# Math Help - Least Squares Regression

1. ## Least Squares Regression

I'll be honest i've not got much idea on where to start with this, could anyone point me in the right direction ?

A sample of data from 40 households in a particular area in 2007 yielded the following information, where X is disposeable income per head and Y is disposeable expenditure per head.

$\sum X_i = 200$ $\sum Y_i= 160$

$\sum X_iY_i=875$ $\sum X$ $^{2}_i =1100$

$\sum (Y_i - \bar{Y})^{2} = 75.25$

a) Show that in the relationship $Y_i = \alpha +\beta X_i + u_i$

where $u_i$ is the disturbance term, the least squares regression estimates of $\alpha, \beta$ are 0.25 and 0.75 respectively.

2. Originally Posted by Apache
I'll be honest i've not got much idea on where to start with this, could anyone point me in the right direction ?

A sample of data from 40 households in a particular area in 2007 yielded the following information, where X is disposeable income per head and Y is disposeable expenditure per head.

$\sum X_i = 200$ $\sum Y_i= 160$

$\sum X_iY_i=875$ $\sum X$ $^{2}_i =1100$

$\sum (Y_i - \bar{Y})^{2} = 75.25$

a) Show that in the relationship $Y_i = \alpha +\beta X_i + u_i$

where $u_i$ is the disturbance term, the least squares regression estimates of $\alpha, \beta$ are 0.25 and 0.75 respectively.
See here

CB

3. Thanks for that think ive cracked it,

$b= \frac {40 * 875 - 200 * 160} {40 * 1100 - 200^2}$

= 0.75

$a = 4 - 0.75 * 5$

= 0.25

The next part of my question asks to give interpretations of $\alpha$ and $\beta$ and discuss wether the signs on the sample estimates accord with your expectations?

Is this referring to the figures of 0.25 and 0.75 and wouldnt they always be positive ?