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 Heteroscedacisity (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 data generating parameters of these models to change. Failure to accommodate these parameter changes in the conditional and unconditional volatility of a series 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 application of GARCH(1,1) model to long time economic processes often lead to 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 observations. 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 invariance 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 studied 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 location-scale time series models with heteroscedasticity and in both studies, the authors 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, research methodology is presented in Section 2. In Section 3, the limiting distribution of the estimator is derived. Section 4 presents results and discussions whilst Section 5 concludes the study.

Consider the model

X t = g ( X t − 1 , X t − 2 , ⋯ , X t − p ; θ t ) + ε t (1)

where the errors ε t = σ t z t , z t has zero expectation and finite variance, σ t follows a GARCH (p,q) model, for instance a standard GARCH model. The conditional mean, g ( X t − 1 , X t − 2 , ⋯ , X t − p ; θ t ) follows an autoregressive function. The test statistic for the change point is constructed by employing the likelihood ratio and derived as follows;

− 2 log Δ k = n log σ ^ n 2 − k log σ ^ k 2 − ( n − k ) log σ ^ k * 2

consider σ ^ k * 2 = σ ^ k 2 + δ and under the null hypothesis take δ → 0 as n → ∞ , we have

− 2 n log Δ k = log ( σ ^ n 2 σ ^ k 2 ) = log ( 1 + σ ^ n 2 − σ ^ k 2 σ ^ k 2 ) = log ( 1 + H k σ ^ k 2 ) (2)

where the variance estimates are given as

σ ^ n 2 ( h ) = 1 ∑ t = 1 n h ( X t ) ∑ t = 1 n     h ( X t ) ( X t − θ ^ n ( h ) ) 2 ,

σ ^ k * 2 ( h ) = 1 ∑ t = k + 1 n h ( X t ) ∑ t = k + 1 n     h ( X t ) ( X t − θ ^ k * ( h ) ) 2 ,

σ ^ k 2 ( h ) = 1 ∑ t = 1 n h ( X t ) ( ∑ t = 1 k     h ( X t ) ( X t − θ ^ n ( h ) ) 2 + ∑ t = k + 1 n     h ( X t ) ( X t − θ ^ k * ( h ) ) 2 ) .

A detailed simplification of Equation (2), and an expression for H k can be found in [1]. The modified weighted test statistic is of the form;

e ˜ n = max 0 < k < n ( ( ∑ t = 1 n h t ) 2 ∑ t = 1 k h t ( ∑ t = 1 n h t − ∑ t = 1 k h t ) ) υ 1 n | S ( k ) − ( ∑ t = 1 k h t ∑ t = 1 n h t ) S ( n ) | (3)

with υ ∈ ( 0, 1 / 2 ) where S ( k ) = ∑ t = 1 k     h t X t and S ( n ) = ∑ t = 1 n     h t X t

Finally, the test statistic as given in [1] is

e ˜ n = max 1 ≤ k < n ( ( ∑ t = 1 n h t ) 2 ∑ t = 1 k h t ( ∑ t = 1 n h t − ∑ t = 1 k h t ) ) υ 1 n | ∑ t = 1 k     h t ε t 2 − ( ∑ t = 1 k h t ∑ t = 1 n h t ) ∑ t = 1 n     h t ε t 2 | (4)

with υ ∈ ( 0, 1 / 2 ) and the estimator given as k ^ = arg max 1 ≤ k < n e ˜ k .

Here we show the limiting distribution of the randomised estimator as described in Equation (4). We note from [16] that the GARCH model, σ t 2 is an α -mixing process, so is h t and hence h t σ t 2 by hereditary.

Assumption 1. The functions h t = g ( X t − 1 , X t − 2 , ⋯ , X t − p , X t + 1 , X t + 2 , ⋯ , X t + p ) and g ( X 1 , X 2 , ⋯ , X 2 p ) are real and positive functions on ℝ 2 p such that E { ( 1 + ‖ X t ‖ 2 + α ) ( h t + h t 2 + α ) } < ∞ for α > 0 .

We note that since ε t 2 is not directly observable we replace with ε ^ t 2 where ε ^ t 2 = ε t 2 + Λ t and Λ t = o P ( 1 ) has been established in [12].

Proposition 1. If assumption (1) holds then under the null hypothesis of no change in variance we have e ˜ n → w σ 1 B 0 ( τ ) ( τ ( 1 − τ ) ) υ − σ 2 E ( h t ε t 2 ) E ( h t ) υ + 1 ( τ ( 1 − τ ) ) υ W 0 ( τ ) as n → ∞ where → w defines the weak convergence of the process, σ 1 2 = ∑ t = − ∞ + ∞     C o v ( h 0 ε 0 , h t ε t ) ≠ 0 and σ 2 2 = ∑ t = − ∞ + ∞     C o v ( h 0 , h t ) ≠ 0 . B 0 ( τ ) and W 0 ( τ ) denote standard brownian motions on [ 0,1 ] .

Proof. Considering Equation (4) we make the following representation;

A k = ( ∑ t = 1 k h ( X t ) ∑ t = 1 n h ( X t ) ) − υ ( ∑ t = k + 1 n h ( X t ) ∑ t = 1 n h ( X t ) ) − υ and

B k = | ∑ t = 1 k h ( X t ) ε t 2 ( ∑ t = 1 n h ( X t ) ) υ − ∑ t = 1 k h ( X t ) ∑ t = 1 n h ( X t ) ∑ t = 1 n h ( X t ) ε t 2 ( ∑ t = 1 n h ( X t ) ) υ |

We re-write

B k = 1 n υ ∑ t = 1 k ( h t ε t 2 − E ( h t ε t 2 ) ) ( 1 n ∑ t = 1 n h t ) υ                 − ∑ t = 1 k h ( X t ) ∑ t = 1 n h ( X t ) | ∑ t = 1 n h ( X t ) ε t 2 ( ∑ t = 1 n h ( X t ) ) υ − ∑ t = 1 n h ( X t ) ∑ t = 1 k h ( X t ) ∑ t = 1 k h ( X t ) ε t 2 ( ∑ t = 1 n h ( X t ) ) υ |           : = L 1 − L 2

We consider L 2 as follows

L 2 = ∑ t = 1 k h ( X t ) ∑ t = 1 n h ( X t ) ( 1 n υ ∑ t = 1 k ( h t ε t 2 − E ( h t ε t 2 ) ) ( 1 n ∑ t = 1 n h t ) υ     + n n υ E ( h t ε t 2 ) ( 1 n ∑ t = 1 n h t ) υ − ∑ t = 1 n h ( X t ) ∑ t = 1 k h ( X t ) ∑ t = 1 k h ( X t ) ε t 2 ( ∑ t = 1 n h ( X t ) ) υ ) = ( ∑ t = 1 k h ( X t ) ∑ t = 1 n h ( X t ) ) ( L ^ 1 + L ^ 2 )

For L ^ 2 we have

L ^ 2 = n E ( h t ε t 2 ) n υ ( 1 n ∑ t = 1 n h t ) υ − ∑ t = 1 n h ( X t ) ∑ t = 1 k h ( X t ) ∑ t = 1 k h ( X t ) ε t 2 n υ ( 1 n ∑ t = 1 n h t ) υ = E ( h t ε t 2 ) ( 1 n ∑ t = 1 n h t ) υ ( n n υ − ∑ t = 1 n h ( X t ) 1 k ∑ t = 1 k h ( X t ) n υ )

= E ( h t ε t 2 ) ( 1 n ∑ t = 1 n h t ) υ ( 1 n υ ( n k ∑ t = 1 k h t − ∑ t = 1 n h t ) 1 k ∑ t = 1 k h t ) = E ( h t ε t 2 ) τ ( 1 n ∑ t = 1 n h t ) υ ( 1 n υ ∑ t = 1 k ( h t − E h t + E h t ) − τ n υ ∑ t = 1 n ( h t − E h t + E h t ) 1 k ∑ t = 1 k h t ) (5)

where τ = k n hence Equation (5) becomes

L ^ 2 = E ( h t ε t 2 ) τ ( 1 n ∑ t = 1 n h t ) υ ( 1 n υ ∑ t = 1 k ( h t − E h t ) − τ n υ ∑ t = 1 n ( h t − E h t ) 1 k ∑ t = 1 k h t )

Consequently we have

L 2 = τ [ 1 n υ ∑ t = 1 k ( h t ε t 2 − E ( h t ε t 2 ) ) ( 1 n ∑ t = 1 n h t ) υ               + E ( h t ε t 2 ) τ ( 1 n ∑ t = 1 n h t ) υ ( 1 n υ ∑ t = 1 k ( h t − E h t ) − τ n υ ∑ t = 1 n ( h t − E h t ) 1 k ∑ t = 1 k h t ) ]

From Equation (4) we have

e ˜ n = ( ∑ t = 1 k h ( X t ) ∑ t = 1 n h ( X t ) ) − υ ( ∑ t = k + 1 n h ( X t ) ∑ t = 1 n h ( X t ) ) − υ ( 1 n ∑ t = 1 k     h t ) υ             × [ 1 n ∑ t = 1 k ( h t ε t 2 − E ( h t ε t 2 ) ) ( 1 n ∑ t = 1 n h t ) υ ] − τ ( 1 n ∑ t = 1 k ( h t ε t 2 − E ( h t ε t 2 ) ) ( 1 n ∑ t = 1 n h t ) υ             + E ( h t ε t 2 ) τ ( 1 n ∑ t = 1 n h t ) υ ( 1 n ∑ t = 1 k ( h t − E h t ) − τ n ∑ t = 1 n ( h t − E h t ) 1 k ∑ t = 1 k h t ) )

We evaluate the limit of the function A k as

( ∑ t = 1 k h ( X t ) ∑ t = 1 n h ( X t ) ) − υ ( ∑ t = k + 1 n h ( X t ) ∑ t = 1 n h ( X t ) ) − υ ( 1 n ∑ t = 1 n     h t ) υ → w E ( h t ) υ ( τ ( 1 − τ ) ) υ

Also from L 1 and L 2 we have

L 1 → w σ 1 B ( τ ) E ( h t ) υ   and   L 2 → w τ σ 1 B ( 1 ) E ( h t ) υ + σ 2 E ( h t ε t 2 ) E ( h t ) υ + 1 ( W ( τ ) − τ W ( 1 ) )

by the invariance principle for dependent variables of [17].

| L 1 − L 2 | = σ 1 E ( h t ) υ ( B ( τ ) − τ B ( 1 ) ) − σ 2 E ( h t ε t 2 ) E ( h t ) υ + 1 ( W ( τ ) − τ W ( 1 ) )

Finally, we have

e ˜ n → w σ 1 B 0 ( τ ) ( τ ( 1 − τ ) ) υ − σ 2 E ( h t ε t 2 ) E ( h t ) ( τ ( 1 − τ ) ) υ W 0 ( τ ) (6)

where B 0 ( τ ) and W 0 ( τ ) denote standard brownian motions on [ 0,1 ] generated by { h t ε t 2 } t = 1 ∞ and { h t } t = 1 ∞ respectively.

We show an analysis of the United States Dollar (USD)-Ghana Cedi(GHS) daily exchange rate from 2008-2019 to illustrate the validity of the estimator. We consider the function of Equation (7) in our study.

h t ( X t , θ ) → ( 1   for     ε t ≤ B B | ε t |   for     ε t > B         B > 0 (7)

In this paper B is chosen to be the 90th quantile of the residual data, ε t .

The negative value of skewness and the small Jarque-Bera test value in Table 1 indicates the return data is not symmetric. The smaller p-value of the Arch test also indicates the presence of Arch effect.

Figure 1 shows the log return of the exchange rate data, the graph displays the typical “stylized fact” such as volatility clustering present in financial log-return series.

In this paper, a randomised change point estimator for GARCH Models is presented. The limiting distribution of the estimator is derived as the sup of a standard Brownian bridge. Data analysis of the estimator was carried using United States Dollar (USD)-Ghana Cedi (GHS) daily exchange rate data. It was observed that the randomised estimator was able to detect and estimate a single change in variance of the data. The illustration shows that ignoring changes in data can lead to a false conclusion in statistical inference. For future studies we recommend other forms of the weight functions, h t be considered. It is worth mentioning that in this study a single change point test popularly referred to as At most one change (AMOC) was considered. We however, advocate for multiple changes (MOSUM) test via the binary segmentation technique.

The authors wish to thank the Pan African Univeristy Institute of Science, Technology and Innovation and the referees for their helpful suggestions.

No external funding has been provided for this research.

The authors used United States Dollar (USD)-Ghana cedi (GHS) daily exchange rate data obtained from the Ghana Stock Exchange (GSE).

The authors declare there is no conflict of interest.

Awiakye-Marfo, G., Mung’atu, J. and Weke, P. (2021) Convergence of a Randomised Change Point Estimator in GARCH Models. Journal of Mathematical Finance, 11, 234-245. https://doi.org/10.4236/jmf.2021.112013