That being said, it would be quite a bold statement if you were to claim that the price > $250 USD/BTC by January 1. That would constitute an all time high by more than 10% and would also require a change in trajectory because BTC has been on decline for about a week now. Certainly >$250 is possible, but possible and most likely are two very different things.
If you take a look at the data I did create two models, one for the long term and one for the short term, and backtest. The models yielded slightly different results. Interestingly, the coefficients for time was not statistically different which may indicate that the two models are roughly consistent. Regardless, I concluded that the short term model is more plausible due to the fact that it seems to constitute a weaker contextual extrapolation.
Thoughts?
First, let's be clear: you extrapolate. With data set 1 (October 30, 2011 until October 30, 2013), extrapolating to January 1st (~2 months), you get 115 to 150 USD. In other words, we'd have to undergo another major correction. With data set 2 (June 1, 2013 until October 30, 2013), extrapolating to the same future date, you get to 175 to 250. Which would basically mean moderate growth or a mild correction in the coming two months.
And that's exactly the problem: if you use data set 1, you belong to the crowd of people that believe in the "long term trend" (i.e. ever since trading started). That's a plausible view, but the recent months simply don't support it. We have been, are and most likely will stay above that long term trend (which is basically what you regression on the entire data models).
If you use the recent data, I think you're getting closer, and personally, I find the numbers you get quite likely to be right. But my remark about "backtesting" went unanswered. What you meant by backtesting, I think, is the confidence with which you can predict the price on January 1st with your best model. What I meant was applying the best model to the truncated data, right before a major price move. It's pretty much impossible that the regression will capture the extreme price move, like for example the extreme rally in March/April. Some would declare such volatile phases "outliers" then, and ignore them, or consider them unimportant. I don't believe in that approach.
Anyway, I don't want to say that your analysis is useless -- it's just a big caveat one should keep in mind: if we continue to grow "as expected", your number will be right. If we will enter one of the (many!) extremely volatile phases, up or down, in the next 2 months, your numbers might well be off by a factor of 2 or more.