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In this paper, the randomised pseudolikelihood ratio change point estimator for GARCH models in [1] is employed and its limiting distribution is derived as the supremum of a standard Brownian bridge. Data analysis to validate the estimator is carried out using the United states dollar (USD)-Ghana cedi (GHS) daily exchange rate data. The randomised estimator is able to detect and estimate a single change in the variance structure of the data and provides a reference point for historic data analysis.

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 [

The change point problems for GARCH models in literature have usually been viewed as deterministic. However, in [

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 [

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 [

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 [

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 [

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 [

| 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

By observing the autocorrelation and partial autocorrelation and comparing the values of AIC and BIC, a more suitable order of p = 1 and q = 1 of the ARMA

Statistical Properties of Return Data | |
---|---|

min | −1.794944e−02 |

max | 1.183120e−02 |

median | 6.214180e−05 |

mean | 2.545361e−04 |

SE.mean | 2.722980e−05 |

var | 2.206591e−06 |

Arch Test | p-value = 0.01 |

Jarque-Bera test (α = 0.05) | p-value = 2.2e−16 |

skewness | −2.189684 |

kutosis | 46.1361 |

function was picked as shown in Equation (8):

X t = 0.0000447210 + 0.8414551500 X t − 1 − 0.6086431547 ε t − 1 + ε t (8)

Based on the assumption of 5% significance level, for the GARCH (1,1) model, when the error term is skewed student-t distribuion, the parameters for the GARCH(1,1) function is given as

σ t 2 = 0.0000000005 + 0.0515904234 ε t − 1 2 + 0.8974842734 σ t − 1 2 (9)

From

We apply the randomised change point estimator to estimate the change point in the USD-GHS exchange rate data under the hypotheses;

H_{0}: there is no change in variance.

H_{a}: there is a change in variance.

In

Statistical Properties of Residual Data | |
---|---|

min | −1.537934e−02 |

max | 1.080608e−02 |

mean | 8.489841e−05 |

SE.mean | 2.413864e−05 |

var | 1.734038e−06 |

Arch Test | p-value = 0.01 |

Jarque-Bera test (α = 0.05) | p-value = 2.2e−16 |

skewness | −0.6262043 |

kurtosis | 35.90743 |

Variable | Estimate | Std. Error | t value | Pr (>|t|) |
---|---|---|---|---|

μ | 0.0000447210 | 0.0000064843 | 6.896791809 | 0.0000000000 |

AR(1) | 0.8414551500 | 0.0191592710 | 43.918954357 | 0.0000000000 |

MA(1) | −0.6086431547 | 0.0359025884 | −16.952626038 | 0.0000000000 |

ω | 0.0000000005 | 0.0000000648 | 0.007050212 | 0.9943747915 |

α 1 | 0.0515904234 | 0.0044876976 | 11.495966890 | 0.0000000000 |

β 1 | 0.8974842734 | 0.0061247224 | 146.534685788 | 0.0000000000 |

e ˜ = 0.07859228 when υ = 0.2 . Consequently

We analyze the mean and the variance of the exchange rate data after the application of randomised change point estimator. We present the mean and variance functions before and after the point of change.

The suitable mean model for the data before the change point is the ARMA(1,1)-GARCH(1,1) and its parameters are given in Equations (10) and (11) when the error term is the skewed student t-distribution.

X t = 0.0000475035 + 0.8441708645 X t − 1 − 0.6457857840 ε t − 1 + ε t (10)

whilst the GARCH(1,1) model and is given as

σ t 2 = 0.0000000004 + 0.0536620794 ε t − 1 2 + 0.8958779068 σ t − 1 2 (11)

The suitable mean and variance model for the data after the point of change is the ARMA(1,1)-GARCH(1,1) and its parameters are given in Equations (12) and (13) when the error term is the skewed student t-distribution.

X t = 0.0000418753 + 0.5358615364 X t − 1 − 0.2512823195 ε t − 1 + ε t (12)

whilst the GARCH(1,1) model is given as

σ t 2 = 0.0000000004 + 0.0501059165 ε t − 1 2 + 0.8992657247 σ t − 1 2 (13)

The standard GARCH model was used as the measure of volatility. The persistence parameter ( α + β ) = 0.94907 of the sample remains relatively stable, though there seems to be an indication of some shifting parameters, the standard GARCH(1,1) appears to support the sample over the IGARCH model of [

The mean (ARMA) function of the data, however, gave a Root Mean Square Error (RMSE) of 0.00007699637 for a 10-period ahead forecast of the after change sample (Aft-ch) as against 0.00008529915 for the full sample. The result illustrates that change point test can be used as a reference point for historic data analysis.

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