Quasi Maximum Likelihood Estimation of Vector Multiplicative Error Model using the ECCC-GARCH Representation

Yongdeng Xu

doi:10.1515/jtse-2022-0018

Published online by De Gruyter January 3, 2024

Quasi Maximum Likelihood Estimation of Vector Multiplicative Error Model using the ECCC-GARCH Representation

Yongdeng Xu

From the journal Journal of Time Series Econometrics

https://doi.org/10.1515/jtse-2022-0018

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Abstract

We introduce an ECCC-GARCH representation for the vector Multiplicative Error Model (vMEM) that enables maximum likelihood estimation using the multivariate normal distribution. We show via Monte Carlo simulations that the QML estimator possesses desirable small sample properties (towards unbiasedness and efficiency). In the empirical application, we firstly use a two-dimensional vMEM for the squared return and realized volatility, which nests the High-frEquency-bAsed VolatilitY (HEAVY) and Realized GARCH model. We show that the Realized GARCH model is a more appropriate specification for the dynamics of the return-volatility relationship. The second empirical application is a four-dimensional vMEM for volatility spillover effects in the four European stock markets. The results confirm interdependence across European markets and the relative strength of volatility spillovers increases in the post-2010 turmoil periods.

Keywords: vector MEM; QML; HEAVY; realized GARCH; spillover effect

JEL Classification: C01; C32; C52

Corresponding author: Yongdeng Xu, Economic Section, Cardiff Business School, Cardiff University, Cardiff, CF10 3EU, UK, E-mail: xuy16@cardiff.ac.uk

Appendix 1: Assumptions for Asymptotic Normality for QML Estimator

The following assumptions are made to ensure the asymptotic normality of the QMLE θ ̂ in (7) (See Ling and McAleer (2003) and Nakatani and Teräsvirta (2009)).

Assumption 1

All the roots of det I K − ∑ i = 1 p A i L i − ∑ i = 1 q B i L i , where I_K denotes the K-dimensional identity matrix, lie outside the unit circle.

Assumption 2

The identifiability conditions are satisfied, such that (1) det A L ≠ 0 and det B L ≠ 0 , (2) A L and B L are coprime, (3) A L or B L is column reduced are met.^[7]

Assumption 3

The spectral radius λ D has a positive lower bound over the parameter space Θ which is a compact subspace of the Euclidean space such that all true parameters lie in the interior of Θ.

Assumption 4

E x i , t 3 < ∞ , i = 1 , … , K .^[8]

In fact, these assumptions have more convenient interpretations: Assumption 1 is a stationary condition. Assumption 2 is an identification condition. Assumption 3 guarantees that the covariance matrix D is positive semi-definite.

Appendix 2: Detailed Simulation Results

Table 8:

Simulation results (T = 1000)

		Mean			RMSE
	True	GMM	QML lognormal	QML normal	GMM	QML lognormal	QML normal
		Panel A: DGP – bivariate gamma copula distribution
α ₁₁	0.05	0.045	0.057	0.043	0.138	0.127	0.130
α ₁₂	0.02	0.029	0.028	0.017	0.131	0.127	0.134
α ₂₁	0.02	0.024	0.028	0.016	0.126	0.137	0.124
α ₂₂	0.05	0.046	0.057	0.041	0.138	0.131	0.133
b ₁₁	0.80	0.771	0.771	0.743	0.327	0.312	0.305
b ₁₂	0.06	0.069	0.072	0.089	0.215	0.209	0.214
b ₂₁	0.04	0.069	0.085	0.054	0.308	0.318	0.309
b ₂₂	0.90	0.861	0.864	0.872	0.255	0.222	0.236
Average		2.39 %	3.16 %	1.96 %	0.205	0.198	0.198
		Panel B: DGP – bivariate lognormal distribution
α ₁₁	0.05	0.049	0.048	0.044	0.070	0.104	0.139
α ₁₂	0.02	0.020	0.020	0.015	0.076	0.101	0.141
α ₂₁	0.02	0.015	0.019	0.023	0.105	0.090	0.137
α ₂₂	0.05	0.065	0.047	0.052	0.089	0.098	0.135
b ₁₁	0.80	0.694	0.738	0.752	0.321	0.265	0.281
b ₁₂	0.06	0.087	0.097	0.076	0.261	0.172	0.203
b ₂₁	0.04	0.084	0.076	0.095	0.292	0.266	0.286
b ₂₂	0.90	0.849	0.867	0.855	0.317	0.192	0.210
Average		2.77 %	1.89 %	2.84 %	0.192	0.161	0.191

Results in this table are based on 1000-repetition Monte Carlo simulations. The true parameter values are reported in the first column. Within each correlation scenario, we report the estimated parameter mean and root mean square error (RMSE) values associated with the GMM estimates and QML estimates under the lognormal and normal distribution. The last row reports the Average bias and Average RMSE values across all parameters.

Table 9:

Simulation results (T = 2000).

		Mean			RMSE
	True	GMM	QML lognormal	QML normal	GMM	QML lognormal	QML normal
		Panel A: DGP – bivariate gamma copula distribution
α ₁₁	0.05	0.050	0.064	0.049	0.064	0.067	0.050
α ₁₂	0.02	0.021	0.027	0.019	0.064	0.059	0.047
α ₂₁	0.02	0.011	0.021	0.020	0.070	0.067	0.053
α ₂₂	0.05	0.048	0.059	0.048	0.081	0.065	0.048
b ₁₁	0.80	0.769	0.769	0.736	0.263	0.256	0.270
b ₁₂	0.06	0.075	0.079	0.101	0.156	0.168	0.169
b ₂₁	0.04	0.064	0.078	0.058	0.267	0.259	0.257
b ₂₂	0.90	0.869	0.872	0.885	0.167	0.161	0.162
Average		1.85 %	2.74 %	1.71 %	0.142	0.138	0.132
		Panel B: DGP – bivariate lognormal distribution
α ₁₁	0.05	0.050	0.049	0.052	0.037	0.049	0.073
α ₁₂	0.02	0.020	0.020	0.021	0.035	0.046	0.071
α ₂₁	0.02	0.015	0.020	0.017	0.052	0.050	0.057
α ₂₂	0.05	0.051	0.049	0.048	0.044	0.047	0.064
b ₁₁	0.80	0.744	0.748	0.760	0.264	0.205	0.218
b ₁₂	0.06	0.088	0.088	0.085	0.210	0.125	0.141
b ₂₁	0.04	0.051	0.057	0.078	0.237	0.198	0.231
b ₂₂	0.90	0.830	0.889	0.872	0.230	0.114	0.155
Average		1.42 %	1.29 %	2.17 %	0.139	0.104	0.126

Results in this table are based on 1000-repetition Monte Carlo simulations. The true parameter values are reported in the first column. Within each correlation scenario, we report the estimated parameter mean and root mean square error (RMSE) values associated with the GMM estimates and QML estimates under the lognormal and normal distribution. The last row reports the Average bias and Average RMSE values across all parameters.

Table 10:

Simulation results (T = 5000).

		Mean			RMSE
	True	GMM	QML lognormal	QML normal	GMM	QML lognormal	QML normal
		Panel A: DGP – bivariate gamma copula distribution
α ₁₁	0.05	0.052	0.065	0.050	0.021	0.038	0.015
α ₁₂	0.02	0.021	0.027	0.020	0.022	0.038	0.008
α ₂₁	0.02	0.021	0.022	0.021	0.036	0.051	0.018
α ₂₂	0.05	0.049	0.060	0.049	0.043	0.060	0.010
b ₁₁	0.80	0.749	0.767	0.769	0.222	0.196	0.171
b ₁₂	0.06	0.082	0.082	0.079	0.108	0.129	0.101
b ₂₁	0.04	0.053	0.057	0.050	0.188	0.209	0.174
b ₂₂	0.90	0.879	0.882	0.892	0.271	0.148	0.103
Average		1.17 %	2.29 %	0.86 %	0.114	0.109	0.075
		Panel B: DGP – bivariate lognormal distribution
α ₁₁	0.05	0.052	0.051	0.051	0.022	0.011	0.014
α ₁₂	0.02	0.020	0.020	0.020	0.008	0.006	0.008
α ₂₁	0.02	0.019	0.020	0.019	0.019	0.013	0.016
α ₂₂	0.05	0.051	0.050	0.050	0.017	0.007	0.010
b ₁₁	0.80	0.775	0.768	0.769	0.237	0.110	0.143
b ₁₂	0.06	0.083	0.079	0.079	0.080	0.064	0.086
b ₂₁	0.04	0.051	0.052	0.053	0.153	0.113	0.138
b ₂₂	0.90	0.889	0.892	0.892	0.148	0.065	0.079
Average		0.93 %	0.86 %	0.95 %	0.085	0.049	0.062

Results in this table are based on 1000-repetition Monte Carlo simulations. The true parameter values are reported in the first column. Within each correlation scenario, we report the estimated parameter mean and root mean square error (RMSE) values associated with the GMM estimates and QML estimates under the lognormal and normal distribution. The last row reports the Average bias and Average RMSE values across all parameters.

vMEM model in data generation process

ω = 0.1 0.1 , A = 0.05 0.02 0.02 0.05 , B = 0.80 0.06 0.04 0.90 .

Appendix 3: Spillover Analysis

Engle, Gallo, and Velucchi (2012) propose a quantitative measure of volatility spillover effects for several markets, based on the measure of spillovers as a response to shocks. Following their lead, we derive similar measures for the vMEM(1,1) and log-vMEM (1,1) model.

The vMEM (1,1) is expressed as:

(20) x t = μ t ⊙ ε t μ t = ω + A x t − 1 + B μ t − 1

where ⊙ denotes the Hadamard (element by element) product.

We will use MEM-based forecasts to derive a spillover balance index later. To this end, we require a formula for E x t + τ | F t , where τ > 0. x_t+τ is not known and needs to be substituted with its corresponding conditional expectation μ t + τ = E x t + τ | F t :

(21) μ t + 1 = ω + A x t + B μ t μ t + 2 = ω + A μ t + 1 + B μ t + 1 = ω + A + B μ t + 1

And then, for τ > 2,

(22) μ t + τ = ω + A + B μ t + 1 − 1

which can be solved recursively for any horizon τ.

The log-vMEM is expressed as:

(23) x t = μ t ⊙ ε t ln μ t = ω + A ⁡ ln x t − 1 + B ⁡ ln μ t − 1 .

We then derive a formula for E ln x t + τ | F t . Under Gaussian assumption^[9]

(24) E ln x t + τ | F t = ln μ t + τ + E ln ε t + τ | F t ≈ ln μ t + τ + ln ⁡ E ε t + τ | F t − var ε t + τ | F t 2 = ln μ t + τ − σ 2 2

where σ² is the variance of the error term and is estimated using an unconditional (sample) variance of ɛ_t. So

(25) μ t + 1 = exp ω + A ⁡ ln x t + B ⁡ ln μ t , μ t + 2 = exp ω + A ln μ t + 1 − σ 2 2 + B ⁡ ln μ t + 1 = exp ω ̄ + A + B ln μ t + 1 ,

where ω ̄ = ω − A σ 2 2 .

And then, for τ > 2,

(26) μ t + τ = exp ω ̄ + A + B ln μ t + 1 − 1 ,

which can be solved recursively for any horizon τ.

Following Engle, Gallo, and Velucchi (2012), let us recall the vMEM in a system,

(27) x t = μ t ⊙ ε t , ⋅ ⋅ ⋅ ε t | F t − 1 ∼ D I , Σ

The innovation vector ɛ_t has a mean vector I with all components’ unity and general variance-covariance matrix Σ, i.e. ε t | F t − 1 ∼ D I , Σ . We can interpret μ t + τ = E x t + τ | F t , ε t = 1 , i.e. the expectation of x_t+τ conditional on ɛ_t, being equal to the unit vector I; this is the basis for the dynamic forecast obtained before. Let us now derive a different dynamic solution, μ t + τ i = E x t + τ | I t , ε t = 1 + s i , for a generic i th element s i . The i th element is equal to the unconditional standard deviation of ɛ_it and the other terms j ≠ i are equal to the linear projection E ε j , t | ε i , t = 1 + σ i = 1 + σ i σ i , j σ i 2 . The element-by-element division (⊘) of the two vectors, ρ t , τ i , is given by

(28) ρ t , τ i = μ t + τ i ⊘ μ t + τ − 1 , τ = 1 , … , K

where K is the number of periods that shocks can last. Given the multiplicative nature of the model, ρ t , τ i gives us the set of responses (relative changes) in the forecast profile starting at time t for a horizon τ brought about by a 1 standard deviation shock in the i th market.

We use ψ t , τ j , i to denote the cumulated impact of the shock from market i to market j:

(29) ψ t , τ j , i = ∑ τ = 1 K ρ t , τ j , i .

So ϕ t , τ j , i is a way to assess the total change induced by the shock. The volatility/illiquidity spillover balance (ζ_i) is expressed as the ratio of the average responses “from” to the average response “to” (excluding one’s own):

(30) ζ i = ∑ j ≠ i ∑ t = 1 T ψ t j , i ∑ j ≠ i ∑ t = 1 T ψ t i , j .

Suppose i is one’s own market, and j (where j ≠ i) are all other markets (excluding its own market), then the numerator is interpreted as the average responses “to”, or the average responses of all other markets to the shocks that happened in one’s own market. The denominator is interpreted as the average response “from”, or the average responses of one’s own market to the shocks that happened in other markets. A value of ζ_i bigger than 1 signals that market as a net creator of spillover. It is notable that the effect of shocks to its own market is not included, so the effect on the size of a shock (i.e. one standard deviation shock) between different markets is trivial.

References

Amendola, A., V. Candila, F. Cipollini, and G. M. Gallo. 2020. “Doubly Multiplicative Error Models with Long-And Short-Run Components.” arXiv preprint arXiv:2006.03458.10.2139/ssrn.3618157Search in Google Scholar

Barndorff-Nielsen, O. E., P. R. Hansen, A. Lunde, and N. Shephard. 2008. “Designing Realized Kernels to Measure the Ex Post Variation of Equity Prices in the Presence of Noise.” Econometrica 76 (6): 1481–536.10.3982/ECTA6495Search in Google Scholar

Bodnar, T., and N. Hautsch. 2012. “Copula-Based Dynamic Conditional Correlation Multiplicative Error Processes.” Available at SSRN 2144741.10.2139/ssrn.2144741Search in Google Scholar

Cattivelli, L., and G. M. Gallo. 2020. “Adaptive Lasso for Vector Multiplicative Error Models.” Quantitative Finance 20 (2): 255–74, https://doi.org/10.1080/14697688.2019.1651451.Search in Google Scholar

Cipollini, F., and G. M. Gallo. 2010. “Automated Variable Selection in Vector Multiplicative Error Models.” Computational Statistics & Data Analysis 54 (11): 2470–86. https://doi.org/10.1016/j.csda.2009.08.007.Search in Google Scholar

Cipollini, F., R. F. Engle, and G. M. Gallo. 2007. “A Model for Multivariate Non-negative Valued Processes in Financial Econometrics.” Technical Report 2007/16. Dipartimento di Statistica, Università di Firenze.Search in Google Scholar

Cipollini, F., R. F. Engle, and G. M. Gallo. 2013. “Semiparametric Vector MEM.” Journal of Applied Econometrics 28 (7): 1067–86. https://doi.org/10.1002/jae.2292.Search in Google Scholar

Cipollini, F., R. Engle, and G. M. Gallo. 2017. “Copula–based vMEM Specifications versus Alternatives: The Case of Trading Activity.” Econometrics 5 (2): 16. https://doi.org/10.3390/econometrics5020016.Search in Google Scholar

Conrad, C., and M. Karanasos. 2010. “Negative Volatility Spillovers in the Unrestricted ECCC-GARCH Model.” Econometric Theory 26 (3): 838. https://doi.org/10.1017/s0266466609990120.Search in Google Scholar

Corsi, F. 2009. “A Simple Approximate Long-Memory Model of Realized Volatility.” Journal of Financial Econometrics 7 (2): 174–96. https://doi.org/10.1093/jjfinec/nbp001.Search in Google Scholar

Diebold, F. X., and K. Yilmaz. 2009. “Measuring Financial Asset Return and Volatility Spillovers, with Application to Global Equity Markets.” The Economic Journal 119 (534): 158–71. https://doi.org/10.1111/j.1468-0297.2008.02208.x.Search in Google Scholar

Engle, R. F. 2002. “New Frontiers for ARCH Models.” Journal of Applied Econometrics 17 (5): 425–46. https://doi.org/10.1002/jae.683.Search in Google Scholar

Engle, R. F., and G. M. Gallo. 2006. “A Multiple Indicators Model for Volatility Using Intra-daily Data.” Journal of Econometrics 131 (1): 3–27. https://doi.org/10.1016/j.jeconom.2005.01.018.Search in Google Scholar

Engle, R. F., and J. R. Russell. 1998. “Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data.” Econometrica 66 (5): 1127–62. https://doi.org/10.2307/2999632.Search in Google Scholar

Engle, R. F., G. M. Gallo, and M. Velucchi. 2012. “Volatility Spillovers in East Asian Financial Markets: A MEM–Based Approach.” The Review of Economics and Statistics 94 (1): 222–3. https://doi.org/10.1162/rest_a_00167.Search in Google Scholar

Francq, C., O. Wintenberger, and J.-M. Zakoïan. 2013. “GARCH Models without Positivity Constraints: Exponential or Log GARCH?” Journal of Econometrics 177 (1): 34–46. https://doi.org/10.1016/j.jeconom.2013.05.004.Search in Google Scholar

Gallo, G. M., and E. Otranto. 2007. “Volatility Transmission across Markets: A Multichain Markov Switching Model.” Applied Financial Economics 17 (8): 659–70. https://doi.org/10.1080/09603100600722151.Search in Google Scholar

Hansen, P. R., Z. Huang, and H. H. Shek. 2012. “Realized GARCH: A Joint Model for Returns and Realized Measures of Volatility.” Journal of Applied Econometrics 27 (6): 877–906. https://doi.org/10.1002/jae.1234.Search in Google Scholar

Hautsch, N. 2008. “Capturing Common Components in High-Frequency Financial Time Series: A Multivariate Stochastic Multiplicative Error Model.” Journal of Economic Dynamics and Control 32 (12): 3978–4015. https://doi.org/10.1016/j.jedc.2008.01.009.Search in Google Scholar

Jeantheau, T. 1998. “Strong Consistency of Estimators for Multivariate ARCH Models.” Econometric Theory 14 (1): 70–86. https://doi.org/10.1017/s0266466698141038.Search in Google Scholar

Ling, S., and M. McAleer. 2003. “Asymptotic Theory for a Vector ARMA-GARCH Model.” Econometric Theory 19 (2): 280–310. https://doi.org/10.1017/s0266466603192092.Search in Google Scholar

Lütkepohl, H. 2005. New Introduction to Multiple Time Series Analysis. New York: Springer Science & Business Media.10.1007/978-3-540-27752-1Search in Google Scholar

Manganelli, S. 2005. “Duration, Volume and Volatility Impact of Trades.” Journal of Financial Markets 8 (4): 377–99. https://doi.org/10.1016/j.finmar.2005.06.002.Search in Google Scholar

Nakatani, T., and T. Teräsvirta. 2009. “Testing for Volatility Interactions in the Constant Conditional Correlation GARCH Model.” The Econometrics Journal 12 (1): 147–63. https://doi.org/10.1111/j.1368-423x.2008.00261.x.Search in Google Scholar

Shephard, N., and K. Sheppard. 2010. “Realising the Future: Forecasting with High‐frequency‐based Volatility (HEAVY) Models.” Journal of Applied Econometrics 25 (2): 197–231. https://doi.org/10.1002/jae.1158.Search in Google Scholar

Sheppard, K. 2013. The Oxford MFE Matlab Toolbox. https://www.kevinsheppard.com/code/matlab/mfe-toolbox/.Search in Google Scholar

Taylor, N. 2017. “Realised Variance Forecasting under Box-Cox Transformations.” International Journal of Forecasting 33 (4): 770–85. https://doi.org/10.1016/j.ijforecast.2017.04.001.Search in Google Scholar

Taylor, N., and Y. Xu. 2017. “The Logarithmic Vector Multiplicative Error Model: An Application to High Frequency NYSE Stock Data.” Quantitative Finance 17 (7): 1021–35. https://doi.org/10.1080/14697688.2016.1260756.Search in Google Scholar

Received: 2022-06-30

Accepted: 2023-12-14

Published Online: 2024-01-03