Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Matrix approach to OLS - Econometrics - Lecture Notes, Study notes of Econometrics and Mathematical Economics

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

2011/2012

Uploaded on 11/10/2012

uzman
uzman 🇮🇳

4.8

(12)

148 documents

1 / 3

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Chapter 2
The matrix approach to the OLS
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:
y1=α+β1+ε1
y2=α+βx2+ε2
…..
……
……
yn=α+βxn+εn
This system of equation can be written as:
Docsity.com
pf3

Partial preview of the text

Download Matrix approach to OLS - Econometrics - Lecture Notes and more Study notes Econometrics and Mathematical Economics in PDF only on Docsity!

Chapter 2

The matrix approach to the OLS

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 =

y 1 y 2 ⋮ ⋮ yn (^) ⎠

, X =

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.