Lecture 7-Simple Linear Regression Model-Hypothesis ...

ACE 562 Fall 2005

Lecture 7: The Simple Linear Regression Model: Hypothesis Testing

by Professor Scott H. Irwin

Required Readings: Griffiths, Hill and Judge. "Statistical Inference II: Interval Estimation and Hypothesis Tests for the Mean of a Normal Population," Ch. 4 and "Hypothesis Testing," Section 7.2 in Learning and Practicing Econometrics

ACE 562, University of Illinois at Urbana-Champaign

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Overview

Many economic problems require some basis for deciding whether a parameter of the statistical model is equal to a specified value

Recall the model for household food expenditure and income,

yt = 1 + 2 xt + et t = 1,...,T

where yt is household food expenditure, xt is household income, and et is the error term

? An important question is whether 2 = 0 or 2 0

? If 2 = 0, then there is no relationship between

household food expenditure and income, which would contradict the underlying economic model

? If 2 0, then there is a relationship between

household food expenditure and income, presumably positive, which would be consistent with the underlying economic model

ACE 562, University of Illinois at Urbana-Champaign

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Hypothesis tests use the information about a population parameter that is contained in a sample of data to draw a conclusion about the hypothesis

? Hypothesis: statement, or conjecture, about a population parameter

? Information: least squares point estimate and estimated standard error

? Example: Is the sample data on food expenditure and income from 40 households

more consistent with 2 = 0 or 2 0?

All hypothesis tests have four basic elements

1. A null hypothesis, H0

2. An alternative hypothesis, H1

3. A test statistic

4. A rejection region

ACE 562, University of Illinois at Urbana-Champaign

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It is helpful to consider the parallel between hypothesis testing and a jury trial,

Statistical Hypothesis Test Jury Trial

Null Hypothesis

Not Guilty

Alternative Hypothesis Guilty

Test Statistic Rejection Region

Evidence Presented in Court of Law

Beyond a Reasonable Doubt

Questions:

? Why don't we conclude "Innocent" instead of "Not Guilty"?

? Why do we make "Not Guilty" the null hypothesis in a jury trial instead of "Guilty"?

? Where is the element of chance in both procedures?

Hint: If hypothesis testing seems confusing, remembering the analogy to a jury trial may be helpful

ACE 562, University of Illinois at Urbana-Champaign

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Null Hypothesis

? Belief we maintain until convinced by the sample evidence that it is not true

? Denoted H0 ("h-naught") Alternative Hypothesis

? Logical alternative to H0 that is accepted if the null hypothesis is rejected

? Denoted H1 or Ha Alternative Hypothesis Setups

Hill, Griffiths, Judge Conventional

H0 : 2 = 0 H1 : 2 0

H0 : 2 = 0 H1 : 2 > 0

H0 : 2 = 0 H1 : 2 < 0

H0 : 2 = 0 H1 : 2 0

H0 : 2 0 H1 : 2 > 0

H0 : 2 0 H1 : 2 < 0

ACE 562, University of Illinois at Urbana-Champaign

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