The effect of ommission of relevant variable in the regression model on adjusted $R^2$ Let's say I have two regression models 
(I) $y_t=\beta_1+\beta_2 x_2+u_t$
(II)  $y_t=\beta_1+\beta_2 x_2+\beta_3 x_3 + u_t$
How the omission of relevant variable (not irrelevant variable) affects adjusted $R^2$? 
That's when I compare Adj-$R^2$ for two models, what can I say? Which one less? 
I see this while studying lecture notes. So I Try to understand the topic. Please explain this.  Thank you:) 
 A: As described in your models, if x3 is a relevant variable Model II will have both a higher R Square and higher Adjusted R Square than Model I.  Also, Model II will have a lower Standard Error than Model I.  Thus, you should keep this x3 variable and chose Model II. 
If x3 is not a relevant variable Model II will have a higher R Square, but a lower Adjusted R Square than Model I.  It also most probably will have a higher Standard Error than Model I.  In this case, you should exclude the x3 variable and stick with Model I. 
That's kind of the basics.  In reality, once you add a few variables the added explanatory power of adding additional variables increasingly diminishes.  That's even though those variables are deemed relevant and that your Adjusted R Square keeps on rising.  However, let's say that adding x3 would cause your Adjusted R Square to increase by 0.15; that's a lot, and you would definitely keep x3.  Now, you add another variable x4.  And, the resulting Adjusted R Square increases by only 0.03.  I think many people would not add x4.  It may not be that worth it.  Adding it may lead to a model that is overfit.  You can test whether a model is overfit by using a Hold Out sample.  The latter is probably more important than the ultimate level of your Adjusted R Square.  
Going back to your two models (I and II).  You should actually test both of them to check their performance in a Hold Out sample.  Only after doing so, can you be sure that Model II is better instead of simply being overfit (which the Adjusted R Square will not capture that). 
