Economic schools of thought VS Econometrics From  Wikipedia:


  In the history of economic thought, a school of economic thought is a group of economic thinkers who share or shared a common perspective on the way economies work


Why is it that different economic schools of thought are  still taught in spite of the fact that we now have data to test their ideas and philosophies of the economy and see if they really line up with reality?
I think I can explain the problem as to why the schools persist.  For simplicity we will limit ourselves to the Marxist, Austrian, Neoclassical and Neo Keynsian schools.  I am not sure where to put the Feminist school except under the broad heading of heterodox with Marxists and Austrians as I don't think of it as a school so much as an attitude or direction of thinking.  The same thing would be true for Green economics, it isn't so much a school as a domain.  There are many small groups out there.

For starters, economics is a broad umbrella that prefers dissent and maybe even vocal and extensive dissent.  Part of economics is part of the humanities and part of economics is in the sciences.

Austrian economics as thought through by von Mises argues that there can be no way to measure what they are saying and so they are part of the humanities.  The neoclassical and the neo Keynesian school say any claim that is stated should be measurable and falsifiable, so it is in the sciences.

That said, any serious review of von Mises and related writers would tend to drive one down the path well laid by Leonard Jimmie Savage and his personalistic statistics and in doing so would have a complete empirical methodology.  I believe the branch of the Austrian school that says things cannot be measured simply hasn't read enough statistics.

The two primary schools, the modern Keynesian and Classical forms, are empirical and the reason you cannot rule one out is that both are empirically supported even though they are contradictory.  This is similar to the problem between quantum physics and relativistic physics.  They make contradictory assertions.  It is impossible that both are true, except where they don't overlap.  Yet they both work everywhere else.

The best tool for model selection in that it is both disciplined and founded on well understood mathematical principles is Bayesian model selection.  Non-Bayesian methods assume that you know the null model and you assert the model is true.  This works great for binary problems but is terrible for non-binary problems.

If we define a model space as indexed on the set $I$, with $i\in{I}$ and a set of models, $M$, so that $\mathcal{M_i}\in{M}$ and with vectors of parameters $\theta_i\in\Theta_i$ in a parameter space $(\theta_i,\mathcal{M}_i)\in\Theta\times{M}$ then we can solve the problem of the best model.  Best in this case being understood as being most probably close to the data generating function.

The probability that a model is the true model, from a set of models and given a data set, $X$, is $$\Pr(\mathcal{M_i}|X)=\int_{\theta_i\in\Theta_i}\frac{\Pr(X|\theta_i;\mathcal{M}_i)\Pr(\theta_i;\mathcal{M}_i)}{\sum_{\mathcal{M}_i\in{M}}\int_{\theta_i\in\Theta_i}\Pr(X|\theta_i;\mathcal{M}_i)\Pr(\theta_i;\mathcal{M}_i)\mathrm{d}\theta_i}\mathrm{d}\theta_i$$.

While this methodology works generally and works especially well on a problem by problem basis it won't answer which school is best since it will provide contradictory solutions across specific problems.

What is really needed is a better, overarching explanation of what triggers Keynesian-like effects and what triggers Classical-like effects.  My suspicion is that it is stochastic shocks in the planetary budget constraint compared to where that constraint is expected to be at by the individual actors.