Multivariate GARCH models. The time varying variance-covariance for the exchange rate

Literature Review, 2020

32 Pages


Table of contents

1. Introduction

2. Objective

3. Literature Review

4. Summary and Conclusions

5. Reference

Abstract - The main purpose of this paper is to review GARCH family’s models. Since the seminal paper of Engle (1982), much advancement has been made in understanding GARCH models and their multivariate extensions. In MGARCH models parsimonious models should be used to overcome the difficulty of estimating the VEC model ensuring MGARCH modeling is to provide a realistic and parsimonious specification of the variance matrix ensuring its positivity. BEKK models are flexible but require too many parameters for multiple time series of more than four elements. BEKK models are much more parsimonious but very restrictive for the cross-dynamics. They are not suitable if volatility transmission is the object of interest, but they usually do a good job in representing the dynamics of variances and covariance. DCC models allow for different persistence between variances and correlations, but impose common persistence in the latter (although this may be relaxed) Student’s t distribution assumption is more proper under negative skewness and high kurtosis of return series.

Index terms-MGARCH,VEC,BEKK,DCC,Volatility, Exchange Rate.


Understanding and predicting the temporal dependence in the second-order moments of asset returns is important for many issues in financial econometrics. It is now widely accepted that financial volatilities move together over time across assets and markets. Recognizing this feature through a multivariate modeling framework leads to more relevant empirical models than working with separate univariate models. From a financial point of view, it opens the door to better decision tools in various areas, such as asset pricing, portfolio selection, option pricing, and hedging and risk management. Indeed, unlike at the beginning of the 1990s, several institutions have now developed the necessary skills to use the econometric theory in a financial perspective.

Since the seminal paper of Engle (1982), traditional time series tools such as autoregressive moving average (ARMA) models (Box and Jenkins, 1970) for the mean have been extended to essentially analogous models for the variance. Autoregressive conditional heteroscedasticity (ARCH) models are now commonly used to describe and forecast changes in the volatility of financial time series. For a survey of ARCH-type models, see Bollerslev et al., (1992, 1994); (Bera and Higgins 1993); (Pagan 1996);(Palm 1996) and (Shephard 1996), among others.

The most obvious application of MGARCH (multivariate GARCH) models is the study of the relations between the volatilities and co-volatilities of several markets. And it answers the question related to is the volatility of a market leading the volatility of other markets, Is the volatility of an asset transmitted to another asset directly (through its conditional variance) or indirectly (through its conditional covariances), Does a shock on a market increase the volatility on another market, and by how much? Is the impact the same for negative and positive shocks of the same amplitude? A related issue is whether the correlations between asset returns change over time.

Are they higher during periods of higher volatility (sometimes associated with financial crises)? Are they increasing in the long run, perhaps because of the globalization of financial markets? Such issues can be studied directly by using a multivariate model, and raise the question of the specification of the dynamics of covariances or correlations. In a slightly different perspective, a few papers have used MGARCH models to assess the impact of volatility in financial markets on real variables like exports and output growth rates, and the volatility of these growth rates. Another application of MGARCH models is the computation of time-varying hedge ratios. Traditionally, constant hedge ratios are estimated by OLS as the slope of a regression of the spot return on the futures return, because this is equivalent to estimating the ratio of the covariance between spot and futures over the variance of the futures. Since a bivariate MGARCH model for the spot and futures returns directly specifies their conditional variance-covariance matrix, the hedge ratio can be computed as a byproduct of estimation and updated by using new observations as they become available. Lien and Tse (2002) for a survey on hedging and additional references.

Asset pricing models relate returns to 'factors', such as the market return in the capital asset pricing model. A specific asset excess return (in excess of the risk-free return) may be expressed as a linear function of the market return. Assuming its constancy, the slope, or if coefficient, may be estimated by OLS. Like in the hedging case, since the is the ratio of a covariance to a variance.

While modeling volatility of the returns has been the main centre of attention, understand the co movement of financial returns is of great practical importance. One of weaknesses of GARCH model is remarkable when we want estimate covariance that depends on interaction of variables, at this time GARCH will no longer be able to deal with the situation, reason why a model that take simultaneously more than one time series into account would be more convenient when estimating time varying covariance and time varying correlation. It is therefore important to extend the univariate GARCH to multivariate GARCH (MGARCH) models.

In recent years, a family of MGARCH models has been developed. The first developed is VEC. The implementation of estimation in MGARCH model with empirical data is of more complex nature since the number of parameters increase rapidly. This issues inspired researchers to propose different models with purpose of arriving at parsimonious model (with minimum number of parameters) such as BEKK, DCC (Dynamic constant correlation) and CCC (constant conditional correlation).

Because of the nature of financial time series data are naturally volatile and it is important to relax modeling them in to more than Gaussian distributional assumption and assuming the models that can capture the volatility natures of series.

The exchange rate volatility is of great concern in financial time series from its impact on inflation and international trade. Furthermore it is used in risk management where its estimation and forecasting is of great value in financial market.

MGARCH model can be used to estimate time-varying coefficients of such volatile natures of higher moments of financial time series data, (Bollerslev et al., 1988); (De Santis and Gerard, 1998); (Hafner and Herwartz, 1998).

Previous research in different Countries exchange market studied the exchange rate fluctuation and raised the problem of volatility effects of exchange rate of major currencies. It is not intellectual to ignore the volatility effect exchange rate that may come from interaction of two or more currencies to the fluctuation of the exchange market. Volatility effect information between currencies is given by the t ime varying second moment's (variance, covariance and correlation) should be appropriately indentified to describe the situation of the exchange rate market.


The objective of this review is to:

- Review and understand some flexible and parsimonious GARCH family models from the theoretical aspects of financial time series.
- Empirically review the exchange rate volatility under GARCH family models.
- Review one exchange rate volatility article based on the theoretical and empirical evidence of GARCH family models.


3.1 Theoretical Review GARCH Type Models

3.1.1 Univariate GARCH

The GARCH family of models was developed primarily to account for the empirical regularities of a certain category of financial data speculative price data. In order to best understand the development of GARCH models we first need to take stock of the empirical features of speculative price data. The following are the most important “stylized facts” regarding such data:

1 Thick tails: The data seem to be leptokurtic with a large concentration of observations around the mean and have more outliers relative to the Normal distribution.
2 Volatility clustering: This is best described by Mandelbrot (1963), “.large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes
3 Bell-shaped symmetry: In general the distributions seem to be bell-shaped and symmetric.
4 Leverage effects: This relates to the tendency of stock returns to be negatively correlated with changes in return volatility.
5 Co-movements in volatilities: Black (1976) observed that “In general it seems fair to say that when stock volatilities change, they all tend to change in the same direction”. This indicates that common factors may explain temporal variation in conditional second moments and also the possibility of linkages or spillovers between markets.

The first attempt to capture volatility clustering through modeling the conditional variance was made by Engle in 1982. He proposed a regression model with errors following an ARCH process to model the means and variances of inflation in United Kingdom. This model can be specified in terms of the first two conditional moments. For a number of applications however, it was found that a rather long lag structure for the conditional variance is required to capture the long memory present in the data. Bollerslev (1986) proposed a GARCH (p, q) model that allows for both long memory as well as a more flexible lag structure. The extension of the ARCH to the GARCH process for the conditional variance in many ways resembles the extension of the AR to the ARMA process for the conditional mean. An extension of ARCH model which is GARCH an important development for capturing volatility persistence, which is common in high frequency data. Other advantages of such models are their parsimonious nature and ease of estimation.

In view of the fact that the Gaussian GARCH model could not explain the leptokurtosis exhibited by asset returns. Bollerslev (1987) suggested replacing the assumption of conditional Normality of the error with that of Conditional Student's t distribution. He argued that this formulation would permit us to distinguish between conditional leptokurtosis and conditional heteroskedasticity as plausible causes of the unconditional kurtosis observed in the data. Another limitation of both the ARCH and the GARCH models was their inability to capture the “leverage effect” since the conditional variance is specified in terms of only the magnitude of the lagged residuals and ignores the signs. This led to an important class of asymmetric models.

Nelson (1991) introduced the EGARCH model, which depends on both the size and the sign of the lagged residuals.

Univariate volatility has concentrated on capturing the empirical regularities of speculative price data by using conditional distributions, and in particular choosing functional forms for the conditional variance which stem from these conditional distributions. It emphasis on capturing volatility clustering (second order dependence) by modeling the dynamic conditional heteroskedasticity. The issue of leptokurticity and thick tails was only addressed through the development of Bollerslev's Student's t GARCH model. The most powerful theoretical justification for the different functional forms for the conditional variance provides a flexible and parsimonious approximation to conditional variance dynamics in the same way that the ARMA models provide a flexible and parsimonious approximation to conditional mean dynamics. Another justification of the Univariate GARCH models is their empirical success that made them extremely popular in the finance literature. While this model is successful in describing the patterns in Univariate series, it leaves a lot of questions unexplored in particular related to the multivariate framework. As already mentioned an important concern with financial data is the fact that financial variables are often interrelated. Therefore studying them in isolation would lead to incorrect hypotheses and erroneous conclusions.

3.1.2 Empirical Review of Univariate GARCH Models

Bala and Asemota (2013) examined exchange rate volatility using GARCH models. They used monthly exchange rate return series for the naira (Nigerian currency) against the US dollar ($), British pound, and euro. To compare the estimates, various GARCH models were estimated with and without volatility breaks. It was revealed that most of the models rejected the existence of a leverage effect, except for those with volatility breaks. Since it was observed that results improved when the volatility models considered breaks, incorporating significant events in the GARCH models was suggested.

Clement and Samuel (2011) also model Nigerian exchange rate volatility. They used the monthly exchange rate of the naira against the US dollar and British pound for the period from 2007 to 2010. They found that the exchange rate return series was non-stationary and that the series residuals were asymmetric. Since return volatility was found to be persistent, then they recommended further investigation of the impact of government policies on foreign exchange rates.

Rofael and Hosni (2015) forecast and estimate exchange rate volatility in Egypt using ARCH and state space (SS) models. Using daily exchange rate data covering about 10 years, they found volatility clustering in Egypt's exchange rate returns, as well as a risk of mismatch between exchange rates and the stock market.

Ahmad etal, (2002), who used GARCH model variants to capture the exchange rate volatility dynamics of the Malaysian ringgit (RM) against the pound sterling,. They used daily data for the period from 1990 to 1997 and concluded that the volatility of the RM-sterling exchange rate was persistent.

Dhamija and Bhalla (2010) argued that conditionally heteroscedastic models can be used to model exchange rate volatility. They found that integrated generalized autoregressive conditional heteroscedsticity (IGARCH) and threshold generalized autoregressive conditional heteroscedsticity (TGARCH) models performed better than others when forecasting the volatility of five daily currencies: the British pound, German mark, Japanese yen, Indian rupee and Euro. Ramasamy and Munisamy (2012) concluded that GARCH models were efficient for predicting exchange rate volatility. They examined the daily exchange rates of four currencies, the Australian dollar, Singapore dollar, Thai bhat, and Philippine peso using GARCH, Glostern - Jagannathan - Runkle GARCH (GJR-GARCH), and exponential generalized autoregressive conditional heteroscedsticity (EGARCH) models. They argued that the improvements made by leveraging in EGARCH and GJR-GARCH models did not improve forecasting accuracy.

Brooks and Burke (1998) used modified information criteria to select models from the GARCH family. Using weekly exchange rate returns for the Canadian dollar, German mark, and Japanese yen against the US dollar for the period from March 1973 to September 1989, they compared the performance of different out of sample models. They found that the out-of­sample forecasting accuracy of the models compared favorably on mean absolute errors but less favorably on mean squared errors with those generated by commonly used GARCH (1, 1) models.

Herwartz and Reimers (2002) analyzed daily exchange rate changes between the Deutsche mark (DM) and the US dollar and the DM and the Japanese yen (JPY) for the period from 1975 to 1998. They used a GARCH (1, 1) model with leptokurtic innovations to capture volatility clustering, and they found that the identified points of structural change were subject to changes in monetary policies in the US and Japan.

Dayioglu et al, (2013) modeled the exchange rate volatility of MIST (Mexico, Indonesia, South Korea, and Turkey) countries against the US dollar using asymmetric GARCH models. They used monthly exchange rate data for the period from 1993 to 2013 to investigate leverage effects and fat-tailed features. They identified the existence of asymmetrical and leveraging effects in the exchange rates of MIST countries against the US dollar.

Vee et al, (2011) also examined the forecasting accuracy of GARCH (1, 1) using Student's t and generalized error distribution (GED). Using daily data for exchange rates between the US dollar and the Mauritian rupee, they compared the mean absolute error (MAE) and root mean squared error (RMSE) of the models based on forecasting estimates. They found that GARCH (1, 1) with GED had better forecasting accuracy compared to that using Student's t- distribution.

Tse (1998) examined the conditional heteroscedsticity of yen-US dollar exchange rates using daily observations for the period from 1978 to 1994. Extending PARCH models to a process that was fractionally integrated; they found that, unlike the stock market, the appreciation and depreciation shock of the yen against the US dollar affected future volatility in a similar fashion. They argued that there is no substantial difference between fractionally integrated models and stable models.

Pelinescu (2014) analyzed exchange rates between the Romanian leu and the euro considering the influence of other macroeconomic variables. Using daily observations for the period from 2000 to 2013, the study found that the exchange rates consisted of ARCH processes, and exchange returns were correlated with volatility

3.2 Multivariate GARCH Models

While Univariate GARCH models have met with widespread empirical success, the problems associated with the estimation of multivariate GARCH models with time varying correlations have constrained researchers to estimating models with either limited scope or considerable restrictions, (Engle R. and K. Sheppard ,2001b).

The generalization of Univariate to multivariate GARCH type models seems quite natural since cross variable interactions play an important role in Macroeconomics and Finance. It is argued that many economic variables react to the same information and hence have non-zero covariance’s conditional on the information set. Thus a multivariate framework is more appropriate in capturing the temporal dependencies in the conditional variances or covariance’s and should lead to significant gains in efficiency.

The extension of Univariate GARCH to multivariate GARCH can be thought as analogous to the extension of ARMA to Vector ARMA (VARMA) models. The first substantive multivariate GARCH model was developed by (Kraft and Engle, 1982) and was simply an extension of the pure ARCH model. However, the first multivariate model that gained popularity was the multivariate GARCH (p, q) model proposed by Bollerslev et al., 1988). They apply the “Diagonal” form of this model to a CAPM model with a market portfolio consisting of three assets stocks, bonds and bills. We now present the most general form of the conditional variance-covariance matrix for the multivariate GARCH model.


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Multivariate GARCH models. The time varying variance-covariance for the exchange rate
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multivariate, garch
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Tekle Bobo (Author), 2020, Multivariate GARCH models. The time varying variance-covariance for the exchange rate, Munich, GRIN Verlag,


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