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Convergence of a Randomised Change Point Estimator in GARCH Models Journal of Mathematical Finance, 2021, 11, 234-245 https://www.scirp.org/journal/jmf ISSN Online: 2162-2442 ISSN Print: 2162-2434 Convergence of a Randomised Change Point Estimator in GARCH Models George Awiakye-Marfo1, Joseph Mung’atu2, Patrick Weke3 1Department of Mathematics and Statistics, Pan African University, Nairobi, Kenya 2Department of Statistics and Actuarial Science, JKUAT, Nairobi, Kenya 3School of Mathematics, University of Nairobi, Nairobi, Kenya How to cite this paper: Awiakye-Marfo, Abstract G., Mung’atu, J. and Weke, P. (2021) Con- vergence of a Randomised Change Point In this paper, the randomised pseudolikelihood ratio change point estimator Estimator in GARCH Models. Journal of for GARCH models in [1] is employed and its limiting distribution is derived Mathematical Finance, 11, 234-245. as the supremum of a standard Brownian bridge. Data analysis to validate the https://doi.org/10.4236/jmf.2021.112013 estimator is carried out using the United states dollar (USD)-Ghana cedi Received: February 7, 2021 (GHS) daily exchange rate data. The randomised estimator is able to detect Accepted: May 9, 2021 and estimate a single change in the variance structure of the data and pro- Published: May 12, 2021 vides a reference point for historic data analysis. Copyright © 2021 by author(s) and Scientific Research Publishing Inc. Keywords This work is licensed under the Creative GARCH, Randomised, Limiting Distribution, Brownian Bridge, Volatility, Commons Attribution International License (CC BY 4.0). CUSUM, IGARCH, Supremum http://creativecommons.org/licenses/by/4.0/ Open Access 1. Introduction Volatility models are becoming increasingly important due to their role in asset pricing and risk management. It is however, not directly observed and hence needs to be estimated. Since the introduction of Autoregressive conditional He- teroscedacisity (ARCH) by [2] and its generalisation, Generalised ARCH (GARCH) by [3], these models have arguably been the most popular and used financial volatility models. It is, however, possible that structural changes such as “shocks” as a result of changes in institutions, financial crises, may cause the da- ta generating parameters of these models to change. Failure to accommodate these parameter changes in the conditional and unconditional volatility of a se- ries in the model may have serious impacts on the forecasting abilities of these models. In fact [4] and [5] showed that neglecting parameter changes in the ap- plication of GARCH(1,1) model to long time economic processes often lead to DOI: 10.4236/jmf.2021.112013 May 12, 2021 234 Journal of Mathematical Finance G. Awiakye-Marfo et al. high persistence and for this reason, the IGARCH model of [6] was introduced. For a comprehensive study on change point problems, we refer to [7]. In GARCH models, the problems of change point estimation have been studied by various authors after the variance change test by [8] for independent observa- tions. Some of the earliest work on GARCH change point estimation was by [9] based on the cumulative sum of squares of [8], and also the standardised residual based test of [10] with the aim of reducing large size distortions and low power of the CUSUM test of [9]. The authors showed that the proposed test statistic has a limiting distribution as the sup of a standard Brownian bridge via the inva- riance principle for mixingale sequences whilst in [11] the point processes theory was utilized to obtain a weak convergence limit of a model order change point process for GARCH models. The authors of [12] considered the problem of testing for parameter change in nonlinear time series with GARCH errors and showed that under some regularity assumptions, the limiting null distribution is a supremum of a standard Brownian Bridge. The authors of [13] proposed a weighted CUSUM test statistic to test for mean change in an AR(p) process and its limiting distribution was obtained via the mixing conditions for linear processes. In [14] estimates-based CUSUM change point test in ARMA-GARCH and non linear Autoregressive Conditional Duration (ACD) models were stu- died and the result of the quasi-maximum likelihood estimator (QMLE) from ACD models was used to derive the limiting null distribution of the CUSUM test. Later [15] considered a modified residual-based CUSUM test for loca- tion-scale time series models with heteroscedasticity and in both studies, the au- thors derived the limiting distribution derived as the supremum of a Brownian bridge. The change point problems for GARCH models in literature have usually been viewed as deterministic. However, in [1], a randomised change point test for GARCH models and its consistency were derived. In this paper, the limiting distribution of the randomised estimator of [1] is derived and validated via the United States dollar (USD)-Ghana cedi (GHS) exchange rate data. The idea of the randomised estimator is to weigh down excessively large observations and hence obtain test statistic values different from the described deterministic cases. Identifying discontinuities helps improve the forecasting abilities of GARCH models. This paper is organised into five sections. Section 1 is the Introduction, re- search methodology is presented in Section 2. In Section 3, the limiting distribu- tion of the estimator is derived. Section 4 presents results and discussions whilst Section 5 concludes the study. 2. Methodology Consider the model Xt= gX( t−−12, X t ,, X tp − ;θε t) + t (1) where the errors εσt= ttz , zt has zero expectation and finite variance, σ t DOI: 10.4236/jmf.2021.112013 235 Journal of Mathematical Finance G. Awiakye-Marfo et al. follows a GARCH (p,q) model, for instance a standard GARCH model. The conditional mean, gX( t−−12, X t ,, X tp − ;θ t) follows an autoregressive function. The test statistic for the change point is constructed by employing the likelihood ratio and derived as follows; 22 2 −2log ∆=kn logσσˆ nk − k log ˆ −( nk −)log σˆ k* 22 consider σˆkk* = σδˆ + and under the null hypothesis take δ → 0 as n →∞, we have 2 σˆ 2 σσˆ 22− ˆ H − ∆=n = +nk = + k logk log222 log 1 log 1 (2) n σσσˆkkkˆ ˆ where the variance estimates are given as 1 n 2 σθˆ 2 = − ˆ n(h) n ∑ hX( t)( X tn( h)) , t=1 ∑t=1 hX( t ) 1 n 2 σθˆ 2 = − ˆ k**(h) n ∑ hX( t)( X tk( h)) , tk= +1 ∑tk= +1 hX( t ) 1 kn22 σ2 = −+ θθˆ −ˆ ˆk(h) n ∑∑hX( t)( X tn( h)) hX( t)( X tk* ( h)) . t=11tk= + ∑t=1 hX( t ) A detailed simplification of Equation (2), and an expression for Hk can be found in [1]. The modified weighted test statistic is of the form; υ n 2 h k (∑t=1 t ) 1 ∑ ht = − t=1 en max Sk( ) n Sn( ) (3) 0<<kn kn k n ∑∑=hhtt = − ∑= h t ∑t=1 ht tt11( t 1) k n with υ ∈(0,12) where S( k) = ∑ hXtt and S( n) = ∑ hXtt t=1 t=1 Finally, the test statistic as given in [1] is υ 2 n k kn (∑t=1 ht ) h 1 22∑t=1 t en= max hhttεε− tt (4) 1≤<knkn k ∑∑n n tt=11= ∑∑=hhtt = − ∑= h t ∑t=1 ht tt11( t 1) ˆ with υ ∈(0,12) and the estimator given as ke= arg max1≤<kn k. 3. Limiting Distribution of the Estimator Here we show the limiting distribution of the randomised estimator as described 2 in Equation (4). We note from [16] that the GARCH model, σ t is an α -mixing 2 process, so is ht and hence httσ by hereditary. Assumption 1. The functions ht= gX( t−−12, X t ,, X tp − , X t ++12 , X t ,, X tp +) 2 p and gX( 12, X ,, X 2p ) are real and positive functions on such that +2+α +2+α <∞ α > {(1 Xt)( hh tt)} for 0 . 2 2 We note that since εt is not directly observable we replace with εˆt where 22 εεˆtt= +Λ t and Λ=tPo (1) has been established in [12]. DOI: 10.4236/jmf.2021.112013 236 Journal of Mathematical Finance G. Awiakye-Marfo et al. Proposition 1. If assumption (1) holds then under the null hypothesis of no 0 2 στB ( ) σε2(htt) w → 1 − 0 τ change in variance we have eWn υυυ +1 ( ) as (ττ(11−−)) (ht ) ( ττ( )) n →∞ where w → defines the weak convergence of the process, σ2 = +∞ εε≠ σ 2 = +∞ ≠ 0 τ 1 ∑t=−∞ Cov( h00,0 htt) and 20∑t=−∞ Cov( h,0 ht ) . B ( ) and W 0 (τ ) denote standard brownian motions on [0,1] . Proof. Considering Equation (4) we make the following representation; −−υυ kn ∑∑hX( tt) hX( ) A = t=11 tk= + and k nn ∑∑tt=11hX( tt) = hX( ) kεε22 kn ∑t=1hX( tt) ∑∑ tt= 11 hX( t) = hX( tt) Bk = υυ− nnn ∑ = hX( t ) (∑∑tt=11hX( tt)) t 1 ( = hX( )) We re-write 1 k εε22− υ ∑t=1(hhtt( tt)) n Bk = υ 1 n ∑ ht n t=1 k nεε22 nk ∑∑t=1hX( t) t= 1 hX( tt) ∑∑ tt= 11 hX( t) = hX( tt) −−υυ nknn ∑∑= hX( tt) = hX( ) tt11(∑∑tt=11hX( tt)) ( = hX( )) :=LL12 − We consider L2 as follows 1 k k εε22− υ ∑t=1(hhtt( tt)) ∑ hX( t ) L = t=1 n 2 n υ hX( ) 1 n ∑t=1 t ∑ = ht n t 1 n 2 ε nk2 υ (htt) hX hX ε n ∑∑tt=11( t) = ( tt) +−υυ k n 1 n ∑ hX( t ) t=1 ∑ = hX( t ) ∑ = ht ( t 1 ) n t 1 k ∑ hX( t ) = t=1 LLˆ + ˆ n ( 12) ∑t=1 hX( t ) ˆ For L2 we have ε 2 nk2 nh( tt) ∑∑hX( t) hX( tt)ε ˆ = − tt=11= L2 υυk nn υυ11∑t=1 hX( t ) nh∑∑tt nh nntt=11 = h ε 2 n ( tt) n ∑t=1 hX( t ) = υυ− n 1 k υ 1 n hX( ) n ∑ ht ∑t=1 t n t=1 k DOI: 10.4236/jmf.2021.112013 237 Journal of Mathematical Finance G.
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