^{1}

^{*}

^{2}

^{2}

^{2}

This paper proposes a multivariate VAR-BEKK-GJR-GARCH volatility model to assess the dynamic interdependence among stock, bond and money market returns and volatility of returns. The proposed model allows for market interaction which provides useful information for pricing securities, measuring value-at-risk (VaR), and asset allocation and diversification, assisting financial regulators for policy implementation. The model is estimated by the maximum likelihood method with Student-t innovation density. The asymptotic chi-square tests for volatility spillovers and leverage effects are constructed and provide predictions of volatility and time-varying correlations of returns. Application of the proposed model to the Australia’s domestic stock, bond, and money markets reveals that the domestic financial markets are interdependent and volatility is predictable. In general, volatility spillovers from stock market to bond and to money markets due to common news. The empirical findings of this paper quantify the association among the security markets which can be utilized for improving agents’ decision-making strategies for risk management, portfolio selection and diversification.

Security traders in the financial markets make their “buy” and “sell” decisions based on the information available in the financial markets. The amount of risk associated with a series of returns, however, depends on the arrival of the so-called “good” and “bad” news that continuously spreads throughout the financial markets in every moment of time. Since “news” is not directly observable, returns are stochastic and volatile. An interesting feature of asset price is that “bad” news seems to have a more pronounced effect on volatility than does the “good” news. This asymmetric “news” is associated with the innovation distribution of losses and gains in the financial markets, which plays a vital role in determining the leverage effect on asset volatility. Black [

In developing dynamic volatility models, there are two strands of modelling conditional volatility―the univariate and multivariate volatility modelling respectively. Engle [

The first two moments respectively called mean and variance of return series have been investigated extensively in the univariate finance literature to understand the trading dynamics of risk and returns in the financial asset markets, for example Bollerslev [

The Second strand of volatility modelling has been emerged from modelling volatilities of returns within the multivariate framework. Within this framework the shocks to volatility from one market is allowed to affect both the risk and return of the other markets. The dynamic dependence of multivariate financial assets provides rich sources of volatility transmission that helps the investors to play active role in financial transactions. Specifically, the multivariate extension to univariate GJR-GARCH (or TGARCH) allows volatility spillovers and leverage effects across markets jointly. Directional causality between assets can be established among the securities by statistical testing. The multivariate extension to univariate model was first introduced by Engle and Granger [

In this paper we take the challenge of fitting our proposed multivariate VAR-BEKK-GJR-GARCH (or, VAR-BEKK-GJR-MGARCH) volatility model and investigate the dynamic interdependence among assets. This model is different from Ling and McAleer [

To apprehend the dynamic interdependence of asset returns and volatility spillovers, we combine Engle and Kroner [

Let r r = ( r 1 t , r 2 t , ⋯ , r N t ) ′ be a vector of returns of N number of assets at time index t ( t = 1 , 2 , 3 , ⋯ , T ). The set of information available at time t is denoted by ℑ t − 1 . We assume that the dynamic multivariate security returns r t can be adequately represented by a vector autoregression of order p conditional on the information set ℑ t − i as

r t | ℑ t − 1 = Φ 0 + ∑ l = 1 p Φ ( l ) r t − l + ε t (1)

where, E ( r t | ℑ t − 1 ) = Φ 0 + ∑ l = 1 p Φ ( l ) r t − l = μ t , say, and Φ ( l ) = ( Φ i j ( l ) ) is the

N × N coefficient matrix of the lagged variables. The N × 1 intercept vector is denoted by Φ 0 and ε t | ℑ t − 1 = H t 0.5 e t , where e t = ( e 1 t , e 2 t , ⋯ , e N t ) ′ is the independent and identically distributed (iid) random vectors of order N × 1 with E e t = 0 and E e t e ′ t = I N , where I n is an Identity matrix . The symmetric matrix H t is of order N × N represents the conditional variance-covariance matrix of innovations defines as follows.

H t = E ( ε t ε ′ t | ℑ t − 1 ) = E ( r t − E ( r t ) ) ( r t − E ( r t ) ) ′ | ℑ t − 1 (2)

Model (1) with (2) can be written more compactly as r t | ℑ t − 1 ~ D ( μ t , H t ) , where D ( . , . ) is some specified probability distribution. Or, equivalently as ε t | ℑ t − 1 ~ D ( 0 , H t ) . Various parameterizations for H t have been proposed in the literature, for example, Bollerslev et al. [

To allow for asymmetric transmission of “good” and “bad” “news” information from one asset to another and /or from one market to another, we define a multivariate indicator vector along the lines with Glosten et al. [

d i t − 1 = { 1 if ε i t − 1 < 0 ( “ bad ” news ) 0 if ε i t − 1 ≥ 0 ( “ good ” news ) (3)

We define the variable η i t − 1 = d i t ε i t − 1 2 to introduce the leverage effects on volatility. Allowing both the news and leverage effects on volatility, we specify (2) as follows.

H t = C ′ C + A ′ ε t − 1 ε ′ t − 1 A + B ′ H t − 1 B + Γ ′ η t − 1 η ′ t − 1 Γ (4)

Combining the leverage effects within BEKK volatility model, we have the following VAR-BEKK-GJR-MGARCH (or VAR-BEKK-TMGARCH) model.

Return: r t | ℑ t − 1 = Φ 0 + ∑ l = 1 p Φ ( l ) r t − l + ε t , ε t | ℑ t − 1 ~ D ( 0 , H t ) (1’)

Volatility: H t | ℑ t − 1 = C C ′ + A ε t − 1 ε ′ t − 1 A ′ + B H t − 1 B ′ + Γ η t − 1 Γ ′ (2’)

where, Γ = ( γ i j ) is an N × N matrix of parameters associated with the individual and cross asset leverage effects. The parameters Φ 0 and Φ ( l ) is the coefficient matrix of the autoregression of lag order l. The matrix C is a N × N lower triangular matrix such that C C ′ is symmetric and positive definite containing the intercept parameters of the volatility model. The matrices A = ( α i j ) and B = ( β i j ) , i , j = 1 , 2 , 3 , ⋯ , N , are both N × N matrices of short-run and long-run parameters, respectively and the innovation ε t is as defined above. The model (2’) provides both quality and quantity effects on volatility jointly. If Γ is a zero matrix then (2’) boils down to Engle-Kroner [

In order to estimate the parameters of the model (1’) and (2’) jointly, we assume that the innovation vector follows a multivariate t-distribution with unknown (but equal) degrees of freedom. The advantage of using the t-distribution is that it nests the normal distribution as a limiting case. The t-distribution with small number of degrees of freedom captures skewness and fat-tailed property of return series. Therefore, a data coherent assumption of t-distribution of innovation is meaningful and useful for modelling volatility clustering and non-normality of the financial asset returns. The multivariate t-distribution with T observations has the following log-likelihood function.

ln L = constant − N T 2 ln ( | H t | ) − ( v + N 2 ) ∑ t ln ( 1 + ( r t − μ t ) H t − 1 ( y t − μ t ) ′ v ) (5)

where μ t = Φ 0 + ∑ l = 1 p Φ ( l ) r t − j is the mean vector of returns (see Equation (1)), | H t | is the determinant of the VCV matrix of the innovation vector, ln ( . ) is the natural logarithm of the argument and, v is the unknown shape parameter and ln L is the log-likelihood function of the parameters given the data. Maximum likelihood method is applied to estimate the parameters of the VAR-BEKK-GJR-MGARCH model under the assumption of multivariate t-distribution using FBGLS optimization routine in RATS. The maximum likelihood estimates (MLE) are consistent and asymptotically normally distributed. This property is useful for developing statistical tests on the parameters.

Refer to the multivariate volatility model of Section 2.1, the following hypotheses are of interest to test for return and volatility spillovers and, leverage effects across assets. Considering three assets portfolio, the following hypotheses can be tested by applying Chi-square tests.

1) Return spillovers from bond and Tbill to stock

H 0 : ϕ 12 = ϕ 13 = 0 against H 1 : ϕ 12 ≠ ϕ 13 ≠ 0

2) Return spillovers from stock and Tbill to bond

H 0 : ϕ 21 = ϕ 23 = 0 against H 0 : ϕ 21 ≠ ϕ 23 ≠ 0

3) Return spillovers from stock and bond to Tbill

H 0 : ϕ 31 = ϕ 32 = 0 against H 1 : ϕ 31 ≠ ϕ 32 ≠ 0

1) Volatility spillovers from bond and Tbill to stock

H 0 : α 12 = α 13 = β 12 = β 13 = 0 against H 1 : α 12 ≠ α 13 ≠ β 12 ≠ β 13 ≠ 0

2) Volatility spillovers from stock and Tbill to bond

H 0 : α 21 = α 23 = β 21 = β 23 = 0 against H 1 : α 21 ≠ α 23 ≠ β 21 ≠ β 23 ≠ 0

3) Volatility spillovers from stock and bond to Tbill

H 0 : α 31 = α 32 = β 31 = β 32 = 0 against H 1 : α 31 ≠ α 32 ≠ β 31 ≠ β 32 ≠ 0

We perform the following hypothesis tests for the presence of leverage effects of own shock and shocks due to the other assets on volatility by testing the leverage parameters γ i j ( i = 1 , 2 , 3 ; j = 1 , 2 , 3 ) representing the simultaneous occurrence of the asymmetric news and leverage in model (4).

1) Leverage effect for stock volatility

H 0 : γ 11 = γ 12 = γ 13 = 0 against H 1 : γ 11 ≠ γ 12 ≠ γ 13 ≠ 0

2) Leverage effect for bond volatility

H 0 : γ 21 = γ 22 = γ 23 = 0 against H 1 : γ 21 ≠ γ 22 ≠ γ 23 ≠ 0

3) Leverage effect for Tbill volatility

H 0 : γ 31 = γ 32 = γ 33 = 0 against H 1 : γ 31 ≠ γ 32 ≠ γ 33 ≠ 0

The above hypotheses tests of Section 2.3 were performed by employing Chi-square tests in RATS programing. We have reported the Chi-square test results in the empirical section 4.

Historical data on stock, bond, and Tbill of Australia’s domestic market from 4 April 2006 to 20 June 2016, for a total 883 observations are used for analysis. The data was retrieved from Bloomberg database. The daily returns, in percentages, for stock (all ordinaries), bond (5-year maturity rate), and Tbill (90 day bank accepted bank accepted bill are) are constructed by the following growth rate form.

r i t = 100 × ln ( p i t p i t − 1 ) , i = 1 , 2 , ⋯ , N ; t = 1 , 2 , ⋯ , T (6)

The variable p i t denote the nominal price of the i-th asset at time t and the variable r i t is the percentage log returns (or the growth rate) of the i-th asset at time t, p i t − 1 is the one-period lag of p i t , and ln ( . ) is the natural logarithm of the argument. N is the number of asset and T is the time index.

Data Property and Preliminary ResultsIn this section we provide graphical means to explore the data properties. First we plot the return series and the squared return series. Then we provide the serial correlations and cross correlation of the variables to determine the data dependencies by employing ACF and PACF graphs and Ljung-Box [

All of the tests results indicate that the series are not unit root processes. The test results of

Statistics | Stock | Bond | Tbill |
---|---|---|---|

Mean (%) Yearly mean (%) | −0.034 (0.337) −8.806 | −0.095 (0.045) −24.61 | 0.003 (0.216) 0.78 |

Stdev (%) Yearly stdev (%) | 1.062 17.09 | 1.413 22.74 | 0.065 1.05 |

Min | −4.249 | −6.278 | −0.389 |

Max | 5.529 | 4.667 | 0.740 |

Skewness | −0.203 (0.0140) | −0.273 (0.0009) | 1.414 (0.0000) |

Excess kurtosis | 2.241 (0.0000) | 1.816 (0.0000) | 20.675 (0.000) |

LB (20) LB^{2} (20) | 22.591 (0.309) 504.046 (0.0000) | 58.436 (0.000) 469.484 (0.0000) | 21.441 (0.372) 40.316 (0.0040) |

JB-χ^{2} (2) Test | 190.841 (0.0000) | 132.348 (0.0000) | 16020 (0.0000) |

Tsay Ori-F (10,865) Test (lags 4) | 4.442 (0.0000) | 3.139 (0.0006) | 2.619 (0.0028) |

McLeod and Li Test (lags 4) | 331.257 (0.0000) | 254.087 (0.0000) | 21.067 (0.0206) |

ARCH (LM) Test (lags 4) | 36.360 (0.0000) | 16.974 (0.0000) | 4.404 (0.0000) |

Note: p-value is in parentheses.

Return series | ADF test with lag = 5 | PP test with lag = 5 | KPSS test with lag = 5 |
---|---|---|---|

Stock | −11.783*** | −28.572*** | 0.138 |

Bond | −10.953*** | −31.442*** | 0.456 |

Tbill | −12.277*** | −30.040*** | 0.257 |

***Significant at 1% level. Note: The Null hypothesis for KPSS is stationary while ADF and PP tests the null hypothesis of non-stationarity.

We apply AIC, BIC, and HQ criteria to select the order of the VAR of mean model. We select order 1 for VAR because among the three criteria both BIC and HQ select VAR of order 1. In the univariate case, there was overwhelming support to GARCH (1, 1) (Bollerslev) [

The nonlinear maximum likelihood with t-innovation is used to estimate the parameters of the model of interest. Estimated parameters with the corresponding standard error and the p-value of tests are reported in

The parameter ( ϕ i j ) ( i , j = 1 , 2 , 3 ) is the (i, j)-th element of the ( 3 × 3 ) matrix of the coefficient of the first order VAR and ϕ 0 i ( i = 1 , 2 , 3 ) is a ( 3 × 1 ) vector of intercept parameters of the mean model. The parameter c i , j where i > j = 1 , 2 , 3 , are the lower-triangular elements of intercept of the Variance-Covariance (VCV) matrix; the ( 3 × 3 ) matrix of the ARCH and GARCH parameters are ( α i j ) and ( β i j ) respectively. The ( γ i , j ) is the ( 3 × 3 ) matrix of leverage parameters associated with the threshold variables ( η i j t − 1 ) . Similarly, h i j t − 1 is element of the symmetric variance-covariance matrix of lagged volatility and ε i j t − 1 ε ′ i j t − 1 is the squared lagged innovation. The parameter v is the degrees of freedom parameter of the t-innovation density. The diagonal elements of α i i , β i i and γ i i are all found to be positive. Many of the parameters ( α i j ) , ( β i j ) , ( γ i j ) and ( ϕ i j ) for i ≠ j are significant in the full VAR-BEKK-GJR-MGARCH model. These results of

The shape parameter is estimated to v = 6 (approximately). This result is based on the assumption that the trivariate t-distribution has common but unknown degrees of freedom. The long-run parameters β ^ i i ’s are significant at the 0.01 level. The shot-run volatility parameters, α ^ i i ’s are significant between 0.01 and 0.10 levels. Many of the leverage effect coefficients are significant indicating the existence of asymmetric news effects on volatility. The value of R^{2} is not reported in the table because the model is highly non-linear therefore R^{2} is not a meaningful measure of goodness of fit.

The parameter stability of the model is tested by using the Nyblom score test. All of the estimated parameters, except ϕ 10 and ϕ 21 , are found to be stable by the Nyblom test. The Nyblom joint score test statistic is found to be 8.2095 with a p-value of 0.28 implying that the parameters of VAR-BEKK-GJR-MGARCH are jointly stable. Parameter stability of a model is a requirement for efficient prediction of econometric models. Further, the multivariate model is tested for model adequacy using the Ljung-Box (LB) [

The LB test results fail to suggest any model inadequacy of serial dependence of the model errors.

Parameter number | Parameter | Variable | Coefficient | Std. error | p-value | Nyblom score | p-value of score |
---|---|---|---|---|---|---|---|

1 | ϕ 11 | S t o c k t − 1 | −0.059** | 0.0295 | 0.0464 | 0.241 | 0.19 |

2 | ϕ 12 | B o n d t − 1 | 0.094*** | 0.0282 | 0.0009 | 0.413 | 0.07 |

3 | ϕ 13 | T b i l l t − 1 | −1.956*** | 0.6240 | 0.0017 | 0.199 | 0.26 |

4 | ϕ 10 | Constant | 0.046* | 0.0275 | 0.0920 | 0.047 | 0.89 |

5 | ϕ 21 | S t o c k t − 1 | −0.019 | 0.0263 | 0.4722 | 0.035 | 0.96 |

6 | ϕ 22 | B o n d t − 1 | −0.014 | 0.0382 | 0.7152 | 0.188 | 0.28 |

7 | ϕ 23 | T b i l l t − 1 | 1.120 | 0.7263 | 0.1230 | 0.486 | 0.04 |

8 | ϕ 20 | Constant | −0.064** | 0.0304 | 0.0365 | 0.062 | 0.79 |

9 | ϕ 31 | S t o c k t − 1 | 0.0002 | 0.0013 | 0.8887 | 0.051 | 0.86 |

10 | ϕ 32 | B o n d t − 1 | 0.0019 | 0.0012 | 0.1122 | 0.301 | 0.13 |

11 | ϕ 33 | T b i l l t − 1 | 0.0165 | 0.0356 | 0.6435 | 0.166 | 0.33 |

12 | ϕ 30 | Constant | 0.0007 | 0.0014 | 0.6109 | 0.084 | 0.65 |

13 | c 11 | C (1,1) | 0.202*** | 0.0540 | 0.0002 | 0.371 | 0.09 |

14 | c 21 | C (2, 1) | −0.018 | 0.0350 | 0.6160 | 0.069 | 0.75 |

15 | c 22 | C (2, 2) | 0.008 | 0.0591 | 0.8880 | 0.157 | 0.36 |

16 | c 31 | C (3, 1) | −0.0002 | 0.0024 | 0.9458 | 0.074 | 0.71 |

17 | c 32 | C (3, 2) | −0.0042 | 0.0031 | 0.2369 | 0.242 | 0.19 |

18 | c 33 | C (3, 3) | 0.00001 | 0.0365 | 1.0000 | 0.227 | 0.22 |

19 | α 11 | ε 1 t − 1 2 | 0.1424* | 0.0760 | 0.0611 | 0.438 | 0.06 |

20 | α 12 | ε 1 t − 1 ε 2 t − 1 | −0.0342 | 0.0470 | 0.4663 | 0.081 | 0.67 |

21 | α 13 | ε 1 t − 1 ε 3 t − 1 | −0.0017 | 0.0024 | 0.4664 | 0.160 | 0.35 |

22 | α 21 | ε 2 t − 1 ε 1 t − 1 | 0.0132 | 0.0403 | 0.7436 | 0.069 | 0.75 |

23 | α 22 | ε 2 t − 1 2 | 0.2789*** | 0.0368 | 0.0000 | 0.283 | 0.15 |
---|---|---|---|---|---|---|---|

24 | α 23 | ε 2 t − 1 ε 3 t − 1 | −0.0011 | 0.0011 | 0.3155 | 0.094 | 0.6 |

25 | α 31 | ε 3 t − 1 ε 1 t − 1 | 0.6959 | 0.9528 | 0.4652 | 0.087 | 0.63 |

26 | α 32 | ε 3 t − 1 ε 1 t − 1 | 0.4807 | 0.7197 | 0.5042 | 0.142 | 0.4 |

27 | α 33 | ε 3 t − 1 2 | 0.1980*** | 0.0358 | 0.0000 | 0.084 | 0.65 |

28 | β 11 | h 11 t − 1 | 0.9334*** | 0.0266 | 0.0000 | 0.376 | 0.08 |

29 | β 12 | h 12 t − 1 | −0.0165 | 0.0158 | 0.2944 | 0.034 | 0.96 |

30 | β 13 | h 13 t − 1 | 0.002* | 0.0009 | 0.0771 | 0.094 | 0.6 |

31 | β 21 | h 21 t − 1 | 0.0010 | 0.0103 | 0.9199 | 0.047 | 0.89 |

32 | β 22 | h 22 t − 1 | 0.951*** | 0.0097 | 0.0000 | 0.359 | 0.09 |

33 | β 23 | h 23 t − 1 | 0.0001 | 0.0003 | 0.6932 | 0.163 | 0.34 |

34 | β 31 | h 31 t − 1 | −0.286 | 0.2995 | 0.3406 | 0.056 | 0.83 |

35 | β 32 | h 32 t − 1 | −0.441* | 0.2360 | 0.0619 | 0.150 | 0.38 |

36 | β 33 | h 33 t − 1 | 0.970*** | 0.0123 | 0.0000 | 0.203 | 0.25 |

37 | γ 11 | η 11 t − 1 | −0.361*** | 0.0559 | 0.0000 | 0.175 | 0.31 |

38 | γ 12 | η 12 t − 1 | −0.151*** | 0.0549 | 0.0059 | 0.149 | 0.38 |

39 | γ 13 | η 13 t − 1 | 0.0095*** | 0.0027 | 0.0005 | 0.204 | 0.25 |

40 | γ 21 | η 21 t − 1 | −0.0201 | 0.0593 | 0.7343 | 0.271 | 0.16 |

41 | γ 22 | η 22 t − 1 | 0.1743*** | 0.0460 | 0.0002 | 0.351 | 0.1 |

42 | γ 23 | η 23 t − 1 | −0.0016 | 0.0013 | 0.2237 | 0.120 | 0.48 |

43 | γ 31 | η 31 t − 1 | −0.2331 | 1.2646 | 0.8538 | 0.136 | 0.42 |

44 | γ 32 | η 32 t − 1 | 1.0941 | 1.1152 | 0.3266 | 0.129 | 0.44 |

45 | γ 32 | η 33 t − 1 | 0.0194 | 0.0786 | 0.8055 | 0.160 | 0.35 |

46 | v | Shape | 5.8344*** | 0.6185 | 0.0000 | 0.096 | 0.58 |

Note: ***1%, **5%, and *10% level of significance.

Ljung-Box statistic | Stock market | Bond market | Money market | Multivariate model |
---|---|---|---|---|

LB-Q (10) LB-Q^{2} (10) | 14.012 (0.172) 11.527 (0.318) | 7.357 (0.691) 8.836 (0.548) | 7.983 (0.631) 11.228 (0.340) | 97.26806 (0.282) 95.97079 (0.314) |

LB-Q (20) LB-Q^{2} (20) | 21.746 (0.354) 20.514 (0.426) | 26.784 (0.141) 24.187 (0.234) | 17.986 (0.588) 20.407 (0.433) | 197.09522 (0.182) 165.92558 (0.766) |

Note: p-value of the LB-test is in parentheses.

In this section we report the spillover and leverage effects of return and volatilities of returns. Based on the MLE estimates of the VAR-BEKK-GJR-MGARCH model parameters we conduct the spillovers effect of returns, volatility and leverage effects. The tests are based on Wald Chi-square statistic.

The test results in

The test results of

Return spillover | Chi-square | |
---|---|---|

From | To | |

Bond and Tbill | Stock | 51.379 (0.000) |

Stock and Tbill | Bond | 3.292 (0.000) |

Stock and bond | Tbill | 2.534 (0.193) |

Volatility spillover | Chi-square | |
---|---|---|

From | To | |

Bond and Tbill | Stock | 11.708 (0.000) |

Stock and Tbill | Bond | 4.960 (0.291) |

Stock and Bond | Tbill | 7728.681 (0.000) |

Asset | Effects of own shock and shocks of the other assets | Test statistic with p-value |
---|---|---|

Stock | H 0 : γ 11 = λ 12 = γ 13 = 0 H 1 : γ 11 ≠ λ 12 ≠ γ 13 ≠ 0 | χ 2 = 61.6435 ( 0.000 ) |

Bond | H 0 : γ 21 = λ 22 = γ 23 = 0 H 1 : γ 21 ≠ λ 22 ≠ γ 23 ≠ 0 | χ 2 = 16.734 ( 0.000 ) |

Tbill | H 0 : γ 31 = λ 32 = γ 33 = 0 H 1 : γ 31 ≠ λ 32 ≠ γ 33 ≠ 0 | χ 2 = 2.651 ( 0.449 ) |

Next we test the leverage in volatility transmission across domestic asset markets. In the context of multivariate asset market trade dynamics, it is important for asset management to know how the “news” spread over to other assets and increase the risk of holding risky assets. Since the negative news have the greater influence on future volatility than do the gains, we therefore, test for the leverage effects of the asset’s own shock and shocks due to the other assets in the multiple financial markets. This has been empirically investigated by testing the leverage parameters across assets jointly by utilizing the Wald Chi-square test. The test results of the leverage parameters γ i j are provided in

The estimated model satisfies most of the desirable properties, namely model adequacy, parameter consistency, volatility clustering and leverage effects. As mentioned before, a good forecast model must capture all stylized facts of the data. In this regard, our VAR-BEKK-GJR-MGARCH (t) model can be used for modelling and predicting volatility and correlation of return volatilities. The graph below displays time plot of the predicted time varying volatility and correlations.

In this paper we investigate the impact of news on volatility in the multiple asset markets using VAR-BEKK-GJR-MGARCH model. Although this model contains a large number of parameters, its statistical second order moment property holds. This model is capable to capture both asymmetric error distributions (measuring news effects) and “volatility leverage”. To our knowledge, application of the simultaneous occurrence of asymmetry and leverage effects on volatility in the Australia’s domestic financial markets is the first. This paper contributes to both methodology and real application within the multivariate financial volatility modelling context. The new modelling strategy of this paper provides important additional information about the sources and linkages among the domestic asset markets of Australia. The results of this paper show that the Australian’s domestic asset markets are interdependent in general. Significant volatility spillovers from stock market to the bond and to money markets simultaneously due to common news information which is supported by the Wald chi-square tests. Time plot of the daily log returns highlights that the domestic bond market is affected most by the global financial crises (GFC), while Tbill is least affected as Tbill is more liquid than the bond market. We also find significant volatility leverage effects from bond and money markets to the stock market and from stock and money markets to the bond market. However, no significant volatility leverage effects are found from stock and bond markets to Tbill. The correlation between Tbill and bond returns volatility is negative, indicating that there is a trade-off between bond and Tbill. This information is useful and vital for asset management and portfolio diversification strategies. Stock and bond volatility correlation is a mix of both positive and negative but with that some noticeable negative correlation is reported between these two assets during 2011 and 2012. Volatility correlations between asset returns are important for policy makers’ asset allocation through diversification during trading under uncertainty. In general, the model adequately fits the data by the LB and Nyblom tests. Significant simultaneous presence of “news” and leverage effects and volatility spillovers determine the sources of volatility transmission across domestic asset markets of Australia. The short and long run volatility parameters are found to be significant with some reservation. The dynamic interactions affect investors’ expectation of trading securities in the Australia’s domestic financial markets simultaneously. The approach of this paper can be extended to investigate spatial dependence of volatility & correlation spillovers across countries for modelling and predicting returns and volatilities simultaneously in the international financial markets for global financial investment policy decision purposes.

The authors declare no conflicts of interest regarding the publication of this paper.

Aftab, H., Beg, R.A., Sun, S.Z. and Zhou, Z.Y. (2019) Testing and Predicting Volatility Spillover―A Multivariate GJR-GARCH Approach. Theoretical Economics Letters, 9, 83-99. https://doi.org/10.4236/tel.2019.91008