Log likelihoods of subsamples don't sum up to full sample log-likelihood When splitting the sample into segments, I would expect the likelihood of the segments to sum up to at least the likelihood of the full sample (constraining the parameters to be the same). Or did I misunderstand?
The code example is in R, using the rugarch package. The rmgarch package is only used to get the data.
First I calculated the log-likelihood of a GARCH model that was fitted to the full sample.
Then I split the sample in two parts and and calculated the log-likelihood for each segment while fixing the parameters to be the same as the ones obtained by step 1. I would expect the subsample log likelihoods to sum up to the log likelihood of the full sample. But this doesn't happen. 
When the segments-GARCH models are not filtered with fixed parameters, but instead allowed to estimate their own parameter estimates, their sum is still below the likelihood of the full sample. Actually it should be equal or bigger than the full likelihood.
Did I misunderstand the concept of ML???
Best, Jo
require('rugarch')
require('rmgarch')
data(dji30retw)
dat <- dji30retw$AA

##### 1: get log-likelihood and parameters of full-sample model
spec1 <- ugarchspec(
     variance.model = list(model = "sGARCH", garchOrder = c(1,1), submodel = NULL)
     , mean.model = list( armaOrder =c(0,0) ) , distribution.model = "norm" )
garch1 <- ugarchfit(spec=spec1, data=dat )
l1 = likelihood(garch1)

spec2 = spec1
setfixed(spec2)<-as.list(coef(garch1))

##### 2: filter two sample-segments individually using the parameters from step 1
garch2 = ugarchfilter(spec = spec2, data = dat[1:344])
l2 = likelihood(garch2)

garch3 <- ugarchfilter(spec=spec2, data=dat[345:length(dat)]  )
l3 = likelihood(garch3)

l1        # 1901.706
l2+l3     # 1898.931
# should be the same in theory

##### 3: now fit the individual segments
garch5 <- ugarchfit(spec=spec1, data=dat[1:344] )
l5 = likelihood(garch5)

garch6 <- ugarchfit(spec=spec1, data=dat[345:length(dat)] )
l6 = likelihood(garch6)

l1        # 1901.706
l5+l6     # 1899.254
# the sum of the individual log-likelihoods shouldn't be smaller than the full-sample log-likelihood!

 A: Firstly, I had to edit numerous instances where you said "likelihood" when you meant "log likelihood". The likelihood of a sample is the multiplicative combination of likelihoods for individual observations, and the loglikelihood of a sample is combined additively across such loglikelihoods of samples. You also referred to a partial likelihood which is very interesting in its own right, but not used correctly in this instance.
The key misunderstanding in your problem description is that when models are run in subsamples, the parameters are calculated, then the likelihood is presented for the MLE of that particular subsample. The MLE of a subsample will be different than in the whole sample. 
Take a braindead-example: population proportion estimation with Bernoulli probability model. If you create subsamples of positive and negative responses, the MLE in either subsample will be exactly 0 or 1, meaning each observation has a loglikelihood of exactly 0. However, in the combined sample, it's obvious that the estimated proportion will be the sample mean and the loglikelihood computed from that will be nonzero.
Take home: different subsamples yield different parameter estimates, so no the loglikelihoods will not be equal unless you fix the parameters to take a specific value in both calculation of the full and subsample likelihood.
