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Matrix approach to the OLS, Data set, Systematic part, Regression Line, Coefficient of regression, Nature of relationship, Elasticity, Solution of the OLS parameters are points you can learn about Econometric in this lecture.
Typology: Study notes
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Chapter 2
Consider the data set given by the fig. As before we can
write the relationship as:
yi = α + βxi + εi
Here the RHS consist of systematic part (regression line, the
straight line for predictions) and stochastic part.
More explicitly, we can write as:
y 1 = α + β1 + ε 1
y 2 = α + βx 2 + ε 2
…..
……
……
yn = α + βxn + εn
This system of equation can be written as:
y 1 y 2 ⋮ ⋮ yn (^) ⎠
1 x 1 1 x 2 ⋮ ⋮ 1
xn ⎠
α β�^ +
⎝
ε 1 ε 2 ⋮ ⋮ εn (^) ⎠
Or
Y = XΒ+∈
Where
y 1 y 2 ⋮ ⋮ yn (^) ⎠
1 x 1 1 x 2 ⋮ ⋮ 1
xn ⎠
α β�^ ,^ ∈=
⎝
ε 1 ε 2 ⋮ ⋮ εn (^) ⎠
Or
∈= Y − XΒ
Now
∈ ′ ∈= (ε 1 ε 2 …^ …^ εn)
ε 1 ε 2 ⋮ ⋮ εn (^) ⎠
= ε 12 + ε 22 + ⋯ εn^2 = ∑ εi^2 =RSS
So minimizing ∈ ′ ∈ is equivalent to minimizing RSS which we
have done in lecture one. However here we have to take
derivative of matrices w.r.t. a matrix in order to apply first
order condition. I skip the intermediate detail and write the
solution of the OLS parameters:
B�^ = (X′X)−1^ X′Y
This formula will give estimates of OLS coefficients.
Note: The estimates of this matrix formula must be similar to
the estimates obtained by the formula you have learnt in
previous lecture.
