# Randomised Pseudolikelihood Ratio Change Point Estimator in Garch Models

Authors: George Awiakye-Marfo; Joseph Mung’atu; Patrick O. Weke
Journal: Journal of Mathematics
 (2020)

---

## Abstract

In this paper, a randomised pseudolikelihood ratio change point estimator for GARCH model is presented. Derivation of a randomised change point estimator for the GARCH model and its consistency are given. Simulation results that support the validity of the estimator are also presented. It was observed that the randomised estimator outperforms the ordinary CUSUM of squares test, and it is optimal with large variance change ratios.


---

## Body

## 1. Introduction

Volatility plays a very important role in financial derivatives; hence, a good prediction of volatility will provide a more accurate pricing model for financial assets. The autoregressive integrated moving average (ARIMA) models developed by Box and Jenkins [1] assume a constant conditional variance of the errors; however, financial data do not obey this assumption. For this reason, Engle [2] proposed the Autoregressive Conditional Heteroscedastic (ARCH) model; this model has found wide applications in finance modeling. Bollerslev [3] generalised the ARCH model and called it the GARCH (Generalised ARCH) model. Other variations of the GARCH model [4, 5], among others referred to as Assymetric GARCH, have been developed to address some limitations of the standard GARCH model. However, due to structural changes, modeling economic processes over long periods of time may undermine the important assumption of stationarity in these models. In fact, Lamoureux and Lastrapes and Hillebrand [6, 7] reported that the application of GARCH models to long time series of stock returns yields a high measure of persistence because of the presence of shifts in the data generating parameters of these models, leading to false results. The problem of structural changes prompted the idea of change point detection and estimation and has found applications in quality control, climate change, finance, etc. For example, Hsu [8] studied a single change in variance in a sequence of independent random variables. Later, Inclan and Tiao [9] explored variance change for independent observations to detect multiple changes. Chen and Gupta [10], among several other authors, have explored variance change in independent variables and counts [11]. For early works and in-depth reviews, we refer to [12]. For the GARCH models, Kim et al. [13] proposed an analogy of [9] test statistic and derived its limiting distribution as a supremum of a Brownian bridge. Kokoszka and Leipus [14] proposed a similar test for which the main difference was to analyse the existence of structural break in the unconditional variance of an ARCH(p) model. To address the problem of size distortions as reported by Lee et al. [15] on Kim et al. [13], the authors based their test on standardised residuals instead of residuals. Later, Lee et al. [16] studied variance change based on Inclan and Tiao [9] test for errors in ARp models and kernel-type estimator regression model. Lee and Lee [17] considered parameter change problem in nonlinear time series models with GARCH errors via [9] test statistic. Zhou and Liu [18] proposed a weighted CUSUM test statistic to test for mean change in an AR(p) process. The change point problem for variance change has usually been viewed as deterministic. Contrary to this notion, we study a randomised change point estimator in GARCH models, where we allow the test statistic to be data driven.


## 2. Methodology

We derive the test statistic for change point detection in GARCH processes, under the null hypothesis of no change, and the alternative that the variance has changed at some unknown time,k. Let the observation Yt, t=1,…,n, be a time series fulfilling Yt1,t≤k,andYt2,t>k, and the two series are based on stationary time series. Consider the model(1)Yt=fYt−1,Yt−2⋯Yt−p;θt+εt,where the errors εt=σtzt and zt has zero mean and finite second moment. In our study, we consider an autoregressive model for the conditional mean of Yt, fYt−1,Yt−2,…,Yt−p;θt, and σt2 is approximated by the standard GARCH (p, q) model as(2)σt2=ω+∑i=1qαiεt−i2+∑j=1pβjσt−j2,t∈ℤ,where ω>0, α≥0, and β≥0. For mean estimates, we focus on procedures based on M-procedures of [19], where the parameter estimator θ^t is a solution to the equation(3)∑t=1nψYt,θt=0,such that ψ2Yt,θ<∞ and ψYt,θt is nondecreasing in θt. Consider ψx,θ=hyy−θ and also define(4)Fhn,θ≔∑t=1nhYtYt−θn.Proposition 1.
To obtain the residuals,εt, we consider a weighted sum of least-squares estimate for the parameter, θn, under H0 as follows:(5)∑t=1nhYtYt−θn2=minθ∑t=1nhYtYt−θn2.
Then,(6)θ^nh=∑t=1nhYtYt∑t=1nhYt.Likewise, the weighted least squares estimators underH1 after and before the change point, respectively, are(7)θ^k∗h=∑t=k+1nhYtYt∑t=k+1nhYt,θ^kh=∑t=1khYtYt∑t=1khYt.To construct the test statistic, we employ the likelihood ratio test derived as(8)−2logΔk=nlogσ^n2−klogσ^k2−n−klogσ^k∗2.Takeσ^k∗2=σ^k2+δ, and under the null hypothesis consider the case δ⟶0 as n⟶∞, and we have(9)−2nlogΔk=logσ^n2σ^k2=log1+σ^n2−σ^k2σ^k2=log1+Hkσ^k2,where(10)σn2h=∑t=1nhYtYt−θ^nh2,σk2h∑t=1khYtYt−θ^kh2,σk∗2h=∑t=k+1nhYtYt−θ^k∗h2,and the variance estimates are now given as(11)σ^n2h=1∑t=1nhYt∑t=1nhYtYt−θ^nh2,σ^k∗2h=1∑t=k+1nhYt∑t=k+1nhYtYt−θ^k∗h2,σ^k2h=1∑t=1nhYt∑t=1khYtYt−θ^nh2+∑t=k+1nhYtYt−θ^k∗h2.SimplifyingHk, we have(12)Hk=σ^n2h−σ^k2h=1∑t=1nht∑t=1khtθ^k−θ^nYt−θ^k+Yt−θ^n+1∑t=1nht∑t=k+1nhtθ^k∗−θ^nYt−θ^k∗+Yt−θ^n.Substitutingθ^n, θ^k, and θ^k∗ into equation (12), we have(13)Hk=Sk−∑t=1kht∑t=1nhtSn×1∑t=1kht∑t=1nht∑t=1khtYt−θ^k−1∑t=1nht∑t=1nht−∑t=1kht∑t=k+1nhtYt−θ^k∗+1∑t=1kht∑t=1nht∑t=1khtYt−θ^n−1∑t=1nht∑t=1nht−∑t=1kht∑t=k+1nhtYt−θ^n≔I+J.Noting thatθ^k∗=θ^k+δ, as δ⟶0, we simplify I and J as follows:(14)I=1∑t=1kht∑t=1nht∑t=1khtYt−1∑t=1nht∑t=1nht−∑t=1kht∑t=k+1nhtYt=1∑t=1kht∑t=1nht−∑t=1khtSk−∑t=1kht∑t=1nhtSn,J=1∑t=1kht∑t=1nht∑t=1khtYt−1∑t=1nht∑t=1nht−∑t=1kht∑t=k+1nhtYt=1∑t=1kht∑t=1nht−∑t=1khtSk−∑t=1kht∑t=1nhtSn.Finally, under the likelihood ratio, we have(15)−2logΔ˜k=∑t=1nhtσ^k2∑t=1kht∑t=1nht−∑t=1khtSk−∑t=1kht∑t=1nhtSn2.Hence, the modified weighted form of the test statistic is of the form(16)e˜n=max0<k<n∑t=1nht2∑t=1kht∑t=1nht−∑t=1khtυ1nSk−∑t=1kht∑t=1nhtSnwithυ∈0,12whereSk=∑t=1khtYtandSn=∑t=1nhtYt.In this paper, we consider observationsY1,Y2,…,Yn with α−mixing errors. Define SGk;θ=∑t=1khtGYt;θandSGn;θ=∑t=1nhtGYt;θ, where GYt;θ=gεt, and in the case of variance change, we take gεt=εt2=Yt−θ2. Finally, we have(17)e˜n=max1≤k<n∑t=1nht2∑t=1kht∑t=1nht−∑t=1khtυ1n∑t=1khtεt2−∑t=1kht∑t=1nht∑t=1khtεt2,with υ∈0,1/2, and the estimator is given as k^=argmax1≤k<ne˜k.


## 3. Consistency of the Estimator

In this section, we establish the consistency of the randomised estimator. The general model involves a GARCH(p,q) model with a possible shift at an unknown time,k. We make the assumption below for the weight, ht.Assumption 1.
The functionsht=gYt−1,Yt−2,…,Yt−p,Yt+1,Yt+2,…,Yt+p and gY1,Y2,…,Y2p are real and positive functions on ℝ2p such that 1+Yt2+αht+ht2+α<∞ for α>0.
We rewrite the test statistic of equation (17) as(18)e˜n=max1≤k<n1∑t=1khYt/∑t=1nhYt∑t=k+1nhYt/∑t=1nhYtυ1n∑t=1khYtεt2∑t=1nhYt−∑t=1khYt∑t=1nhYt∑t=1nhYtεt2∑t=1nhYt∑t=1nhYt.
Under the null hypothesis of no variance change, we have(19)∑t=1khYtεt2∑t=1nhYt=kn1/k∑t=1khYtεt21/n∑t=1nhYt=knσ2/k∑t=1khYtzt21/n∑t=1nhYt⟶Pτσ2hYtzt2hYt.
Also,(20)∑t=1khYt∑t=1nhYt∑t=1nhYtεt2∑t=1nhYt=kn1/k∑t=1khYt1/n∑t=1nhYt1/n∑t=1nhYtεt21/n∑t=1nhYt⟶Pτσ2hYtzt2hYt.
Subtracting equation (19) from equation (20), the difference ⟶P0
Under the alternative hypothesis of a change in varianceσ12≠σ22, we have(21)∑t=1khYtεt2∑t=1nhYt=kn1/k∑t=1khYtεt21/n∑t=1nhYt=knσ12/k∑t=1khYtzt21/n∑t=1nhYt⟶Pτσ12hYtzt2hYt,(22)∑t=1khYt∑t=1nhYt∑t=1nhYtεt2∑t=1nhYt=kn1/k∑t=1khYt1/n∑t=1nhYtσ12/k0k0/n∑t=1khYtzt21/n∑t=1nhYt+σ22/n−k0n−k0/n∑t=1khYtzt21/n∑t=1nhYt=ττ0σ12λ+τ1−τ0σ22λ,whereλ=∑t=1khYtzt2∑t=1nhYt.
Subtracting equation (21) from equation (22), the difference ⟶P1−τ0τλσ12−σ22. Let us also define e^n=e˜n/∑t=1nhYt1/2
Then, the weight function of equation (18) leads to(23)max1≤k<n1∑t=1khYt/∑t=1nhYt∑t=k+1nhYt/∑t=1nhYtυ∑t=1khYtn1/2⟶Pτ−υ1−τ−υhYt1/2.
Therefore,(24)e^n⟶Pτ1−υ1−τ−υhYt1/21−τ0λσ12−σ22,forτ≤τ0,τ−υ1−τ1−υhYt1/2τ0λσ12−σ22,forτ>τ0.
Takee¯k=limn⟶∞e¯n, λ0=hYt1/2, and ρ=σ12−σ22, and we have(25)e¯k⟶Pτ1−υ1−τ−υ1−τ0λλ0ρ,forτ≤τ0,τ−υ1−τ1−υτ0λλ0ρ,forτ>τ0.
Then, forτ=τ0, we obtain(26)e¯k0=τ01−υ1−τ01−υλλ0ρ.
Also,(27)e¯k0−e^n⟶P1−τ01−υτ01−υ−τ1−υ1−τ01−τυλ0λρ,forτ≤τ0,τ01−υ1−τ01−υ−1−τ1−υτ0τυλ0λρ,forτ>τ0,≥1−τ01−υτ01−υ−τ1−υλ0λρ,forτ≤τ0,τ01−υ1−τ01−υ−1−τ1−υλ0λρ,forτ>τ0.
Forτ≤τ0, take gτ=τα and α=1−υ, and applying the mean value theorem, we obtain(28)τ01−υ−τ1−υ≥1−υτ0−υτ0−τ.
Similarly, forτ>τ0, take Hτ=1−τ1−υ and apply the mean value theorem to obtain(29)1−τ01−υ−1−τ1−υ≥1−υ1−τ0−υτ0−τ.
Putting equations (28) and (29) in equation (27), we obtain(30)e¯k0−e^n≥1−τ01−υ1−υτ0−υτ0−τλ0λρ,forτ≤τ0,τ01−υ1−υ1−τ0−υτ−τ0λ0λρ,forτ>τ0,≥τ¯τ0−τλ0λρ,whereτ¯=1−τ01−υ1−υτ0−υ,forτ≤τ0,τ01−υ1−υ1−τ0−υ,forτ>τ0.
We note that(31)e¯k−e¯k0≤2maxnδ≤k≤n1−δe^k−e¯k+e^k0−e^k.
Replacingτ by τ^n and noting that e^k0≤e^nτ^, we have(32)τ¯τ0−τ^n≤e¯k0−e¯k≤2maxnδ≤k≤n1−δe^k−e¯k+e¯k−e¯k0≤2maxnδ≤k≤n1−δe^k−e¯k.
Let us consider(33)e^k−e¯k≤M∑t=1khYt∑t=1nhYt−υ∑t=k+1nhYt∑t=1nhYt−υ−τ−υ1−τ−υλ0+τ−υ1−τ−υMλ0−τ1−τ0λλ0ρ≔S1+S2,where M=∑t=1khYtεt2/∑t=1nhYt−∑t=1khYt/∑t=1nhYt∑t=1nhYtεt2/∑t=1nhYt,(34)S1=M∑t=1khYt∑t=1nhYt−υ∑t=k+1nhYt∑t=1nhYt−υ−τ−υ1−τ−υλ0,S2=τ−υ1−τ−υMλ0−τ1−τ0λλ0ρ.
MaximisingS1, we obtain(35)S1≤M∑t=k+1nhYt∑t=1nhYt−υ∑t=1khYt∑t=1nhYt−υ−τ−υλ0+M∑t=1khYt∑t=1nhYt−υ∑t=k+1nhYt∑t=1nhYt−υ−1−τ−υλ0.
Furthermore, we have(36)S1≤λ0M∑t=k+1nhYt∑t=1nhYt−υ∑t=1khYt∑t=1nhYt−υ−τυ+λ0Mτ−υ∑t=k+1nhYt∑t=1nhYt−υ−1−τ−υ.
We apply the mean value theorem to equation (36) by assuming fx=x−τ at ∑t=1khYt/∑t=1nhYt−τ and ∑t=k+1nhYt/∑t=1nhYt−υ, ∀x∈0,1, to obtain(37)S1≤−υλk1−υ−1λ0M∑t=k+1nhYt∑t=1nhYt−υ∑t=1khYt∑t=1nhYt−τ+−υλk2−υ−1λ0Mτυ∑t=k+1nhYt∑t=1nhYt−1−τ≔K1+K2,where λk1⟶τ and λk2⟶1−τ, such that(38)ℙmaxnδ≤k≤k0λk1−τ>ε≤ℙmaxk>nδλk1−τ>ε⟶P0.Proposition 2.
K1 is given by −υλk1−υ−1λ0M∑t=k+1nhYt/∑t=1nhYt−υ∑t=1khYt/∑t=1nhYt−τ and converges to Ψa/n.Proof.
(39)ℙmaxnδ≤k≤k0K1>ε≤ε+ℙmaxnδ≤k≤k0∑t=1khYt∑t=1nhYt−τ>αε,αε=∑t=k+1nhYt∑t=1nhYtυελk1υ+1λ0−1M−1,ℙmaxnδ≤k≤k0K1>ε≤ε+ℙmaxnδ≤k≤k01k∑t=1khYt−hYt>αε1−δhYt.
We apply Theorem 4.1 of [14] to equation (39) as follows:(40)ℙmaxnδ≤k≤k01k∑t=1khYt−hYt>αε1−δhYt≤1n2δ2∑i,j=1nδCovi,j+∑k=nδk0−11k2−1k+12∑i,j=1kCovi,j+2∑k=nδk0−11k+12Covk+1,k+11/2∑i,j=1kCovi,j1/2+∑k=nδk0−11k+12Covk+1,k+1≤C1nδ+∑k=nδk0−11k2+∑k=nδk0−11k2k1/2+∑k=nδk0−11k2≤Cδn≤Ψan.Proposition 3.
K2 is given by −υλk2−υ−1λ0Mτυ∑t=k+1nhYt/∑t=1nhYt−1−τ and converges to Ψb/n.Proof.
(41)ℙmaxnδ≤k≤k0K2>ε≤ε+ℙmaxnδ≤k≤k0∑t=K+1nhYt∑t=1nhYt−1−τ>βεβε=ετυλk2υ+1λ0−1M−1,ℙmaxnδ≤k≤k0K2>ε≤ε+ℙmaxnδ≤k≤k01n−k∑t=k+1nhYt−hYt>βεnn−khYt≤ℙmaxnδ≤k≤k01n−k∑t=1khYt−hYt>βε1−δhYt.
Next, we apply Theorem 4.1 of [14] to equation (41) as follows:(42)ℙmaxnδ≤k≤k0K2>ε≤1n2δ2∑i,j=1nδCovi,j+∑k=nδk0−11n−k+12−1n−k2∑i,j=1kCovi,j+2∑k=nδk0−11n−k+12Covk+1,k+11/2∑i,j=1kCovi,j1/2+∑k=nδk0−11n−k+12Covk+1,k+1≤δC1n1−δ+C2∑k=nδk0−11n−k2+C3∑k=nδk0−11n−k3/2+C4∑k=nδk0−11n−k2.
Simplifying further, we obtain(43)ℙmaxnδ≤k≤k0K2>ε≤C1n1−δ2+∑k=nδk0−11n−k2+∑k=nδk0−11n−k3/2+∑k=nδk0−11n−k2≤Ψbn.
Similarly,(44)ℙmaxnδ≤k≤k0S2>ε≤Ψcn.
Finally, combiningS1 and S2, we have(45)ℙe^k−e¯k>ε≤Ψen.
Consequently, from equation (32), we have(46)ℙmaxnδ≤k≤n1−δτ0−τ^n≤Ψen⟶P0.
This completes the proof of consistency of the randomised estimator.


## 4. Simulation Study

We present simulation results to illustrate the validity of the estimator. Data were generated from the model,εt=σtzt, where zt∼N0,1, and σt2 is from a standard GARCH (1, 1) model as given in equation (2). The choice of weight htYt,θ greatly improves the efficiency of the estimator. We, however, consider the function of equation (47) in this paper:(47)htYt,θ⟶1,forεt≤B,Bεt,forεt>B,B>0.The number of replications is 1000 with sample sizes set at 500 to 4000. We present in Figures1 and 2 a simulation study of the estimator.Figure 1
Graphs showing changes in volatility: (a) graph showing an early, middle, and late change with a 5% change in volatility of the True(Tru), ordinary CUSUM(Lee) of [20], and the Randomised(Ran) estimator and (b) graph showing an early, middle, and late change with a 200% change in volatility of the True(Tru), ordinary CUSUM(Lee) of [20], and the Randomised(Ran) estimator.
(a)(b)Figure 2
Graph showing an early, middle, and late change with a 400% change in volatility of the True(Tru), ordinary CUSUM(Lee) of [20], and Randomised(Ran) estimator.The graphs of Figures1 and 2 show the mean estimate of test of [20] (Lee), the randomised(Ran) estimator and the True(Tru) value of the estimator. It can be seen that both estimators tend to perform well when the change occurs in the middle of the sample. They, however, tend to underperform when the change occurs at the early (first quarter) or late (third quarter) of the sample, although the randomised estimator outperforms the ordinary CUSUM test of [20]. The estimator is shown to be consistent with increasing sample size, as shown theoretically in Section 3, for situations where there is an early, mid, or late change in the variance structure of the data. The efficiency of the estimator greatly improves with larger variance change ratios.


## 5. Conclusion

In this paper, a randomised change point estimator for GARCH Models is presented. A simulation result of the estimator was carried out and compared to the estimator of [20]. We noticed that depending on the weight function chosen the randomised estimator outperforms the ordinary CUSUM of [20] especially when there is an early or late change in the sample. The consistency of the estimator was also established.



---

*Source: 6671515-2020-12-29.xml*