The Financial Risk Measurement EVaR Based on DTARCH Models

The value at risk based on expectile (EVaR) is a very useful method to measure financial risk, especially in measuring extreme financial risk. The double-threshold autoregressive conditional heteroscedastic (DTARCH) model is a valuable tool in assessing the volatility of a financial asset’s return. A significant characteristic of DTARCH models is that their conditional mean and conditional variance functions are both piecewise linear, involving double thresholds. This paper proposes the weighted composite expectile regression (WCER) estimation of the DTARCH model based on expectile regression theory. Therefore, we can use EVaR to predict extreme financial risk, especially when the conditional mean and the conditional variance of asset returns are nonlinear. Unlike the existing papers on DTARCH models, we do not assume that the threshold and delay parameters are known. Using simulation studies, it has been demonstrated that the proposed WCER estimation exhibits adequate and promising performance in finite samples. Finally, the proposed approach is used to analyze the daily Hang Seng Index (HSI) and the Standard & Poor’s 500 Index (SPI).


Introduction
Scientifically and accurately measuring financial risk is the core part of the financial risk management process.Developing efficient statistical methods of financial risk measurement is essential for effectively controlling financial risk.We aim to develop financial risk models that account for extreme events, thereby enhancing the accuracy and efficacy of risk assessments in the field of finance.Due to the increasing complexity, time-variation and randomness of financial markets, nonlinear time series models are used to provide a more reasonable description of the markets' behaviors or phenomena, the double-threshold autoregressive conditional heteroscedastic (DTARCH) model is one of nonlinear time series models which are designed for this purpose (see [1] for details).A significant characteristic of DTARCH models is that their conditional mean and conditional variance functions both are piecewise linear involving double thresholds.Our investigation will focus on developing an expectile-based value at risk (EVaR) model with a DTARCH structure.
DTARCH models are very useful and flexible in analyzing asymmetric financial time series, making them a subject of considerable attention in recent statistical and econometric papers.Ref. [1] investigated the model identification, estimation and diagnostic checking techniques based on the maximum likelihood principle under the normal assumption of the conditional distribution of the observed data.Ref. [2] investigated robust modeling techniques without a specific form of the conditional distribution, focusing on the L 1 estimation of DTARCH models and deriving limiting distributions for the proposed estimators.Ref. [3] further studied the parameter estimation of DTARCH models using the weighted composite quantile regression procedure, which includes quantile regression as a special case while significantly improving efficiency and inheriting robustness.Ref. [4] investigated DTARCH models with restrictions on parameters and proposed both unrestricted and restricted weighted composite quantile regression estimation for the model parameters, which can be utilized to construct the likelihood ratio-type test statistic.However, these papers are all based on the known threshold and delay parameters of DTARCH models.
The risk measure EVaR proposed by [5] is based on expectile regression theory.Ref. [6] proposed the concept of expectile.The expectile is a one-to-one mapping relationship with the quantile and has similar properties as the quantile.So, the expectile can be regarded as an estimation of quantile; see [7][8][9][10] for details.Expectile has gained popularity in recent years as a subject of interest.Ref. [11] discovered that similar to quantiles, timevarying expectiles can be estimated using a state space signal extraction algorithm.Ref. [12] proposed a new model based on expectile regression-geoadditive expectile regression model.Ref. [13] proposed regularized expectile regression with smoothly clipped absolute deviation (SCAD) penalty for analyzing heteroscedasticity in high dimensions when the error has finite moments.Ref. [14] considered penalized linear expectile regression using SCAD penalty function.Ref. [15] proposed aggregated expectile regression by exponential weighting.Ref. [16] derived joint weighted Gaussian approximations of the tail empirical expectile and quantile processes.Ref. [17]focused on the semi-parametric estimation of multivariate expectiles for extreme levels of risk.Ref. [18] proposed expectHill estimators, which are used as the basis for estimating tail expectiles and expected shortfall.Ref. [19] built a general theory for the estimation of extreme conditional expectiles in heteroscedastic regression models with heavy-tailed noise.Ref. [20] developed a weighted expectile regression approach for estimating the conditional expectile when covariates are missing at random.Ref. [21] studied the problem of the nonparametric estimation of the expectile regression model for strong mixing functional time series.Ref. [22] considered model averaging for expectile regressions.Ref. [23] exploited the fact that the expectiles of a distribution F are in fact the quantiles of another distribution E explicitly linked to F, in order to construct nonparametric kernel estimators of extreme conditional expectiles.Ref. [24] dealt with the problem of the nonparametric estimation of the functional expectile regression, and so on.
Since EVaR is derived from expectile theory and utilizes a squared loss function as its loss function, it exhibits higher sensitivity to extreme values and is mathematically easier to handle.In addition, EVaR is a weighted average of the lower risk (expected shortfall, i.e., ES) and upper risk in conditions.Currently, several research papers on EVaR have been published, exploring various aspects of its application and properties.For example, Ref. [11] proved that EVaR is a consistent risk measure when the confidence level p is less than 0.5.Ref. [25] studied risk measurement EVaR under a variable coefficient model.Ref. [26] proposed a weighted composite expectile regression estimation for autoregressive models.Ref. [27] discussed the financial meaning of EVaR, compared them with VaR and ES, and studied their asymptotic behavior.Ref. [28] considered a new class of conditional dynamic expectile models with partially varying coefficients in assessing the tail risk of asset returns for S&P 500 Index.Ref. [29] proposed a class of semiparametric composite expectile models with varying coefficients.Ref. [30] proposed a semi-parametric model with varyingcoefficients to analyze the EVaR under the assumption of α-mixing.Ref. [31] forecasted the expectile-based risk measures by using the expected-based procedures.Ref. [32] provided a basis for inference on extreme expectiles and expectile-based marginal expected shortfall in a general β-mixing context that encompasses ARMA and GARCH models with heavy-tailed innovations.Ref. [33] developed a single-index approach for modeling the expectile-based value at risk.Ref. [34] studied the estimation of extremal conditional expectile based on quantile regression and expectile regression models.Considering the advantages of EVaR, we will propose the estimation of the DTARCH model based on expectile regression theory.Unlike the existing papers on the DTARCH model, we do not assume that the threshold and delay parameters of DTARCH models are known.
The rest of the paper is organized as follows.Section 2 investigates the estimation problem of DTARCH models based on expectile regression theory.We propose WCER estimation of DTARCH models in Section 2.3, and the proposed expectile regression estimation in Section 2.2 is a special case of WCER estimation, while the least squares estimation in Section 2.1 is a special case of the expectile regression estimation.In Section 2.4, we show that the asymptotic efficiency of WCER estimators calculated using weights obtained through data-driven methods is the same as those of WCER estimators calculated using known weights.We compare the least squares estimation, quantile regression estimation, expectile regression estimation and weighted composite expectile regression estimation of DTARCH models based on the maximum likelihood estimation in Section 3. The proposed methodology is also applied to analyze the daily Hang Seng Index (HSI) and the Standard & Poor's 500 Composite Index (SPI) in Section 4. We summarize our work in Section 5. Also, for readers interested in the theoretical basis of our results, the proofs of our theoretical results are provided in Appendix A. In addition, some of our simulation results are given in Appendix B.

Estimation of the DTARCH Model
Ref. [1] proposed the DTARCH model based on the autoregressive conditional heteroskedasticity model (ARCH) model (see [35]) and the threshold model (see [36]).The DTARCH model can handle situations where both conditional mean and conditional variance specifications are piecewise linear based on previous information.Given a time series y t , t = 1, 2, • • • , n, let F t be the σ-field generated from the realized value {y t , y t−1 , • • • } at time t.Assume that y t is generated by where j = 1, 2, • • • , m; the delay parameter d is a positive integer; the threshold parameters vector of lagged variables; and α (j) = α is a (p j + 1) × 1 parameter vector.The stochastic error satisfies t = h t (γ)u t with where I t,j = I r j−1 < y t−d ≤ r j , and γ = vec γ (1)  2) is an ARCH process, the innovations u t are independently and identically distributed random variables with E(u t ) = 0, Var(u t ) = 1, the parameters γ This is the DTARCH model proposed by [1].A significant characteristic of DTARCH models is that their conditional mean and conditional variance functions both are piecewise linear involving double thresholds.
We have made a slight modification to the DTARCH model under consideration.As reported by [3,4], the stochastic error satisfies t = h t (β)u t with The innovations u t are independently and identically distributed random variables with an unknown distribution F(u) and a density function f (u).
Let α = vec α (1) ), and de- note ∑ m j=1 p j + 1 = p and ∑ m j=1 q j + 1 = q.Then Equations ( 1) and ( 3) can be written as and t = h t (β)u t with h t (β) = z t β, (5) respectively.As in [1,3,4], we denote the model defined by (4) and ( 5) by where p 1 , • • • , p m represent the autoregressive model (AR) orders in the m regimes and q 1 , • • • , q m denote the ARCH orders in the m regimes.We use the DTARCH model with a conditional scale, rather than a conditional variance, because modeling the conditional scale is very important.Previous studies emphasized that such a scale provides a more natural dispersion concept than the variance and offers substantial advantages in terms of robustness.The advantage of such an approach with conditional scale instead of conditional variance can be found in [37][38][39][40][41] and so on.

Least Squares Estimators of the DTARCH Model
Most of the research papers on DTARCH models are based on the condition that the threshold parameters {r 0 , r 1 , • • • , r m } and delay parameter d are known.But in real data analysis, we know that this condition is hard to meet.In the literature on threshold models, there are also a few studies that are based on scenarios where the threshold or delay parameters are unknown.For example, [42] proposed the least squares (LS) estimators for a threshold AR(1) model with an unknown threshold and proved that LS estimators of the threshold parameters were strongly consistent.Ref. [43] proposed the conditional least squares (CLS) estimators for the threshold autoregressive model with unknown threshold and delay parameters and proved that CLS estimators of the threshold parameters were convergent in distribution.In this paper, we propose the parameter estimation methods for the DTARCH model based on expectile regression theory, which includes the expectile regression estimation and the weighted composite expectile regression estimation of the DTARCH model.Note that the expectile regression estimation can be seen as a special case of the weighted composite expectile regression estimation when the expectile takes on a certain value (see Section 2.3 of this paper for details), while the least squares estimation can be seen as a special value of the expectile regression estimation (see Section 2.2 of this paper for details).Under some conditions, we can show that the proposed estimators of the threshold and delay parameters are consistent.
Using the least squares estimation method, we can obtain the least squares estimation of Denote the threshold parameters (r 0 , r 1 , • • • , r m ) = r, the least squares estimator of r by r LS 0 , and the least squares estimator of the delay parameter d by d LS 0 .However, these estimators are biased.Obviously, the distribution of | t | is skewed and the log-transformation is an intuitive mechanism that can make the distribution less skewed; see [44] for details.Thus, in light of [3,4], we introduce a modified form of the model (6) where e t = log{|u t |}, and , β) is equivalent to h t (β), as we can see that h t (β) is also related to α.Therefore, we rename h t (β) as h t (α, β).Apply the least squares method again, we can obtain LS estimators of , r LS and d LS , respectively.We study the properties of the least squares estimators under the following conditions.For j = 1, 2, • • • , m, suppose that x t,j are all Markov chains.Their l-step transition probability is denoted by P l (x j , A j ), where x j ∈ R p and A j are Borel sets.Later on, we will need the following set of regularity conditions.(C1) {x t,j } admits a unique invariant measure π j (•) such that ∃ K j , ρ j < 1, ∀x j ∈ R p , ∀n j ∈ N , Under some additional conditions, we have the following corollary.
Corollary 1. Suppose that the conditions (C1), (C2) and (C4) hold.Then it follows from Theorem 1 that the estimator d LS is strongly consistent, that is, Theorem 2. Suppose that the conditions (C1), (C2) and (C4) hold.Then, the estimator r LS converges to r * in distribution, that is, According to Corollary 1 and Theorem 2, both the threshold and delay parameters converge to their true values.Therefore, after obtaining estimated values for threshold and delay parameters, estimating the remaining parameters of the DTARCH model will yield convergence properties that are equivalent to those obtained by estimating the parameters using the known threshold and delay parameters.In order to simplify the theoretical analysis, without loss of generality, we assume that the threshold and delay parameters are known throughout the remainder of this paper.

Expectile Regression Estimators of the DTARCH Model
The definition of expectile regression proposed by [6] states that the τ-th expectile of a random variable u can be obtained minimizing the following check function, and the derivative where f (•)is the density function of u.Therefore, the τ-th expectile of u is 0. Let t (α) = y t − x t α = h t (α, β)u t and the τ-th expectile of u t be µ(τ).Based on Theorem 1 in [6], the τ-th conditional expectile of t given F t−1 is The τ-th expectile regression (ER) estimator of α and β can be obtained by minimizing over b τ , α and β, where 0 ) and b τ is the τ-th expectile of u t .
Let the resulting estimators from (7) be b ER τ , α ER 0 , β ER 0 .Not surprisingly, these estimators are also biased.To correct the bias, we should still perform expectile regression estimation on the DTARCH model ( 6) that has undergone a logarithmic transformation.
From model ( 6), the τ-th expectile of log| t (α)| given F t−1 is where c τ is the τ-th expectile of e t .Applying the expectile regression scheme, we can obtain the expectile regression estimators of c τ , α and β by minimizing over c τ , α and β.Obviously, when τ = 0.5, the expectile regression estimators are the least square estimators in Section 2.1.
Let the resulting estimators of c τ , α , β be c ER τ , α ER , β ER .To derive the asymptotic property of the proposed estimator, we introduce some notations and conditions.Let c * τ be the τ-th expectile of e t , t = t (α * ), We assume that (C5) Covariance matrix Π is positive definite.
Then we have the following asymptotic results for c ER τ , α ER , β ER .

Weight Composite Expectile Regression Estimators of the DTARCH Model
Refs. [45][46][47] considered the composite quantile regression (CQR) estimation, which is obtained by incorporating the information of multiple quantiles into the objective function.This estimation method incorporates more comprehensive model information.Subsequently, ref. [29] introduced an estimation called composite expectile regression (CER) and established large sample properties of the resulting CER estimator.However, both CQR and CER estimations assign equal weights to different quantiles and expectiles, respectively.Intuitively, using different weights for different quantile regression (QR) and expectile regression (ER) models might lead to improved efficiency.Hence, ref. [48,49] proposed the weighted composite quantile regression (WCQR) estimation method.The standard deviation (SD) of the WCQR estimator is smaller than the SD of the CQR estimator and QR estimator as discussed in [4].Furthermore, ref.
[26] proposed a weighted composite expectile regression (WCER) estimation for AR models and established its large sample'properties.
From model ( 6), the τ k -th expectile of log| t (α)| given F t−1 is where c τ k is the τ k -th expectile of e t .Applying the WCER scheme, we can jointly estimate the AR and ARCH parameters by minimizing over c τ k , α and β, where ω = (ω 1 , • • • , ω K ) is a vector of weights such that ω = 1, with • denoting the Euclidean norm.Without loss of generality, we assume that 0 If ω i = 1/ √ K, the estimation obtained from Equation ( 9) is CER estimation.Obviously, the weight ω k is the contribution rate of the τ k -th expectile.Since may not have a positive correlation, it is possible for the weight component ω to be negative.Therefore, the WCER estimation is not a simple extension of the CER estimation.Due to the limited space, we will not discuss the CER estimation in detail.

Let the resulting estimator of
and c * τ k be the true value of the τ k -th expectile of e t .Under certain conditions, we have the following asymptotic Theorem 4. Suppose that the conditions (C2), (C4) and (C5) hold.Then c where Σ is a block matrix with blocks

Selection of Optimal Weight
By Theorem 4, we have where Σ 22 = σ 2 (ω)Π −1 .Because Π does not contain weight ω, to obtain the optimal weight, we only need to minimize σ 2 (ω) under the condition of ω = 1, which yields of error {e t } can be obtained by the kernel smooth estimation.Then, the nonparametric estimator of ω opt is given by denoted by α 0 β 0 can be obtained by the following formula, min Then, under certain conditions, we have the following asymptotic results for α 0 β 0 .
When ω opt is known, the asymptotic covariance of α 0 β 0 is the same as the covariance variance of α β .In other words, the asymptotic efficiency of WCER estimators calculated using weights obtained through data-driven optimal weighting is the same as those of WCER estimators calculated using known weights.

Comparison of Estimation Methods
In this section, we compare the least squares estimation, quantile regression estimation, expectile regression estimation and weighted composite expectile regression estimation of DTARCH models by using the maximum likelihood estimation (MLE) of DTARCH models as the benchmark.We consider the following DTARCH(2, 2; 2, 2) model: (2) 2 = (0.45, 0.20) and t = h t u t , with We consider three types of innovation variables, which are distributed as N(0, 1), t(6) and χ 2 (4).They are centralized and normalized so that the medians of the absolute innovations are 1, i.e., u t is normalized to satisfy Median(|u t |) = 1.The sample size n is chosen are 100, 300, 800, 1500 and 2500.All the simulation results are based on 500 Monte Carlo replications.Seven equally spaced expectiles in (0, 1) are chosen for each simulation setting when we apply the WCER estimation process.For QR and ER estimation, we take τ = 0.25 and 0.75, respectively.In each simulation, the root mean squared error (RMSE) for different estimators are calculated, and they are reported in Tables 1-3.In addition, the parameter estimators obtained from different estimation methods of the DTARCH model are listed in Tables A1-A3 in Appendix B.  As expected, the oracle MLE performs the best, while the WCER estimators outperform both the QR estimators and the ER estimators.As can be seen from Tables 1 and A1, the WCER estimators slightly underperform LS estimators only when the residual error follows a normal distribution.From Tables 2, 3, A2 and A3, we can see that the WCER estimators greatly outperform the LS estimators in terms of RMSE when the error follows a heavy-tailed or asymmetric distribution.In studies that apply time series models to study financial data, it is more realistic to assume that the error follows a non-normal and heavy-tailed distribution.For example, [50] considered the heavy-tailed nature and extreme volatility of asset returns, and demonstrated these statistical characteristics using financial data.Ref. [51] introduced a new heavy-tailed distribution to characterize errors in the ARCH/GARCH model and applied it to financial data.Furthermore, [52] assumed that there are two different types of heavy-tail distributions for GARCH model errors: the student's t-distribution and the normal reciprocal inverse Gaussian distribution.They compared the application of these distributions to South Korea's daily stock market returns.Therefore, it is possible to obtain a WCER estimator with favorable statistical properties similar to those of the MLE, even when the distribution of the error is unknown.Moreover, the RMSEs of all estimation methods decrease with the increase of sample size n, indicating that all estimators are consistent.
We also make an empirical analysis of the DTARCH model with delay parameter d = 1.Similarly, based on the maximum likelihood estimation, we compare the QR estimation, ER estimation and WCER estimation of this DTARCH model.We present the simulation results in the supplementary materials.

Real Data Analysis
In this section, we use the proposed method to analyze the Hang Seng Index (HSI) and the Standard & Poor's 500 Index (SPI) daily from 7 February 2013, to 6 February 2023.The formula for calculating returns series y t uses the daily returns of the exponential market, which are represented by the first-order difference of the logarithm of the closing prices of the index on adjacent days, where x t represents the closing price of the HSI or SPI on day t.The sample size for SPI is n = 2516, and the sample size for HSI is n = 2453.We are interested in the asymmetry of the conditional mean and conditional variance of the stock market.
First, we need to identify the values of m, the delay parameter d and the threshold parameters r i .Applying the same method as in [3], we obtain the values of m, d and r i as d = 1, m = 2 and (r 0 , r 1 , r 2 ) = (−∞, 0, +∞).This aligns with stock market observations and supports our goal of examining the asymmetry in conditional mean and variance.Similar to [3,4], we employ the generalized Akaike information criterion (GAIC) and the generalized Bayesian information criterion (GBIC) methods to determine the orders of DTARCH models before fitting the models to HSI and SPI.We find out that both GAIC and GBIC reach their minimum values for HSI and SPI with p = 2 and q = 4.The minimum GAIC and GBIC values for HSI are 0.3144 and 0.3876, respectively.The minimum GAIC and GBIC values for SPI are 0.2975 and 0.3463, respectively.This designates a DTARCH(2, 2; 4, 4) model for the return series.Thus, the following DTARCH model is taken into account for the return series y t , y t = α (1)  To evaluate the predictive performance of the models we built, we split the dataset into two parts: a larger part for model building and a smaller part for model validation.For example, we split the SPI dataset with sample size n = 2516 into two subsets: sample 1 (the sample from 7 February 2013 to 21 December 2022) of size n 1 = 2486 and sample 2 (the sample from 22 December 2022 to 6 February 2023) of n 2 = 30.We build a DARCH model using the sample 1, and then apply the model to predict the dataset from 22 December 2022 to 6 February 2023.We compare the predicted values with the sample 2 to evaluate the performance of different estimation methods.The chosen evaluation metric is Median Absolute Percentage Error (MAPE), calculated as the median absolute difference between predicted values y i and observed values y i (i = 1, 2, . . ., n 2 ).This approach is similar to that used in [29].We perform the same procedure on the SPI and HSI datasets with sample 2 of different sizes, specifically n 2 = 10, 20 and 30.The results obtained are shown in Table 6 and Table 7, respectively.
Based on the MAPE from Tables 6 and 7, it can be see that the WCER estimation consistently produces lower MAPE values compared to the other methods.Therefore, we conclude that the WCER estimation method outperforms other methods.

Concluding Remarks
In this paper, we develop an estimation method for DTARCH models based on the expectile theory.We propose the WCER estimators for DTARCH models and derive the large sample properties of the proposed estimators.Unlike the existing papers on the study of DTARCH models, we do not need to know the threshold and delay parameters.We conduct a simulation study to test the proposed theory and find that our WCER estimator outperforms the LS estimator in terms of RMSE, particularly when the errors follow a heavy-tailed or asymmetric distribution.The simulation results are consistent with our theoretical results.Even if the common distribution of errors is unknown, we can still obtain a WCER estimator with good statistical properties like the MLE.Furthermore, we apply the proposed WCER estimation method to estimate the parameters of DTARCH models using daily returns data for HSI and SPI.
It is noted that the proposed WCER estimation method is more effective for DTARCH models when the errors follow a non-normal heavy-tail distribution.This finding is consistent with real data examples, which adds to the practical significance of our study.Therefore, our future work will focus on further practical data analysis using the proposed methods.In addition, considering the high dimensionality of many real datasets, one of our next key steps is to come up with estimates based on expectile regression theory in such a scenario.This is one of the important research tasks that we will undertake.
Lemma A1.Suppose that the conditions (C2), (C4) and (C5) hold.Then, which is defined in (A11), can be written as , is a block matrix with each block being Proof of Lemma A1.To facilitate the proof, we denote by where , and Note that for arbitrary positive number a, we have Then, it follows from Taylor's expansion for the natural logarithm log (A1) And by Taylor's expansions for the natural logarithm log{h t (α, β)}, we have where (j) .
By substituting Equations (A1) and (A2) into ∆ tk , where the definition of ∆ tk is given in the proof of Theorem 4, we can obtain that where , δ, u) converges pointwise to its conditional expectation is enough, and the convergence is uniformly valid on any compact set of (v , δ , u ) , which is because of the convexity lemma in [53]. where From the above equations, we can obtain that By the Chebyshev's weak law of large numbers, we obtain that By substituting the above equations into Equation (A4), we can obtain that where = 0, by the Chebyshev's weak law of large numbers, we obtain that Subsequently, we obtain that (A7) According to Lemma A2, we can obtain that which combining with (A7) and performing some straightforward calculations, we arrive at the conclusion of Lemma A1.
Proof of Lemma A2.We can manipulate Equation (A4) to obtain the following form According to Equations (A6) and (A11), we obtain that .
Lemma A3.Suppose that the conditions (C2), (C4) and (C5) hold.Then, where L n is defined in Lemma A1, and is a block matrix with blocks Σ 0 11 is a K × K matrix with the (j, k)th element being Proof of Lemma A3.Note that .
By the Cramer-Wald device and the Central Limit Theorem, we obtain that We now calculate µ L and Σ 0 , respectively.It is easy to show that µ L = 0 (K+p+q)×1 . Let where which is a K × K matrix with the (j, k)th element being We can obtain the expressions of Σ 12 , Σ 21 and Σ 22 similarly.Thus, which completes the proof of Lemma A3.

Proof of Theorem
Then minimizing the objective in ( 9) is equivalent to minimizing n (v, δ, u).By Lemma A1, we obtain that where θ = v , δ , u ; L n = L n1 , L n2 , L n3 , with is a block matrix, with each block being We further perform transformation calculations on matrix G, resulting in , that is to say, G is a positive matrix.Furthermore, according to Lemma A3, we obtain that is a block matrix, and each block is, respectively: Σ 0 11 is a K × K matrix with the (j, k)th element being . Therefore, by Theorem 2 in [54], we can obtain that , by applying the inverse operation rules of block matrices, we can obtain that Substituting the above formula into G 22.1 yields So, we can obtain the expressions for G 11 , G 12 , G 21 and G 22 as follows: (A13) (A14) So, by substituting the results of (A18)-(A21) into the expression for Σ 11 in (A17), we can obtain the value of Σ 11 as where ξ is a K × K matrix with the (j, k)th element being with Λ is a K × K matrix with the (j, k)th element being Λ(j, k),

, β, d and r, respectively
where • and | • | denote the total variation norm and the Euclidean norm, respectively.(C2) E|y t | 2+δ < +∞ for δ > 0, and {y t } is strictly stationary and ergodic., d * and r * be the true values of α.Then, we can obtain the following theorems and corollary.

Table 4 .
Estimates of parameters for HSI.
Note: Estimated SEs are given in parentheses, and all SEs are multiplied by 10 2 .

Table 5 .
Estimates of parameters for SPI.
Note: Estimated SEs are given in parentheses, and all SEs are multiplied by 10 2 .

Table 6 .
Comparing the fitted values and predicted values of HSI using MAPE.

Table 7 .
Comparing the fitted values and predicted values of SPI using MAPE.