y = Xb + e
where y is the dependent variable or vector, X is a matrix of independent variables, b is a vector of parameters to be estimated, and e is a vector of errors with mean zero that make the equations equal.
The estimator of b is: (X'X)-1X'y
A common derivation of this estimator from the model equation (1) is:
y = Xb + e
Multiply through by X'. X'y = X'Xb + X'e
Now take expectations. Since the e's are assumed to be uncorrelated to the X's the last term is zero, so that term drops. So now:
E[X'Xb] = E[X'y]
Now multiply through by (X'X)-1
E[(X'X)-1X'Xb] = E[(X'X)-1X'y]
E = E[(X'X)-1X'y]
Since the X's and y's are data the estimate of b can be calculated.(Econterms)
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