Estimating $b_1 x_1+b_2 x_2$ instead of $b_1 x_1+b_2 x_2+b_3x_3$ I have a theoretical economic model which is as follows,
$$ y = a + b_1x_1 + b_2x_2 + b_3x_3 + u $$
So theory says that there are $x_1$, $x_2$ and $x_3$ factors to estimate $y$.
Now I have the real data and I need to estimate $b_1$, $b_2$, $b_3$. The problem is that the real data set contains only data for $x_1$ and $x_2$; there are no data for $x_3$. So the model I can fit actually is:
$$y = a + b_1x_1 + b_2x_2 + u$$


*

*Is it OK to estimate this model? 

*Do I lose anything estimating it?  

*If I do estimate $b_1$, $b_2$,  then where does the $b_3x_3$ term go? 

*Is it accounted for by error term $u$?


And we would like to assume that $x_3$ is not correlated with $x_1$ and $x_2$.
 A: Let's think of this in geometric terms.  Think of a "ball", the surface of a ball.  It is described as $ r^2 = ax^2+by^2+cz^2 + \epsilon$.  Now if you have the values for $ x^2$, $ y^2$, $ z^2$, and you have measurements of  $ r^2$ then you can determine your coefficients "a", "b", and "c".  (You could call it ellipsoid, but to call it a ball is simpler.)
If you have only the  $ x^2$, and $ y^2$ terms then you can make a circle.  Instead of defining the surface of a ball, you will describe a filled in circle.  The equation you instead fit is $ r^2 \le ax^2 + by^2 + \epsilon$.  
You are projecting the "ball", whatever shape it is, into the expression for the circle.  It could be a diagonally oriented "ball" that is shaped more like a sewing needle, and so the $ z$ components utterly wreck the estimates of the two axes.  It could be a ball that looks like a nearly crushed m&m where the coin-axes are "x" and "y", and there is zero projection.  You can't know which it is without the "$ z$" information.
That last paragraph was talking about a "pure information" case and didn't account for the noise.  Real world measurements have the signal with noise.  The noise along the perimeter that is aligned to the axes is going to have a much stronger impact on your fit.  Even though you have the same number of samples, you are going to have more uncertainty in your parameter estimates.  If it is a different equation than this simple linear axis-oriented case, then things can go "pear shaped".  Your current equations are plane-shaped, so instead of having a bound (the surface of the ball), the z-data might just go all over the map - projection could be a serious problem.
Is it okay to model?  That is a judgment call.  An expert who understands the particulars of the problem might answer that.  I don't know if someone can give a good answer if they are far from the problem.
You do lose several good things, including certainty in parameter estimates, and the nature of the model being transformed.
The estimate for $ b_3$ disappears into epsilon and into the other parameter estimates.  It is subsumed by the whole equation, depending on the underlying system.
A: The other answers, while not wrong, over complicate the issue a bit.
If $x_3$ is truly uncorrelated with $x_1$ and $x_2$ (and the true relationship is as specified) then you can estimate your second equation without an issue. As you suggest, $\beta_3 x_3$ will be absorbed by the (new) error term. The OLS estimates will be unbiased, as long as all the other OLS assumptions hold.
