It is important to remember that statistics is a branch of rhetoric and not a branch of math. When you are performing a regression analysis you are engaging in argument. It follows that the regression should follow from your argument and does not stand alone as an independent concept.
It is also important to remember that a p-value does not tell you if anything is true or that your model makes sense. A low p-value just tells you that you should be investigating the phenomenon.
Let us assume, for a moment, that you are suffering neither from a false positive or a false negative in your significance tests.
Without dummy variables, there is no underpricing. This is a finding. It is, in fact, a very important finding. With a dummy variable, it is significant. This implies that at least one, but not all of the exchanges have assets coming into them that are mispriced. If all had been mispriced, then the regression slope would have been significant, but the dummies would not have been.
This is also a finding.
There is a problem with your methodology, however. With Frequentist methods, you are assuming that your model is the true model in nature. You have two models. Both cannot be true. One is, or the other is. If the first model is true, then the exchanges do not matter. If the second is true, then the exchanges drive everything. There are specific differences in how the exchanges are structured so that those differences are facilitating mispricing.
You should perform either an AIC or BIC on your model, only do one. Choose it based on its properties and not by convenience. If the xIC chooses the model without the exchanges this would imply that while the exchanges were significant, they are a poor fit to the data generating function in nature. This would likely indicate that an outlier or set of outliers are driving the statistical significance. They are probably very large outliers. On the other hand, if the xIC chooses the later model, it would imply that the market microstructure is causing the mispricing. Please do note, this mispricing may not be real.
An implication that the market microstructure is causing a mispricing may imply that your understanding of what it means to be mispriced is incorrect. It may be that there exist market specific risks, probably through liquidity and price formation, that are present and that your thesis is incorrect. Instead, it would imply you would need to perform a far deeper investigation.
If that was the case, I would recommend reading:
Measures of Discount for Lack of Marketability and Liquidity, pp.474-507. The Valuation Handbook; Wiley Finance, John Wiley & Sons. 2010. Authors: Ashok Abbott
I would also look at the work out there on market making and price formation.
