameters for HSVR-GARCH model consist of four elements: (i) the regularization parameter C in (3), (ii) two parameters that comprise the ϵ-insensitive Huber-loss function ϵ and γ, (iii) the kernel tuning parameter s² for the Gaussian kernel in Section 2.1 (16) and (iv) the regularization parameter δ for the likelihood-based loss (15). In this study, we initially fix δ, then seek for the remaining tuning parameters that minimize via the particle swarm optimization. The specifications regarding the tuning parameters are presented in Sections 4 and 5 below.

One of the prominent advantages of utilizing the HSVR-GARCH model over the SVR-GARCH model of [27] is that it significantly stabilizes the estimated volatility, thus liberates from the necessity of any posterior treatment of residuals when constructing the control chart. Due to the unpredictable nature of GARCH models, conditional volatility is prone to have abrupt anomalies, even when no structural breaks exist. As such, one of the innate drawbacks of the SVR-GARCH model is that it occasionally returns incorrectly estimated conditional variance, as witnessed in Fig 2, which results in the underestimation of in regions where the time series is relatively stable. This may prevent practitioners from directly utilizing the residuals based on the SVR-GARCH model because some residuals are computed to be explosively large. This phenomenon might have occurred as the SVR-GARCH method uses the mean absolute error-type loss as shown below that compares the estimated with the proxy in (14) in optimizing tuning parameters: where k and m respectively denote the length of the training and the total length of the time series. In fact, two aforementioned figures concurrently depict that estimated from the SVR-GARCH model does not heavily depart from , and fails to resemble the true in some occasions. On the other hand, by employing the likelihood-based loss in (15) with an optimal set of tuning parameters, the HSVR-GARCH method can escape from comparing against the proxy variables which were already deployed in obtaining . This alteration substantially stabilizes the estimation process producing , which more closely resembles the true , as reflected in Fig 2, and thereby, results in producing well-behaved residuals.

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Fig 2.

Plot of the estimated conditional volatility obtained from fitting HSVR-GARCH with s = 20 (top) and SVR-GARCH models with s = 5 (center), 20 (bottom), respectively, when the underlying model is GARCH(1,1) in (17) with (ω, α, β) = (0.1, 0.1, 0.8). Orange and black lines respectively denote the estimated () and true volatility (), while the translucent blue line depicts with the specified s.

https://doi.org/10.1371/journal.pone.0299120.g002

Algorithm 1 HSVR-GARCH

 Input: A time series , a kernel K₁,

    a collection of tuning parameters

 Output: A conditional volatility estimator

1: Set 1 ≤ n_i ≤ n (i = 1, …, k), m₀ ≡ 0, n_k ≡ n, n_i < n_j (i < j)

2: (i = 1, …, k)

3: for each pair of do

4:  for 2 ≤ i ≤ k do

5:   

6:   With Y_train,

7:    

8:    ;

9:    g_z ← K₁(x_i, A^T)w_z + b_z

10:    

11:   With ,

12:    

13:  end for

14:  

15: end for

16:

17:

   return

3.2 Constructing OCC-based control chart

In this study, OCC-based control chart refers to the control chart that exploits the properties of ρ ∈ (0, 1) in (11) mentioned in Section 2.2 to prescribe a desired level of the control limit. Given model residuals and a set of tuning parameters (ρ, κ), where is estimated by using any suitable estimation method, we fit SVDD with the Gaussian kernel presented in (16) to the squared residuals obtained from the training time series, and regard R² as the control limit. Moreover, when a new time series y_n+k (k ≥ 1) is observed, we obtain its squared residuals and then measure the distance defined in (10) against the origin of hypersphere for each k, and declare the process out-of-control if it exceeds R².

The control limit is determined by leveraging two tuning parameters, namely, the inclusion rate ρ and the kernel tuning parameter κ, which settle the in-control average run length (ARL) to a desired level. To find the optimal tuning parameters, we use a method that hybridizes the standard grid searching method and quasi-Newton methods, such as the limited-memory BFGS algorithm [50]. This is because solely relying on the latter method or other first-order optimization methods may render tuning parameters not to converge when the provided length of the training time series is insufficient.

Although the general procedure to construct the OCC-based control chart in this study resembles those of [47], our OCC-based control chart is designed to preserve the original structure of the given time series compared to the latter. In implementation, OCC charts of [47] must sequentially perform the Yeo-Johnson transformation and the moving-average smoothing to the residuals to alleviate the instability caused by the inaccurate estimation of the model parameters. Despite these efforts, OCC charts had a relatively inferior performance in terms of ARL₁ compared to other charts, such as CUSUM or EWMA charts, when monitoring the conditional volatility using squared residuals, see Section 4 of [47]. Furthermore, as our simulation study indicates, smoothing residuals undermines the overall detection ability when the underlying model is contaminated with noises, see Section 4 of this paper for more detail. We actually suspect that such posterior manipulations on the residuals ignited this phenomenon by contaminating the information that the time series originally contained. However, as our OCC-based chart directly utilizes the squared residuals without any alterations, it not only facilitates the model residuals to retain the original structure, but also enhances the detection ability, as witnessed in the simulation results in Section 4.

The gist of constructing the OCC-based control chart, given a time series and some adequate models, is summarized in Algorithms 2 and 3 below. Specifically, Algorithm 2 describes the procedure of implementing SVDD for building an OCC-based control chart, given two tuning parameters ρ and κ. Meanwhile, Algorithm 3 provides an instruction to formulate OCC-based control charts using SVDD.

Remark 1 Under the circumstance in which obtaining copies of in-control time series is infeasible, we can empirically optimize tuning parameters by performing a wild-bootstrap method as specified below:

1. Estimate from the training time series y₁, …, y_m, then recursively compute using (13). When parametric models are employed, we instead estimate the model parameters;
2. Generate an iid sequence of standard normal random variables of length n, namely, , t = 1, …, n, b = 1, …, B, and construct a sequence of bootstrap samples ;
3. Obtain the residuals of bootstrap samples by computing , where ;
4. Based on , fit SVDD and adjust its tuning parameters accordingly to obtain the control limit R² that sets the in-control ARL to the desired level.

The validity of this method is discussed in the simulation experiments in Section 4. Also, this method is adopted when analyzing financial time series in Section 5, where only a small amount of training sample is available.

Algorithm 2 SVDD_b for OCC-based control charts

 Input: A sequence of residuals , a kernel K₂, a vector of tuning parameters b = (C₂, κ)

 Output: Dual variables α = (α₁, ⋯, α_n), a control limit candidate R²

1:

2: Given and b ≔ (ρ, κ),

3:  

4:   (: a support vector)

   return ,

Algorithm 3 OCC-based control chart for nonlinear GARCH models

 Input: A training set , an estimated model ,

    N streams of in-control time series {z_tj}_t≥1 (j = 1, …, N),

    a kernel K₂, s ≥ 1, a desired level of in-control ARL c*, a tolerance ϵ_tol > 0,

    a collection of tuning parameters for SVDD

 Output: A control limit R²

1: for 1 ≤ t ≤ n do

2:  ;

3:  

4: end for

5:

6: With ,

7: for each pair of do

8:  

9:  With ,

10:  for 1 ≤ j ≤ N do

11:   for t ≥ 1 do

12:    ;

13:    

14:    

15:    if then

16:     ; break

17:    end if

18:   end for

19:  end for

20:  

21: end for

22:

23:

    return

4.1 Specifications of the experiment

For the experiment, we consider the cases where the underlying models are GJR-GARCH(p, q) [51] and log-GARCH(p, q) specified below: where ω ≥ 0 and {ϵ_t} is an iid random process. Here, the log-GARCH(1,1) model is a variation of the exponential GARCH model (i.e., EGARCH(p, q); see [42, 52]), which has a high degree of nonlinearity. In this experiment, we set p = q = 1 and set (ω, α₁, α₂, β) = (0.3, 0.1, 0.5, 0.3) for the GJR-GARCH model, and (ω, α, β) = (0.3, 0.3, 0.3) for the log-GARCH model.

Moreover, to further reflect the circumstance where the underlying model is highly volatile and unstable, we additionally consider the cases where the initial distribution of ϵ_t in the training time series is specified as follows:

• Case 1. ϵ_t ∼ N(0, 1), the standard normal distribution;
• Case 2. ϵ_t ∼ t(5), t-distribution with 5 degrees of freedom;
• Case 3. ϵ_t ∼ Z_k = 0.9X + 0.1Y, X ∼ N(0, 1), Y ∼ N(0, k²).
• Case 4. ϵ_t ∼ Z_{l, m} = 0.9X + 0.1W, X ∼ N(0, 1), W ∼ N(l, m²).

The latter three distributions symbolize the scenarios in which the observed time series is either inherently heavy-tailed or occasionally unstable. In particular, the latter two distributions are variants of a normal mixture distribution that represents the underlying model being highly volatile due to the innovational outliers. Specifically, Z_k and Z_{l, m} respectively denote that the underlying structure of time series is systematically unstable and asymmetric, which are some traits that can be witnessed in various financial time series. In this study, we consider the case of k = 3 and (l, m) = (1, 2).

In addition to considering innovational outliers, we also contaminate the observed time series with additive outliers in some experiments. Namely, we consider the case where we observe rather than directly observing y_t, where η_t follows some prescribed distribution function. For this experiment, we respectively set η_t analogous to Cases 1 and 4 for GJR-GARCH model, and to Cases 3 and 4 for log-GARCH model when additive outliers are taken into consideration in the model.

When comparing the performance, we evaluate ARLs of three control charts, namely, the proposed OCC-based control chart in Section 3, CUSUM chart, and EWMA chart, that uses squared residuals. Here, CUSUM chart refers to the control chart that utilizes where K = kσ, H = hσ, σ denotes the standard deviation of squared residuals obtained from the training time series, and μ denotes the mean of . On the other hand, EWMA chart denotes the control chart that uses with the upper and lower control limits computed as where λ ∈ (0, 1) is a smoothing parameter, and Z₀ is defined to be the sample mean of . Notice that the control limit of CUSUM and EWMA charts are respectively controlled by tuning k ≥ 0, h ≥ 0, or L_e ≥ 0, to dictate control charts to achieve the desired level of ARL₀. For an overview of these control charts, we refer to [53–55].

Moreover, as we regard the underlying structure of the time series is unknown, we compare the detection ability of control charts using HSVR-GARCH against those that utilizes the standard GARCH(1,1) model specified below: (17) where ω, α, and β are all nonnegative, and α + β < 1 is fulfilled to ensure the stationarity. Notice that fitting GARCH(1,1) models to some other conditionally heteroscedastic time series is customarily accepted in the presence the model uncertainty, see [56].

The general procedure of the experiment is summarized as follows. We first generate a time series of length n for training either the HSVR-GARCH model stated in Section 3 or the standard GARCH(1,1) model to obtain the residuals. We subsequently use squared residuals to formulate the control chart, as they are solely designed to target the change of variance. We then sequentially generate 1,000 independent streams of time series without any structural changes in order to find adequate tuning parameters that fix ARL₀ to the desired level, for example, the frequently used 370, though this number must be reduced significantly in a practical situation, as mentioned later in the real data analysis. To evaluate the detection ability, we generate another 1,000 independent streams of time series that includes a structural change, then apply the control charts to obtain ARL₁.

We set the length of the training time series as n = 2, 000 and examine ARL₁ of control charts when some parameters, innovational distribution of ϵ_t, or the magnitude of the additive noise experience a change, respectively. Particularly, we specify the way in which the structural change occurs in Tables 1–8 below, alongside with the respective tuning parameters for each used control charts. When fitting HSVR-GARCH to time series, we initially fix δ = 0.1 and the number of chunks k to 5, then obtain other tuning parameters by employing the particle swarm optimization method.

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Table 1. A comparison of control charts using and when the underlying model is GJR-GARCH(1,1) with the specified parameters without any additive outliers.

Results of the first two columns directly compares our OCC-based control chart with that of [47].

https://doi.org/10.1371/journal.pone.0299120.t001

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Table 2. ARLs of control charts when the underlying model is GJR-GARCH(1,1) with the specified parameters, where no additive outliers are present.

“HSVR” and “GARCH” denotes that the chart is constructed using residuals obtained from fitting HSVR and GARCH(1,1) models, respectively.

https://doi.org/10.1371/journal.pone.0299120.t002

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Table 3. ARLs of control charts when the underlying model is GJR-GARCH(1,1) with the specified parameters, where the dataset is contaminated with a N(0, 1)-distributed noise.

https://doi.org/10.1371/journal.pone.0299120.t003

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Table 4. ARLs of the OCC-based, CUSUM, and EWMA control charts when the underlying model is GJR-GARCH(1,1) with the specified parameters, where the dataset is contaminated with a Z_1,2-distributed noise.

https://doi.org/10.1371/journal.pone.0299120.t004

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Table 5. ARLs of control charts when the underlying model is log-GARCH(1,1) with the specified parameters, where no additive outliers are present.

https://doi.org/10.1371/journal.pone.0299120.t005

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Table 6. ARLs of control charts when the underlying model is log-GARCH(1,1) with the specified parameters, where the dataset is contaminated with a Z₃-distributed noise.

https://doi.org/10.1371/journal.pone.0299120.t006

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Table 7. ARLs of control charts when the underlying model is log-GARCH(1,1) with the specified parameters, where the dataset is contaminated with a Z_1,2-distributed noise.

https://doi.org/10.1371/journal.pone.0299120.t007

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Table 8. ARLs of control charts when trained with samples obtained with the wild-bootstrap method, where η_t ∼ N(0, 1), and no additive outliers are present.

https://doi.org/10.1371/journal.pone.0299120.t008

4.2 Experiment results

In this subsection, we report the results of three experiments as follows. First, we compare the performance of our proposed OCC-based control chart with that of [47] by comparing control charts constructed from either or , where the latter denotes the residuals employed in [47]. For fairness, we consider the control charts constructed from HSVR-GARCH residuals for both models to avoid the model misspecification issue in this particular experiment. Finally, we present the performance of control charts constructed from wild-bootstrap samples introduced in Remark 1 of Section 3.2. For the latter experiment, we set the number of bootstrap samples as B = 1000.

Table 1 report the comparison of OCC, CUSUM, and EWMA control charts that utilize the squared residuals against those that uses , where the underlying model is GJR-GARCH(1,1) with only innovational outliers of Z_1,2 being present. Note that for OCC-based control charts, directly using yields significantly enhanced results compared to those using . Moreover, in most circumstances using , a small improvement is observed regarding the detection ability in the latter two control charts. We speculate that this phenomenon occurred because diluted the information contained in the residuals, therefore undermining the overall detection ability. Indeed, as HSVR-GARCH model possesses the robustness on the estimated conditional volatility, squared residuals also become stabilized, thereby allowing to circumvent the constraint of using heuristic methods that struggle to suppress numerical instabilities. The result also implies that our proposed OCC-based control chart is superior to that of [47] in terms of the ARL₁ performance.

Tables 2–4 depict the ARLs of control charts when the underlying model is GJR-GARCH model, where the experiment of the latter two tables particularly contains additive outliers of Cases 1 and 4, respectively. The overall performance of charts with HSVR-GARCH residuals is relatively superior especially when the dataset is contaminated with either innovational outliers or additive noises, although the detection ability of HSVR-GARCH-based control charts somewhat recedes when neither of them are present. Specifically, ARL₁ of CUSUM and EWMA charts with HSVR-GARCH residuals consistently detect a change which is 5-10 percent faster than GARCH(1,1) counterparts. In particular, control charts with HSVR-GARCH residuals commonly exhibit stellar detection ability over their GARCH(1,1) residual counterparts when α_j (j = 1, 2) experience a change. This finding validates HSVR-GARCH models to be highly effective in capturing a structural change when the time series is systematically more convoluted.

Furthermore, unlike [47], our OCC-based chart is shown to compare well with the CUSUM and EWMA charts that use identical squared residuals in most cases and to benefit the most from using HSVR-GARCH residuals among the three control charts. Most notably, all results regarding GJR-GARCH model excluding the case of no outliers indicate the loss of the detection ability of the OCC-based control chart when GARCH(1,1) residuals are used. This phenomenon is most likely resulting from the ill-behaved residuals due to the model bias. Therefore, it is practically advised to jointly deploy the OCC-based control charts with the HSVR-GARCH model to avoid this model bias problem.

Tables 5–7 portray the ARL₀ and ARL₁ of control charts when time series follows the log-GARCH model. Note that the latter two tables specifically depict the dataset contaminated with additive outliers of Cases 3 and 4, respectively. Here, HSVR-GARCH model-based control charts are observed to be more competent, as they detect a structural change up to twice as fast especially when only innovational outliers are present. Indeed, the effect of model misspecification becomes more apparent as control charts with the parametric GARCH(1,1) residuals lose detection ability in some circumstances like the case of α changing from 0.3 to 0.5. Although HSVR-GARCH residuals-based charts do not outperform when the innovations structurally become heterogeneous, HSVR-GARCH residuals still continue to be a superior choice for OCC-based control charts as those residuals significantly shorten ARL₁ compared to the other two charts, while still providing a stable chart.

In addition, unlike the results of GJR-GARCH models, control charts in log-GARCH models without any external additive outliers is shown to be much more stable and immensely powerful, especially in the cases when ω or the innovational distribution of η_t experience a change. In particular, when either α or β increases to make a time series to be nearly nonstationary, most control charts with GARCH(1,1) residuals are observed to respond relatively poorly or even fail to respond. In contrast, control charts with HSVR-GARCH residuals not only successfully capture a change under those circumstances, and, on average, detect them twice as fast in the case of the parameter changes. In a nutshell, these results all fortify the strength of constructing control charts with HSVR-GARCH residuals when the underlying model is nonlinear and possibly unknown.

Finally, Table 8 denotes the performance of control charts when they are trained with samples generated by the wild-bootstrap method for both HSVR-GARCH and GARCH(1,1) models, when no outliers exist. Although some degree of performance drop is noticeable in certain cases, the three control charts with HSVR residuals retained the strong detection ability across all settings. Bootstrap methods applied to HSVR-GARCH models can be advantageous particularly when ample samples of in-control time series are unavailable. Note, however, that the bootstrap method can be less effective, for example, when the observed time series is impaired with innovational outliers, as the bootstrap samples are generated from iid standard normal innovations. Therefore, in practice, bootstrap methods can be especially beneficial in real-world scenarios when the source of observations do not possess any serious structural noises.