ch03 - Chapter 3 Interval Estimation and Hypothesis Testing Walter R Paczkowski Rutgers University Principles of Econometrics 4th Edition Chapter 3

# ch03 - Chapter 3 Interval Estimation and Hypothesis Testing...

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Principles of Econometrics, 4t h Edition Page 1 Chapter 3: Interval Estimation and Hypothesis Testing Chapter 3 Interval Estimation and Hypothesis Testing Walter R. Paczkowski Rutgers University
Principles of Econometrics, 4t h 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 3.6 Linear Combinations of Parameters Chapter Contents
Principles of Econometrics, 4t h Edition Page 3 Chapter 3: Interval Estimation and Hypothesis Testing 3.1 Interval Estimation
Principles of Econometrics, 4t h 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 often called confidence intervals . We prefer to call them interval estimates because the term ‘‘confidence’’ is widely misunderstood and misused 3.1 Interval Estimation
Principles of Econometrics, 4t h Edition Page 5 Chapter 3: Interval Estimation and Hypothesis Testing The normal distribution of b 2 , the least squares estimator of β 2 , is A standardized normal random variable is obtained from b 2 by subtracting its mean and dividing by its standard deviation: 3.1.1 The t - Distribution Eq. 3.1 3.1 Interval Estimation 2 2 2 2 , ~ x x N b i 1 , 0 ~ 2 2 2 2 N x x b Z i
Principles of Econometrics, 4t h 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 95 . 0 96 . 1 2 96 . 1 2 2 2 x x b P i 95 . 0 96 . 1 96 . 1 Z P 95 . 0 96 . 1 96 . 1 2 2 2 2 2 2 2 x x b x x b P i i
Principles of Econometrics, 4t h 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 β 2 . This easy derivation of an interval estimator is based on the assumption SR6 and that we know the variance of the error term σ 2 . 3.1 Interval Estimation 3.1.1 The t - Distribution 2 2 2 96 . 1 x x b i
Principles of Econometrics, 4t h Edition Page 8 Chapter 3: Interval Estimation and Hypothesis Testing Replacing σ 2 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 ˆ 2 2 2 2 2 2 2 2 2 2 2 ~ r a ˆ v N i t b se b b b x x b t 2 2 2 b se b t 2 ~ N t t
Principles of Econometrics, 4t h Edition Page 9 Chapter 3: