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:)

• 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 $R^2$.
– Stat
Apr 4 '14 at 22:19
• 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.
– 1190
Apr 4 '14 at 22:36
• 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?
– Stat
Apr 4 '14 at 22:57
• Adding irrelevant variable increases SSE. I am begining level. So I have No enough knowledge to answer your questions properly. @Stat
– 1190
Apr 5 '14 at 0:05