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I have estimated sGARCH, EGARCH and TGARCH, which some for particular models are significant. For others, the alpha remain insignificant using various innovations such as the skewed variants of the normal or student $t$ distributions.

I am tempted to rely on the models that give significant estimates and then ignore the ones that do not. For instance, if the EGARCH-std is not significant in terms of alpha and the EGARCH-sstd is significant, I prefer to focus on the later for further analysis.

Please let me know if I am on the right path.

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What are you trying to do? If you're doing an empirical study and you decide to go with the one model that generates statistical significance then this sounds like data snooping bias. –  Jase Dec 29 '12 at 8:26
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The answer depends on what you will use the models for. For example, if you care about prediction, you should use different metrics (model fit, prediction accuracy on hold-out samples, etc) to determine what works best.

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