Section 3. Simple Regression – OLS Estimators · 2008-01-26 · Section 3. Simple Regression –...

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Section 3. Simple Regression – OLS Estimators 1. Last time: Lines, Population Linear Regression 2. Estimators of β 0 ,β 1 3. Least Squares Estimators 4. OLS Assumptions

Transcript of Section 3. Simple Regression – OLS Estimators · 2008-01-26 · Section 3. Simple Regression –...

Section 3. Simple Regression –OLS Estimators

1. Last time: Lines, Population Linear Regression

2. Estimators of β0,β1

3. Least Squares Estimators4. OLS Assumptions

Copyright © 2003 by Pearson Education, Inc. 4-2

1. Review: Lines, Clouds and Pop. Linear Regression

• e.g. Demand for Coffee, Global warming, CA test scores and student-teacher ratios

• Decide which parameters in population we care about (β0,β1)

- just like we did with µ• Draw a sample and estimate parameters

- just like we did with µ• Construct CI for parameters, test

hypotheses, make predictions.- just like..

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Coffee Example (with a line)

cup

s

price

quantity demanded qhat

0 .5 1 1.5 2

0

5

10

15

20cu

ps

price

quantity demanded qhat

0 .5 1 1.5 2

0

5

10

15

20

What’s the slope of the line?About how many cups would you sell at $1?

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Global Warming ExampleEarth's temperature since 1880

year1880 1993

-1.2

.8

Is the slope statistically different from zero?

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2. Estimators of β0,β1

• Decide which parameters in population we care about (β0,β1)

- we chose Pop. Linear Regression • Draw a sample and estimate parameters• Which estimates to use?• We’d like something with nice properties:

- unbiased, - consistent, - efficient, - with an approximately normal distribution.

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3. The Ordinary Least Squares (OLS) Estimators

• Try b0 and b1 that minimize the sum of ei2

in

where ei = Yi – (b0+b1 x)• Why this one? Well, it’s analogous to what

we asked for in the population.• And it seems like a nice property in the

sample.• Deriving formulae for OLS estimators..

(S&W p. 143)

∑=

n

iie

1

2

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Formula for OLS estimators

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4. Assumptions

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Next time..

• Confidence intervals for β0 and β1• Examples