Diagonal VECH GARCH models

Stata’s mgarch command estimates the parameters of multivariate generalized autoregressive conditional-heteroskedasticity (GARCH) models. Stata’s mgarch dvech command estimates diagonal vech GARCH models. Diagonal vech GARCH models allow the conditional covariance matrix of the dependent variables to follow a flexible dynamic structure. mgarch dvech allows each element of the current conditional covariance matrix of the dependent variables to depend 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 thru 1998w25                            Number of obs =  1,499
Distribution: Gaussian                                  Wald chi2(2)  = 275.66
Log likelihood = -5972.053                              Prob > chi2   = 0.0000




             
 
 Coefficient  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

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

. tsline v_anvil_acme