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Diagonal VECH GARCH models

Stata’s mgarch dvech command estimates the parameters of multivariate generalized autoregressive conditional-heteroskedasticity (GARCH) models. Diagonal VECH GARCH models allow the conditional covariance matrix of the dependent variables to follow a flexible dynamic structure. mgarch dvech estimates the parameters of diagonal vech GARCH models in which each element of the current conditional covariance matrix of the dependent variables depends only on its own past and on past shocks.

Here we analyze some fictional weekly data on the percentages of bad widgets found in the factories of Acme Inc. and Anvil Inc. We model the levels as a first-order autoregressive process. We believe that the adaptive management style in these companies causes the variances to follow a diagonal vech GARCH process with one ARCH term and one GARCH term.

. webuse acme . mgarch dvech (acme = L.acme) (anvil = L.anvil), arch(1) garch(1) nolog Diagonal vech MGARCH model Sample: 1969w35 - 1998w25 Number of obs = 1499 Distribution: Gaussian Wald chi2(2) = 275.66 Log likelihood = -5972.053 Prob > chi2 = 0.0000
Coef. Std. Err. z P>|z| [95% Conf. Interval]
acme
acme
L1. .3388568 .0254684 13.30 0.000 .2889397 .388774
_cons 1.120651 .0598184 18.73 0.000 1.003409 1.237893
anvil
anvil
L1. .3159488 .0263346 12.00 0.000 .264334 .3675636
_cons 1.216031 .0642366 18.93 0.000 1.09013 1.341933
Sigma0
1_1 2.399669 .2847566 8.43 0.000 1.841556 2.957781
2_1 .4358023 .1059507 4.11 0.000 .2281429 .6434618
2_2 1.802058 .2685172 6.71 0.000 1.275774 2.328342
L.ARCH
1_1 .2958368 .0433673 6.82 0.000 .2108384 .3808352
2_1 .1808973 .0334578 5.41 0.000 .1153212 .2464734
2_2 .2782433 .0400212 6.95 0.000 .1998032 .3566833
L.GARCH
1_1 -.0244739 .0827987 -0.30 0.768 -.1867563 .1378085
2_1 .1305242 .1315984 0.99 0.321 -.127404 .3884524
2_2 .2280066 .0830777 2.74 0.006 .0651773 .3908359

mgarch dvech supports constraints, so if we recognized that these close competitors might follow essentially the same process, we could have imposed the constraints that the ARCH coefficients are the same for the two companies and that the GARCH coefficients are also the same. We could estimate that model by typing

. constraint 1 [L.ARCH]1_1 = [L.ARCH]2_2 . constraint 2 [L.GARCH]1_1 = [L.GARCH]2_2 . mgarch dvech (acme = L.acme) (anvil = L.anvil), arch(1) garch(1) constraints(1 2)

In addition to predicting the dependent variables, we can predict the conditional variance to observe the modeled volatility. Here we make one-step predictions of volatility over the sample and graph the results.

. predict v*, variance . tsline v_acme_acme v_anvil_anvil

Graph

In the prediction above, we also predicted the conditional covariance between the two companies. Let’s graph that now,

. tsline v_anvil_acme

Graph

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