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=β_{1}+ β_{2}xi

Yi=(Y- β_{2}Xi) + β_{2}Xi

Yi=Y – β_{2}X + β_{2}Xi

Yi=Y+ β_{2}(x-X)

Summering both sides

∑Yi=nY+ β_{2}∑(x-X) or

∑Yi =nY+ β_{2}∑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- β_{1}– β_{2}∑Xi) = 0

But since Ui=yi- β_{1}– β_{2}∑Xi, the preceding equation reduce to -2∑Ui=0

Where Ui=0

As a result of the proceeding property, the sample regression

Yi= β_{1}+ β_{2}∑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 β_{1}+ β_{2}∑xi +∑Ui

∑yi=nβ_{1}+ β_{2}∑xi+0 since ∑Ui=0

∑yi=nβ_{1}+ β_{2}∑xi

Y= β_{1}+ β_{2}X after dividing by n

If we subtracting the Y= β_{1}+β_{2}X from

yi= β_{1}+ β_{2}xi+Ui then we find the Yi=yi

yi-Y= β_{2}(xi-X) + Ui

yi= β_{2}xi+Ui SRF in deviation form

Yi= β_{2}xi 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= β_{2}∑xiUi

∑YiUi= β_{2}∑xi(yi-β_{2}xi)

∑YiUi= β_{2}∑xiyi-β_{2}^{2}∑xi^{2}

∑YiUi= β_{2}^{2}∑xi^{2}– β_{2}^{2}∑xi^{2}

∑YiUi=0

where use is made of the fact that

∑xy= β_{2}

So ∑YiUi= β_{2}∑xiyi – β_{2}^{2}∑xi^{2}

∑YiUi=β_{2} (∑xy/∑xi^{2}) – β_{2}^{2}∑xi^{2}

∑YiUi= β_{2} (β_{2}∑xi^{2}) – β_{2}^{2}∑xi^{2}

∑YiUi= β_{2}^{2}∑xi^{2} – β_{2}^{2}∑xi^{2}

∑YiUi=0

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

∂∑Ui^{2}/∂β_{2} = -2∑(yi-β_{1}-β_{2}xi)xi

= -2∑Uixi

= 0