Lecture 3 lecture notes

Lecture 3 lecture notes - Lecture 3. A Simple Linear...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 3. A Simple Linear Regression Model A simple (one-variable) linear regression (SLR) model is given by Y j = β 0 + β 0 X j + ² j , j = 1 ,...,n, (3.1) where Y j is a dependent variable and X j is an independent (explanatory) variable; β 0 and β 1 are called an intercept and a slop respectively. The Gauss-Markov Theorem . If ² j are uncorrelated random variables with common variance, then of all possible estimators β * 0 and β * 1 that are linear functions of Y t , the least squares (LS) estimators have the smallest variance. Using historical data Y 1 ,...,Y N we want to obtain optimal estimates β * 0 and β * 1 and further use this information to obtain the forecasts of Y . Let us denote the estimated (historical) values of Y j by Y * j for j = 1 ,...,n , i.e. Y * = β * 0 + β * 1 X j . Then using the LS approach, we minimize the sum of squares of estimated residuals (SSE) SSE = n X j =1 Y j - Y * j · 2 = n X j =1 ( Y j - β * 0 - β * 1 X j ) 2 (3.2) Hence, we need to obtain the first partial derivatives of SSE with respect to each of β * 0 and β * 1 and set them both equal to 0, solving simultaneously. The solution is given by β * 0 = Y n - β * 1 ¯ x n (3.3) and β * 1 = S XY SS X = n j =1 X j - X n ·‡ Y j - Y n · n j =1 X j - X n · 2 (3.4) It is clear from 3.3 that the two estimates
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 06/21/2009 for the course STAT 331 taught by Professor Yuliagel during the Fall '08 term at Waterloo.

Page1 / 4

Lecture 3 lecture notes - Lecture 3. A Simple Linear...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online