# The OLS Estimators: Properties and formula

The OLS Estimators: Properties and formula

• It passes through the sample means of Y and X. this fact is obvious from the latter can be written as Y=B1+B2X which is proved.
• The mean value of the estimated y=Y is equal to the mean value of the actual y for

Yi=β1+ β2xi

Yi=(Y- β2Xi) + β2Xi

Yi=Y – β2X + β2Xi

Yi=Y+ β2(x-X)

Summering both sides

∑Yi=nY+ β2∑(x-X)          or

∑Yi =nY+ β2∑xi              where ∑xi =0

∑Yi=nY

Dividing by n both sides

∑Yi/n        =          nY/n

Yi              =          Y

• The mean value of the residuals Ui is zero the first equation is

-2∑ (yi- β1– β2∑Xi)           =          0

But since Ui=yi- β1– β2∑Xi, the preceding equation reduce to -2∑Ui=0

Where Ui=0

As a result of the proceeding property, the sample regression

Yi= β1+ β2∑xi +Ui

Can be expressed in an alternative form where both y and x are expressed as a deviation from their mean values.

To see this, sum on both sides to give.

### The OLS Estimators: Properties and formula

∑yi=n β1+ β2∑xi +∑Ui

∑yi=nβ1+ β2∑xi+0                       since ∑Ui=0

∑yi=nβ1+ β2∑xi

Y= β1+ β2X after dividing by n

If we subtracting the Y= β12X from

yi= β1+ β2xi+Ui    then we find the Yi=yi

yi-Y= β2(xi-X) + Ui

yi= β2xi+Ui           SRF in deviation form

Yi= β2xi                SRL

• The residuals Ui are uncorrelated with the predicted Y. This statement can be verified as follows

Using the deviation form we can write.

∑YiUi= β2∑xiUi

∑YiUi= β2∑xi(yi-β2xi)

∑YiUi= β2∑xiyi-β22∑xi2

∑YiUi= β22∑xi2– β22∑xi2

∑YiUi=0

where use is made of the fact that

∑xy= β2

So       ∑YiUi= β2∑xiyi – β22∑xi2

∑YiUi=β2 (∑xy/∑xi2) – β22∑xi2

∑YiUi= β22∑xi2) – β22∑xi2

∑YiUi= β22∑xi2 – β22∑xi2

∑YiUi=0

• The residuals Ui are uncorrelated with xi: that is , ∑Uixi=0

∂∑Ui2/∂β2       = -2∑(yi-β12xi)xi

= -2∑Uixi

= 0