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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:)

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  • $\begingroup$ Can you tell me why we normally prefer to use an Adj-$R^2$ rather than the $R^2$? Hint: write down the formula for the Adj-$R^2$ and see what it has more than $R^2$. $\endgroup$ – Stat Apr 4 '14 at 22:19
  • $\begingroup$ Because adjusted one inçlerden penalty. That's when we add irrelevant variable, adjusted one decreases but $R^2$ increases. So we prefer adjusted. @Stat but when the relevant variable is omitted, I am confused. $\endgroup$ – user315 Apr 4 '14 at 22:36
  • $\begingroup$ OK here is the problem. You said that "when we add irrelevant variable ... $R^2$ increases. This is true. But what is the reason for that increase? Is this because the variable was irrelevant or is it because when you add a variable (whether relevant or irrelevant), you are actually having a more complex model (with more parameters) that reduces your $SSE$? Which one? $\endgroup$ – Stat Apr 4 '14 at 22:57
  • $\begingroup$ Adding irrelevant variable increases SSE. I am begining level. So I have No enough knowledge to answer your questions properly. @Stat $\endgroup$ – user315 Apr 5 '14 at 0:05
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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).

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