Statistical test for significant change in regression parameter with new data? Is there a statistical test that determines if there is a significant change in a regression coefficient parameter's value when more data is added?
For example, you have 11 months of data, fit a regression model for variable x1 which has an estimated coefficient of 0.8. Next month, you get another set of data and refit the model using the full 12 months of data and get a coefficient of 0.9.
How do I tell if there is a significant change in this parameter value between the two models?

I'll take a stab at it, using the example of 11 months and 1 extra month.  I assume from your OP, as opposed to later comments, that you are analyzing cross-sectionally.
You wouldn't want to conduct a significance test in which the 11 months of data are evaluated twice -- both on their own and as the majority of the larger data set.  Significance tests involve statements about "the probability that chance would produce...." and to me it wouldn't make sense to ask how often chance would produce certain results when a large chunk of data are repeated in two samples.  (How would chance have produced those 11 but also grouped those 11 with that extra 1?)
So treat these two data sets as distinct, while still keeping them in the same regression.  A dummy variable can be used to mark all cases as either part of the 11 or part of the extra 1.  That dummy variable can be a predictor in your regression and you can test an interaction between it and x1.  This interaction's p-value will help answer the question, "Does the value of the x1 coefficient significantly differ when the dummy is 1 vs. when the dummy is 0?"
