# What are the concepts in Linear Algebra that model the idea of Identification Strategy in Econometrics?

I just would like to know what concepts one should know before talking about identification strategies in econometrics. I see people studying such concepts but I'm not sure they realize (or even know) that probably there are concepts in Linear Algebra (or Multivariate Mathematical Analysis if that is the case) that probably would make their life easier had they known that beforehand. Since I'm currently studying a more rigorous approach to linear algebra (nothing fancy, but definitely proving everything), I would like to understand this "Identification Strategy" concept that I remember "learning" in an introductory econometrics course where I'm pretty sure no one even mentioned what was the math behind it, which I think is extremely important.

If you could be clear and didactic I would appreciate a lot.

Thank you!

• Do you want to know what the concept of identification is or the linear algebra concepts that go into econometrics? Identification strategy has nothing to do with linear algebra. What do you mean “model the idea of Identification Strategy”? Apr 27 '20 at 21:27
• From Regio's answer, I think I know the direction now. Thanks. Apr 28 '20 at 23:09

To be more specific, suppose that you are studying outcome $$y$$ which depends (stochastically) on $$x$$ so that the data is generated according to $$F(y|x)$$ where $$F$$ is some distribution function. Then you have an econometric model that depends on a parameter, $$\theta$$ (this can be a vector of parameters), and for each parameter, it spells out a conditional distribution of $$y$$ in terms of $$x$$. That is your econometric model is a function $$M(\theta)=G(y|x; \theta)$$. We say that the model is identified if there exists a unique parameter, $$\theta^*$$, in the domain of $$M$$ such that $$M(\theta^*)=F(y|x)$$.
In reduced-form work, $$M$$ is assumed to be linear (or in general to be an injective function and have a relatively simple form), so ensuring identification has more to do with your data, $$F$$, having the right properties (mainly having enough exogenous variation).
In contrast, structural work typically deals with data that is not experimental or quasi-experimental so $$M$$ cannot credibly be assumed linear. Instead, you develop a rich enough model and identification requires $$M$$ to be invertible. If $$M$$ is invertible, then you have that $$M^{-1}(F(y|x))=\theta^*$$.
I think that in relation with identification, linear algebra tools are mainly useful to prove the invertibility of $$M$$. However, it can prove also useful when it comes to designing an estimation strategy, and when implementing the estimation strategy in the computer, etc.