# What does R, R square , Adjusted R square, STD error of estimates helps for ?

#### hanialbarni

##### New Member
Hello

After I run out the regression analysis, can someone enlighten me what can I get from R, R square, STD Error estimates and the adjusted r square ?!

P.S: I am not professional in stats, I have only taken one course in university and that was back 4 years ago so I don't have any knowledge about that except watching couple videos in youtube.

#### Dragan

##### Super Moderator
Well, I'll help you with R. That is, R is the Pearson correlation between the actual values of the dependent variable (Y) and the predicted values of Y (Y-hats) - regardless of the number of predictors (X's) in the regression model. As such, R is an index of the strength of the linear association between Y and Y-hats. Note that R is bounded between 0 and 1.

#### Dragan

##### Super Moderator
Between -1 and 1.
Well, no...it is not....rather, the correlation between Y and the Y-hats is bounded between 0 and 1 as I stated above.

#### hlsmith

##### Omega Contributor
R^2 is the R value squared, which is bounded between 0 and 1. It is typically interpreted as the amount of variance predicted by a variable (or variables, if there are more than 1 in a model).

#### bruin

##### New Member
Based on this...

R is the Pearson correlation between the actual values of the dependent variable (Y) and the predicted values of Y (Y-hats)
...wouldn't a negative R only occur if your regression line had the wrong slope? How could a least-squares line have the wrong slope?

#### Dragan

##### Super Moderator
Based on this...

...wouldn't a negative R only occur if your regression line had the wrong slope? How could a least-squares line have the wrong slope?
Technically speaking, it is conceivable to obtain a negative value of R^2. This (unusual) case can occur when one conducts a regression without an intercept term i.e. regressing through the origin. However, the OLS estimate of the slope coefficient is still unbiased - unless you force the error terms to sum to zero.

#### spunky

##### Smelly poop man with doo doo pants.
Well, no...it is not....rather, the correlation between Y and the Y-hats is bounded between 0 and 1 as I stated above.
i thought this could be provable so i decided to give it a try. i haven't done one of this in a while so plz point out potential mistakes:

to show that $$0<Corr(\mathbf{Y},\mathbf{\widehat{Y}})<1$$ we could show that $$Cov(\mathbf{Y},\mathbf{\widehat{Y}})>0$$.

to do this, it is convenient to use $$\mathbf{H}$$, the 'hat matrix' that we know $$\mathbf{H}=\mathbf{X(X^{t}X)^{-1}X^{t}}$$ so that:

$$\mathbf{\widehat{Y}}=\mathbf{X\widehat{\beta}}=\mathbf{X(X^{t}X)^{-1}X^{t}Y}=\mathbf{HY}$$

so since we've established $$\mathbf{\widehat{Y}}=\mathbf{HY}$$

we can do:

$$Cov(\mathbf{Y,\widehat{Y}})=Cov(\mathbf{Y,HY})=\mathbf{H}Cov(\mathbf{Y,Y})\mathbf{H^{t}}$$

now, we know: $$Cov(\mathbf{Y,Y})=\sigma^{2}\mathbf{I}$$

by assumption of independence (or we'd introduce heteroskedasticity and violate the typical assumptions of OLS Regression)

so we substitute that back in:

$$\mathbf{H}Cov(\mathbf{Y,Y})\mathbf{H^{t}}=\mathbf{H}\sigma^{2}\mathbf{I}\mathbf{H}^{t}=\sigma^{2}\mathbf{HH^{t}}$$

which is clearly greater than 0 because $$\sigma^{2}$$ is positive (it's a variance) and $$\mathbf{HH^{t}}$$ is positive-definite.

so that (i think) shows $$Cov(\mathbf{Y},\mathbf{\widehat{Y}})>0$$ which i'm sure, when standardized, bounds $$Corr(\mathbf{Y},\mathbf{\widehat{Y}})$$ to be between 0 and 1.