Is high ICC essential when using HLM? I have a question related to 'whether HLM is necessary or not.' 
My data has regions and firms with the time range of 2005 and 2010.
I set each firm (with multiple observations within it) as the first level and regions as the second level. 
However, ICC turned out to be very low at the region level (0.008) while the LR test(chi2(4) =  2981.71) said that the use of the hierarchical model was proper.
Q. Is ICC the best indicator when deciding the use of HLM? Or that clash between ICC and LR test imply something? 
Q. What criteria or tests should I conduct to decide whether to use HLM other than ICC? 
Thank you. 
 A: Nezlek (2008) on p.857 puts forward a case that

analysts should use multilevel modeling when they have a multilevel data structure – pure and simple. When I am asked for advice regarding whether or not multilevel modeling is appropriate, my first question concerns the nature of the data structure. If there is a meaningful nested hierarchy to the data, my advice is to use multilevel modeling, irrespective of distracting arguments about ICCs and so forth.

On the same page he acknowledges

that researchers sometimes use a
low or zero ICC to justify a decision not to use multilevel modeling – on
the grounds that because there is no (or very little) between-group variance
in the dependent measure, the grouped (or nested) structure of the data
can be ignored. This is a dangerous assumption that is not justifiable.
Frequently (or almost invariably), researchers are interested in relationships
between measures. The fact that there is little or no between-group
variance in a measure does not mean that the relationship between this
measure and another measure is the same across all groups, something that
is assumed if one conducts and analysis that ignores the grouped structure
of the data. By extension, even if there is no between-group variance for
all of the measures of interest, it cannot be assumed that relationships
between or among these measures do not vary across groups.

I believe that answers your main question: high ICC is not essential.
As for the circumstances that might cause people to not use HLM even when they have a hierarchical structure, I refer you to this more general question. One factor that might push people away from using HLM is if there is a very low number of units at Level 2, or if there is a low ICC. But neither of those need preclude using HLM.
Nezlek, J. B. (2008). An introduction to multilevel modeling for social and personality psychology. Social and Personality Psychology Compass, 2(2), 842-860.
A: The higher the ICC the greater the variability between and within the regions. You may also want to take a look at Schwartz Bayesian Criteria to see how well the model fits. This explanation of HLM may help.
