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Multivariate GARCH


MGARCH stands for multivariate GARCH, or multivariate generalized autoregressive conditional heteroskedasticity. MGARCH allows the conditional-on-past-history covariance matrix of the dependent variables to follow a flexible dynamic structure.

Stata fits MGARCH models. mgarch implements diagonal vech and conditional correlation models. Conditional correlation models use nonlinear combinations of univariate GARCH models to represent the conditional covariances. mgarch provides estimators for three popular conditional correlation models—CCC, DCC, VCC—also known as constant, dynamic, and varying conditional correlation.

Below we analyze daily data on returns of Toyota, Nissan, and Honda stocks. We include the lag of the Nissan stock in the mean equation for Honda. We specify one ARCH term and one GARCH term for the conditional variance equation of each company.

. webuse stocks (Data from Yahoo! Finance) . mgarch ccc (toyota nissan = , noconstant) (honda = L.nissan, noconstant), arch(1) garch(1)

And the results are

Constant conditional correlation MGARCH model Sample: 1 - 2015 Number of obs = 2,014 Distribution: Gaussian Wald chi2(1) = 1.81 Log likelihood = 17439.87 Prob > chi2 = 0.1781
Coef. Std. Err. z P>|z| [95% Conf. Interval]
ARCH_toyota
arch
L1. .0619603 .0087942 7.05 0.000 .0447239 .0791967
 
garch
L1. .9208963 .0110995 82.97 0.000 .8991417 .942651
 
_cons 4.43e-06 1.13e-06 3.94 0.000 2.23e-06 6.64e-06
ARCH_nissan
arch
L1. .0773094 .012328 6.27 0.000 .053147 .1014718
 
garch
L1. .9060882 .0147303 61.51 0.000 .8772173 .934959
 
_cons 6.77e-06 1.85e-06 3.66 0.000 3.14e-06 .0000104
honda
nissan
L1. .0186628 .0138575 1.35 0.178 -.0084975 .045823
ARCH_honda
arch
L1. .0433738 .0069961 6.20 0.000 .0296618 .0570858
 
garch
L1. .9391101 .0100707 93.25 0.000 .9193718 .9588484
 
_cons 5.02e-06 1.31e-06 3.83 0.000 2.45e-06 7.60e-06
corr(toyota, nissan)
.6522989 .0128271 50.85 0.000 .6271583 .6774396
corr(toyota, honda)
.718953 .0108005 66.57 0.000 .6977844 .7401217
corr(nissan, honda)
.6284349 .0135653 46.33 0.000 .6018474 .6550224

Having estimated our model, we can now forecast the conditional variances 50 time periods into the future.

. tsappend, add(50) . predict H*, variance dynamic(2016)

We can graph the result:

Above, we fit a CCC model. We could instead fit a DCC model in which the correlation matrix at each time period is modeled as a weighted average of its own past and recent shocks.

Below we fit a bivariate model of stock returns and specify that the error term follows a multivariate Student’s t distribution:

. webuse stocks (Data from Yahoo! Finance) . mgarch dcc (toyota honda =) , arch(1) garch(1) distribution(t)

And the results are

Dynamic conditional correlation MGARCH model Sample: 1 - 2015 Number of obs = 2,015 Distribution: t Wald chi2(.) = . Log likelihood = 11697.67 Prob > chi2 = .
  Coef. Std. Err. z P>|z| [95% Conf. Interval]
toyota
_cons .0003445 .0002998 1.15 0.251 -.0002431 .0009321
ARCH_toyota
arch
L1. .0557317 .0096148 5.80 0.000 .0368871 .0745764
 
garch
L1. .9284291 .0121769 76.25 0.000 .9045629 .9522953
 
_cons 4.16e-06 1.25e-06 3.34 0.001 1.71e-06 6.60e-06
honda
_cons .0005349 .0003277 1.63 0.103 -.0001074 .0011771
ARCH_honda
arch
L1. .0496351 .0089487 5.55 0.000 .0320959 .0671743
 
garch
L1. .9342399 .0117626 79.42 0.000 .9111857 .9572942
 
_cons 4.89e-06 1.46e-06 3.36 0.001 2.04e-06 7.75e-06
corr(toyota, honda)
.734078 .0144722 50.72 0.000 .7057129 .762443
Adjustment
lambda1 .0322827 .0162193 1.99 0.047 .0004933 .064072
lambda2 .8483622 .1025461 8.27 0.000 .6473754 1.049349
df
_cons 8.597491 1.024617 8.39 0.000 6.589279 10.6057

The DCC model reduces to the CCC model when the adjustment parameters that govern the dynamic correlation process are jointly equal to zero. We can perform a Wald test to test this hypothesis.

. test _b[Adjustment:lambda1] = _b[Adjustment:lambda2] = 0 ( 1) [Adjustment]lambda1 - [Adjustment]lambda2 = 0 ( 2) [Adjustment]lambda1 = 0 chi2( 2) = 346.91 Prob > chi2 = 0.0000

We have fit a CCC model and a DCC model. We could fit a VCC model in which the correlation matrix is modeled as a weighted average of its own past and averages of recent shocks.

To illustrate the flexibility of the conditional correlation estimators, we specify each variance equation separately in the mgarch command below. We include two ARCH terms, one GARCH term, and an independent variable in the variance equation of Honda and one ARCH term for the variance equation of Toyota.

Varying conditional correlation MGARCH model Sample: 1 - 2015 Number of obs = 2,014 Distribution: Gaussian Wald chi2(.) = . Log likelihood = 11258.81 Prob > chi2 = .
  Coef. Std. Err. z P>|z| [95% Conf. Interval]
toyota
_cons .0003512 .0003839 0.91 0.360 -.0004012 .0011036
ARCH_toyota
arch
L1. .1438126 .0248111 5.80 0.000 .0951839 .1924414
 
_cons .0002859 .0000107 26.81 0.000 .000265 .0003068
honda
_cons .0005797 .0003707 1.56 0.118 -.0001469 .0013064
ARCH_honda
arch
L1. .0361065 .0071381 5.06 0.000 .0221161 .050097
L5. -.0291568 .0075789 -3.85 0.000 -.0440111 -.0143025
 
garch
L1. .9771737 .0073828 132.36 0.000 .9627038 .9916436
 
nissan
L1. -22.75917 6.287591 -3.62 0.000 -35.08262 -10.43571
 
_cons -12.37179 .3394628 -36.45 0.000 -13.03713 -11.70646
corr(toyota, honda)
.7236678 .0127599 56.71 0.000 .6986588 .7486768
Adjustment
lambda1 .1220534 .0398213 3.07 0.002 .0440051 .2001017
lambda2 .0337092 .2747204 0.12 0.902 -.504733 .5721514

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