JPSJournal of Probability and Statistics1687-95381687-952XHindawi10.1155/2021/66971206697120Research ArticleAdjusted Extreme Conditional Quantile Autoregression with Application to Risk Measurementhttps://orcid.org/0000-0002-0449-3776KithinjiMartin M.1https://orcid.org/0000-0002-4543-6313MwitaPeter N.2https://orcid.org/0000-0002-9947-7954KubeAnanda O.3ThavaneswaranAera1Department of MathematicsPan African University Institute for Basic Sciences, Technology and InnovationP.O. Box 62000Nairobi 00200Kenya2Department of MathematicsMachakos UniversityP.O. Box 136Machakos 90100Kenyamksu.ac.ke3Department of Statistics and Actuarial SciencesKenyatta UniversityP.O. Box 43844Nairobi 00100Kenyaku.ac.ke202184202120211611202063202122320218420212021Copyright © 2021 Martin M. Kithinji et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we propose an extreme conditional quantile estimator. Derivation of the estimator is based on extreme quantile autoregression. A noncrossing restriction is added during estimation to avert possible quantile crossing. Consistency of the estimator is derived, and simulation results to support its validity are also presented. Using Average Root Mean Squared Error (ARMSE), we compare the performance of our estimator with the performances of two existing extreme conditional quantile estimators. Backtest results of the one-day-ahead conditional Value at Risk forecasts are also given.

Pan-African University of Basic Sciences, Technology and Innovation (PAUSTI)
1. Introduction

Correct specification of a loss/returns distribution is key to the accuracy of a risk measure such as Value at Risk. As noted in , the major difference among many estimators of Value at Risk lies in estimation of the distribution of returns. Complexity in modeling financial data is due to its failure to exhibit standard statistical properties such as normality, independence, and identical distribution . Statistical tests have revealed that returns exhibit fat-tails, time-varying volatility, and volatility clustering. Moreover,  showed that returns exhibit serial correlation over long time horizons. Models based on mean autoregression coupled with results from extreme value theory such as the AR(1)-GARCH(1,1) model in  incorporated most of the aforementioned characteristics of financial data but suffer from lack of robustness due to the effect of extreme observations on the mean. Extreme quantile autoregression in [2, 24, 25] among others leads to a more robust model. This is because they combine regression quantiles introduced by  in an autoregressive fashion while using extreme value techniques on the resulting residuals to capture the tail behaviour. A major challenge of this approach is possible quantile crossing.

The challenge of quantile crossing has been addressed by smoothing suggestions in [5, 6, 16] among others in a nonparametric setting. Equally, the conditional location-scale model used in obtaining Restricted Regression Quantiles (RRQ) in  averts possible crossing in extreme quantiles but can suffer from the same when estimating the median. To avert quantile crossing even at the middle,  added a forced ordering constraint in the estimation of multiple quantiles. Simulation results revealed that, based on standard error of the estimates, noncrossing quantile regression in  produces better estimates in the middle. However, the RRQ estimator produced better estimates at the tails, especially when the sample size was large.

The lack of monotonicity in estimation of conditional quantiles is addressed in  through sorting of originally estimated nonmonotone quantile curves using a functional delta method. The monotonic quantile functions obtained were found to be closer to the true quantile than the nonmonotonic quantiles. Function limit theory for the rearranged estimators was also derived. The resulting monotonic quantile functions were then used in estimating economic functions using Vietnam data. However, the model was not extended to extreme cases to cover heavy tails beyond the sample.

Parametric quantile regression is used in  to estimate percentiles in positive valued datasets. Specifically, a linear quantile regression model was used with the error term assumed to follow a generalized gamma distribution. The idea of quantile regression was achieved by allowing parameters for the error distribution to depend on the univariate covariate leading to a location-scale model. The four-, five-, and six-parameter generalized conditional gamma distributions were considered and likelihood ratio test was used in selecting the best-fit model for each dataset. Asymptotics for the three resulting models were also derived. However, the use of generalized gamma distribution limits the model to applications where the covariate is greater than zero. This together with the fact that some financial datasets have heavier tails than the Gamma distribution limits the application of the model in finance.

We seek to improve the extreme conditional quantile estimator in  using an interquantile dispersion from the central conditional quantile.

2. Methods and Estimation

Let St,t+U0 be a real valued financial time series on a complete probability space Ω,,P. We assume that St+ and it is t-measurable where t,t+ is an increasing sequence of σ-algebras representing information available up to time t. In particular, let St be the value of a portfolio at trading time t. The return on the portfolio at time t, used to quantify the gain in value of the portfolio from trading time tΔt to trading time t, is given by(1)Rt=StStΔtStΔt,so that(2)Xt=Rt,is the corresponding loss return of St.

Definition 1.

(Risk Measure). A risk measure is a function ρ from a set of risks in a financial position (in this case, the loss distribution) to ; that is, ρ:.

We assume that Xt can be expressed using a linear heteroscedastic model of the following form:(3)Xt=μt+et,where μtf:d is the conditional mean function of Xt given t1 and it is defined as μt=Ytβ. et are errors and Yt is a d-dimensional process which is t1-measurable. In particular, Yt has 1 as the first element and a collection of the last observed returns up to time t1; that is, Yt=1,Xt1,,Xtd. To ensure that the model is smooth and obeys some of the financial norms such as clustering of shocks, we further assume that et can be decomposed into(4)et=ϵtσt,where ϵt are independent and identically distributed random variables and σt>0 is the conditional volatility. In this case, Xt is said to assume a location-scale model of the form(5)Xt=μt+σtϵt.

The corresponding α quantile of Xt under this formulation is given by(6)μt,αm=μt+σtqαϵ,where qαϵ is the α-quantile of ϵt. Let us now define a conditional quantile autoregressive model on Xt of the form(7)Xt=μt,θ+εt,where μt,θ is the central conditional θ-quantile of Xt and εt are errors with zero θ-quantile. Let εt=σt,θZt, where σt,θ is the central conditional scale of Xt and Zt are i.i.d. residuals. Using an approach similar to Points Over Threshold (POT), we propose an extreme conditional quantile of the form given in equation (8) and refer to it as the adjusted extreme conditional quantile. That is, suppose that we are interested in an extreme quantile, μt,θ,α, for some α1or 0; the idea is to estimate the central quantile, μt,θ, and scale σt,θ, for some level θ in the middle and approximate the extreme quantile as(8)μt,θ,α=μt,θ+σt,θqαzqθz,where qαz and qθz are the α,θ-quantile of Zt, respectively, for α,θ0,1. If the parametric distribution of Zt is known, then qαz and qθz are easily determined as the inverse of the cumulative distribution of Zt at probability levels α and θ, respectively; otherwise, appropriate estimates are determined. Note that μt,θ,α is μt,αm in conditional quantile autoregressive form. Observe that when α=θ, equation (8) reduces to(9)μt,θ,θ=μt,θ,which is the central conditional quantile of Xt given Yt=y. This confirms that indeed εt has a zero θ-quantile. From equation (8), we obtain the following estimator for the extreme conditional quantile:(10)μ^t,θ,α=μ^t,θ+σ^t,θq^αzq^θz,where q^θz and q^αz are appropriate estimates of the θ,α-quantiles of the i.i.d. residuals, respectively. We compare this estimator with(11)μ^t,θ,αm=μ^t,θ+σ^t,θq^αz,proposed in , and(12)μ^t,θ,αh=μ^t,θ+σ^t,θcα,where cα is the resulting coefficient from quantile regression of εt against εt^ at 100α%. Note that μ^t,θ,αh is the estimator proposed in .

2.1. Estimation of Central Quantiles

Let μt:d be an unknown smooth function and define the loss function Mθ:(13)Mθx,μt=θxμt++1θxμt=xμtθIxμt0,where xμt+ and xμt represent absolute positive and negative values, respectively, and Ixμt0 is the indicator function. Assuming that the conditional quantile process is well defined, we expect(14)EMθXt,μt,θ|Yt=y=0.So the θ-conditional quantile of Xt is given by(15)μt,θ=argminμtEMθXt,μt|Yt=y.

Note that equation (14) can be used to check whether the conditional quantile process is correctly specified or not. We impose the following regularity assumptions to ensure consistency of the conditional quantiles.

Assumption 1.

Xt,Yt are identically distributed with the joint probability density fX,Yx,y and a continuous conditional probability density fX|Yx|y of Xt given Yt=y.

Assumption 2.

There exists δ>1 such that EYtδ<.

Assumption 3.

μt,θf:B, where B is a compact subset of d.

Assumption 4.

fε|Yε|x>0x, where fε|Yε|x is the conditional probability density of εt=Xtμt,θ given Yt=y.

From the sample analog of equation (15), we obtain(16)μ^t,θ=argminμt1nt=1nMθXt,μt|Yt=y,which is the θ-conditional quantile estimator for a sample of size n. To overcome the limitation of quantile crossing, we used the approach in  where required quantiles were estimated simultaneously with a noncrossing constraint using the optimization problem:(17)minμti=1kwθit=1nMθiXt,μti|Yt=ysubject toμtiμti1,YtB,i=1,,k,for some weight function wθi>0. A conveniently used practical choice of the weight function which was also adopted for this study is wθi=1,i=1,,k.

Lemma 1 (consistency central quantiles’ estimator).

Given that Assumptions 1, 2, 3, and 4 hold, μ^t,θμt,θ=op1.

2.1.1. The Scale Function

To still maintain dependence and ensure positivity of the scale function, this study incorporated a scale function in the form of a quantile autoregressive (QAR) function on the absolute of the nonstandardized residuals. This was achieved by replacing μt,θ with its corresponding estimate in equation (7) so that εt=Xtμ^t,θ and(18)εt=σt,θ+ϑt,θηt,where σt,θ is the central conditional θ-quantile of εt, ϑt,θ is the conditional scale of εt, ηt are i.i.d. residuals, and ϖt=ϑt,θηt are errors with zero θ-quantile. Similarly, as is the case in Section 2.1, we let σt:d+ be an unknown smooth function and define the loss function Mθx,σt. We assume that the QAR process in equation (18) obeys the four regularity assumptions given earlier so that, using noncrossing quantile regression approach, we obtain the following estimate of the scale:(19)σ^t,θ=argminσt1nt=1nMθεt,σt|Yt=y,where Mθεt,σt|Yt=y is a loss function of the form given in equation (13).

Lemma 2 (consistency of the scale estimator).

Given that the QAR process defined in equation (18) satisfies Assumptions 1, 2, 3, and 4, σ^t,θσt,θ=op1.

2.2. Extreme Value Theory (EVT)

Most of the financial datasets are heavy-tailed . Therefore, it is fundamentally important to incorporate extreme value theory in the estimation of extreme quantiles. A basic requirement for the application of EVT is independence in the particular distribution. The study in  observed that it is appropriate to assume that, at high (low) levels of α, the standardized residuals in equation (7) given by(20)Zt=εtσ^t,θ,where σ^t,θ is the estimate of the scale, are approximately independent, which allows us to apply EVT. Let Z1,Z2,, follow a common distribution function F. Consider a sample Zii=1,,n from which Mn=maxZ1,,Zn and is such that MnzF (a.s) where zFsupz|Fz<1. Pursuant to Fisher–Tippett’s theorem in , the random variable Z (or alternatively the distribution F of Z) is said to belong to the Maximum Domain of Attraction (MDA) of the extreme value distribution H if there exist norming constants cn>0 and dn such that(21)MndncndH.

Consequently, H is referred to as the Generalized Extreme Value (GEV) distribution.

This study applied Points Over Threshold (POT) method because it uses more data leading to better estimates compared to the Block Maxima method. POT models the distribution(22)Fuz=PZuz|Z>u=Fz+uFu1Fu.of all excesses above a particular threshold u, where 0zzFu.

Theorem 1 (Pickands–Balkema–de Haan) (see [<xref ref-type="bibr" rid="B22">22</xref>]).

We can find (positive-measurable function) βu such that(23)limuzFsup0z<zFuFuzGλ,βuz=0,if and only if FMDAHλ,λ, and Gλ,βuz is the Generalized Pareto Distribution (GPD) given by(24)Gλ,βuz=Gλ,βzu=11+λzuβ1/λ,λ0,1expzuβ,λ=0,with zu when λ>0 and uzuβ/λ when λ<0. λ and β are the shape and scale parameters, respectively.

A major challenge in POT is accuracy in choosing a threshold to separate extreme observations from the center of the distribution . Among the methods discussed in , most authors such as those of [21, 28, 30] prefer the conventional method in which a threshold that ensures that between 5% and 10% of the sample data is classified as extreme observations is chosen. Although conventional method is subjective, the choice of the threshold can be checked for appropriateness using a mean excess plot. The mean excess function for the GPD is given by(25)EZu|Z>u=βλu1+λ,where 1+λ>0. This implies that an optimal threshold corresponds to start of approximate linearity of the mean excess plot with the sign of the slope, λ/1λ, indicating the specific family of the GPD. A positive sign corresponds to the Frechet family, while a negative sign implies the Weibull family .

For convenience in making inferences on variability of the estimated quantiles, a recommendation in  on the use of Probability Weighted Moments (PWM) method in estimating parameters of the GPD was adopted. Using the first and second PWMs, we obtain the corresponding parameters estimates as(26)λ=M0M02M12,β=2M0M1M02M1,where M0 and M1 are obtained by replacing for k=0 and k=1 in(27)M1,0,k=Mk=EX1FXk=βk+1k+1+λ,which is the PWM of GPD with λ>1. See [10, 23] for details on PWM method. For a sample of size n, the corresponding PWM estimates are given by(28)M^k=1ni=1n1Pi:nkxi:n,where x1:nxn:n is the ordered sample and Pj:n=i+γ/n+δ for suitable constants γ and δ. As recommended in , γ=0.35 and δ=0.

The overall distribution of the standardized residuals was obtained by splicing the GPD with the empirical bulk distribution at the threshold using the approach in [21, 29, 30] among others to obtain(29)Fz=1FuGλ,βzu+Fu,for z>u and Gλ,βzu the Generalized Pareto Distribution. When Fu is approximated empirically, we obtain the following estimate of Fz:(30)F^z=1mN1+λ^zuβ^1/λ^,where N is the sample of size, m is the number of exceedances above the threshold u, and β^ together with λ^ are the estimated GPD parameters.

Lemma 3 (consistency of Probability Density Function (PDF) estimator).

Let Z1,Z2,,Zn be i.i.d. random variables from a Cumulative Distribution Function (CDF) Fz belonging to the MDA of Hλ. Suppose that Fz has a right endpoint at zF; then supzF^zFz0 as n.

From equation (30), we get the following estimate for the quantile of the standardized residuals at level α:(31)q^αz=u+β^λ^Nm1αλ^1.

Lemma 4 (consistency of error quantiles).

Let Z1,Z2,,Zn be i.i.d. random variables from a CDF F belonging to the MDA of Hλ satisfying α<Fqαz+ϵ for any ϵ>0. Then, for every ϵ>0 and n=1,2,,(32)limnPq^αzqαz<ϵ=1.

Using equation (10), we obtained the one-step VaR predictions as(33)VaR^t+1α=μ^t+1,θ,α=μ^t+1,θ+σ^t+1,θq^αzq^θz,where μ^t+1,θ and σ^t+1,θ are the corresponding one-step θ-quantile and scale estimates, respectively, from the linear conditional quantile process.

Theorem 2 (consistency of extreme quantile estimator).

Given that the QAR processes defined satisfy Assumptions 1, 2, 3, and 4 and Fz belongs to the MDA of Hλ, μ^t,θ,αμt,θ,α=op1.

3. Simulations

To evaluate the accuracy of our estimators, a sample of size T = 4250 was generated using the model Xt=0.5+0.3Xt1+1+0.35Xt12Zt, where Zt follows Student’s t-distribution with 4 degrees of freedom. The sample was partitioned into design data of size n and test data of size Tn. Figure 1 represents a sample path of the model superimposed with the median. Note that, for simulation purposes, θ=0.5 was used in estimating the central quantile.

Sample path superimposed with median.

Clearly, from the sample path, there is some level of volatility clustering, which is common in most financial data. An ACF plot of the resulting standardized residuals in Figure 2 confirms that indeed they are independent.

Autocorrelation plot of resulting residuals.

z=2.3283 was chosen as the threshold to ensure that 10% of the resulting ordered standardized errors were classified as extremes. This was confirmed by approximate linearity of the mean excess plot after the threshold as shown in Figure 3.

Mean excess plot.

The corresponding shape and scale parameter estimates from the GPD fit were 0.1156182 and 1.272937, respectively. Table 1 outlines the sample statistics of the estimates of the various quantiles together with the data.

Sample statistics.

α00.500.750.950.99
Minimum−15.81−3.58−2.80−0.731.76
Median Q1,Q30.67 (0.01, 1.38)0.65 (0.48, 0.83)1.15 (0.99, 1.32)2.47 (2.34, 2.61)4.07 (3.97, 4.17)
Mean ± SD0.69 ± 1.410.66 ± 0.361.16± 0.342.48 ± 0.274.07 ± 0.20
Maximum10.693.223.554.425.47

Q1, lower quartile; Q3, upper quartile; SD, standard deviation.

Figure 4 shows the corresponding quantile estimates at different levels of α.

Quantile estimates at various α-levels for θ=50%.

The accuracy of our extreme quantile estimator was evaluated using the Average Root Mean Squared Error (ARMSE). The RMSE seeks to return the Mean Squared Error (MSE) to the original scale of the sample. For k sample paths of simulations of size n of the extreme quantiles, the average RMSE is given by(34)A.R.M.S.E=1mi=1k1nt=1nμ^t,θ,αμt,αm2,where μ^t,θ,α is replaced by μ^t,θ,αm or μ^t,θ,αh when considering extreme conditional quantiles or restricted regression quantiles, respectively. To check how our model behaves under different choices of central quantile, we computed ARMSE for the extreme conditional quantile at α=0.95, where θ=0.25,0.5 and 0.75 were considered. 1000 sample paths, each of sizes 250, 500, 1000, 2000, and 4000, were used in the computation of ARMSE. Table 2 reports the obtained ARMSE values.

ARMSE for μ^t,θ,α for α=95%.

Sample size (n)θ=0.25θ=0.5θ=0.75
2500.968440.744930.62015
5000.796320.607880.56179
10000.719740.540560.53254
20000.714160.495850.50037
30000.697510.478620.49713
40000.696080.464390.48776

We note that, for a large enough sample (2000 observations and above), ARMSE is lowest when θ=0.5 and thus this choice of theta is maintained in investigating the accuracy of our estimator and forecasting the one-day-ahead VaR. Table 3 outlines the obtained ARMSE for the extreme conditional quantile at α=0.95 under three different models. The sample sizes and number or replications are still maintained.

ARMSE for μ^t,θ,α under different models for θ=50% and α=95%.

Sample size (n)Without noncrossing constraintWith noncrossing restriction
RRQECQAECQRRQECQAECQ
2500.582360.645520.733070.582000.642300.72728
5000.569890.613360.607660.568160.610830.60341
10000.533860.586540.533860.557800.585290.53186
20000.560510.582040.495620.558000.578800.49360
30000.554780.576630.482350.552350.573690.48073
40000.552710.576190.470320.550130.572950.46832

RRQ, restricted regression quantiles in . ECQ, extreme conditional quantiles in . AECQ, proposed adjusted extreme conditional quantiles.

Based on ARMSE, both RRQ and ECQ perform better than AECQ for small samples. However, as the sample size increases, AECQ outperforms both RRQ and ECQ. The decreasing ARMSE with increase in the sample size for AECQ and ECQ confirms that both are consistent estimators of the extreme conditional quantile. Also, for sample size above 2000, the rate of convergence of the AECQ estimator is higher than that of the ECQ estimator. It was not possible to comment on the consistency of the RRQ estimator, since its ARMSE fluctuated with increase in sample size. The consistent reduction in ARMSE when noncrossing constraint is added during estimation, confirming that indeed this constraint increases accuracy of resulting estimators.

4. Evaluating VaR Forecasts

In Section 3, we evaluated accuracy of our in-sample quantile estimates. In this section, we extend this by evaluating the out-of-sample VaR forecasts from our quantile estimator. To achieve this, we carry out backtests on 250 one-day-ahead VaR forecasts (as recommended in the Basel Accord) using coverage tests in [4, 31]. Coverage tests were adopted due to their popularity in literature and practice . Consider the failure process Itα=IXt>VaRtα, where I is the indicator function such that(35)Itα=1,if Xt>VaRtα,0,otherwise,and t=n+1,n+2,,T. By Lemma 1 in , Itαiid Binomα which is tested using the conditional coverage test that combines both the unconditional coverage test in  and the test for independence in  to under the null hypothesis EItα|It1α=α. The likelihood under the null hypothesis is(36)Lα;In+1α,In+2α,,ITα=1αn0αn1,where n1 is the number of VaR exceedances and n0=Tnn1. Now consider the first-order Markov chain generated by the transition probabilities of Itα:(37)Π1=π00π01π10π11=1π01π011π11π11,where πij=PrItα=j|It1α=i. This has an approximate likelihood function:(38)LΠ1;In+1α,In+2α,,ITα=π00n00π01n01π10n10π11n11,where nij is the number of times observation i is followed by j in the failure process Itα. From equation (38), we obtain the maximum likelihood estimate of πij as(39)π^ij=nijj=01nij.

Therefore, the conditional coverage hypothesis can be assessed using the likelihood ratio:(40)LRcc=2  logLα;In+1α,In+2α,,ITαLΠ^1;In+1α,In+2α,,ITαasyχ22.

Table 4 reports the obtained P values for the likelihood ratios of the three tests in . The tests were conducted on 250 one-day-ahead 5% VaR forecasts from the three considered models.

P values for the likelihood ratio tests.

Sample size (n)RRQ modelECQ modelAECQ model
LRucLRindLRccLRucLRindLRccLRucLRindLRcc
2500.0000.0010.0000.0140.0130.0020.0010.0060.000
5000.0000.0010.0000.0360.0100.0030.0090.0050.000
10000.0000.0000.0000.0420.0080.0030.0190.0040.000
20000.0000.0010.0000.0550.0100.0050.0510.0050.002
30000.0000.0010.0000.0620.0120.0060.0620.0050.003
40000.0000.0000.0000.0670.0090.0050.0660.0040.002

LRuc denotes likelihood ratio for the unconditional coverage test. LRind denotes likelihood ratio for the independence test. LRcc denotes likelihood ratio for the conditional coverage test. Models accepted at 5% level of significance are highlighted in bold. Note that n is size of the sample used in estimation, while testing was done using a sample of size 250 for all n.

Observe that, as a consequence of consistency, the accuracy of ECQ and AECQ forecasts improves with increase in sample size. It can also be seen that all the three models perform poorly under LRind and LRcc due to dependence in autoregression. Based on LRuc, the RRQ model performed poorly when it comes to forecasting. This can be attributed to the failure to incorporate extreme value theory in estimating residual quantiles in the RRQ model.

5. Conclusions and Recommendations

We have derived the extreme conditional quantile estimator and used it to obtain the one-step-ahead conditional Value at Risk forecast for a simulated financial distribution. Consistency of our estimators has been proved and illustrated through Monte Carlo simulations. We noticed that adding the noncrossing restriction during estimation improves accuracy of the resulting extreme conditional quantile estimator. Backtesting results from the one-step-ahead conditional Value at Risk forecasts indicate that independence and conditional coverage tests in  are not appropriate for our estimators due to dependence in autoregressive models.

6. ProofsProof of Lemma 1.

Let ζμt,θ=MθXt,μtMθXt,μt,θ, ζnμt,θ=1/nμt,θnζμt,θ, and ζ¯μt,θ=Eζnμt,θ. Note that, by Assumption 1, ζ¯nμt,θ does not depend on n. Since MθXt,μt does not depend on μt,θ, then μ^t,θargminμt1/nt=1nζμt,θ. We need to show that the objective function ζμt,θ satisfies the following conditions for application of Theorem 12.2 in :

ζμt,θ is measurable for each βB

ζμt,θ is continuous on B almost surely

a measurable function m:d such that

ζμt,θmYtβB

δ>0:EmYt1+δM

ζ¯μt,θ has a unique minimum at μt,θ

The functional form of ζμt,θ and Assumption 3 guarantees measurability of MθXt,μt,θ. To prove condition (2), we first show that ζμt,θ is Lipschitz continuous. By definition,(41)ζμt,θ=XtμtθIXtμt0Xtμt,θθIXtμt,θ0=θμt,θμtXtμtIXtμt0Xtμt,θIXtμt,θ0.

Considering the possible range Xt, we have the following:

For μt<Xt<μt,θ, it follows that IXtμt0=0 and IXtμt,θ0=1; hence, equation (41) reduces to(42)ζμt,θ=θμt,θμt+Xtμt,θ=θμt,θμt+Xtμt,θ+μtμt=Xtμt1θμt,θμt.

Since Xtμt>0 and Xtμt,θ<0, we have(43)1θμt,θμtζμt,θθμt,θμt,

and so ζμt,θ is bounded above by either 1θμt,θμt or θμt,θμt.

Similarly, when μtμt,θ<Xt, IXtμt0=0and IXtμt,θ0=0; hence,(44)ζμt,θ=θμt,θμt,

Xt<μtμt,θ; then IXtμt,θ0=1and IXtμt,θ1; hence,

(45)ζμt,θ=θ1μt,θμt.

Combining equations (43)–(45), we have(46)ζμt,θmaxθ,1θμt,θμtμtμt,θ.

Thus, ζμt,θ is Lipschitz continuous for K=1 and hence is differentiable almost everywhere by Rademacher’s theorem in  which implies that ζμt,θ is continuous everywhere.

To prove Condition 3(i), let mYt=YtLβ¯2θ+1, where β¯=maxβBβ. Existence of β¯ is guaranteed by Assumption 3. Clearly, mYt is measurable and ζμt,θmYt. Assumption 2 ensures that Condition 3(ii) is satisfied.

To verify Condition 4, let δ=μtμt,θ; we need to show that Eζμt,θ>0 for any δ>0. By Knight’s identity in , we have(47)Eζμt,θ=EMθXt,μtMθXt,μt,θ=Eμtμt,θIXtμt,θ0θ+0μtμt,θIXtμt,θtIXtμt,θ0dt=0μtμt,θFε|YtFε|Y0dt.

Thus, Eζμt,θ>0 for all δ>0 by monotonicity of the CDF. Therefore, μ^t,θμt,θ=op1 by Theorem 12.2 in .

Proof of Lemma 2.

The proof proceeds in a similar way to the proof of Lemma 1.

Proof of Lemma 3.

Observe that(48)supzF^zFz=supz1F^uG^λ,βzu+F^u1FuGλ,βzuFu=supzF^uFu+G^λ,βzuGλ,βzuF^uG^λ,βzu+FuGλ,βzu+F^uGλ,βzuF^uGλ,βzusupuF^uFu+supzG^λ,βzuGλ,βzuGλ,βzusupuF^uFuF^usupzG^λ,βzuGλ,βzu.

Note that supuF^uFu0 as n by Glivenko-Cantelli theorem in  and supzG^λ,βzuGλ,βzu0 as n heuristically from consistency of GPD parameters, Lemma 4.1 in , and asymptotic normality of PWM estimators in . Therefore, since 0Gλ,βzu1 and 0F^u1, we have the result.

Proof of Lemma 4.

Let ϵ>0 and δ=minFqαz+ϵα,αFqαzϵ and note that, for any CDF F defined on Fzk, if zF1k, then(49)Pq^τzqτz<ɛ=Pq^τz>qτz+ɛ=PF^qτz>F^qτz+ɛ=Pα>F^qτz+ɛ=PFqτz+ɛF^qτz+ɛ>Fqτz+ɛαPsupzFzF^z>δ,which tends to 0 as n by Lemma 3.

Proof of Theorem 2.

Combining Lemma 1, Lemma 2, and Lemma 4, we have the desired result.

Data Availability

The data used in the article were simulated, and the data generating process (DGP) is included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors acknowledge the Pan-African University of Basic Sciences, Technology and Innovation (PAUSTI) for funding this research.

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