Question

How do I perform linear regression on data?

Answer 1

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Explanation:

I don't fully know what you mean first you get your data set where you regress y on x to find how a change in x effects y.

x y
1 4
2 6
3 7
4 6
5 2

And you want to find the relationship between x and y so say you believe the model is like

`y=mx+c`

or in stats

`y=beta_0+beta_1x+u`

these `beta_0,beta_1` are the parameters in the population and `u` is the effect of unobserved variables otherwise called the error term so you want estimators `hatbeta_0,hatbeta_1`

So `haty=hatbeta_0+hatbeta_1x`

This tells you that the predicted coefficents will give you the predicted y value.

So now you want to find the best estimates for these co-efficents we do this by finding the lowest difference between the actual y value and predicted.

`min sum_(i=1)^nhatu_i^2~hatbeta_0,hatbeta_1`

This basically says that you want the minimum of the sum of the differences between the acutal y values and predicted y values for your regression line

So the formulas for finding them are

`hatbeta_1=(sum_(i=1)^n(x_i- barx)(y_i-bary))/(sum_(i=1)^n(x_i-barx)^2)`

`hatbeta_0=bary-hatbeta_1barx`