Chapter3 - Chapter3 Testing ,4thEdition Chapter3:Interval Estimation and Hypothesis Testing Page1 ChapterContents 3.1 Interval Estimation 3.2 Hypothesis

# Chapter3 - Chapter3 Testing ,4thEdition Chapter3:Interval...

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Principles of Econometrics, 4th Edition Page 1 Chapter 3:  Interval Estimation and Hypothesis Testing Chapter 3 Interval Estimation and Hypothesis  Testing Principles of Econometrics, 4th Edition Page 2 Chapter 3:  Interval Estimation and Hypothesis Testing 3.1 Interval Estimation 3.2 Hypothesis Tests 3.3 Rejection Regions for Specific Alternatives 3.4 Examples of Hypothesis Tests 3.5 The p -value Chapter Contents Principles of Econometrics, 4th Edition Page 3 Chapter 3:  Interval Estimation and Hypothesis Testing 3.1 Interval Estimation Principles of Econometrics, 4th Edition Page 4 Chapter 3:  Interval Estimation and Hypothesis Testing There are two types of estimates Point estimates The estimate b 2 is a point estimate of the unknown population parameter in the regression model. Interval estimates Interval estimation proposes a range of values in which the true parameter is likely to fall Providing a range of values gives a sense of what the parameter value might be, and the precision with which we have estimated it Such intervals are called confidence intervals or interval estimates 3.1 Interval Estimation Principles of Econometrics, 4th Edition Page 5 Chapter 3:  Interval Estimation and Hypothesis Testing Let us assume that assumptions SR1-SR6 hold. The normal distribution of b2, the least squares estimator of β2, is A standardized normal random variable is obtained from b2 by subtracting its mean and dividing by its standard deviation: 3.1.1 The  t -Distribution Eq. 3.1 3.1 Interval Estimation ( 29 - 2 2 2 2 , ~ x x N b i σ β ( 29 ( 29 1 , 0 ~ 2 2 2 2 N x x b Z i - - = σ β Principles of Econometrics, 4th Edition Page 6 Chapter 3:  Interval Estimation and Hypothesis Testing We know that: Substituting: Rearranging: 3.1 Interval Estimation 3.1.1 The  t -Distribution ( 29 95 . 0 96 . 1 2 96 . 1 2 2 2 = - - - x x b P i σ β ( 29 95 . 0 96 . 1 96 . 1 = - Z P ( 29 ( 29 95 . 0 96 . 1 96 . 1 2 2 2 2 2 2 2 = - + - - x x b x x b P i i σ β σ Principles of Econometrics, 4th Edition Page 7 Chapter 3:  Interval Estimation and Hypothesis Testing The two end-points provide an interval estimator. In repeated sampling 95% of the intervals constructed this way will contain the true value of the parameter β2. This easy derivation of an interval estimator is based on the assumption SR6 and that we know the variance of the error term σ2. 3.1 Interval Estimation 3.1.1 The  t -Distribution ( 29 - ± 2 2 2 96 . 1 x x b i σ Principles of Econometrics, 4th Edition Page 8 Chapter 3:  Interval Estimation and Hypothesis Testing Replacing σ2 with creates a random variable t: The ratio has a t -distribution with ( N – 2) degrees of freedom, which we denote as: Eq. 3.2 3.1 Interval Estimation 3.1.1 The  t -Distribution 2 σ ˆ ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 2 2 2 - - = - = - - = N i t ~ b se b b r a ˆ v b x x ˆ b t β β σ β ( 29 2 2 2 b se ) b ( t β - = ( 29 2 ~ - N t t Principles of Econometrics, 4th Edition Page 9 Chapter 3:  Interval Estimation and Hypothesis Testing In general we can say, if assumptions SR1-SR6 hold in the simple linear regression model, then The t  • • • 