multi-collinearity in a time series? Say I have a set of time series data spanning 2000-2016
I code my years as the variable time, starting in 2000 as 0, 1, 2,....15
Say I want to compare the bush presidency to the obama presidency and so code the bush variable as 0, 1, 2, 3, 4, 5, 6, 7, 0, 0, 0.... and then the obama as 0, 0, ... 0, 1, 2, 3, 4, 5, 6, 7.  
Would I expect multi-collinearity between these time variables? They are linear combinations. 
Is it necessary to have the variable time when bush+obama=time
To more accurately model 
GDP = bush+obama+time
or 
GDP = bush+obama
where the goal is to find the annual change in GDP (as opposed to the mean change). 
my thoughts are -
time accounts for the overall serial correlation
the coefficients for bush and obama provide their effect on GDP with the serial correlation controlled for. 
 A: We can probably help more if we know more about what you're trying to do. To answer your specific question, you probably only need 2 variables. Time will be distinct from Bush/Obama, as time will continue to increase while Bush/Obama flips.  
You probably only need 2 variables, assuming that you only want to know whether in a given year Bush or Obama was president. One binary variable would be good enough-- if it is a 1 when Bush is president, then its reference to zero will be a comparison to when Obama is president.  
If you are interested in some aspect of the year of the presidency of each, then you could have a single variable that repeats through 0-7. I would not recommend doing 3 variables, with one each for Bush and Obama where it goes 0-7.  
All of that said if you'd like to explain your goal a bit more we may be able to give some prescriptive advice. 
A: You have bigger issues than multicollinearity here. The variables that you described are called linear splines, by the way, they are not a problem per se, but if you have them together with time you get perfect multicollinearity. These splines are used for piece-wise interpolation.
I'm going to state that nothing of any use can be accomplished by the regression you'r thinking about indeed. There's so much confounding and all kinds of other statistical problems in this sort of setup.
