# Forecasting the Covolatility of Coffee Arabica and Crude Oil Prices: A Multivariate GARCH Approach with High-Frequency Data

Authors: Dawit Yeshiwas; Yebelay Berelie
Journal: Journal of Probability and Statistics
 (2020)

---

## Abstract

Forecasting the covolatility of asset return series is becoming the subject of extensive research among academics, practitioners, and portfolio managers. This paper estimates a variety of multivariate GARCH models using weekly closing price (in USD/barrel) of Brent crude oil and weekly closing prices (in USD/pound) of Coffee Arabica and compares the forecasting performance of these models based on high-frequency intraday data which allows for a more precise realized volatility measurement. The study used weekly price data to explicitly model covolatility and employed high-frequency intraday data to assess model forecasting performance. The analysis points to the conclusion that the varying conditional correlation (VCC) model with Student’st distributed innovation terms is the most accurate volatility forecasting model in the context of our empirical setting. We recommend and encourage future researchers studying the forecasting performance of MGARCH models to pay particular attention to the measurement of realized volatility and employ high-frequency data whenever feasible.


---

## Body

## 1. Introduction

Forecasting the covolatility of asset return series is becoming the subject of extensive research among academics, practitioners, and portfolio managers [1]. This has been used in risk management, derivative pricing and hedging, portfolio selection, and policy making. Similarly, the analysis of volatility spillovers between commodity and asset prices has a profound implication for risk management and portfolio maximization by the government and investors [2]. In view of the current bearish behavior of oil price in the international markets, it is arguably of special interest to study the relationship between oil and coffee prices [2]. Recently, a number of papers have studied the comovement of oil prices with equities, agricultural commodities, and precious metals prices.Prior studies provide evidence on the connectedness between oil and one or more markets. However, the bulk of these studies has focused on price connectedness. Relatively little research has been devoted to volatility comovement. This is surprising since price volatilities are the main driving forces in options and commodity future markets. This paper seeks to bridge this gap in the literature. In particular, it undertakes a systematic analysis of the volatility interconnectedness between two important commodity markets—oil and coffee markets—using multivariate GARCH (MGARCH) family of models with the view of providing some rigorous price covolatility forecasting models.Naturally, different papers in the literature also tend to focus on different commodity pairs when studying volatility interconnectedness. The focus of this work is on the covolatility between Brent crude oil and Coffee Arabica prices in the commodity futures markets which are markets where one can buy specific quantities of a commodity at a specified price with delivery set at a specified time in the future. The choice of these commodities is motivated by the fact that these are the most actively traded commodities in the world. Existing work appears to focus mostly on in-sample modeling of covolatility of commodity prices giving less attention to model forecasting performance. This work will also seek to bridge this gap in the literature.A number of recent papers have investigated the covolatility of energy and commodity price mainly due to increased interest for understanding the drivers and dynamics of such covolatilities in these highly volatile times. Alternative estimation approaches are used in the literature, but thus far there is a dearth of work undertaking a thorough “MGARCH” analysis—a statistical framework which is particularly suited for modeling asset prices covolatility. Moreover, existing work appears to focus mostly on in-sample modeling of covolatility of commodity prices giving less attention to model forecasting performance. Accordingly, this work will seek to contribute to the literature by undertaking a systematic analysis of volatility forecasting performance.Most of the existing research on volatility spillovers employ statistical models in order to estimate realized volatilities which turned out to be oftentimes poor approximations of true volatilities. An attractive alternative to model-based statistical volatility is to compute realized volatility based on high-frequency intraday or “tick” data. Realized volatility has been found to be more accurate than model-based volatilities in predicting latent volatility [3]. Thus, the results provided by previous studies can be potentially misleading in that they may have underestimated or overestimated the extent of the true volatility.This work avoids this potential pitfall by using realized volatility based on 30-minute intraday data. We argue that doing so represents an important contribution to the literature of covolatility measurement within commodity markets.The work makes the following contributions to the literature:(1)
From an applied econometrics stand point, the work provides a robust and systematic investigation of the statistical and econometric properties of the covolatility of the two important commodities.(2)
The direction and magnitude of correlation between the prices of coffee and crude oil which are documented in this work would help the design of a more informed policy making on the part of financial authorities as well as other stakeholders and market participants interested in foreseeing the volatility of these two important commodities.


## 2. Methods

### 2.1. Multivariate GARCH Models

A large part of the literature deals with univariate model price series. But markets interact, and therefore a generalization from the univariate model to a multivariate one is important. Multivariate GARCH models can be categorized into four types [4].(i)
Models of the conditional covariance matrix: in these models, the conditional covariance is computed in a direct way. The vector error correction (VEC) and Baba–Engle–Kraft–Kroner (BEKK) models are among this type of models.(ii)
Factor models: the return process is assumed to consist of a small number of unobservable heteroskedastic factors. This approach benefits from that the dimensionality of the problem reduces when the number of factors compared to the dimension of the return vector is small.(iii)
Models of conditional variances and correlations: at first, the univariate conditional variances and correlations are computed and then used to get the conditional covariance matrix. Some models are, for example, the constant conditional correlation (CCC) model and the dynamic conditional correlation (DCC) model.(iv)
Nonparametric and semiparametric approaches: models in this class form an alternative to parametric estimation of the conditional covariance structure. The advantage of these models is that they do not impose a particular distribution (that can be misspecified) on the data.In this paper, conditional variance and correlation models are used. Before detailed discussion of the MGARCH models, some definitions of the series are presented. This study uses two commodity price series, resulting in a bivariate approach.(1)rt=r1,tr2,t,where rt is the vector of returns which can be decomposed as(2)rt=µt+εt,εt=Ht1/2zt,µt=ErtIt−1,zt∼iidN0¯,IN,varrtIt−1=Ht1/2Ht1/2′=∑t,where rt is the N×1 vector of returns; µt is the N×1 vector of expected return at time t given the available information set It−1; Ht1/2N×N is the Cholesky factor of time-varying conditional covariance matrix ∑t; and zt is the N×1 vector of independent errors with mean zero and variance one.The variance of the conditional unpredictable component can also be defined as(3)varrtIt−1=∑t=h11,th12,th21,th22,t.In equation (3), the matrix obtained should be symmetric. Furthermore, the matrix ∑t has to be positive definite for all t. The approach of multivariate modeling brings complications. Firstly, it may be hard to ensure that ∑t is positive definite for all t. Secondly, the number of parameters becomes too large to estimate, and at last, it may be difficult to obtain the stationarity condition for ∑t.The various MGARCH models proposed in the literature differ in how they trade off flexibility and parsimony in their specifications for∑t. Increased flexibility allows a model to capture more complex ∑t. The following sections will describe the theory of the models, where the VEC and DVEC models which are the predecessors to the CCC, VCC, and DCC models are first described.


#### 2.1.1. The Constant Conditional Correlation (CCC) Model

The constant conditional correlation model was suggested by Bollerslev [5], where the time-varying covariance matrix ∑t at time t is expressed as(4)∑t=DtRtDt.The right hand side consists of the conditional correlation matrixR that is time invariant, i.e., Rt=R.D is a diagonal matrix of (h1t,h2t,…,hkt) such as(5)Dt=h1,th2,th2,t⋮hk−1,thk,t,where each hi,t follows a univariate GARCH process. The conditional correlation matrix is given by R=ρi,j, and the nondiagonal elements of ∑t are(6)∑ij,t=hi,t1/2hj,t1/2ρi,j,∀i≠j.The desired conditional variances can be expressed in vector form:(7)∑t=C+∑i=1qAiεt−j⊙εt−j′+∑j=1pBj∑t−j⊙∑t−j.′The first termC is a vector of the intercepts with a dimension of n×1 and the matrices of the coefficients are ×n.The advantage of the CCC model is that the computational procedure is more easily performed since the correlation matrixR is constant. However, this means that the model may be too restrictive [6].


#### 2.1.2. The Dynamic Conditional Correlation (DCC) Model

The dynamic conditional correlation model is given as(8)∑t=DtRtDt,where ∑t is the covariance matrix and Rt is n×n matrix of the conditional correlation of the returns which is symmetric by definition. The diagonal matrix Dt is expressed as(9)Dt=diagh1t,h2t,…,hkt,where each hi,t follows a univariate GARCH process. Furthermore, ∑t has to be positive definite, which is automatically obtained while Rt is a correlation matrix that is symmetric by definition. When this matrix is defined, two requirements are needed. Firstly, ∑t needs to be positive definite, since it is a covariance matrix. Secondly, the parts that belong to Rt need to be less than one. These requirements are met through decomposition:(10)Rt=diagqiit−1Qtdiagqiit−1for,ii=1,&,n,where(11)Qt=1−π1−π2Q¯+π1εt−1εt−1′+π2Qt−1,Q¯=covεtεt′=Eεtεt′,where Q¯ is the unconditional covariance and will be finite as the model contains finite parameters and the vector εt has finite variance. Additionally, the parameters π1 and π2 are scalars and diagQ is used to rescale the parts of Qt in order to fulfill that ρij=qijt/qiitqjjt≤1, where qij is the element of the matrix diagQ. An estimate of Q¯ is(12)Q^=1T∑t=1Tεtεt′.Moreover, the scalarsπ1 and π2 must be larger than zero, but the sum has to be less than one. One may note that these are conditions of the univariate GARCH to be stationary, but which is applied in the DCC model [7].


#### 2.1.3. The Varying Conditional Correlation (VCC) Model

The varying conditional correlation (VCC) is a multivariate generalized autoregressive conditionally heteroskedastic (MGARCH) model in which the conditional variances are modeled as univariate GARCH models and the conditional covariance is modeled as nonlinear functions of the conditional variances. The conditional correlation parameters that weight the nonlinear combinations of the conditional variance follow the GARCH-like process specified by Tse and Tsui [6]. The VCC MGARCH model is about as flexible as the closely related dynamic conditional correlation MGARCH model and more flexible than the constant conditional correlation model and more parsimonious than the diagonal vector error conditionally heteroskedastic (DVECH) models.MGARCH models differ in the parsimony and flexibility of their specifications for a time-varying conditional covariance matrix of the disturbances, denoted by∑t. In the conditional correlation family of MGARCH models, the diagonal elements of ∑t are modeled as univariate GARCH models, whereas the off-diagonal elements are modeled as nonlinear functions of the diagonal terms. In the VCC MGARCH model,(13)hij,t=ρij,thii,thjj,t,where the diagonal elements hii,t and hjj,t follow univariate GARCH processes and ρij,t follows the dynamic process specified in Tse and Tsui [6] and discussed below. Because ρij,t varies with time, this model is known as the VCC GARCH model.The VCC-GARCH(1, 1) model proposed by Tse and Tsui [6] can be written as(14)rt=θxt+ϵt,ϵt=Ht1/2zt,∑t=DtRtDt,Rt=1−π1−π2R+π1ϵt−1+π2Rt−1,where xt is a k×1 vector of independent variables, which may contain lags of rt; θ is an m×k matrix of parameters; rt is an m×1 vector of dependent variables; Ht1/2 is the Cholesky factor of the time-varying conditional covariance matrix ∑t; zt is an m×1 vector of independent and identically distributed innovations; and Dt is a diagonal matrix of conditional standard deviation. Dt=diagh1t,h2t,…,hkt. Rt is a matrix of conditional correlations, Rt=1ρ12,t⋯ρ1m,tρ21,t1⋯ρ2m,t⋮⋮⋮ρm1,t⋯ρm2,t1.R is the matrix of means to which the dynamic process in equation (14) reverts; π1 and π2 are parameters that govern the dynamics of conditional correlations, and they are nonnegative and satisfy 0 ≤π1 + π2 < 1.


### 2.2. Determining the Conditional Distribution

When fitting a GARCH model based on financial data, the conditional distribution of the returns has to be defined. Studies, for example Bollerslev [8], illustrate that returns are not normally distributed. Instead, the Student t distribution captures the observed kurtosis in empirical returns in a more sufficient way than the normal distribution. Returns have excess kurtosis and fatter tails than the normal distribution. Therefore, the Student t distribution is more suitable [9]. There are three assumptions about the conditional distribution of the error term commonly employed when working with GARCH models: normal (Gaussian) distribution, Student’s t distribution, and the generalized error distribution (GED).


### 2.3. Estimation Evaluation

The evaluation is done with several methods. Firstly, the estimations are compared using the method of three measures: MAE, RMSE, and the explanatory power from an ordinary least squares regression. Secondly, an attempt to determine the goodness of fit of the residuals is done through univariate approach consisting of an analysis with the Ljung–Box test.


#### 2.3.1. Defining a Proxy

In evaluating volatility forecasts, the usual proxy for “true” volatility is “ex post” squared returns or the squared errors. However, as noted by Andersen and Bollerslev [5], although the use of squared returns or the squared errors is justifiable as an unbiased estimate of the volatility process, it provides a very noisy measure due to a large idiosyncratic term. Specifically, the returns innovation can be written as εt=σtzt with zt an independent zero mean and unit variance stochastic process and σt is the volatility process. If the model for σt2 is correctly specified, then the conditional expectation E(Eεt2It−1=Eσt2zt2It−1=σt2, and the squared error is an unbiased estimate of the volatility process; however, it still contains the noisy idiosyncratic term, zt2. This typically resulted in a poor performance, which instigated a discussion of the practical relevance of volatility models.Hansen et al. [10] provided another important argument for using the realized variance rather than the squared return. They showed that substituting the squared returns for the conditional variance can severely distort the comparison, in the sense that the empirical ranking of models may be inconsistent for the true (population) ranking. So, an evaluation that is based on squared returns may select an inferior model as the “best” with a probability that converges to one as the sample size increases. For this reason, our evaluation is based on the realized variance. Building upon this line of research, Andersen et al. [11] defined the so-called “realized volatility” on day t as(15)htrv=∑i=1Nrt,i2,where N is the number of equally spaced intervals within a day and rt,i is a logarithmic return on day t at time interval i with i = 1, 2…, N and t = 1,…,T.Thus, realized volatility is the sum of squared intraday returns. In principle, lettingN tend to large, i.e., continuous time sampling, the measure approaches to the true integrated volatility of the underlying continuous time process and is theoretically free from measurement error. Further, this measure allows a market participant to essentially treat volatility as an observed variable and to allow direct estimation.


### 2.4. Forecast Evaluation

#### 2.4.1. The MZ Regression

A popular way to evaluate volatility models’ out-of-sample forecasting performance is in terms ofR2 from a Mincer–Zarnowitz (MZ) regression,(16)σt2=a+bht2+vt,that is, squared returns are regressed on the model forecasts of σt2 and a constant. Here, σt2 is ex post volatility (e.g., realized volatility) at time t, ht2 is estimated (in-sample) or forecasted (out-of sample) volatility at time t, and vt is the error term which is independent and identically distributed; vi∼N (0,1). a and b are parameters to be estimated.If the model for conditional variance is well specified, we should havea = 0, b = 1. According to specific features of financial data series, the value of R2 is usually low (even less than 5%) [11].


#### 2.4.2. Mean Absolute Error (MAE)

Another way of determining the goodness of the estimations and forecasts is calculating theMAE. The approach is to measure how the received conditional covariance is close to its corresponding realized value. The formula is(17)MAE=1n∑t=1nσt2−ht2,where σt2 is used as a proxy and ht2 is the estimated covariance. By comparing the MAE between the estimated models, it can give an indication of which model that makes the best estimations.


#### 2.4.3. Root Mean Square Error (RMSE)

The third measure is the root mean square error (RMSE), which is defined as(18)RMSE=1n∑t=1nσt2−ht22.Using these methods, the estimated models can be compared, and using the same measurements for estimations and forecasts, one can determine if the relatively best estimation model also makes the best forecast.


## 2.1. Multivariate GARCH Models

A large part of the literature deals with univariate model price series. But markets interact, and therefore a generalization from the univariate model to a multivariate one is important. Multivariate GARCH models can be categorized into four types [4].(i)
Models of the conditional covariance matrix: in these models, the conditional covariance is computed in a direct way. The vector error correction (VEC) and Baba–Engle–Kraft–Kroner (BEKK) models are among this type of models.(ii)
Factor models: the return process is assumed to consist of a small number of unobservable heteroskedastic factors. This approach benefits from that the dimensionality of the problem reduces when the number of factors compared to the dimension of the return vector is small.(iii)
Models of conditional variances and correlations: at first, the univariate conditional variances and correlations are computed and then used to get the conditional covariance matrix. Some models are, for example, the constant conditional correlation (CCC) model and the dynamic conditional correlation (DCC) model.(iv)
Nonparametric and semiparametric approaches: models in this class form an alternative to parametric estimation of the conditional covariance structure. The advantage of these models is that they do not impose a particular distribution (that can be misspecified) on the data.In this paper, conditional variance and correlation models are used. Before detailed discussion of the MGARCH models, some definitions of the series are presented. This study uses two commodity price series, resulting in a bivariate approach.(1)rt=r1,tr2,t,where rt is the vector of returns which can be decomposed as(2)rt=µt+εt,εt=Ht1/2zt,µt=ErtIt−1,zt∼iidN0¯,IN,varrtIt−1=Ht1/2Ht1/2′=∑t,where rt is the N×1 vector of returns; µt is the N×1 vector of expected return at time t given the available information set It−1; Ht1/2N×N is the Cholesky factor of time-varying conditional covariance matrix ∑t; and zt is the N×1 vector of independent errors with mean zero and variance one.The variance of the conditional unpredictable component can also be defined as(3)varrtIt−1=∑t=h11,th12,th21,th22,t.In equation (3), the matrix obtained should be symmetric. Furthermore, the matrix ∑t has to be positive definite for all t. The approach of multivariate modeling brings complications. Firstly, it may be hard to ensure that ∑t is positive definite for all t. Secondly, the number of parameters becomes too large to estimate, and at last, it may be difficult to obtain the stationarity condition for ∑t.The various MGARCH models proposed in the literature differ in how they trade off flexibility and parsimony in their specifications for∑t. Increased flexibility allows a model to capture more complex ∑t. The following sections will describe the theory of the models, where the VEC and DVEC models which are the predecessors to the CCC, VCC, and DCC models are first described.


### 2.1.1. The Constant Conditional Correlation (CCC) Model

The constant conditional correlation model was suggested by Bollerslev [5], where the time-varying covariance matrix ∑t at time t is expressed as(4)∑t=DtRtDt.The right hand side consists of the conditional correlation matrixR that is time invariant, i.e., Rt=R.D is a diagonal matrix of (h1t,h2t,…,hkt) such as(5)Dt=h1,th2,th2,t⋮hk−1,thk,t,where each hi,t follows a univariate GARCH process. The conditional correlation matrix is given by R=ρi,j, and the nondiagonal elements of ∑t are(6)∑ij,t=hi,t1/2hj,t1/2ρi,j,∀i≠j.The desired conditional variances can be expressed in vector form:(7)∑t=C+∑i=1qAiεt−j⊙εt−j′+∑j=1pBj∑t−j⊙∑t−j.′The first termC is a vector of the intercepts with a dimension of n×1 and the matrices of the coefficients are ×n.The advantage of the CCC model is that the computational procedure is more easily performed since the correlation matrixR is constant. However, this means that the model may be too restrictive [6].


### 2.1.2. The Dynamic Conditional Correlation (DCC) Model

The dynamic conditional correlation model is given as(8)∑t=DtRtDt,where ∑t is the covariance matrix and Rt is n×n matrix of the conditional correlation of the returns which is symmetric by definition. The diagonal matrix Dt is expressed as(9)Dt=diagh1t,h2t,…,hkt,where each hi,t follows a univariate GARCH process. Furthermore, ∑t has to be positive definite, which is automatically obtained while Rt is a correlation matrix that is symmetric by definition. When this matrix is defined, two requirements are needed. Firstly, ∑t needs to be positive definite, since it is a covariance matrix. Secondly, the parts that belong to Rt need to be less than one. These requirements are met through decomposition:(10)Rt=diagqiit−1Qtdiagqiit−1for,ii=1,&,n,where(11)Qt=1−π1−π2Q¯+π1εt−1εt−1′+π2Qt−1,Q¯=covεtεt′=Eεtεt′,where Q¯ is the unconditional covariance and will be finite as the model contains finite parameters and the vector εt has finite variance. Additionally, the parameters π1 and π2 are scalars and diagQ is used to rescale the parts of Qt in order to fulfill that ρij=qijt/qiitqjjt≤1, where qij is the element of the matrix diagQ. An estimate of Q¯ is(12)Q^=1T∑t=1Tεtεt′.Moreover, the scalarsπ1 and π2 must be larger than zero, but the sum has to be less than one. One may note that these are conditions of the univariate GARCH to be stationary, but which is applied in the DCC model [7].


### 2.1.3. The Varying Conditional Correlation (VCC) Model

The varying conditional correlation (VCC) is a multivariate generalized autoregressive conditionally heteroskedastic (MGARCH) model in which the conditional variances are modeled as univariate GARCH models and the conditional covariance is modeled as nonlinear functions of the conditional variances. The conditional correlation parameters that weight the nonlinear combinations of the conditional variance follow the GARCH-like process specified by Tse and Tsui [6]. The VCC MGARCH model is about as flexible as the closely related dynamic conditional correlation MGARCH model and more flexible than the constant conditional correlation model and more parsimonious than the diagonal vector error conditionally heteroskedastic (DVECH) models.MGARCH models differ in the parsimony and flexibility of their specifications for a time-varying conditional covariance matrix of the disturbances, denoted by∑t. In the conditional correlation family of MGARCH models, the diagonal elements of ∑t are modeled as univariate GARCH models, whereas the off-diagonal elements are modeled as nonlinear functions of the diagonal terms. In the VCC MGARCH model,(13)hij,t=ρij,thii,thjj,t,where the diagonal elements hii,t and hjj,t follow univariate GARCH processes and ρij,t follows the dynamic process specified in Tse and Tsui [6] and discussed below. Because ρij,t varies with time, this model is known as the VCC GARCH model.The VCC-GARCH(1, 1) model proposed by Tse and Tsui [6] can be written as(14)rt=θxt+ϵt,ϵt=Ht1/2zt,∑t=DtRtDt,Rt=1−π1−π2R+π1ϵt−1+π2Rt−1,where xt is a k×1 vector of independent variables, which may contain lags of rt; θ is an m×k matrix of parameters; rt is an m×1 vector of dependent variables; Ht1/2 is the Cholesky factor of the time-varying conditional covariance matrix ∑t; zt is an m×1 vector of independent and identically distributed innovations; and Dt is a diagonal matrix of conditional standard deviation. Dt=diagh1t,h2t,…,hkt. Rt is a matrix of conditional correlations, Rt=1ρ12,t⋯ρ1m,tρ21,t1⋯ρ2m,t⋮⋮⋮ρm1,t⋯ρm2,t1.R is the matrix of means to which the dynamic process in equation (14) reverts; π1 and π2 are parameters that govern the dynamics of conditional correlations, and they are nonnegative and satisfy 0 ≤π1 + π2 < 1.


## 2.1.1. The Constant Conditional Correlation (CCC) Model

The constant conditional correlation model was suggested by Bollerslev [5], where the time-varying covariance matrix ∑t at time t is expressed as(4)∑t=DtRtDt.The right hand side consists of the conditional correlation matrixR that is time invariant, i.e., Rt=R.D is a diagonal matrix of (h1t,h2t,…,hkt) such as(5)Dt=h1,th2,th2,t⋮hk−1,thk,t,where each hi,t follows a univariate GARCH process. The conditional correlation matrix is given by R=ρi,j, and the nondiagonal elements of ∑t are(6)∑ij,t=hi,t1/2hj,t1/2ρi,j,∀i≠j.The desired conditional variances can be expressed in vector form:(7)∑t=C+∑i=1qAiεt−j⊙εt−j′+∑j=1pBj∑t−j⊙∑t−j.′The first termC is a vector of the intercepts with a dimension of n×1 and the matrices of the coefficients are ×n.The advantage of the CCC model is that the computational procedure is more easily performed since the correlation matrixR is constant. However, this means that the model may be too restrictive [6].


## 2.1.2. The Dynamic Conditional Correlation (DCC) Model

The dynamic conditional correlation model is given as(8)∑t=DtRtDt,where ∑t is the covariance matrix and Rt is n×n matrix of the conditional correlation of the returns which is symmetric by definition. The diagonal matrix Dt is expressed as(9)Dt=diagh1t,h2t,…,hkt,where each hi,t follows a univariate GARCH process. Furthermore, ∑t has to be positive definite, which is automatically obtained while Rt is a correlation matrix that is symmetric by definition. When this matrix is defined, two requirements are needed. Firstly, ∑t needs to be positive definite, since it is a covariance matrix. Secondly, the parts that belong to Rt need to be less than one. These requirements are met through decomposition:(10)Rt=diagqiit−1Qtdiagqiit−1for,ii=1,&,n,where(11)Qt=1−π1−π2Q¯+π1εt−1εt−1′+π2Qt−1,Q¯=covεtεt′=Eεtεt′,where Q¯ is the unconditional covariance and will be finite as the model contains finite parameters and the vector εt has finite variance. Additionally, the parameters π1 and π2 are scalars and diagQ is used to rescale the parts of Qt in order to fulfill that ρij=qijt/qiitqjjt≤1, where qij is the element of the matrix diagQ. An estimate of Q¯ is(12)Q^=1T∑t=1Tεtεt′.Moreover, the scalarsπ1 and π2 must be larger than zero, but the sum has to be less than one. One may note that these are conditions of the univariate GARCH to be stationary, but which is applied in the DCC model [7].


## 2.1.3. The Varying Conditional Correlation (VCC) Model

The varying conditional correlation (VCC) is a multivariate generalized autoregressive conditionally heteroskedastic (MGARCH) model in which the conditional variances are modeled as univariate GARCH models and the conditional covariance is modeled as nonlinear functions of the conditional variances. The conditional correlation parameters that weight the nonlinear combinations of the conditional variance follow the GARCH-like process specified by Tse and Tsui [6]. The VCC MGARCH model is about as flexible as the closely related dynamic conditional correlation MGARCH model and more flexible than the constant conditional correlation model and more parsimonious than the diagonal vector error conditionally heteroskedastic (DVECH) models.MGARCH models differ in the parsimony and flexibility of their specifications for a time-varying conditional covariance matrix of the disturbances, denoted by∑t. In the conditional correlation family of MGARCH models, the diagonal elements of ∑t are modeled as univariate GARCH models, whereas the off-diagonal elements are modeled as nonlinear functions of the diagonal terms. In the VCC MGARCH model,(13)hij,t=ρij,thii,thjj,t,where the diagonal elements hii,t and hjj,t follow univariate GARCH processes and ρij,t follows the dynamic process specified in Tse and Tsui [6] and discussed below. Because ρij,t varies with time, this model is known as the VCC GARCH model.The VCC-GARCH(1, 1) model proposed by Tse and Tsui [6] can be written as(14)rt=θxt+ϵt,ϵt=Ht1/2zt,∑t=DtRtDt,Rt=1−π1−π2R+π1ϵt−1+π2Rt−1,where xt is a k×1 vector of independent variables, which may contain lags of rt; θ is an m×k matrix of parameters; rt is an m×1 vector of dependent variables; Ht1/2 is the Cholesky factor of the time-varying conditional covariance matrix ∑t; zt is an m×1 vector of independent and identically distributed innovations; and Dt is a diagonal matrix of conditional standard deviation. Dt=diagh1t,h2t,…,hkt. Rt is a matrix of conditional correlations, Rt=1ρ12,t⋯ρ1m,tρ21,t1⋯ρ2m,t⋮⋮⋮ρm1,t⋯ρm2,t1.R is the matrix of means to which the dynamic process in equation (14) reverts; π1 and π2 are parameters that govern the dynamics of conditional correlations, and they are nonnegative and satisfy 0 ≤π1 + π2 < 1.


## 2.2. Determining the Conditional Distribution

When fitting a GARCH model based on financial data, the conditional distribution of the returns has to be defined. Studies, for example Bollerslev [8], illustrate that returns are not normally distributed. Instead, the Student t distribution captures the observed kurtosis in empirical returns in a more sufficient way than the normal distribution. Returns have excess kurtosis and fatter tails than the normal distribution. Therefore, the Student t distribution is more suitable [9]. There are three assumptions about the conditional distribution of the error term commonly employed when working with GARCH models: normal (Gaussian) distribution, Student’s t distribution, and the generalized error distribution (GED).


## 2.3. Estimation Evaluation

The evaluation is done with several methods. Firstly, the estimations are compared using the method of three measures: MAE, RMSE, and the explanatory power from an ordinary least squares regression. Secondly, an attempt to determine the goodness of fit of the residuals is done through univariate approach consisting of an analysis with the Ljung–Box test.


### 2.3.1. Defining a Proxy

In evaluating volatility forecasts, the usual proxy for “true” volatility is “ex post” squared returns or the squared errors. However, as noted by Andersen and Bollerslev [5], although the use of squared returns or the squared errors is justifiable as an unbiased estimate of the volatility process, it provides a very noisy measure due to a large idiosyncratic term. Specifically, the returns innovation can be written as εt=σtzt with zt an independent zero mean and unit variance stochastic process and σt is the volatility process. If the model for σt2 is correctly specified, then the conditional expectation E(Eεt2It−1=Eσt2zt2It−1=σt2, and the squared error is an unbiased estimate of the volatility process; however, it still contains the noisy idiosyncratic term, zt2. This typically resulted in a poor performance, which instigated a discussion of the practical relevance of volatility models.Hansen et al. [10] provided another important argument for using the realized variance rather than the squared return. They showed that substituting the squared returns for the conditional variance can severely distort the comparison, in the sense that the empirical ranking of models may be inconsistent for the true (population) ranking. So, an evaluation that is based on squared returns may select an inferior model as the “best” with a probability that converges to one as the sample size increases. For this reason, our evaluation is based on the realized variance. Building upon this line of research, Andersen et al. [11] defined the so-called “realized volatility” on day t as(15)htrv=∑i=1Nrt,i2,where N is the number of equally spaced intervals within a day and rt,i is a logarithmic return on day t at time interval i with i = 1, 2…, N and t = 1,…,T.Thus, realized volatility is the sum of squared intraday returns. In principle, lettingN tend to large, i.e., continuous time sampling, the measure approaches to the true integrated volatility of the underlying continuous time process and is theoretically free from measurement error. Further, this measure allows a market participant to essentially treat volatility as an observed variable and to allow direct estimation.


## 2.3.1. Defining a Proxy

In evaluating volatility forecasts, the usual proxy for “true” volatility is “ex post” squared returns or the squared errors. However, as noted by Andersen and Bollerslev [5], although the use of squared returns or the squared errors is justifiable as an unbiased estimate of the volatility process, it provides a very noisy measure due to a large idiosyncratic term. Specifically, the returns innovation can be written as εt=σtzt with zt an independent zero mean and unit variance stochastic process and σt is the volatility process. If the model for σt2 is correctly specified, then the conditional expectation E(Eεt2It−1=Eσt2zt2It−1=σt2, and the squared error is an unbiased estimate of the volatility process; however, it still contains the noisy idiosyncratic term, zt2. This typically resulted in a poor performance, which instigated a discussion of the practical relevance of volatility models.Hansen et al. [10] provided another important argument for using the realized variance rather than the squared return. They showed that substituting the squared returns for the conditional variance can severely distort the comparison, in the sense that the empirical ranking of models may be inconsistent for the true (population) ranking. So, an evaluation that is based on squared returns may select an inferior model as the “best” with a probability that converges to one as the sample size increases. For this reason, our evaluation is based on the realized variance. Building upon this line of research, Andersen et al. [11] defined the so-called “realized volatility” on day t as(15)htrv=∑i=1Nrt,i2,where N is the number of equally spaced intervals within a day and rt,i is a logarithmic return on day t at time interval i with i = 1, 2…, N and t = 1,…,T.Thus, realized volatility is the sum of squared intraday returns. In principle, lettingN tend to large, i.e., continuous time sampling, the measure approaches to the true integrated volatility of the underlying continuous time process and is theoretically free from measurement error. Further, this measure allows a market participant to essentially treat volatility as an observed variable and to allow direct estimation.


## 2.4. Forecast Evaluation

### 2.4.1. The MZ Regression

A popular way to evaluate volatility models’ out-of-sample forecasting performance is in terms ofR2 from a Mincer–Zarnowitz (MZ) regression,(16)σt2=a+bht2+vt,that is, squared returns are regressed on the model forecasts of σt2 and a constant. Here, σt2 is ex post volatility (e.g., realized volatility) at time t, ht2 is estimated (in-sample) or forecasted (out-of sample) volatility at time t, and vt is the error term which is independent and identically distributed; vi∼N (0,1). a and b are parameters to be estimated.If the model for conditional variance is well specified, we should havea = 0, b = 1. According to specific features of financial data series, the value of R2 is usually low (even less than 5%) [11].


### 2.4.2. Mean Absolute Error (MAE)

Another way of determining the goodness of the estimations and forecasts is calculating theMAE. The approach is to measure how the received conditional covariance is close to its corresponding realized value. The formula is(17)MAE=1n∑t=1nσt2−ht2,where σt2 is used as a proxy and ht2 is the estimated covariance. By comparing the MAE between the estimated models, it can give an indication of which model that makes the best estimations.


### 2.4.3. Root Mean Square Error (RMSE)

The third measure is the root mean square error (RMSE), which is defined as(18)RMSE=1n∑t=1nσt2−ht22.Using these methods, the estimated models can be compared, and using the same measurements for estimations and forecasts, one can determine if the relatively best estimation model also makes the best forecast.


## 2.4.1. The MZ Regression

A popular way to evaluate volatility models’ out-of-sample forecasting performance is in terms ofR2 from a Mincer–Zarnowitz (MZ) regression,(16)σt2=a+bht2+vt,that is, squared returns are regressed on the model forecasts of σt2 and a constant. Here, σt2 is ex post volatility (e.g., realized volatility) at time t, ht2 is estimated (in-sample) or forecasted (out-of sample) volatility at time t, and vt is the error term which is independent and identically distributed; vi∼N (0,1). a and b are parameters to be estimated.If the model for conditional variance is well specified, we should havea = 0, b = 1. According to specific features of financial data series, the value of R2 is usually low (even less than 5%) [11].


## 2.4.2. Mean Absolute Error (MAE)

Another way of determining the goodness of the estimations and forecasts is calculating theMAE. The approach is to measure how the received conditional covariance is close to its corresponding realized value. The formula is(17)MAE=1n∑t=1nσt2−ht2,where σt2 is used as a proxy and ht2 is the estimated covariance. By comparing the MAE between the estimated models, it can give an indication of which model that makes the best estimations.


## 2.4.3. Root Mean Square Error (RMSE)

The third measure is the root mean square error (RMSE), which is defined as(18)RMSE=1n∑t=1nσt2−ht22.Using these methods, the estimated models can be compared, and using the same measurements for estimations and forecasts, one can determine if the relatively best estimation model also makes the best forecast.


## 3. Results and Discussion

### 3.1. Data

The data considered in this paper were the weekly time series of Brent crude oil and Coffee Arabica futures market closing price, given in US dollars per barrel and US dollars per pound, respectively. Both series contain data spanning between first week of January 2005 and last week of October 2016 extracted from Bloomberg database. The full sample is split in to two parts: in-sample data, in order to estimate the parameters of models, and out-of-sample data, in order to make forecasts. The in-sample period spans from first week of January 2005 up to last week of December 2015 and the out-of-sample period spans from first week of January 2016 through last week of October 2016. For the out-of-sample period, we have also extracted 30-minute intraday data for realized volatility measuring purpose.


### 3.2. Test of Stationarity and Features of Log Return Series

Our data consist of two commodity prices: crude oil and Arabica coffee. The two commodity prices present the same trend and direction during the entire study period. Prices for both of the commodities have risen from 2005 up to 2008 and fallen from 2009 up to 2010 with a similar pattern. Another impression is that both commodity level series are nonstationary (see Figure1).Figure 1
Time series plot for level series, Arabica coffee and crude oil.To visualize the returns series for these two markets, we depict the return time series plots in Figure2. The weekly return series display volatility persistence properties, indicating that large changes tend to be followed by large changes of both signs and small changes tend to be followed by small changes, and the differenced series suggests stationarity. The plots further show that the volatilities of these two commodity returns not only have a volatility clustering phenomenon during the selected sample period but also have certain relation on their return volatility processes. That is, when the fluctuation of the Arabica coffee price grew larger, the volatility of crude oil return also became larger for most of the times. This is the main motive for discussing the relationships of coffee price returns and crude oil price returns.Figure 2
Time series plot for differenced series, Arabica coffee and crude oil.Table1 summarizes the DF test results of the two commodity level prices, respectively. As can be seen in the table, p-value is greater than 0.05. Hence, the null hypothesis of unit root would not be rejected, that is, there is a unit root problem in each of the series.Table 1
DF unit root test of stationarity for level prices, with trend.
Series
t-statistic
MacKinnon approximatep-value
Crude oil
−1.675
0.4562
Arabica coffee
−2.053
0.3896If time series data are nonstationary, it is necessary to look for possible transformations that might bring stationarity. In practice, econometricians usually transform financial prices in to return forms. This is because often return series are found to be stationary such that analysis is possible. The log return series is obtained by(19)rt=52∗logptpt−1,where rt is the log return series of the real price multiplied by 52 (which is simply a scaling factor), to annualize as we are using weekly data and each year contains 52 weeks, and pt is the originalprice series and log is the natural logarithm in base ten.Table2 summarizes the DF test of the log return series for each of the commodities. The table shows that both p-values are too small (p-value <0.05) at 5% level of significance. These indicate that the null hypothesis of unit root would be rejected in both cases. Hence, the log return series are stationary for each of the items as shown by the time series plots.Table 2
DF unit root test of stationarity for log returns.
Series
t-statistic
MacKinnon approximatep-value
Crude oil
−5.791
0.001
Arabica coffee
−5.877
0.001Looking at Table3, the average prices for Arabica coffee (per Pound) and crude oil (per Barrel) were $145.8668 and $80.9327, respectively, over the entire period. The average log returns were then computed, and they are 0.0458 and 0.0119 for coffee and oil, respectively. The average returns for crude oil price were smaller than those of Arabica coffee price return in the sample period considered. One more impression here is that the two commodities’ return is positive, which is an implication of the increasing trend in the two commodity price. The estimate of the standard deviation is 2.229 for coffee and 2.3639 for oil which is nearly similar. Hence, we can say that they exhibit approximately similar variations in their prices.Table 3
General summary statistics for Arabica coffee and crude oil series.
Statistics
Coffee market
Crude oil market
P
t
r
t
P
t
r
t
Mean
145.8668
0.0458
80.9327
0.0119
Median
132.7250
0
76.1950
0.1327
Maximum
299.3500
9.6800
144.4900
10.449
Minimum
88.2000
−7.538
28.9400
−15.451
Std.dev.
43.3636
2.229
26.1545
2.3639
Skewness
1.4120
0.0359
0.1347
−0.6890
Kurtosis
4.4545
3.6941
1.7781
7.2470
Jarque–Bera
144.8500 (0.001)
8.2600 (0.016)
488.0200 (0.001)
107.4100 (0.001)
∗
∗Values below the Jarque–Bera statistics are p-values at 5% of significance level.


### 3.3. Volatility Modeling

To build a volatility model for the log return series, the first step is specifying the mean equation. Once we specify the mean equation, we have to test for ARCH effects using the residuals of the mean equation. If ARCH effects are statistically significant, specifying a volatility model and carrying out a joint estimation of the mean and volatility equations are necessary. The conditional mean specification is, in general, arbitrary for GARCH models of the conditional volatility.


#### 3.3.1. Test for ARCH Effects

Considering a financial return model and on fitting this model, if there is no volatility clustering in each of the returns, the random disturbance termεt should be a white noise process. The standardized residual plot can be an initial insight to judge the heteroskedastic characteristics of the error term. The standardized residual plot from each commodity return series is shown in Figure 3. The figure depicts the residual plots of the two commodity return series generated from the mean equation.Figure 3
Residual plots of the Arabica coffee and crude oil return series from the mean.We see from the figure that for both of the series, there is a prolonged period of low volatility and prolonged period of high volatility. For example, for Arabica coffee return, there is a long period of low volatility from the first week of 2005 to the end of 2008 and there exist long periods of high volatility from first week of 2014 to the end week of 2016. Crude oil return also exhibits a prolonged period of low volatility than the coffee series, which is from the first week of 2005 to the first week of 2009. In other words, periods of high volatility are followed by periods of high volatility and periods of low volatility tend to be followed by periods of low volatility, which is known as volatility clustering. This suggests that the residuals or error terms are conditionally heteroskedastic and can be represented by GARCH models.Table4 shows the results of ARCH LM test for the two commodity returns. The last column of the table includes the p-values that indicate rejection of the null hypothesis that “there is no ARCH effect” up to the fourth lag at 5% level of significance. The results indicate that the two commodities’ price log return series are volatile and need to be modeled using GARCH models.Table 4
ARCH LM test summary statistics.
Series
Obs∗R2
χ
2 (4)
Lag (p)
Prob.
Arabica coffee
12.6477
9.488
4
0.004
Crude oil
105.6419
9.488
4
0.001


### 3.4. Multivariate GARCH Modeling

At the beginning of reviewing different formulations of MGARCH models, one should consider what specification of a MGARCH model should be imposed in contrast to the univariate case. On the one hand, it should be flexible enough to state the dynamics of the conditional variances and covariance. On the other hand, as the number of parameters in a MGARCH model increases rapidly along with the dimension of the model, the specification should be parsimonious to simplify the model estimation and also reach the purpose of easy interpretation of the model parameters. However, parsimony may reduce the number of parameters, in which situation the relevant dynamics in the covariance matrix cannot be captured.A number of alternative MGARCH formulations for modeling long-run conditional heteroskedasticity are extensively used in relevant literature. The most widely used are the time-varying conditional correlation (VCC), the constant conditional correlation (CCC), and the dynamic conditional correlation (DCC) models [5]. A more elaborate discussion of these models is presented in section2.1 of chapter two . One complexity of MGARCH models is that the number of parameters grows rapidly and researchers often use a single ARCH and GARCH term in the model specification, and often it is enough in capturing the variation. So, in this paper, we use bivariate-GARCH (1,1) models.One of the major objectives of this study is choosing an appropriate multivariate GARCH model from the competing models. Using the Mincer-Zarnowitz regression and the forecast error measures, a model with minimum forecast error measure value and maximum MZR2 value is selected as an appropriate model. We can chose a single bivariate GARCH model and then estimate the parameters of the selected model. For the distribution of the innovation term, the multivariate normal distribution and the multivariate version of Student’s t distribution with eight degree of freedom are considered. Like the previous analysis, we also consider two volatility measures, that is, the weekly return square and the 30-minute intraday return square.Looking at Tables5 and 6 for each of the models, the R2 obtained from Mincer-Zarnowitz regression (R2) value is larger for the high-frequency data. Moreover, the R2 is larger under Student t distribution assumption of the innovation term. Now combining the high-frequency data and Student t distribution, the time-varying conditional correlation (VCC) model is a better model for the variance covariance estimation of the Arabica coffee return and crude oil returns.Table 5
Bivariate GARCH model comparison under normal distribution for covolatility of coffee and oil returns.
Statistics
Realized volatility measure
Weekly returns squared
30-minute interval returns(high freq.)
CCC
DCC
VCC
CCC
DCC
VCC
MZ.reg.R2
0.0145
0.0037
0.0070
0.2575
0.1740
0.3929
MSE
29.2503
29.9553
29.313629
0.7865
0.9161
0.8590
RMSE
5.1442
5.4730
5.4083
0.8868
0.9571
0.9268
MAE
3.7900
3.7452
3.7307
0.879
0.8517
0.8000
AIC
4965.981
4957.583
4962.655
4965.981
4962.655
4957.583
BIC
5022.474
5022.768
5027.84
5022.474
5022.768
5027.84
LL
−2469.99
−2463.79
−2466.32
−2469.99
−2463.79
−2466.32Table 6
Bivariate GARCH model comparison under Student’st distribution for covolatility of coffee and oil returns.
Statistics
Realized volatility measure
Weekly return square
High frequency
CCC
DCC
VCC
CCC
DCC
VCC
MZ.reg.R2
0.0937
0.0747
0.060
0.2372
0.1614
0.4014
MSE
29.36
29.99
29.28
0.7876
0.8877
0.8634
RMSE
5.4184
5.4763
5.4110
0.8874
0.9421
0.9291
MAE
3.7953
3.7532
3.742
0.8819
0.8423
0.8372
AIC
4945.74
4940.492
4943.54
4945.74
4943.54
4940.492
BIC
5002.24
5005.677
5008.725
5002.24
5005.677
5008.725
LL
−2459.84
−2455.24
−2456.77
−2459.84
−2455.24
−2456.77The results in Table7 of the variance equations actually indicated volatility effects in the returns series across the sample period considered. With regard to the estimates of the time-varying correlation ρ^coffee,oil, the correlation was found to be positive and significant. This implies positive relationship between the conditional volatility series of the two commodity series and indicates that the returns on these commodities rise or fall together.Table 7
Variance parameter estimates from time-varying conditional correlation (VCC-GARCH) model undert distribution (v=8).
Parameters
Coefficients
z-statistics
p-value
ω
^
coffee
0.1647
1.04
0.300
α
^
coffee
0.0305
1.60
0.109
β
^
coffee
0.9393
22.16
0.001
ω
^
oil
0.0835
1.60
0.109
α
^
oil
0.0774
3.60
0.001
β
^
oil
0.9092
38.48
0.001
ρ
^
coffee,oil
0.2886
1.60
0.001
π
^
1
0.02919
1.70
0.034
π
^
2
0.919
15.88
0.001
α
^
coff.
+
β
^
coff.
0.9698
—
—
α
^
oil
+
β
^
oil
0.9866
—
—The estimates of the parametersπ^1 and π^2, that is, the parameters dealing with the effects of previous shocks and previous dynamic conditional correlations on the current dynamic conditional correlation, were significantly different from zero, with π^1 = 0.0291 and π^2 = 0.919. The time-varying conditional correlation (VCC) model reduces to constant conditional correlation (CCC) model when π^1=π^2=0 [1]. Hence, employing CCC is too restrictive for these series.Concerning the estimates of persistence (α^i+β^i), the persistence estimates were 0.9698 and 0.9866 at coffee and oil markets, respectively, which is closer to one, implying that volatility is persistent in these markets. The sufficient conditions for the variance matrix to be positive definite are the usual GARCH restrictions, i.e., ω,α,β>0 for the univariate GARCH (1,1) model. Also, this univariate GARCH process needs to be stationary, i.e., 1−α−β > 0. These conditions are satisfied in the above output table.


### 3.5. Discussion

The aim has been modeling and forecasting the covolatility dynamics of weekly time series of Brent crude oil and Coffee Arabica futures market closing price using bivariate GARCH models. From the preliminary analysis over the time period considered, both of the price series show an increasing trend. To determine whether the series are stationary or not, the augmented Dickey–Fuller (ADF) test was carried out. Often, raw data of commodity prices are nonstationary, which was also the case in this study. For both level time series, the tests indicate to the existence of unit root I (1). The first log difference of each time series was considered as stationary (Table3).In the two commodity returns, it appears that the ARCH term coefficients were significant at 5% level of significance. The result is an implication of the presence of volatility clustering, i.e., large changes followed by large changes and small changes followed by small changes.Noemi and Junior [12] investigated the volatility of two tropical commodities, i.e., world coffee and crude oil, and found that the two commodities were volatile and concluded that fluctuations in coffee and cocoa price follow the oil price fluctuation. Our results shows that Arabica coffee and crude oil volatilities are correlated (in fact, not highly). In addition, the covariance of Arabica coffee and crude oil is found to be similar to the variance of crude oil which indicates that the variation in the Arabica coffee price follows crude oil price, a similar finding with Noemi and Junior.Overall, the positive relationship between these commodity returns is consistent with previous research studies on energy and commodity covolatility (see for example, [13, 14]).


## 3.1. Data

The data considered in this paper were the weekly time series of Brent crude oil and Coffee Arabica futures market closing price, given in US dollars per barrel and US dollars per pound, respectively. Both series contain data spanning between first week of January 2005 and last week of October 2016 extracted from Bloomberg database. The full sample is split in to two parts: in-sample data, in order to estimate the parameters of models, and out-of-sample data, in order to make forecasts. The in-sample period spans from first week of January 2005 up to last week of December 2015 and the out-of-sample period spans from first week of January 2016 through last week of October 2016. For the out-of-sample period, we have also extracted 30-minute intraday data for realized volatility measuring purpose.


## 3.2. Test of Stationarity and Features of Log Return Series

Our data consist of two commodity prices: crude oil and Arabica coffee. The two commodity prices present the same trend and direction during the entire study period. Prices for both of the commodities have risen from 2005 up to 2008 and fallen from 2009 up to 2010 with a similar pattern. Another impression is that both commodity level series are nonstationary (see Figure1).Figure 1
Time series plot for level series, Arabica coffee and crude oil.To visualize the returns series for these two markets, we depict the return time series plots in Figure2. The weekly return series display volatility persistence properties, indicating that large changes tend to be followed by large changes of both signs and small changes tend to be followed by small changes, and the differenced series suggests stationarity. The plots further show that the volatilities of these two commodity returns not only have a volatility clustering phenomenon during the selected sample period but also have certain relation on their return volatility processes. That is, when the fluctuation of the Arabica coffee price grew larger, the volatility of crude oil return also became larger for most of the times. This is the main motive for discussing the relationships of coffee price returns and crude oil price returns.Figure 2
Time series plot for differenced series, Arabica coffee and crude oil.Table1 summarizes the DF test results of the two commodity level prices, respectively. As can be seen in the table, p-value is greater than 0.05. Hence, the null hypothesis of unit root would not be rejected, that is, there is a unit root problem in each of the series.Table 1
DF unit root test of stationarity for level prices, with trend.
Series
t-statistic
MacKinnon approximatep-value
Crude oil
−1.675
0.4562
Arabica coffee
−2.053
0.3896If time series data are nonstationary, it is necessary to look for possible transformations that might bring stationarity. In practice, econometricians usually transform financial prices in to return forms. This is because often return series are found to be stationary such that analysis is possible. The log return series is obtained by(19)rt=52∗logptpt−1,where rt is the log return series of the real price multiplied by 52 (which is simply a scaling factor), to annualize as we are using weekly data and each year contains 52 weeks, and pt is the originalprice series and log is the natural logarithm in base ten.Table2 summarizes the DF test of the log return series for each of the commodities. The table shows that both p-values are too small (p-value <0.05) at 5% level of significance. These indicate that the null hypothesis of unit root would be rejected in both cases. Hence, the log return series are stationary for each of the items as shown by the time series plots.Table 2
DF unit root test of stationarity for log returns.
Series
t-statistic
MacKinnon approximatep-value
Crude oil
−5.791
0.001
Arabica coffee
−5.877
0.001Looking at Table3, the average prices for Arabica coffee (per Pound) and crude oil (per Barrel) were $145.8668 and $80.9327, respectively, over the entire period. The average log returns were then computed, and they are 0.0458 and 0.0119 for coffee and oil, respectively. The average returns for crude oil price were smaller than those of Arabica coffee price return in the sample period considered. One more impression here is that the two commodities’ return is positive, which is an implication of the increasing trend in the two commodity price. The estimate of the standard deviation is 2.229 for coffee and 2.3639 for oil which is nearly similar. Hence, we can say that they exhibit approximately similar variations in their prices.Table 3
General summary statistics for Arabica coffee and crude oil series.
Statistics
Coffee market
Crude oil market
P
t
r
t
P
t
r
t
Mean
145.8668
0.0458
80.9327
0.0119
Median
132.7250
0
76.1950
0.1327
Maximum
299.3500
9.6800
144.4900
10.449
Minimum
88.2000
−7.538
28.9400
−15.451
Std.dev.
43.3636
2.229
26.1545
2.3639
Skewness
1.4120
0.0359
0.1347
−0.6890
Kurtosis
4.4545
3.6941
1.7781
7.2470
Jarque–Bera
144.8500 (0.001)
8.2600 (0.016)
488.0200 (0.001)
107.4100 (0.001)
∗
∗Values below the Jarque–Bera statistics are p-values at 5% of significance level.


## 3.3. Volatility Modeling

To build a volatility model for the log return series, the first step is specifying the mean equation. Once we specify the mean equation, we have to test for ARCH effects using the residuals of the mean equation. If ARCH effects are statistically significant, specifying a volatility model and carrying out a joint estimation of the mean and volatility equations are necessary. The conditional mean specification is, in general, arbitrary for GARCH models of the conditional volatility.


### 3.3.1. Test for ARCH Effects

Considering a financial return model and on fitting this model, if there is no volatility clustering in each of the returns, the random disturbance termεt should be a white noise process. The standardized residual plot can be an initial insight to judge the heteroskedastic characteristics of the error term. The standardized residual plot from each commodity return series is shown in Figure 3. The figure depicts the residual plots of the two commodity return series generated from the mean equation.Figure 3
Residual plots of the Arabica coffee and crude oil return series from the mean.We see from the figure that for both of the series, there is a prolonged period of low volatility and prolonged period of high volatility. For example, for Arabica coffee return, there is a long period of low volatility from the first week of 2005 to the end of 2008 and there exist long periods of high volatility from first week of 2014 to the end week of 2016. Crude oil return also exhibits a prolonged period of low volatility than the coffee series, which is from the first week of 2005 to the first week of 2009. In other words, periods of high volatility are followed by periods of high volatility and periods of low volatility tend to be followed by periods of low volatility, which is known as volatility clustering. This suggests that the residuals or error terms are conditionally heteroskedastic and can be represented by GARCH models.Table4 shows the results of ARCH LM test for the two commodity returns. The last column of the table includes the p-values that indicate rejection of the null hypothesis that “there is no ARCH effect” up to the fourth lag at 5% level of significance. The results indicate that the two commodities’ price log return series are volatile and need to be modeled using GARCH models.Table 4
ARCH LM test summary statistics.
Series
Obs∗R2
χ
2 (4)
Lag (p)
Prob.
Arabica coffee
12.6477
9.488
4
0.004
Crude oil
105.6419
9.488
4
0.001


## 3.3.1. Test for ARCH Effects

Considering a financial return model and on fitting this model, if there is no volatility clustering in each of the returns, the random disturbance termεt should be a white noise process. The standardized residual plot can be an initial insight to judge the heteroskedastic characteristics of the error term. The standardized residual plot from each commodity return series is shown in Figure 3. The figure depicts the residual plots of the two commodity return series generated from the mean equation.Figure 3
Residual plots of the Arabica coffee and crude oil return series from the mean.We see from the figure that for both of the series, there is a prolonged period of low volatility and prolonged period of high volatility. For example, for Arabica coffee return, there is a long period of low volatility from the first week of 2005 to the end of 2008 and there exist long periods of high volatility from first week of 2014 to the end week of 2016. Crude oil return also exhibits a prolonged period of low volatility than the coffee series, which is from the first week of 2005 to the first week of 2009. In other words, periods of high volatility are followed by periods of high volatility and periods of low volatility tend to be followed by periods of low volatility, which is known as volatility clustering. This suggests that the residuals or error terms are conditionally heteroskedastic and can be represented by GARCH models.Table4 shows the results of ARCH LM test for the two commodity returns. The last column of the table includes the p-values that indicate rejection of the null hypothesis that “there is no ARCH effect” up to the fourth lag at 5% level of significance. The results indicate that the two commodities’ price log return series are volatile and need to be modeled using GARCH models.Table 4
ARCH LM test summary statistics.
Series
Obs∗R2
χ
2 (4)
Lag (p)
Prob.
Arabica coffee
12.6477
9.488
4
0.004
Crude oil
105.6419
9.488
4
0.001


## 3.4. Multivariate GARCH Modeling

At the beginning of reviewing different formulations of MGARCH models, one should consider what specification of a MGARCH model should be imposed in contrast to the univariate case. On the one hand, it should be flexible enough to state the dynamics of the conditional variances and covariance. On the other hand, as the number of parameters in a MGARCH model increases rapidly along with the dimension of the model, the specification should be parsimonious to simplify the model estimation and also reach the purpose of easy interpretation of the model parameters. However, parsimony may reduce the number of parameters, in which situation the relevant dynamics in the covariance matrix cannot be captured.A number of alternative MGARCH formulations for modeling long-run conditional heteroskedasticity are extensively used in relevant literature. The most widely used are the time-varying conditional correlation (VCC), the constant conditional correlation (CCC), and the dynamic conditional correlation (DCC) models [5]. A more elaborate discussion of these models is presented in section2.1 of chapter two . One complexity of MGARCH models is that the number of parameters grows rapidly and researchers often use a single ARCH and GARCH term in the model specification, and often it is enough in capturing the variation. So, in this paper, we use bivariate-GARCH (1,1) models.One of the major objectives of this study is choosing an appropriate multivariate GARCH model from the competing models. Using the Mincer-Zarnowitz regression and the forecast error measures, a model with minimum forecast error measure value and maximum MZR2 value is selected as an appropriate model. We can chose a single bivariate GARCH model and then estimate the parameters of the selected model. For the distribution of the innovation term, the multivariate normal distribution and the multivariate version of Student’s t distribution with eight degree of freedom are considered. Like the previous analysis, we also consider two volatility measures, that is, the weekly return square and the 30-minute intraday return square.Looking at Tables5 and 6 for each of the models, the R2 obtained from Mincer-Zarnowitz regression (R2) value is larger for the high-frequency data. Moreover, the R2 is larger under Student t distribution assumption of the innovation term. Now combining the high-frequency data and Student t distribution, the time-varying conditional correlation (VCC) model is a better model for the variance covariance estimation of the Arabica coffee return and crude oil returns.Table 5
Bivariate GARCH model comparison under normal distribution for covolatility of coffee and oil returns.
Statistics
Realized volatility measure
Weekly returns squared
30-minute interval returns(high freq.)
CCC
DCC
VCC
CCC
DCC
VCC
MZ.reg.R2
0.0145
0.0037
0.0070
0.2575
0.1740
0.3929
MSE
29.2503
29.9553
29.313629
0.7865
0.9161
0.8590
RMSE
5.1442
5.4730
5.4083
0.8868
0.9571
0.9268
MAE
3.7900
3.7452
3.7307
0.879
0.8517
0.8000
AIC
4965.981
4957.583
4962.655
4965.981
4962.655
4957.583
BIC
5022.474
5022.768
5027.84
5022.474
5022.768
5027.84
LL
−2469.99
−2463.79
−2466.32
−2469.99
−2463.79
−2466.32Table 6
Bivariate GARCH model comparison under Student’st distribution for covolatility of coffee and oil returns.
Statistics
Realized volatility measure
Weekly return square
High frequency
CCC
DCC
VCC
CCC
DCC
VCC
MZ.reg.R2
0.0937
0.0747
0.060
0.2372
0.1614
0.4014
MSE
29.36
29.99
29.28
0.7876
0.8877
0.8634
RMSE
5.4184
5.4763
5.4110
0.8874
0.9421
0.9291
MAE
3.7953
3.7532
3.742
0.8819
0.8423
0.8372
AIC
4945.74
4940.492
4943.54
4945.74
4943.54
4940.492
BIC
5002.24
5005.677
5008.725
5002.24
5005.677
5008.725
LL
−2459.84
−2455.24
−2456.77
−2459.84
−2455.24
−2456.77The results in Table7 of the variance equations actually indicated volatility effects in the returns series across the sample period considered. With regard to the estimates of the time-varying correlation ρ^coffee,oil, the correlation was found to be positive and significant. This implies positive relationship between the conditional volatility series of the two commodity series and indicates that the returns on these commodities rise or fall together.Table 7
Variance parameter estimates from time-varying conditional correlation (VCC-GARCH) model undert distribution (v=8).
Parameters
Coefficients
z-statistics
p-value
ω
^
coffee
0.1647
1.04
0.300
α
^
coffee
0.0305
1.60
0.109
β
^
coffee
0.9393
22.16
0.001
ω
^
oil
0.0835
1.60
0.109
α
^
oil
0.0774
3.60
0.001
β
^
oil
0.9092
38.48
0.001
ρ
^
coffee,oil
0.2886
1.60
0.001
π
^
1
0.02919
1.70
0.034
π
^
2
0.919
15.88
0.001
α
^
coff.
+
β
^
coff.
0.9698
—
—
α
^
oil
+
β
^
oil
0.9866
—
—The estimates of the parametersπ^1 and π^2, that is, the parameters dealing with the effects of previous shocks and previous dynamic conditional correlations on the current dynamic conditional correlation, were significantly different from zero, with π^1 = 0.0291 and π^2 = 0.919. The time-varying conditional correlation (VCC) model reduces to constant conditional correlation (CCC) model when π^1=π^2=0 [1]. Hence, employing CCC is too restrictive for these series.Concerning the estimates of persistence (α^i+β^i), the persistence estimates were 0.9698 and 0.9866 at coffee and oil markets, respectively, which is closer to one, implying that volatility is persistent in these markets. The sufficient conditions for the variance matrix to be positive definite are the usual GARCH restrictions, i.e., ω,α,β>0 for the univariate GARCH (1,1) model. Also, this univariate GARCH process needs to be stationary, i.e., 1−α−β > 0. These conditions are satisfied in the above output table.


## 3.5. Discussion

The aim has been modeling and forecasting the covolatility dynamics of weekly time series of Brent crude oil and Coffee Arabica futures market closing price using bivariate GARCH models. From the preliminary analysis over the time period considered, both of the price series show an increasing trend. To determine whether the series are stationary or not, the augmented Dickey–Fuller (ADF) test was carried out. Often, raw data of commodity prices are nonstationary, which was also the case in this study. For both level time series, the tests indicate to the existence of unit root I (1). The first log difference of each time series was considered as stationary (Table3).In the two commodity returns, it appears that the ARCH term coefficients were significant at 5% level of significance. The result is an implication of the presence of volatility clustering, i.e., large changes followed by large changes and small changes followed by small changes.Noemi and Junior [12] investigated the volatility of two tropical commodities, i.e., world coffee and crude oil, and found that the two commodities were volatile and concluded that fluctuations in coffee and cocoa price follow the oil price fluctuation. Our results shows that Arabica coffee and crude oil volatilities are correlated (in fact, not highly). In addition, the covariance of Arabica coffee and crude oil is found to be similar to the variance of crude oil which indicates that the variation in the Arabica coffee price follows crude oil price, a similar finding with Noemi and Junior.Overall, the positive relationship between these commodity returns is consistent with previous research studies on energy and commodity covolatility (see for example, [13, 14]).


## 4. Conclusion and Recommendations

Forecasting the volatility dynamics of asset returns has been the subject of extensive research among academics, practitioners, and portfolio managers. This thesis estimates a variety of multivariate GARCH models using weekly closing price (in USD/barrel) of Brent crude oil and weekly closing prices (in USD/per pound) of Coffee Arabica and compares the forecasting performance of these models based on high-frequency intraday data which allows for a more precise realized volatility measurement. The analysis points to the conclusion that varying conditional correlation (VCC) model with Student’st distributed innovation terms is the most accurate volatility forecasting model in the context of our empirical setting.We recommend and encourage future researchers studying the forecasting performance of MGARCH models to pay particular attention to the measurement of realized volatility and employ high-frequency data whenever feasible.We also recommend that public policy makers interested in foreseeing the price volatility of these two major commodities in the context of the Ethiopian economy consider using the information documented in this study as input in their deliberations given that it is based on some robust econometric work and highly appropriate data.The scope of the analysis in this study has been limited to the covolatility between the two commodities. In order to overcome this limitation and provide a more nuanced analysis, it might be profitable for future researchers to consider incorporating stock, currency, and bond market volatilities into the analysis. The potential complexity of such a research agenda notwithstanding, the results are likely to be rewarding in light of the deeper integration between global financial and commodity markets in recent years, a phenomenon which came to be known as the financialization of commodities.



---

*Source: 1424020-2020-04-04.xml*