Should VAR(1) and VAR(1)-GARCH(1,1) give equal point forecasts out of sample?

Question

I have a VAR(1) with heteroscedastic errors, so I used the rmgarch package for R to estimate a VAR(1)-GARCH(1,1). After that I performed an out-sample forecast for the mean equation with both models. They give me the exact same result with GARCH or without. Is that suppose to happen?

I will provide the code that I'm using:

    ## VAR(1) ##

library("vars")

Data <- betas[-c(163,164), ]

 var1 <- VAR(Data, p = 1, type = "const",
    season = NULL, exogen = NULL, lag.max = NULL,
    ic = c("AIC", "HQ", "SC", "FPE"))


 var.predict <- predict(var1, n.ahead = 2, ci = 0.95)

 ## VAR(1)-GARCH(1,1) ##

 library(rmgarch)

 uspec = ugarchspec(mean.model = list(armaOrder = c(0,0), include.mean = 
                                        FALSE), 
                    variance.model = list(garchOrder = c(1,1), model = "sGARCH"), 
                    distribution.model = "norm") 

 spec = dccspec(uspec = multispec( replicate(3, uspec) ), VAR = TRUE, 
                lag = 1, dccOrder = c(1,1),  model = "DCC", distribution = "mvnorm") 

 fit = dccfit(spec, data = Data) 

 forecast = dccforecast(fit, n.ahead = 2, n.roll = 0)

And also the results:

> forecast@mforecast$mu
, , 1

         [,1]      [,2]      [,3]
[1,] 1.339556 -1.828901 -3.290908
[2,] 1.331527 -1.814802 -3.290787

> var.predict
$beta_1
         fcst     lower    upper        CI
[1,] 1.339556 0.8548577 1.824255 0.4846984
[2,] 1.331527 0.6600307 2.003022 0.6714958

$beta_2
          fcst     lower     upper        CI
[1,] -1.828901 -2.390441 -1.267361 0.5615401
[2,] -1.814802 -2.597114 -1.032490 0.7823119

$beta_3
          fcst     lower     upper       CI
[1,] -3.290908 -4.471721 -2.110096 1.180812
[2,] -3.290787 -4.871586 -1.709988 1.580799

Answer 1

If the model is estimated in stages (conditional mean equations first, conditional variance equations second), the conditional variance specification will not affect estimates of the parameters in the conditional mean model. Then point forecasts equal to predicted conditional mean will not be affected. Interval forecasts, however, will be affected since the spread around the conditional mean will vary depending on the conditional variance model. Other types of point forecasts such as predicted quantiles will also be affected by that.

If the model is estimated in one stage (both the conditional mean and variance equations together), even the point forecasts should differ from a model with constant conditional variance specification -- unless the effects of the nonconstant conditional variance on the estimates of the parameters in the conditional mean equation happen to cancel out exactly (which is very unlikely).